Class 10 Mathematics Previous Year Questions, Chapter-wise (2022-2026)

If HCF (850, 325) is 25, then LCM (850, 325) is:

1. 442
2. 11050
3. 8450
4. 2210

7 × 29 × 23 + 1 is:

1. a prime number
2. divisible by 23
3. an odd number
4. a composite number

Assertion (A): 4ⁿ can not end with the digit zero. Reason (R): Prime factorisation of 4ⁿ is unique.

1. Both, Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A)
2. Both, Assertion (A) and Reason (R) are true, but Reason (R) is not correct explanation for Assertion (A)
3. Assertion (A) is true, but Reason (R) is false
4. Assertion (A) is false, but Reason (R) is true

Probability of getting an irrational number at random from the numbers 3, 4, ∛9, ∛8, √5, 0, ∛(42/3) is:

1. 0
2. 4/7
3. 3/7
4. 1

HCF of two consecutive natural numbers is:

1. 2
2. 1
3. 0
4. smaller number

7 × 11 × 13 + 5 is

1. a prime number
2. an odd number
3. a composite number
4. a multiple of 5

√17 × √11 × √13 + √11 is

1. a prime number
2. multiple of 17
3. a composite number
4. an odd number

If HCF of 66 and 99 is expressible in the form of 55m – 132, then the value of m is :

1. 4
2. 2
3. 1
4. 3

The product of the H.C.F. and L.C.M. of two numbers 50 and 20 is

1. 100
2. 1000
3. 50
4. 20

Assertion (A): H.C.F. (36m², 18m) = 18m, where m is a prime number. Reason (R): H.C.F. of two numbers is always less than or equal to the smaller number.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

If the HCF of two positive integers a and b is 1, then their LCM is:

1. a + b
2. a
3. b
4. ab

The number 3 + √2 is:

1. a rational number
2. an irrational number
3. an integer
4. a natural number

Assertion (A): For any two natural numbers a and b, the HCF of a and b is a factor of the LCM of a and b. Reason (R): HCF of any two natural numbers divides both the numbers.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

√2 + √3 is :

1. a rational number
2. an irrational number
3. an integer
4. a natural number

√2 is:

1. a rational number
2. an irrational number
3. an integer
4. a natural number

The number 2n, where n is a natural number, cannot end with the digit:

1. 4
2. 6
3. 2
4. 0

Assertion (A): The prime numbers which divide 36 also divide 6. Reason (R): Any number which divides p² also divides p.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

The value of (HCF × LCM) for the two numbers 3 and 5 is:

1. 2
2. 4
3. 14
4. 15

The number 3ⁿ, where n is a natural number cannot end with the digit:

1. 1
2. 3
3. 5
4. 7

If the number aⁿ, where n is a natural number, always ends with digit a, then a is:

1. 2
2. 4
3. 6
4. 8

Assertion (A): The prime numbers which divide 36 also divide 6. Reason (R): Any number which divides p² also divides p. Select the correct answer from the codes: (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A). (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). (C) Assertion (A) is true, but Reason (R) is false. (D) Assertion (A) is false, but Reason (R) is true.

1. (A)
2. (B)
3. (C)
4. (D)

If p = 23 × 32 × 5 and q = 22 × 33, then the LCM of p and q is:

1. 23 × 33
2. 22 × 32
3. 22 × 32 × 5
4. 23 × 33 × 5

3n, where n is a natural number, cannot end with the digit:

1. 3
2. 5
3. 7
4. 9

A prime number has:

1. exactly two prime factors
2. exactly one prime factor
3. at least one prime factor
4. at least two prime factors

If HCF(98, 28) = m and LCM(98, 28) = n, then the value of n − 7m is:

1. 0
2. 28
3. 98
4. 198

Which of the following is a rational number between 3 and π?

1. 1.4142387954012....
2. √2
3. π
4. 1.857142

Which of the following is a rational number between √2 and √3?

1. 1.4142387954012....
2. √2.5
3. √2.8
4. 1.857142

The greatest number which divides 70 and 125, leaving remainders 5 and 8 respectively, is:

1. 13
2. 65
3. 875
4. 1750

If (−1)ⁿ + (−1)⁸ = 0, then n is:

1. any positive integer
2. any negative integer
3. any odd number
4. any even number

The least number which is a perfect square and is divisible by each of 16, 20 and 50, is:

1. 1200
2. 100
3. 3600
4. 2400

The sum of the exponents of prime factors in the prime factorisation of 4004 is:

1. 5
2. 4
3. 3
4. 2

The HCF of 40, 110 and 360 is:

1. 40
2. 110
3. 360
4. 10

If 1080 = 2^p × 3^q × 5, then (p - q) is equal to:

1. 6
2. -1
3. 1
4. 0

If a^b = ab is:

1. 72
2. 5√10
3. 2√10
4. 512

If x is the LCM of 4, 6, 8 and y is the LCM of 3, 5, 7 and p is the LCM of x and y, then which of the following is true?

1. p = 35x
2. p = 4y
3. p = 8x
4. p = 16y

The LCM of 24, 36 and 60 in terms of their prime factors is:

1. 2² × 3 × 5
2. 2³ × 3²
3. 2³ × 3² × 5
4. 2³ × 3³ × 5

If the HCF (2520, 6600) = 40 and LCM (2520, 6600) = 252 × k, then the value of k is

1. 1650
2. 1600
3. 165
4. 1625

If a = 2² × 3ˣ, b = 2² × 3 × 5, c = 2² × 3 × 7 and LCM (a, b, c) = 3780, then x is equal to

1. 1
2. 2
3. 3
4. 0

Given HCF (2520, 6600) = 40, LCM (2520, 6600) = 252 × k, then the value of k is:

1. 1650
2. 1600
3. 165
4. 1625

A pair of irrational numbers whose product is a rational number is:

1. (√16, √4)
2. (√5, √2)
3. (√3, √27)
4. (√36, √2)

The smallest irrational number by which √20 should be multiplied so as to get a rational number, is:

1. √20
2. √2
3. √5
4. 5

The HCF of two numbers 65 and 104 is 13. If LCM of 65 and 104 is 40x, then the value of x is:

1. 5
2. 13
3. 40
4. 8

If 3825 = 3ˣ × 5ʸ × 17ᶻ, then the value of x + y – 2z is:

1. 0
2. 1
3. 2
4. 3

The greatest number which divides 281 and 1249, leaving remainder 5 and 7 respectively, is:

1. 23
2. 276
3. 138
4. 69

The LCM of three numbers 28, 44, 132 is:

1. 258
2. 231
3. 462
4. 924

If the product of two co-prime numbers is 553, then their HCF is:

1. 1
2. 553
3. 7
4. 79

If the prime factorisation of 2520 is 2³ × 3ᵃ × b × 7, then the value of a + 2b is:

1. 12
2. 10
3. 9
4. 7

The LCM of the smallest prime number and the smallest odd composite number is:

1. 10
2. 6
3. 9
4. 18

The HCF of smallest 2–digit number and the smallest composite number is:

1. 2
2. 20
3. 40
4. 4

If p and q are natural numbers and p is a multiple of q, then what is the HCF of p and q?

1. pq
2. p
3. q
4. p + q

If n is a natural number, then which of the following numbers ends in zero (0) ?

1. (3 - 2)n
2. (2 - 5)n
3. (6 - 2)n
4. (5 - 3)n

The ratio of HCF to LCM of the least composite number and the least prime number is :

1. 1:2
2. 2:1
3. 1:1
4. 1:3

Assertion (A): The number 5ⁿ cannot end with the digit 0, where n is a natural number. Reason (R): Prime factorisation of 5 has only two factors, 1 and 5.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

Find the H.C.F. and L.C.M. of 1530 and 2040.

If H.C.F. of 135x² and 189x³ is 108, then find the value of x.

Find the H.C.F. and L.C.M. of 408 and 312.

Two numbers are in the ratio 3 : 5 and their LCM is 180. Find the HCF of these two numbers.

Find the smallest 5-digit number exactly divisible by 24 and 36.

Prove that 2 – 5√3 is an irrational number given that √3 is irrational.

Find the smallest number which is divisible by both 644 and 462.

(a) Find the smallest number which is divisible by both 644 and 462. OR (b) Two numbers are in the ratio 4 : 5 and their HCF is 11. Find the LCM of these numbers.

Find the smallest number that is divisible by each of 8, 9 and 10.

Can the number (15)ⁿ, n being a natural number, end with the digit 0? Give reasons.

Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.

Can the number 8ⁿ, n being a natural number, end with the digit 0? Give reasons.

Three bells toll at intervals of 9, 12 and 15 minutes respectively. If they start tolling together, after what time will they next toll together?

In a school, there are two sections of class X. There are 40 students in the first section and 48 students in the second section. Determine the minimum number of books required for their class library so that they can be distributed equally among students of both sections.

Find the HCF of 84 and 144 by prime factorisation method.

Prove that 2 + √3 is an irrational number, given that √3 is an irrational number.

Prove that 4n can never end with digit 0, where n is a natural number.

Prove that 6 - √7 is irrational number, given that √7 is an irrational number.

Two numbers are in the ratio 2 : 3 and their LCM is 180. What is the HCF of these numbers ?

Using prime factorisation, find HCF and LCM of 96 and 120.

Find the greatest 3-digit number which is divisible by 18, 24 and 36.

Prove that 2 + 3√5/7 is an irrational number, given that √5 is an irrational number.

The dimensions of a window are 156 cm × 216 cm. Arjun wants to put grill on the window creating complete squares of maximum size. Determine the side length of the square and hence find the number of squares formed.

(a) Prove that √5 is an irrational number. OR (b) Find the LCM of 36 and 54.

Find which among the following numbers a, b and c is/are composite numbers. a = 7 × 11 × 13 + 13, b = 6 × 5 × 4 + 4, c = 7 × 13 + 6

Three sets of Physics, Chemistry and Mathematics books have to be stacked in such a way that all the books are stored subject-wise and the height of each stack is the same. The number of Physics books is 144, the number of Chemistry books is 180 and the number of Mathematics books is 192. Assuming that the books are of same thickness, determine the number of stacks of Physics, Chemistry and Mathematics books.

Prove that √7 is an irrational number.

Prove that √2 is an irrational number.

Prove that √3 - √2 is an irrational number given that √6 is an irrational number.

Prove that √3 is an irrational number given that √2 is an irrational number.

Prove that 2 – 3√5 is an irrational number, given that √3 is an irrational number.

Prove that 5√3 – 2 is an irrational number, given that √3 is an irrational number.

Find by prime factorisation the LCM of the numbers 18180 and 7575. Also, find the HCF of the two numbers.

Prove that √5 is an irrational number.

Prove that √3 is an irrational number.

(a) Prove that √3 is an irrational number. OR (b) The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively. If they change simultaneously at 7 a.m., at what time will they change together next ?

The value of k for which sum of the zeroes of the polynomial p(x) = 3x² – kx + 6 is 2, is

1. 2
2. –6
3. –2
4. 6

If the zeroes of the polynomial p(x) = 2x² – 7x + 6 are α and β, then the value of 1/α + 1/β is

1. 7/2
2. 6/7
3. –7/6
4. 7/6

The graph of a quadratic polynomial f(x) passes through (5,0), (0, –1) and (–2, 0). The two factors of the polynomial are

1. ( x + 2), (x – 5)
2. ( x + 5), (x – 2)
3. ( x + 1), (x – 5)
4. ( x – 1), (x + 2)

If α and β are two zeroes of the quadratic polynomial p(x) = x² – 11x + 30, then 1/α + 1/β is equal to:

1. 30/11
2. 11/30
3. –11/30
4. –30/11

If the zeroes of a polynomial p(x) are –3 and 8, then p(x) equals

1. x² + 5x – 4
2. (x + 3)(–x + 8)
3. a(x² + 5x – 24)
4. x² – 24

If (−1)ⁿ + (−1)⁸ = 0, then n is:

1. any positive integer
2. any negative integer
3. any odd number
4. any even number

If α and β are the zeroes of polynomial 3x² + 6x + k such that α + β + αβ = −4/3, then the value of k is:

1. -8
2. 8
3. -4
4. 4

The sum of the zeroes of the polynomial p(x) = 5x³ − 7x² + 3 is:

1. 7/5
2. -7/5
3. 3
4. -3

If 4 is a zero of the polynomial p(x) = x² - x - (2 + 2k), then the value of k is:

1. 3
2. 9
3. 6
4. -9

If one zero of the polynomial q(x) = (p² + 4)x² + 65x + 4p is reciprocal of the other, then the value of p is:

1. -1
2. 1
3. -2
4. 2

If α and β are the zeroes of the polynomial p(x) = x² - ax - b, then the value of (α + β + αβ) is equal to:

1. a + b
2. -a - b
3. a - b
4. -a + b

Zeroes of the polynomial p(x) = x² - 3x + 4 are:

1. 2, -2
2. 2, 1/2
3. 4, 1/4
4. 1/2, 2

The graph of a polynomial intersects the y-axis at one point and the x-axis at two points. The number of zeroes of this polynomial are:

1. 1
2. 2
3. 3
4. 0

If a and b are zeroes of the polynomial 5x² + 3x – 7, the value of 1/a + 1/b is

1. –3/7
2. 3/5
3. 3/7
4. –5/7

If a and b are zeroes of the polynomial 2x² – 9x + 5, then value of a² + b² is

1. 1/4
2. 61/4
3. 1
4. 71/4

If a and b (a > b) are the zeroes of the polynomial –x² + 8x + 9, then (a – b) is equal to

1. –10
2. 10
3. ±10
4. 8

The zeroes of the quadratic polynomial 2x² – 3x – 9 are:

1. 3, –3/2
2. –3, –3/2
3. –3, 3/2
4. 3, 3/2

What should be subtracted from the polynomial x² – 16x + 30, so that 15 is the zero of the resulting polynomial?

1. 30
2. 14
3. 15
4. 16

What should be added from the polynomial x² – 5x + 4, so that 3 is the zero of the resulting polynomial?

1. 1
2. 2
3. 4
4. 5

If a polynomial p(x) is given by p(x) = x² – 5x + 6, then the value of p(1) + p(4) is:

1. 0
2. 4
3. 2
4. –4

A quadratic polynomial, one of whose zeroes is 2 + √5 and the sum of whose zeroes is 4, is:

1. x² + 4x – 1
2. x² – 4x – 1
3. x² – 4x + 1
4. x² + 4x + 1

If a and b are the zeroes of the polynomial p(x) = kx² – 30x + 45k and a + b = ab, then the value of k is:

1. -3/2
2. -2/3
3. 2/3
4. 3/2

A quadratic polynomial, the sum of whose zeroes is –5 and their product is 6, is

1. x² + 5x + 6
2. x² – 5x + 6
3. x² – 5x – 6
4. –x² + 5x + 6

If one zero of the polynomial 6x² + 37x - (k - 2) is reciprocal of the other, then what is the value of k?

1. -4
2. -6
3. 6
4. 4

The zeroes of the polynomial p(x) = x² + 4x + 3 are given by:

1. -1, -3
2. 1, 3
3. -1, 3
4. 1, -3

If α and β are the zeroes of the quadratic polynomial p(x) = x² - ax - b, then the value of α² + β² is:

1. a² - 2b
2. a² + 2b
3. b² - 2a
4. b² + 2a

Assertion (A): The polynomial p(x) = x² + 3x + 3 has two real zeroes. Reason (R): A quadratic polynomial can have at most two real zeroes.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

The sum of zeroes of the polynomial 2x² - 17 are given as :

1. 22/17
2. -22/17
3. 0
4. 1

If α, β are zeroes of the polynomial x² – 1, then value of (α + β) is :

1. 2
2. 1
3. –1
4. 0

If α, β are the zeroes of the polynomial p(x) = 4x² – 3x – 7, then 1/α + 1/β is equal to :

1. 7/3
2. -7/3
3. 3/7
4. -3/7

The number of polynomials having zeroes 3 and 5 is :

1. only one
2. infinite
3. exactly two
4. at most two

The number of polynomials having zeroes 1 and 2 is:

1. exactly 2
2. only 1
3. at most 2
4. infinite

If α, β are the zeroes of polynomial p(x) = 6x² – 5x – 3, then find the value of 1/α + 1/β.

If α and β are the zeroes of polynomial p(x) = – 9x² – 6x + 1. Find the value of α² + β².

Form a quadratic polynomial whose sum and product of the zeroes are 1/2 and –1/9 respectively. Hence, find the zeroes of the polynomial.

Find the quadratic polynomial the sum of whose zeroes is 1 and their product is –12. Hence find the zeroes of the polynomial.

Find the zeroes of the polynomial p(x) = 15x² – 19x + 6.

Find the zeroes of the polynomial p(x) = x² + √2x − √6.

If the zeroes of the polynomial x² + ax + b are in the ratio 3 : 4, then prove that 12a² = 49b.

Find the zeroes of the polynomial p(x) = x² + (7/6)x − 5.

If p and q are zeroes of the polynomial p(y) = 21y² - y - 2, then find the value of (1 - p)·(1 - q).

Find a quadratic polynomial whose zeroes are 2 and 1/2.

If the sum of the zeroes of the polynomial p(x) = (p + 1)x² + (2p + 3)x + (3p + 4) is -1, then find the value of p

If a, b are zeroes of the polynomial p(x) = 5x² – 6x + 1, then find the value of a + b + ab.

If a and b are zeroes of the quadratic polynomial p(x) = x² – 5x + 4, then find the value of 1/a + 1/b – 2ab.

If one zero of the polynomial p(x) = 6x² + 37x – (k – 2) is reciprocal of the other, then find the value of k.

If α and β are zeroes of the polynomial p(x) = 5x² – 7x – 3, then form a quadratic polynomial whose zeroes are α² and β².

(a) Find the zeroes of the polynomial p(x) = 4x² – 8x + 3 and verify the relationship between its zeroes and co-efficients. OR (b) Form a polynomial whose zeroes are 2α and 2β, where α and β are zeroes of the polynomial p(x) = x² – 3√2x + 4.

If α, β are zeroes of the polynomial p(x) = 5x² – 7x – 3, then form a quadratic polynomial whose zeroes are α/2 and β/2.

Find a quadratic polynomial whose sum and product of zeroes are 0 and -9, respectively. Also, find the zeroes of the polynomial so obtained.

Find a quadratic polynomial, sum and product of whose zeroes are 5 and –6, respectively. Also, find the zeroes of the polynomial so obtained.

Determine a quadratic polynomial, sum and product of whose zeroes are -10 and 24, respectively. Also, determine the zeroes of the polynomial so obtained.

Find the zeroes of the polynomial p(x) = 3x² - 2x - 1 and verify the relationship between the zeroes of p(x) and the coefficients of p(x).

Find the zeroes of the polynomial p(x) = 2x² + 5x + 2 and verify the relationship between zeroes of p(x) and its coefficients.

Find the zeroes of the polynomial q(x) = 6x² - 5x - 1 and verify the relationship between the zeroes of q(x) and its coefficients.

Find the zeroes of the polynomial 9s² - 6s + 1 and verify the relationship between the zeroes and the coefficients of the given polynomial.

Find the zeroes of the polynomial 4x² + 4x + 1 and verify the relationship between the zeroes and the coefficients of the given polynomial.

Find the zeroes of the polynomial 25a2 - 10a + 1 and verify the relationship between the zeroes and coefficients of the given polynomial.

If a and b are the zeroes of the quadratic polynomial f(x) = 6x² + 11x – 10, find the value of a/b + b/a.

Find the zeroes of the quadratic polynomial x² – 15 and verify the relationship between the zeroes and the coefficients of the polynomial.

(a) Find the zeroes of the polynomial 4x² + 4x – 3 and verify the relationship between zeroes and coefficients of the polynomial. OR (b) If a and b are the zeroes of the polynomial x² + x – 2, then find the value of a/b + b/a.

(a) Find the zeroes of the polynomial 4x² + 4x – 3 and verify the relationship between zeroes and coefficients of the polynomial.

(a) Zeroes of the quadratic polynomial x² + x – 6 are 'a' and 'b'. Construct a quadratic polynomial whose zeroes are 1/a and 1/b.

If pair of linear equations given by a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 has infinitely many solutions, then which of the following is definitely true ?

1. a1/a2 = c1/c2
2. a1a2 ≠ b1b2
3. a1/b1 = a2/b2
4. a1b2 = a2b1

The value of k for which the pair of linear equations kx – 3y = 5, 4x – 6y = 10 has infinitely many solutions, is:

1. 4
2. 3
3. 2
4. 1

Assertion (A): The system of linear equations 3x – 5y + 7 = 0 and –6x + 10y + 14 = 0 is inconsistent. Reason (R): When two linear equations don't have unique solution, they always represent parallel lines.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

Assertion (A): The value of p for which the system of equations 4x + py + 8 = 0 and 2x + 2y + 2 = 0 is consistent is 4. Reason (R): The system of equations a₁x + b₁y = c₁ and a₂x + b₂y = c₂ is consistent with infinitely many solutions, if a₁/a₂ = b₁/b₂ = c₁/c₂.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

If (0, 0) is the solution of the equation x + y = c - 1, then the value of c is:

1. 0
2. 1
3. -1
4. any real number

The number of solutions of the system of equations x = 3, y = 1 is:

1. 0
2. 1
3. 2
4. Infinite

The number of solutions of the system of equations x = a, x = b (a ≠ b) is:

1. 0
2. 1
3. 2
4. Infinite

For what value(s) of k does the system of equations kx + 2y = 3 and 2x + y = 5 have a unique solution?

1. k = a real number
2. k ≠ 8
3. k ≠ 4
4. k = 4

For what value(s) of k, is the system of equations kx + 2y = 3 and 2x + y = 5 inconsistent?

1. k = Any real number
2. k ≠ 2
3. k ≠ 4
4. k = 4

The system of linear equations given by x = a and y = b is:

1. Consistent with a unique solution.
2. Consistent with infinitely many solutions.
3. Consistent with two solutions.
4. Inconsistent.

If x = 1 and y = 2 is a solution of the pair of linear equations 2x − 3y + a = 0 and 2x + 3y − b = 0, then:

1. a = 2b
2. 2a = b
3. a + 2b = 0
4. 2a + b = 0

Assertion (A): The pair of linear equations px + 3y + 59 = 0 and 2x + 6y + 118 = 0 will have infinitely many solutions if p = 1. Reason (R): If the pair of linear equations px + 3y + 19 = 0 and 2x + 6y + 157 = 0 has a unique solution, then p ≠ 1.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

The value of k for which the system 6x + y = 3k and 36x + 6y = 3 have infinitely many solutions is:

1. 6
2. 1/2
3. 3/2
4. 1/6

A system of two linear equations in two variables is inconsistent, if the lines in the graph are:

1. coincident
2. parallel
3. intersecting at one point
4. intersecting at right angles

The solution of the pair of linear equations 2x/3 – y/2 = –16 and x/2 + 2y/3 = 3 is:

1. x = 2, y = –3
2. x = –2, y = 3
3. x = 2, y = 3
4. x = –2, y = –3

The value of k for which the system of equations 3x – y + 8 = 0 and 6x – ky + 16 = 0 has infinitely many solutions, is

1. –2
2. 2
3. 1/2
4. –1/2

The pair of linear equations x + 2y + 5 = 0 and –3x = 6y – 1 has

1. unique solution
2. exactly two solutions
3. infinitely many solutions
4. no solution

If a pair of linear equations in two variables is consistent, then the lines represented by the two equations are:

1. always intersecting
2. parallel
3. always coincident
4. intersecting or coincident

If ax + by = a² – b² and bx + ay = 0, then the value of x + y is:

1. a² – b²
2. a + b
3. a – b
4. a² + b²

Two lines are given to be parallel. The equation of one of these lines is 5x – 3y = 2. The equation of the second line can be:

1. –15x – 9y = 5
2. 15x + 9y = 5
3. 9x – 15y = 6
4. –15x + 9y = 5

The pair of linear equations y = 0 and y = –7 have

1. exactly one solution
2. two solutions
3. infinitely many solutions
4. no solution

Which out of the following type of straight lines will be represented by the system of equations 3x + 4y = 5 and 6x + 8y = 7?

1. Parallel
2. Intersecting
3. Coincident
4. Perpendicular to each other

The graph of a pair of linear equations a₁x + b₁y = c₁ and a₂x + b₂y = c₂ in two variables x and y represents parallel lines, if

1. a₁/a₂ ≠ b₁/b₂
2. a₁/a₂ = b₁/b₂ = c₁/c₂
3. a₁/a₂ = b₁/b₂ ≠ c₁/c₂
4. a₁/a₂ ≠ b₁/b₂ = c₁/c₂

The value of 'k' for which the pair of linear equations x + y – 4 = 0 and 2x + ky – 8 = 0 has infinitely many solutions, is

1. k ≠ 2
2. k ≠ –2
3. k = 2
4. k = –2

If the pair of equations 3x - y + 8 = 0 and 6x - ry + 16 = 0 represent coincident lines, then the value of r is:

1. 2
2. -2
3. 1/2
4. -1/2

The pair of equations x = a and y = b graphically represents lines which are:

1. parallel
2. intersecting at (b, a)
3. coincident
4. intersecting at (a, b)

If the pair of equations 3x - y + 8 = 0 and 6x - ry + 16 = 0 represent parallel lines, then the value of r is:

1. -2
2. 2
3. -2/1
4. 2/1

If 2x + 3y = 15 and 3x + 2y = 25, then the value of x - y is :

1. -10
2. 8
3. 10
4. -8

If the pair of equations 3x - y + 8 = 0 and 6x - ry + 16 = 0 represent coincident lines, then r equals :

1. 1/2
2. -1/2
3. 2
4. -2

The pair of linear equations 2x = 5y + 6 and 15y = 6x – 18 represents two lines which are :

1. intersecting
2. parallel
3. coincident
4. either intersecting or parallel

The pair of equations ax + 2y = 9 and 3x + by = 18 represent parallel lines, where a, b are integers, if :

1. a = b
2. 3a = 2b
3. 2a = 3b
4. ab = 6

The solution of the pair of equations x + y = a + b and ax - by = a² - b² is :

1. x = b, y = a
2. x = -a, y = b
3. x = a, y = b
4. x = a, y = -b

Solve the following system of equations for x and y: 1/x + 1/y = 1 and x - y = 3

Solve for x and y: 0·1x + 0·3y = 1 and 0·2x – 0·1y = –0·1

Solve for x and y: 3x + 5y = 8; 5x - 3y = 2

(a) Find the value of c for which the following pair of linear equations has infinitely many solutions: cx + 3y = c - 3 and 12x + cy = c

(a) Find the value of c for which the following pair of linear equations has infinitely many solutions: cx + 3y = c - 3, 12x + cy = c

(a) Find the value of c for which the following pair of linear equations has infinitely many solutions: cx + 3y = c - 3 and 12x + cy = c. OR (b) Solve for x and y: 3x + 2y = 65 and 2x + 3y = 60

Solve the following system of equations graphically: 2x - 3y = -6 and x = 3

Solve the following system of equations graphically: x + 2y = 10 and y = 3

Solve the following system of linear equations graphically: x + y = 5 and x - y = 3

If 2x + y = 13 and 4x – y = 17, find the value of (x – y).

(a) The sum of two natural numbers is 70 and their difference is 10. Find the natural numbers.

Rohan's age 5 years ago and his age 7 years from now, when multiplied together, is one more than twice his present age. Find his present age.

Solve the following pair of linear equations: 31x + 43y – 117 = 0; 43x + 31y = 105

Solve the system of linear equations: x = 4 and 3x – 2y = 6 graphically.

(a) Solve the following system of equations graphically: x + 3y = 6; 2x - 3y = 12 OR (b) x and y are complementary angles such that x : y = 1 : 2. Express the given information as a system of linear equations in two variables and hence solve it.

Solve the following system of equations graphically: x + 3y = 6; 2x – 3y = 12

(a) Solve graphically the following pair of linear equations: 2x - y = 2 and 4x - y = 4. Also, write the coordinates of the points where the lines represented by these equations cut the y-axis.

(a) Solve graphically the following pair of linear equations: 2x - y = 2 and 4x - y = 4. Also, write the coordinates of the points where the lines represented by these equations cut the y-axis. OR (b) An academy offering cricket coaching bought 10 bats and 5 balls for ₹32,500. Later, the academy bought 2 bats and 8 balls for ₹10,000. If there is no change in the cost of the bat and of the ball, find the cost of 1 bat and 1 ball.

A fraction becomes 1/3, when 1 is subtracted from the numerator and it becomes 1/4, when 8 is added to its denominator. Find the fraction.

Find the value of k for which the following pair of linear equations will have infinitely many solutions: kx + 3y - (k - 3) = 0 and 12x + ky - k = 0. Hence, find any two solutions of the given pair of equations.

The monthly incomes of A and B are in the ratio 8 : 7 and their expenditures are in the ratio 19 : 16. If each saves ₹2500 per month, find the monthly income of each.

In a 2-digit number, the digit at the unit's place is 5 less than the digit at the ten's place. The product of the digits is 36. Find the number.

A dealer sells an article for ₹75 and gains as much percent as the cost price of the article. Find the cost price of the article.

Solve the following system of linear equations graphically: x – y + 1 = 0 and x + y = 5

Rehana went to a bank to withdraw ₹2,000. She asked the cashier to give her ₹50 and ₹100 notes only. Rehana got 25 notes in all. Find how many notes of ₹50 and ₹100 did she receive.

A part of monthly hostel charges is fixed and the remaining depends on the number of days one has taken food in the mess. When a student A takes food for 20 days, she has to pay ₹1,500 as hostel charges while another student B, who takes food for 26 days, pays ₹1,800. Find the fixed charges and the cost of food.

If the system of linear equations 2x + 3y = 7 and 2ax + (a + b)y = 28 has infinitely many solutions, then find the values of a and b.

If 217x + 131y = 913 and 131x + 217y = 827, then solve the equations for the values of x and y.

Two people are 16 km apart on a straight road. They start walking at the same time. If they walk towards each other with different speeds, they will meet in 2 hours. Had they walked in the same direction with same speeds as before, they would have met in 8 hours. Find their walking speeds.

A 2-digit number is seven times the sum of its digits. The number formed by reversing the digits is 18 less than the given number. Find the given number.

A fraction becomes 3/1 when 1 is subtracted from the numerator. It becomes 4/1 when 8 is added to the denominator. Find the fraction.

Rohan's age 5 years ago and his age 7 years from now, if multiplied together, is one more than twice his present age. Find his present age.

The difference between two numbers is 12. The greater number is 6 less than twice the smaller one. (i) Representing the above situation, frame two linear equations in two variables. (ii) Show that the equations have unique solution. (iii) Solve the equations and hence find the numbers.

Venkat can row a boat in still water at the speed of 12 km/h. He ferries tourists 15 km upstream and 18 km downstream in 3 hours. Find the speed of the stream.

Vijay invested certain amounts of money in two schemes A and B, which offer interest at the rate of 8% per annum and 9% per annum, respectively. He received ₹1,860 as the total annual interest. However, had he interchanged the amounts of investments in the two schemes, he would have received ₹20 more as annual interest. How much money did he invest in each scheme?

A bag contains some red and blue balls. Ten percent of the red balls, when added to twenty percent of the blue balls, give a total of 24. If three times the number of red balls exceeds the number of blue balls by 20, find the number of red and blue balls.

A man lent a part of his money at 10% p.a. and the rest at 15% p.a. His income at the end of the year is ₹1,900. If he had interchanged the rate of interest on the two sums, he would have earned ₹200 more. Find the amount lent in both cases.

The students of a class are made to stand equally in rows. If 3 students are extra in each row, there would be 1 row less. If 3 students are less in a row, there would be 2 more rows. Find the number of students in the class.

Using graphical method, solve the following system of equations: 3x + y + 4 = 0 and 3x – y + 2 = 0

If the length of a rectangle is reduced by 5 cm and its breadth is increased by 2 cm, then the area of the rectangle is reduced by 80 cm². However, if we increase the length by 10 cm and decrease the breadth by 5 cm, its area is increased by 50 cm². Find the length and breadth of the rectangle.

Using graphical method, solve the following system of equations: 3x – 2y = 10 and 5x + 3y = 4

The value of k for which the equation 2kx² – 6x + 3 = 0 has real and equal roots, is:

1. 3/2
2. 1/2
3. –3/2
4. 2

The value of k for which the equation 5x² – 2x + k = 0 has equal real roots, is:

1. 5
2. – 1/5
3. 1/5
4. 0

The roots of the quadratic equation x² + 9 = 0 are

1. real and equal
2. not real
3. real and negative of each other
4. rational numbers

The value of k for which the equation kx² – 6x – 4 = 0 has real and equal roots, is

1. 9/4
2. –4
3. –9/4
4. –2

The value of p for which roots of the quadratic equation x² – px + 6 = 0 are rational, is

1. 1
2. –5
3. 25
4. 5

The discriminant of the quadratic equation x² - 3x - 2 = 0 is:

1. 1
2. 17
3. -1
4. -17

The equation x + 1/x = 3 (x ≠ 0) is expressed as a quadratic equation in the form of ax² + bx + c = 0. The value of a - b + c is:

1. 5
2. 2
3. 1
4. -1

The discriminant of the quadratic equation x² – 5x + 6 = 0 is :

1. 1
2. –1
3. 49
4. 7

The discriminant of the quadratic equation 2x² - 3x - 5 = 0 is:

1. -31
2. 49
3. 7
4. -9

ax² + 3x + 1 = 0 has real and equal roots is:

1. a = 9/4
2. a = 4/9
3. a = 3/4
4. a = 4/3

Which of the following equations is a quadratic equation?

1. x² = (x + 1)²
2. (x - 1)(x + 2) = 2x + 1
3. (x + 2)³ = 2x(x² - 1)
4. x³/x = x²

The value of p for which px² + 4x + p = 0 has real and equal roots is:

1. 4
2. -4
3. 2
4. 0

Assertion (A): Every quadratic equation has two real roots. Reason (R): A quadratic polynomial can have at most two zeroes. Select the correct answer from the codes: (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A). (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). (C) Assertion (A) is true, but Reason (R) is false. (D) Assertion (A) is false, but Reason (R) is true.

1. (A)
2. (B)
3. (C)
4. (D)

Assertion (A): Every quadratic equation has two real roots. Reason (R): A quadratic polynomial can have at most two zeroes. Select the correct answer from the codes given.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

Assertion (A): Every quadratic equation has two real roots. Reason (R): A quadratic polynomial can have at most two zeroes.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

The quadratic equation whose roots are 7 and 1/7 is:

1. 7x² - 50x + 7 = 0
2. 7x² - 50x + 1 = 0
3. 7x² + 50x - 7 = 0
4. 7x² + 50x - 1 = 0

A quadratic equation whose roots are α and 1/α respectively is:

1. ax² - ax + 1 = 0
2. ax² - a²x + 1 = 0
3. ax² + ax + 1 = 0
4. ax² + a²x - 1 = 0

If (x-6)/(x-3) = 0, then the values of x are:

1. ±6
2. ±4
3. ±12
4. ±3

The roots of the quadratic equation x² + x – p (p + 1) = 0 are:

1. p, p + 1
2. –p, p + 1
3. –p, –(p + 1)
4. p, –(p + 1)

The quadratic equation x² + x + 1 = 0 has ______ roots.

1. real and equal
2. irrational
3. real and distinct
4. not-real

Value of k for which x = 2 is a solution of the equation 5x² – 4x + (2 + k) = 0, is

1. 10
2. –10
3. 14
4. –14

The roots of the quadratic equation 4x² – 5x + 4 = 0 are

1. irrational
2. rational and distinct
3. not real
4. rational and equal

If the discriminant of the quadratic equation 3x² – 2x + c = 0 is 16, then the value of c is:

1. 1
2. 0
3. –1
4. 2

If y = 1 is one of the solutions of the quadratic equation py² + py + 3 = 0, then the value of p is:

1. –3
2. 2
3. –3/2
4. –2

The ratio of the sum and product of the roots of the quadratic equation 5x² – 6x + 21 = 0 is:

1. 5 : 21
2. 2 : 7
3. 21 : 5
4. 7 : 2

If the quadratic equation ax² + bx + c = 0 has real and equal roots, then the value of c is:

1. a²/b
2. –a²/b
3. a⁴/b²
4. –a⁴/b²

Which of the following quadratic equations has –1 as a root?

1. x² – 4x – 5 = 0
2. –x² – 4x + 5 = 0
3. x² + 3x + 4 = 0
4. x² – 5x + 6 = 0

Which of the following quadratic equations has sum of its roots as 4 ?

1. 2x² - 4x + 8 = 0
2. x² + 4x + 4 = 0
3. 2x² - 24x + 1 = 0
4. 4x² - 4x + 4 = 0

The roots of the equation x² + 3x – 10 = 0 are :

1. 2, –5
2. –2, 5
3. 2, 5
4. –2, –5

A quadratic equation whose roots are (2 + √3) and (2 - √3) is :

1. x² - 4x + 1 = 0
2. x² + 4x + 1 = 0
3. 4x² - 3 = 0
4. x² - 1 = 0

A quadratic equation whose roots are (3 - √2) and (3 + √2) is :

1. x² - 6x + 7 = 0
2. x² + 6x + 7 = 0
3. 9x² - 2 = 0
4. x² - 7 = 0

For what value of k does the quadratic equation 4x² + kx + 1 = 0 have real and equal roots?

(a) 4x² + kx + 1 = 0 has real and equal roots. OR (b) If α and β are zeroes of p(y) = y² − 5y + 3, then find the value of 4/α³ + 3/β⁴.

Find the sum and product of the roots of the quadratic equation 2x² – 9x + 4 = 0.

(A) Find the sum and product of the roots of the quadratic equation 2x² – 9x + 4 = 0. OR (B) Find the discriminant of the quadratic equation 4x² – 5 = 0 and hence comment on the nature of roots of the equation.

For what value of m, the quadratic equation mx² - 2(m - 1)x + (m + 2) = 0 has two real and equal roots?

Find the nature of the roots of the quadratic equation x² - 5x + 9 = 0.

Three consecutive integers are such that sum of the square of second and product of other two is 161. Find the three integers.

Find the value of 'p' for which the quadratic equation px(x – 2) + 6 = 0 has two equal real roots.

Find the value of 'p' for which one root of the quadratic equation px² – 14x + 8 = 0 is 6 times the other.

Find the value of m for which the quadratic equation (m - 1) x² + 2 (m - 1) x + 1 = 0 has two real and equal roots.

Rohan's father is 5 years older than Rohan's mother. Rohan's mother is 3 times as old as Rohan. The product of Rohan's mother's age 5 years from now and Rohan's age 7 years from now is one more than twice his present age. Find his present age.

Solve the quadratic equation for x: x² - 2ax - (4b² - a²) = 0

Solve the quadratic equation 2x² - 5x - 1 = 0 for x.

Solve the quadratic equation x² + 3x - 9 = 0 for x.

For what value of p, does the quadratic equation px² + 2x + p = 0 have real and equal roots?

ABCD is a rectangle of dimensions 80 cm × 60 cm. Another rectangle PQRS is drawn inside ABCD leaving space of equal width x cm along the edges of ABCD. If area PQRS is half of the area ABCD, then find the value of x.

(a) ABCD is a rectangle of dimensions 80 cm × 60 cm. Another rectangle PQRS is drawn inside ABCD leaving space of equal width x cm along the edges of ABCD. If area PQRS is half of the area ABCD, then find the value of x. OR (b) A train covers a distance of 90 km at a uniform speed. Had the speed been 15 km/h more, it would have taken 30 minutes less for the same journey. Find the original speed of the train.

The sum of numerator and denominator of a fraction is 4 less than twice the denominator. If each of the numerator and denominator is decreased by 1, the fraction becomes 1/3. Find the fraction.

Solve for x: (x – 2)/(x – 3) + (x – 4)/(x – 5) = 10/3 ; x ≠ 3, 5

(a) The difference of the squares of two positive numbers is 180. The square of the smaller number is 8 times the greater number. Find the two numbers. OR (b) Find the value(s) of k for which the equation 2x² + kx + 3 = 0 has real and equal roots. Hence, find the roots of the equations so obtained.

The difference of the squares of two positive numbers is 180. The square of the smaller number is 8 times the greater number. Find the two numbers.

(a) The sum of areas of two squares is 2650 cm². If the sum of their perimeters is 280 cm, find the sides of the two given squares.

(a) The sum of areas of two squares is 2650 cm². If the sum of their perimeters is 280 cm, find the sides of the two given squares. OR (b) Express the equation (x - 0, 2) as a quadratic equation in standard form. Hence, find the roots of the quadratic equation so obtained.

Find two consecutive odd integers, sum of whose squares is 290.

(a) The perimeter of a right triangle is 60 cm and its hypotenuse is 25 cm. Find the lengths of other two sides of the triangle. OR (b) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. Find the speed of the train.

A two-digit number is such that the product of its digits is 12. When 36 is added to this number, the digits interchange their places. Find the number.

The perimeter of a right triangle is 60 cm and its hypotenuse is 25 cm. Find the lengths of other two sides of the triangle.

The sum of the areas of two squares is 52 cm² and difference of their perimeters is 8 cm. Find the lengths of the sides of the two squares.

(a) The numerator of a fraction is 3 less than its denominator. If 2 is added to both numerator and denominator, then the sum of the new fraction and the original fraction is 29/20. Find the original fraction.

The sum of two numbers is 18 and the sum of their reciprocals is 1/4. Find the numbers.

In a flight of 2800 km, an aircraft was slowed down due to bad weather. Its average speed is reduced by 100 km/h and by doing so, the time of flight is increased by 30 minutes. Find the original duration of the flight.

(a) A train travels a distance of 90 km at a constant speed. Had the speed been 15 km/h more, it would have taken 30 minutes less for the journey. Find the original speed of the train. OR (b) Find the value of 'c' for which the quadratic equation (c + 1)x² – 6(c + 1)x + 3(c + 9) = 0; c ≠ –1 has real and equal roots.

(a) Find the value of 'k' for which the quadratic equation (k + 1) x² – 2(3k + 1) x + (8k + 1) = 0 has real and equal roots.

(a) A train travels a distance of 90 km at a constant speed. Had the speed been 15 km/h more, it would have taken 30 minutes less for the journey. Find the original speed of the train.

(a) Find the value of 'k' for which the quadratic equation (k + 1)x² – 6(k + 1)x + 3(k + 9) = 0, k ≠ –1 has real and equal roots.

A 2-digit number is such that the product of the digits is 14. When 45 is added to the number, the digits are reversed. Find the number.

A cottage industry produces a certain number of pottery articles in a day. It was observed that on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was ₹ 90, find the number of articles produced and the cost of each article.

A train travels at a certain average speed for a distance of 54 km and then travels a distance of 63 km at an average speed of 6 km/h more than the first speed. If it takes 3 hours to complete the journey, what was its first average speed ?

(a) A train travels at a certain average speed for a distance of 54 km and then travels a distance of 63 km at an average speed of 6 km/h more than the first speed. If it takes 3 hours to complete the journey, what was its first average speed ? OR (b) Two pipes together can fill a tank in 8/15 hours. The pipe with larger diameter takes 2 hours less than the pipe with smaller diameter to fill the tank separately. Find the time in which each pipe can fill the tank separately.

The sum of first n terms of an A.P. is 502. If the first and the last terms are 2 and 192 respectively, then the value of n is:

1. 10
2. 5
3. 15
4. 20

nth term of an A.P. is 5n – 15. The common difference of the A.P. is:

1. 5n
2. 5
3. – 5
4. 10

nth term of the A.P.: 4/3, –1/3, 3, ... is

1. 9/3 – 5n
2. 6/3 – 5n
3. 4/3 – 3n
4. 2/3 + 3n

If –26, x, 2 are in A.P., then the value of x is

1. 14
2. –13
3. –12
4. –14

If 16/9, –2/9, –20/9, .... are in A.P., then next term of the sequence is

1. 32/9
2. 46/9
3. 2/9
4. 34/9

If 14th term of an A.P. is 4 and its 15th term is zero, then its first term is

1. –48
2. –56
3. 56
4. 48

The 31st term of the A.P. : –5/6, –3/4, –2/3, –7/12, … is

1. –5/3
2. 5/3
3. 12/20
4. –12/20

If a, b, c and d are consecutive terms of an A.P., then c – b is equal to :

1. d – a
2. d – b
3. d – c
4. c – a

The nth term of an A.P. is 3n + 2. The common difference is:

1. 8
2. 2
3. 5
4. 3

If aₙ represents nth term of the A.P. –15/4, –10/4, –5/4, …… then value of a₁₆ – a₁₂ is

1. 4
2. 5/4
3. 5
4. 25/4

If the sum of first m terms of an AP is 2m² + 3m, then its second term is:

1. 10
2. 9
3. 12
4. 4

The 11th and 13th term of an AP are 39 and 45, respectively. What is the common difference of the AP?

1. 42
2. 21
3. 6
4. 3

The 10th term of the AP 5, 7/2, 4, 5/2, ... is:

1. -5/2
2. -1/2
3. 1/2
4. 5/2

Assertion (A): Common difference of the AP: 5, 1, -3, -7, ... is -4. Reason (R): Common difference of the AP: a₁, a₂, a₃, ..., aₙ is obtained by d = aₙ - aₙ₋₁.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

The 9th term from the end (towards first term) of the AP

1. 135
2. 125
3. 115
4. 39

Assertion (A): Common difference of the AP: 5, 1, -3, ... is 4. Reason (R): Common difference of the AP: a₁, a₂, a₃, ... aₙ is obtained by d = aₙ - aₙ₋₁.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

The 10th term of the AP 5, ..., ..., 10

1. Option A
2. Option B
3. Option C
4. Option D

The sum of the first three terms of an AP is 30 and the sum of the last three terms is 36. If the first term is 9, then the number of terms is:

1. 10
2. 5
3. 6
4. 13

Assertion (A): The sum of the first fifteen terms of the AP 21, 18, 15, 12, … is zero. Reason (R): The sum of the first n terms of an AP with first term 'a' and common difference 'd' is given by Sₙ = n/2 [a + (n – 1) d].

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

The common difference of an A.P. in which a₁₅ – a₁₁ = 48, is

1. 12
2. 16
3. –12
4. –16

Which term of the A.P. –29, –26, –23, ....., 61 is 16?

1. 11th
2. 16th
3. 10th
4. 31st

The sum of first 200 natural numbers is

1. 2010
2. 2000
3. 20100
4. 21000

The common difference of an A.P. in which a₂₀ – a₁₅ = 20, is

1. 4
2. 5
3. 4d
4. 5d

The common difference of the A.P. 1/(2x), (1 – 4x)/(2x), (1 – 8x)/(2x), .............. is:

1. –2x
2. –2
3. 2
4. 2x

If the mean of five observations x, x + 2, x + 4, x + 6 and x + 8 is 11, then the value of x is:

1. 4
2. 7
3. 11
4. 6

If the sum of first n terms of an A.P. is 3n² + 4n and its common difference is 6, then its first term is:

1. 7
2. 4
3. 6
4. 3

The common difference of the A.P. 1/(12x), (1 – 4x)/(2x), (1 – 8x)/(2x), ................ is:

1. –2x
2. –2
3. 2
4. 2x

Three numbers in A.P. have the sum 30. What is its middle term?

1. 4
2. 10
3. 16
4. 8

If the first three terms of an A.P. are 3p – 1, 3p + 5, 5p + 1 respectively; then the value of p is:

1. 2
2. –3
3. 4
4. 5

The next (4th) term of the A.P. 98, 50, 18 … is:

1. 128
2. 140
3. 162
4. 200

The number of terms in the A.P. 3, 6, 9, 12, …, 111 is:

1. 36
2. 40
3. 37
4. 30

The next (4th) term of the A.P. ..., 63, 28, 7 is:

1. 70
2. 84
3. 97
4. 112

If the mean of the first n natural numbers is 9n/5, then the value of n is:

1. 5
2. 4
3. 9
4. 10

The 14th term from the end of the A.P. –11, –8, –5, ..., 49 is:

1. 7
2. 10
3. 13
4. 28

If the first term of an AP is –3 and common difference –2, then the seventh term is

1. –9
2. 9
3. –17
4. –15

The next term of the A.P. : 6, 24, 54 is :

1. 60
2. 96
3. 72
4. 216

Assertion (A): a, b, c are in A.P. if and only if 2b = a + c. Reason (R): The sum of first n odd natural numbers is n².

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true and Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true but Reason (R) is false.
4. Assertion (A) is false but Reason (R) is true.

If k + 2, 4k – 6 and 3k – 2 are three consecutive terms of an A.P., then the value of k is:

1. 3
2. – 3
3. 4
4. – 4

The next term of the A.P. : 7, 28, 63 is :

1. 70
2. 80
3. 97
4. 112

The common difference of the A.P. whose nth term is given by an = 3n + 7, is :

1. 7
2. 3
3. 3n
4. 1

The 11th term from the end of the A.P. : 10, 7, 4, ......., -62 is :

1. 25
2. 16
3. -32
4. 0

The 13th term from the end of the A.P. : 20, 13, 6, 1, ....., 148 is:

1. 57
2. -57
3. 64
4. -64

The common difference of the A.P. whose nth term is given by aₙ = 5n - 7 is:

1. -7
2. 7
3. 5
4. 2

The sum of first n terms of an A.P. is given by Sₙ = 4n² – n. Find the 25th term of this A.P.

In which term are the 5th terms of the APs: 9, 7, 5, ..... and 15, 12, 9, ..... the same?

In an A.P., it is given that a = 2, d = 8 and Sn = 90. Find the value of n.

(a) How many terms of the A.P. 3, 5, 7, 9, ... must be added to get the sum 80? OR (b) Find three consecutive terms in A.P. whose sum is 21 and their product is 231.

The 4th and 10th term of an A.P. are 13 and 25 respectively. Find its 24th term.

In an A.P., 15th term exceeds the 8th term by 21. If sum of first 10 terms is 55, then form the A.P.

A sum of ₹2,000 is invested at 7% per annum simple interest. Calculate the interests at the end of 1st, 2nd and 3rd year. Do these interests form an AP? If so, find the interest at the end of the 27th year.

Find the sum of all 3-digit natural numbers which are divisible by 11.

If the sum of first m terms of an A.P. is same as sum of its first n terms (m ≠ n), then show that the sum of its first (m + n) terms is zero.

If the sum of first 7 terms of an A.P. is 49 and that of first 17 terms is 289, find the sum of its first 20 terms.

A man starts his job with a certain monthly salary and earns a fixed increment every year. If his salary was ₹15,000 after 4 years of service and ₹18,000 after 10 years of service, what was his starting salary and what was the annual increment?

(a) If the sum of the first 14 terms of an A.P. is 1050 and the first term is 10, then find the 20th term and the nth term.

How many terms are there in an A.P. whose first and fifth terms are – 14 and 2, respectively and the last term is 62.

(A) The sum of first 15 terms of an A.P. is 750 and its first term is 15. Find its 20th term. OR (B) Rohan repays his total loan of ₹1,18,000 by paying every month starting with the first instalment of ₹1,000. If he increases the instalment by ₹100 every month, what amount will be paid by him in the 30th instalment? What amount of loan has he paid after 30th instalment?

How many terms are there in A.P. whose first and fifth term are – 14 and 2, respectively and the last term is 62.

If pth term of an A.P. is q and qth term is p, then prove that its nth term is (p + q - n).

In an A.P., the sum of the first n terms is given by Sₙ = 6n - n². Find its 30th term.

Find the common difference of an A.P. whose first term is 8, the last term is 65 and the sum of all its terms is 730.

Find the sum of first 30 terms of AP: 30, 24, 18, .....

Find the sum of the first twelve 2-digit numbers which are multiples of 6.

Find the sum of first 20 terms of an AP in which d = 5 and a₂₀ = 135.

For which value of n are the nth terms of the APs: 9, 7, 5, ..... and 15, 12, 9, ..... the same?

In an AP, the sum of the first 10 terms is 1505. Find the common difference if the sum of the first 14 terms is 1505.

Which terms of the APs: 9, 7, 5, ..... and 15, 12, 9, ..... are the same?

In an AP, if a = 50, d = -4 and Sₙ = 0, then find the value of n.

For an AP with common difference 6, the sum of first ten terms is same as four times the sum of first five terms. Determine the first term of the AP.

Find the 15th term of an AP whose first term is 17 and fourth term is 44.

Find the 8th term of an AP whose first term is 24 and 11th term is 21.

The third and ninth term of an A.P. are 4 and – 8 respectively. (i) Which term of the A.P. is zero? (ii) Find the value of n if Sₙ = – 36.

The sum of the third term and the seventh term of an AP is 6 and their product is 8. Find the sum of the first sixteen terms of the AP.

The nth term is 4 and the common difference is 2. Also, find the sum of the first 20 terms.

(a) The sum of the third term and the seventh term of an AP is 6 and their product is 8. Find the sum of the first sixteen terms of the AP. OR (b) The minimum age of children eligible to participate in a painting competition is 8 years. It is observed that the age of the youngest boy was 8 years and the ages of the participants, when seated in order of age, have a common difference of 4 months. If the sum of the ages of all the participants is 168 years, find the age of the eldest participant in the painting competition.

How many terms of the arithmetic progression 45, 39, 33, ........ must be taken so that their sum is 180? Explain the double answer.

Solve the equation for x: 1 + 4 + 7 + 10 + .... + x = 287

Prerna saves ₹32 during the first month, ₹36 in the second month and ₹40 in the third month. If she continues to save in this manner, in how many months will she save ₹2,000?

Which of the following statements is not always true?

1. Two circles are similar
2. Two isosceles right triangles are similar
3. Two rectangles are similar
4. Two equilateral triangles are similar

ABCD is a trapezium in which AB||DC and E, F are points on AD and BC respectively such that EF||DC. If ED = 36 cm, BF = 70 cm and FC = 30 cm, then the length of AD is :

1. 124 cm
2. 120 cm
3. 110 cm
4. 114 cm

If DEF ~ PQR such that 3 DE = PQ and EF = 6 cm, then the length of QR is :

1. 12 cm
2. 3 cm
3. 2 cm
4. 18 cm

If in a ABC, AB = 6 cm and DE||BC such that AE = 1/3 AC, then the length of BD is

1. 2 cm
2. 3 cm
3. 4 cm
4. 5 cm

If in two triangles ABC and DEF, AB/EF = BC/DE = CA/DF; then

1. DEF ~ BCA
2. DEF ~ CBA
3. ABC ~ DEF
4. ABC ~ DFE

In ABC, P is a point on AB and Q is a point of AC such that PQ || BC. If AP : PB = 3 : 2, then PQ : BC is equal to:

1. 3 : 2
2. 2 : 5
3. 3 : 5
4. 5 : 3

Which of the following is not the criterion for similarity of triangles?

1. AAA
2. SSS
3. SAS
4. RHS

Assertion (A): All congruent triangles are similar. Reason (R): In congruent triangles, the ratio of corresponding sides is 1 : 1.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

Assertion (A): All congruent triangles are similar. Reason (R): In congruent triangles, the ratio of corresponding sides is 1 : 1. Select the correct answer from the codes: (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A). (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). (C) Assertion (A) is true, but Reason (R) is false. (D) Assertion (A) is false, but Reason (R) is true.

1. (A)
2. (B)
3. (C)
4. (D)

Which types of triangles are always similar?

1. Right-angled triangles
2. Acute-angled triangles
3. Isosceles triangles
4. Equilateral triangles

In triangles ABC and DEF, ∠B = ∠E, ∠F = ∠C and AB = 3 DE. Then, the two triangles are:

1. congruent but not similar
2. congruent as well as similar
3. neither congruent nor similar
4. similar but not congruent

The points (-5, 0), (5, 0) and (0, 4) are the vertices of a triangle which is a/an:

1. right-angled triangle
2. isosceles triangle
3. equilateral triangle
4. scalene triangle

Which of the following statements is incorrect?

1. Two congruent figures are always similar.
2. A square and a rhombus of the same area are always similar.
3. Two equilateral triangles are always similar.
4. Two similar triangles need not be congruent.

If in two triangles DEF and PQR, angle D = angle Q and angle R = angle E, then which of the following is not true?

1. DE/PQ = DF/PR
2. EF/QR = DE/PQ
3. DE/QR = DF/PQ
4. EF/PR = DE/PQ

If in triangles ABC and PQR, AB/QR = BC/PR, then they will be similar, when:

1. ∠B = ∠Q
2. ∠A = ∠R
3. ∠B = ∠R
4. ∠C = ∠Q

A line l intersects the sides PQ and PR of a △PQR at L and M respectively such that LM ∥ QR. If PL = 5·7 cm, PQ = 15·2 cm and MR = 5·5 cm, then the length of PM (in cm) is:

1. 3
2. 1·8
3. 2·5
4. 3·3

The perimeters of two similar triangles ABC and PQR are 56 cm and 48 cm respectively. PQ/AB is equal to

1. 7/8
2. 6/7
3. 7/6
4. 8/7

AB and CD are two chords of a circle intersecting at P. Choose the correct statement from the following:

1. ΔADP ~ ΔCBA
2. ΔADP ~ ΔBPC
3. ΔADP ~ ΔBCP
4. ΔADP ~ ΔCBP

If the diagonals of a quadrilateral divide each other proportionally, then it is a:

1. parallelogram
2. rectangle
3. square
4. trapezium

Assertion (A): A line drawn parallel to any one side of a triangle intersects the other two sides in the same ratio. Reason (R): Parallel lines cannot be drawn to any side of a triangle.

1. Both Assertion (A) and Reason (R) are correct and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are correct but Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

If △ABC ~ △PQR, A = 32° and R = 65°, then the measure of ∠B is:

1. 32°
2. 65°
3. 83°
4. 97°

In ∆ ABC, PQ || BC. If PB = 6 cm, AP = 4 cm, AQ = 8 cm, find the length of AC.

1. 12 cm
2. 20 cm
3. 6 cm
4. 14 cm

In triangles ABC and DEF, if FD || BC and DE || AB. Which of the following makes the two triangles similar ?

1. ∠A = ∠D
2. ∠B = ∠D
3. ∠B = ∠E
4. ∠A = ∠F

In triangles ABC and DEF, DE || AB and FD || BC. Which of the following makes the two triangles similar?

1. ∠A = ∠D
2. ∠B = ∠D
3. ∠B = ∠E
4. ∠A = ∠F

In triangles ABC and DEF, ∠F = ∠B and DE/AB = EF/BC. Which of the following makes the two triangles similar ?

1. ∠A = ∠D
2. ∠B = ∠D
3. ∠B = ∠E
4. ∠A = ∠F

D is a point on side BC of triangle ABC such that angle ADC = angle BAC. Prove that (CA)² = CB · CD.

D is a point on side BC of triangle ABC such that ∠ADC = ∠BAC. Prove that (CA)² = CB . CD.

If △ABC ~ △PQR in which AB = 6 cm, BC = 4 cm, AC = 8 cm and PR = 6 cm, then find the length of (PQ + QR).

In a △ABC, D and E are points on the sides AB and AC respectively such that BD = CE. If ∠B = ∠C, then show that DE ∥ BC.

S is any point on the side QR of a triangle PQR such that ∠PSR = ∠QPR. Prove that QR/RP = RP/RS.

Prove that a rectangle circumscribing a circle is a square.

Prove that, if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

D is the mid-point of side BC of ABC. CE and BF intersect at O, a point on AD. AD is produced to G such that OD = DG. Prove that (i) OBGC is a parallelogram. (ii) EF || BC (iii) AEF ~ ABC

Prove that: A line through the mid-point of one side of a triangle, parallel to another side, bisects the third side.

Prove the following: Line segment joining mid-points of any two sides of a triangle is parallel to the third side.

(a) The diagonal BD of a parallelogram ABCD intersects the line segment AE at the point F, where E is any point on the side BC. Prove that DF · EF = FB · FA. OR (b) In △ABC, if AD ⊥ BC and AD² = BD · DC, then prove that ∠BAC = 90°.

The diagonal BD of a parallelogram ABCD intersects the line segment AE at the point F, where E is any point on the side BC. Prove that DF × EF = FB × FA.

The perimeter of an isosceles triangle is 32 cm. If each equal side is 5/6 th of the base, find the area of the triangle.

ABCD is a trapezium with AB ∥ DC. AC and BD intersect at E. If △AED ~ △BEC, then prove that AD = BC.

If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides are divided in the same ratio.

State and prove Basic Proportionality theorem.

Through the mid-point M of the side CD of a parallelogram ABCD, the line BM is drawn intersecting AC in L and AD produced in E. Prove that EL = 2BL.

In △ABC, if AD ⊥ BC and AD² = BD × DC, then prove that ∠BAC = 90°.

(a) E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that ∆ ABE ~ ∆ CFB. OR (b) Sides AB, BC and the median AD of ∆ ABC are respectively proportional to sides PQ, QR and the median PM of another ∆ PQR. Prove that ∆ ABC ~ ∆ PQR.

(a) E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that ∆ABE ~ ∆CFB.

Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of another triangle PQR. Show that △ABC ~ △PQR.

Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of triangle PQR. Show that ∠ABC ∼ ∠PQR.

D is a point on the side BC of a triangle ABC such that ∠ADC = ∠BAC, prove that CA² = CB.CD

In a ∆PQR, N is a point on PR, such that QN ⊥ PR. If PN × NR = QN², prove that ∠PQR = 90°.

A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC are of lengths 10 cm and 8 cm respectively. Find the lengths of the sides AB and AC, if the area of triangle ABC is 90 cm².

The distance between the points (– 4, 2) and (1, 0) is:

1. √13 units
2. 3 units
3. 9 units
4. √29 units

The distance between the points (– 5, 1) and (2, 2) is:

1. 2√5 units
2. 10 units
3. 5√2 units
4. 3√2 units

The distance between the points (–1, 22) and (2, 2) is:

1. 5 units
2. 11 units
3. 13 units
4. 7 units

The distance between the points (–2, 5) and (5, –2) is

1. 7√2
2. 14
3. 2√7
4. 7

The distance between the points (–4, 5) and (–1, 2) is

1. 5
2. 3√2
3. 6
4. 2√3

The fourth vertex D of a parallelogram ABCD whose three vertices are A(–4, 1), B(4, 5) and C(6, 1) is:

1. (–2, –3)
2. (3, –2)
3. (0, –1)
4. (0, 1)

The ratio in which the x-axis divides the line segment joining the points A(–8, 4) and B(–6, –2) is:

1. 5 : 1
2. 3 : 1
3. 2 : 1
4. 1 : 2

The line segment joining the points P(–4, –2) and Q(10, 4) is divided by y-axis in the ratio

1. 2 : 5
2. 1 : 2
3. 2 : 1
4. 5 : 2

For a point (3, -5), the value of (abscissa - ordinate) is:

1. -8
2. -2
3. 2
4. 8

The mid-point of a line segment divides the line segment in the ratio:

1. 1 : 2
2. 2 : 1
3. 1 : 1
4. 3 : 2

The distance of a point P(3, –7) from y-axis is :

1. 3
2. 7
3. –7
4. 10

For a point X(a, b) where (b > a > 0), the value of its [distance from x-axis - distance from y-axis] is:

1. a - b
2. b - a
3. a² - b²
4. b² - a²

The distance of the point (2, 3) from the origin is:

1. 2
2. 3
3. 5
4. √13

The mid-point of the line segment joining points (1, 3) and (1, -3) lies:

1. at the origin
2. in the second quadrant
3. on x-axis
4. on y-axis

If point (1, 2) is the mid-point of the line segment joining the points (3, 5) and (2a, b), then (a, b) =

1. (1, 1)
2. (-1, -1)
3. (2, 2)
4. (-1, 1)

The distance of point (3, 4) from y-axis is:

1. 3
2. -3
3. 4
4. 5

If point (a, 2b) is the mid-point of the line segment joining the points (3, 5) and (1, 1), then (a, b) is equal to:

1. (1, 2)
2. (2, 2)
3. (2, 1)
4. (1, 1)

The mid-point of the line segment joining the points P(−4, 5) and Q(4, 6) lies on:

1. x-axis
2. y-axis
3. origin
4. neither x-axis nor y-axis

The end points of a diameter of circle are (2, 4) and (−3, −1). The length of its radius is:

1. √26 units
2. √(26/2) units
3. √(25/2) units
4. 5√2 units

The coordinates of the end points of a diameter of a circle are (5, -2) and (5, 2). The length of the radius of the circle is:

1. ± 2
2. ± 4
3. 4
4. 2

The line represented by the equation x - y = 0 is:

1. parallel to x-axis
2. parallel to y-axis
3. passing through the origin
4. passing through the point (3, 2)

The equation of a line parallel to the x-axis and at a distance of 3 units below x-axis is:

1. x = 3
2. x = -3
3. y = -3
4. y = 3

The perimeter of the triangle formed by the vertices (0, 0), (2, 0) and (0, 2) is:

1. 4 units
2. 6 units
3. 6√2 units
4. (4 + 2√2) units

The equation of a line parallel to y-axis and at a distance of 5 units to the right of y-axis is:

1. x = 5
2. x = -5
3. y = 5
4. y = -5

The line represented by x/2 + y/3 = 1 intersects x-axis and y-axis respectively at P and Q. The coordinates of the mid-point of line segment PQ are:

1. (2, 3)
2. (3, 2)
3. (2, 0)
4. (0, 3)

Two of the vertices of triangle PQR are P(1, 5) and Q(5, 2). The coordinates of a point which divides PQ in the ratio 2 : 1 are:

1. (3, -3)
2. (5, 5)
3. (3, 3)
4. (5, 1)

AOBC is a rectangle whose three vertices are A(0, 2), O(0, 0) and B(4, 0). The square of the length of its diagonal is equal to:

1. 36
2. 20
3. 16
4. 4

If the mid-point of the line segment joining the points (a, 4) and (2, 2b) is (2, 6), then the value of (a + b) is given by:

1. 6
2. 7
3. 8
4. 16

Two of the vertices of △PQR are P(1, 5) and Q(5, 2). The coordinates of a point which divides PQ in the ratio 2 : 1 are:

1. (3, −3)
2. (5, 5)
3. (3, 3)
4. (5, 1)

The line represented by x/a + y/b = 1, intersects x-axis and y-axis respectively at P and Q. The coordinates of the mid-point of line segment PQ are:

1. (2, 3)
2. (3, 2)
3. (2, 0)
4. (0, 3)

The perpendicular bisector of the line segment joining the points A(–1, 3) and B(2, 4) cuts the y-axis at:

1. (0, 5)
2. (0, –5)
3. (0, 4)
4. (0, –4)

Point P divides the line segment joining the points A(4, –5) and B(1, 2) in the ratio 5:2. Co-ordinates of point P are

1. (5/2, –3/2)
2. (11/7, 0)
3. (13/7, 0)
4. (0, 13/7)

XOYZ is a rectangle with vertices X(–3, 0), O(0, 0), Y(0, 4) and Z(x, y). The length of its each diagonal is

1. 5 units
2. √5 units
3. x² + y² units
4. 4 units

The distance between the points (a cos θ, – a sin θ) and (a sin θ, a cos θ) is

1. a
2. a²
3. 0
4. 2a

Assertion (A): The point which divides the line segment joining the points A (1, 2) and B(–1, 1) internally in the ratio 1 : 2 is (–1/3, 5/3). Reason (R): The coordinates of the point which divides the line segment joining the points A (x₁, y₁) and B(x₂, y₂) in the ratio m₁ : m₂ are ((m₁x₂ + m₂x₁)/(m₁ + m₂), (m₁y₂ + m₂y₁)/(m₁ + m₂))

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

Two lines are given to be parallel. The equation of one of these lines is 5x – 3y = 2. The equation of the second line can be:

1. –15x – 9y = 5
2. 15x + 9y = 5
3. 9x – 15y = 6
4. –15x + 9y = 5

The fourth vertex D of a parallelogram ABCD whose three vertices are A(–2, 3), B(6, 7) and C(8, 3) is:

1. (0, 1)
2. (0, –1)
3. (–1, 0)
4. (1, 0)

The distance of the point (3, 4) from the origin is

1. 25
2. 5
3. 7
4. 1

Assertion (A): The point (0, 4) lies on y–axis. Reason (R): The x-coordinate of a point, lying on y–axis, is zero.

1. Both Assertion (A) and Reason (R) are correct and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are correct but Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

The coordinates of the vertex A of a rectangle ABCD whose three vertices are given as B(0, 0), C(3, 0) and D(0, 4) are:

1. (4, 0)
2. (0, 3)
3. (3, 4)
4. (4, 3)

The pair of equations x = a and y = b graphically represents lines which are:

1. parallel
2. intersecting at (b, a)
3. coincident
4. intersecting at (a, b)

The area of the triangle formed by the line ax + by = 1 with the coordinate axes is:

1. ab
2. 1/(2ab)
3. 1/(4ab)
4. 2ab

The ratio in which the x-axis divides the line segment joining the points (2, 3) and (6, 7) is:

1. 1:3
2. 3:7
3. 7:3
4. 1:2

The distance of the point (–1, 7) from x-axis is :

1. –1
2. 7
3. 6
4. 50

The distance of the point (– 6, 8) from origin is :

1. 6
2. – 6
3. 8
4. 10

The points (– 4, 0), (4, 0) and (0, 3) are the vertices of a :

1. right triangle
2. isosceles triangle
3. equilateral triangle
4. scalene triangle

The distance between the points P(5, 3√11) and Q(5, 3√2) is :

1. 6 units
2. 4 units
3. 2 units
4. 3 units

Assertion (A): If the points A(4, 3) and B(x, 5) lie on a circle with centre O(2, 3), then the value of x is 2. Reason (R): Centre of a circle is the mid-point of each chord of the circle.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

If end points of a diameter of a circle are (5, 4) and (1, 0), then the radius of the circle is:

1. √13/2 units
2. √13 units
3. 2√4 units
4. 2√2 units

A (– 2, 3) and B (4, 1) are end points of the diameter of a semi-circle. The semi-circle intersects y-axis at point P. Find the co-ordinates of the point P.

The vertices of a ABC are A(–1, 3), B(2, –3) and C(4, 5). Find the coordinates of a point P on median AD such that AP : PD = 2 : 3.

If A(a, 0), B(1, 1) and C(0, b) form a triangle, right angled at B when joined, then establish a relation between a and b.

Vertices of a right triangle ABC with ∠B = 90° are A(3, 4), B(1, 1) and C(–8, 7). Find the value of tan A.

The coordinates of the centre of a circle are (2a, a − 7). Find the value(s) of a if the circle passes through the point (11, −9) and has diameter 10 units.

The coordinates of the centre of a circle are (2a, a − 7). Find the value(s) of a if the circle passes through the points (−9, 9) and has diameter 10 units.

Find the length of the median through the vertex B of △ABC with vertices A(9, −2), B(−3, 7) and C(1, 10).

Find the type of triangle ABC formed whose vertices are A(1, 0), B(–5, 0) and C(–2, 5).

In what ratio is the line segment joining the points (3, –5) and (–1, 6) divided by the line y = x?

The vertices of a ΔABC are A(–2, 4), B(4, 3) and C(1, –6). Find length of the median BD.

Find a relation between x and y such that the point P(x, y) is equidistant from the points A(7, 1) and B(3, 5).

(a) Find the ratio in which the point P(–4, 6) divides the line segment joining the points A(–6, 10) and B(3, –8).

Show that the points (-2, 3), (8, 3) and (6, 7) are the vertices of a right-angled triangle.

The line segment joining the points A(4, -5) and B(4, 5) is divided by the point P such that AP : AB = 2 : 5. Find the coordinates of P.

Find the ratio in which y-axis divides the line segment joining the points (5, -6) and (-1, -4).

Find the ratio in which line y = x divides the line segment joining the points (6, 3) and (1, 6).

(a) The line segment joining the points A(4, -5) and B(4, 5) is divided by the point P such that AP : AB = 2 : 5. Find the coordinates of P. OR (b) Point P(x, y) is equidistant from points A(5, 1) and B(1, 5). Prove that x = y.

The vertices of a rhombus ABCD are A(– 3, – 4), B(5, – 3), C(1, 4) and D(– 7, 3). Find the length of both the diagonals. Hence, find area of the rhombus ABCD.

Points P(6, 0), Q(2, 8) and R(–2, 4) are vertices of △PQR. It is given that MN || QR such that PM/MQ = 1/3. Using distance formula and ratio formula, show that MN/QR = 1/4.

Determine the ratio in which the line 2x + y = 6 divides the line segment joining the points (1, 3) and (2, 5).

Find the coordinates of the points of trisection of the line segment joining the points P(5, –4) and Q(–4, 2).

Find a relation between x and y such that the point P(x, y) is equidistant from the points A(5, 3) and B(1, 7).

A circle centered at (2, 1) passes through the points A(5, 6) and B(–3, K). Find the value(s) of K. Hence find length of chord AB.

Find the ratio in which the y-axis divides the line segment joining the points (5, −6) and (−1, −4). Also find the point of intersection.

If the mid-point of the line segment joining the points A(3, 4) and B(k, 6) is P(x, y) and x + y - 10 = 0, then find the value of k.

If (a, b) is the mid-point of the line segment joining the points A(10, -6) and B(k, 4) and a - 2b = 18, then find the value of k.

If the mid-point of the line segment joining the points A(3, 4) and B(k, 6) is P(x, y) and x + y - 10 = 0, find the value of k.

(a) If the mid-point of the line segment joining the points A(3, 4) and B(k, 6) is P(x, y) and x + y − 10 = 0, find the value of k. OR (b) Find the coordinates of the points which divide the line segment joining A(−2, 2) and B(2, 8) into four equal parts.

Find the ratio in which the line segment joining the points (5, 3) and (–1, 6) is divided by Y-axis.

In what ratio does the X-axis divide the line segment joining the points (2, –3) and (5, 6)? Also, find the coordinates of the point of intersection.

If Q(0, 1) is equidistant from P(5, -3) and R(x, 6), find the values of x.

Draw a line segment AB = 7 cm. Divide it in the ratio 3 : 2.

Draw a line segment AB of length 8 cm and locate a point P on AB such that AP : PB = 1 : 5.

Draw a line segment PQ = 7·5 cm. Divide it in the ratio 3 : 1.

The vertices of a quadrilateral ABCD are A(6, –2), B(9, 2), C(5, –1) and D(2, –5). Prove that ABCD is a rhombus, and not a square.

The value of sin 90° cos 90° – sin² 60° is:

1. 1/4
2. 3/4
3. –3/4
4. 5/4

If sin θ = 1/7, then tan θ is:

1. 1/(4√3)
2. 1/(2√3)
3. (4√3)/7
4. 6/7

The value of tan 30° tan 60° – sin 90° cos 90° is:

1. 1
2. 0
3. 2
4. not defined

The value of 2 tan 45° sin² 60° – cos 90° is:

1. –1/4
2. 3/2
3. 3/4
4. 1/4

If value of cot θ is 5, then sin θ equals

1. 1/√6
2. √6
3. 5/√6
4. 1/2

Assertion (A): For an acute angle θ, cos θ is always less than 1. Reason (R): In a right-angled triangle, hypotenuse is the longest side and cos θ = Base/Hypotenuse.

1. Both, Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A)
2. Both, Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A)
3. Assertion (A) is true, but Reason (R) is false
4. Assertion (A) is false, but Reason (R) is true

If sin θ = 1/11, then cot θ equals

1. 11/√10
2. √10/11
3. √10
4. 11

The length of a pendulum is 70 cm and it describes an arc of length 88 cm when swings. The angle subtended by the arc at the centre is

1. 36°
2. 70°
3. 72°
4. 80°

If xtan 45sin 30 = cos 30cot 60, then x is equal to

1. 3
2. 1/3
3. 1
4. 1/2

sin 2θ = 2sinθ is true, when θ is equal to

1. 90°
2. 60°
3. 45°
4. 0°

If cos A = 3/5, then the value of tan A is:

1. 4/5
2. 5/4
3. 3/4
4. 4/3

If sin θ = √3/2, then the value of 2√3 cos² θ is:

1. 3
2. 2√3
3. √3/2
4. √3

For an acute angle θ, if sin θ = 1/9, then value of (9 cosec θ + 1)/(9 cosec θ – 1) is

1. 0
2. 80/81
3. 1
4. 82/80

Simplest form of sec A/√(sec²A – 1) is

1. sin A
2. tan A
3. cosec A
4. cos A

Which of the following statements is false?

1. tan 45° = cot 45°
2. sin 90° = tan 45°
3. sin 30° = cos 30°
4. sin 45° = cos 45°

The value of sin² 45° + cos² 45° is:

1. more than 1
2. 1
3. 0
4. -1

The value of sin² 30° + cos² 30° is:

1. more than 1
2. 1
3. 0
4. -1

The value of sin 30° + cos 60° is:

1. 3/2
2. 1/2
3. 1
4. √3/2

The value of (sin²A + cos²A) + (sec²A - tan²A) - (cot²A - cosec²A) is:

1. 1
2. -1
3. 3
4. 2

If angle of elevation changes from 30° to 60°, the length of shadow will:

1. become shorter
2. become longer
3. remain same
4. be doubled

(sin θ + cos θ)² + (sin θ - cos θ)² =

1. 1
2. 2
3. 2 + 2 sin θ cos θ
4. 2 + 4 sin θ cos θ

The value of sin²30° + cos²60° is:

1. 3
2. 1/2
3. 1
4. 3/4

(sec θ - cos θ)² + sin²θ - tan²θ = ?

1. 0
2. 1
3. 2
4. 4

Assertion (A): For sin θ = 1, cos θ must be 0. Reason (R): sin²θ - cos²θ = 1. Select the correct answer from the codes: (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A). (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A). (C) Assertion (A) is true, but Reason (R) is false. (D) Assertion (A) is false, but Reason (R) is true.

1. (A)
2. (B)
3. (C)
4. (D)

Assertion (A): For an acute angle θ, cot θ = 1 implies cosec θ = √2. Reason (R): cosec² θ - cot² θ = 1. Select the correct answer from the codes given.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

The value of θ for which sin 2θ = tan 45° is:

1. 22·5°
2. 30°
3. 45°
4. 90°

Assertion (A): For an angle θ, sec θ = 1 and tan θ = 0. Reason (R): sec2θ + tan2θ = 1.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

If θ is an acute angle and 7 + 4 sin θ = 9, then the value of θ is:

1. 90°
2. 30°
3. 45°
4. 60°

The value of tan² 45° − cos² 60° is:

1. 1
2. 0
3. −1
4. 2

If α + β = 90° and tan α = 2, then cos² β + sin² α is equal to:

1. 0
2. 1/2
3. 1
4. 2

The value of tan² 45° − cot² 60° is:

1. 1
2. 0
3. -1
4. 2

The value of (tan A cosec A)² − (sin A sec A)² is:

1. 0
2. 1
3. -1
4. 2

If 7 cos² θ + 3 sin² θ = 4, then the value of θ is:

1. 30°
2. 45°
3. 60°
4. 90°

If sin 30° tan 45° = k/2, then the value of k is:

1. 4
2. 3
3. 2
4. 1

If sin 30° / tan 45° = k, then the value of k is:

1. 4
2. 3
3. 2
4. 1

If tan 3θ = 1, then θ equals:

1. 60°
2. 30°
3. 20°
4. 10°

(cot θ + tan θ) equals:

1. cosec θ sec θ
2. sin θ sec θ
3. cos θ tan θ
4. sin θ cos θ

If sin 4θ = 1, then θ equals:

1. 60°
2. 20°
3. 15°
4. 5°

If tan 3θ = √3, then θ equals:

1. 60°
2. 30°
3. 20°
4. 10°

If sin²θ − 3sin θ = 0, then the values of θ are:

1. ±6°
2. ±4°
3. ±12°
4. ±3°

If tan A = 3 cot A, then the measure of the angle A is:

1. 15°
2. 30°
3. 45°
4. 60°

(sec θ + tan θ) (1 – sin θ) is equal to:

1. sec θ
2. sin θ
3. cosec θ
4. cos θ

If sin θ = cos θ, (0° < θ < 90°), then value of (sec θ · sin θ) is:

1. 1/2
2. 2
3. 1
4. 0

Assertion (A): If sin A = 1/3 (0° < A < 90°), then the value of cos A is 2√2/3. Reason (R): For every angle θ, sin² θ + cos² θ = 1.

1. Both Assertion (A) and Reason (R) are true. Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true. Reason (R) does not give correct explanation of (A).
3. Assertion (A) is true but Reason (R) is not true.
4. Assertion (A) is not true but Reason (R) is true.

Assertion (A): If sin A = 1/3 (0° < A < 90°), then the value of cos A is 2√2/3. Reason (R): For every angle θ, sin² θ + cos² θ = 1. Select the correct option.

1. Both Assertion (A) and Reason (R) are true. Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true. Reason (R) does not give correct explanation of (A).
3. Assertion (A) is true but Reason (R) is not true.
4. Assertion (A) is not true but Reason (R) is true.

If cos θ = √3/2 and sin φ = 1/2, then tan (θ + φ) is:

1. √3
2. 1/√3
3. 1
4. not defined

If sin α = √3/2, cos β = √3/2, then tan α × tan β is:

1. √3
2. 1/√3
3. 1
4. 0

If sin θ = 1, then the value of 2^(1/2) sin(θ/2) is:

1. 2^(1/2)/2
2. 1/2
3. √(1/2)
4. 0

At some time of the day, the length of the shadow of a tower is equal to its height. Then, the Sun's altitude at that time is:

1. 30°
2. 45°
3. 60°
4. 90°

The value of θ for which 2 sin² θ = 1/2; 0° ≤ θ ≤ 90° is:

1. 30°
2. 60°
3. 45°
4. 90°

If 3x = 2 sin A, 3y = 2 cos A, then the value of x² + y² is:

1. 36
2. 9
3. 6
4. 18

If 4 sec θ – 5 = 0, then the value of cot θ is:

1. 4/3
2. 5/4
3. 3/5
4. 3/4

If x = a cos θ and y = b sin θ, then the value of b²x² + a²y² is:

1. a²b²
2. ab
3. a⁴b⁴
4. a² + b²

If 5 tan θ – 12 = 0, then the value of sin θ is:

1. 12/5
2. 13/12
3. 13/5
4. 5/12

tan²A + cot²A + 1 is equal to

1. sec²A
2. –1
3. cot²A
4. tan²A

If sin²θ = 3/4, then θ is

1. 30°
2. 45°
3. 60°
4. 90°

Which of the following is true for all values of θ (0° ≤ θ ≤ 90°)?

1. cos²θ – sin²θ = 1
2. cosec²θ – sec²θ = 1
3. sec²θ – tan²θ = 1
4. cot²θ – tan²θ = 1

(sec² θ – 1) (cosec² θ – 1) is equal to :

1. – 1
2. 1
3. 0
4. 2

If tan θ = 12/5, then the value of (cos θ - sin θ)/(cos θ + sin θ) is :

1. 7/17
2. -7/17
3. 13/17
4. 1/3

tan 30° × tan 130° - tan 22° is equal to :

1. sin 60°
2. cos 60°
3. tan 60°
4. sin 30°

tan 30° × tan 13° / tan 122° is equal to:

1. sin 60°
2. cos 60°
3. tan 60°
4. cos 30°

If sec θ - tan θ = 1/3, then the value of (sec θ + tan θ) is :

1. 3/4
2. 3/2
3. 3/1
4. 3

tan 30° × tan 13° × tan 22° × tan 60° is equal to :

1. sin 60°
2. cos 60°
3. tan 60°
4. sin 30°

If sin (A – B) = 1/2 and tan (A + B) = √3, 0° ≤ A + B < 90°, A > B then find the values of A and B.

If sin 2A = √3/2 and 2 tan B + 1 = 3, then find the value of (A + B).

If sin A = 1/2 and tan B = √3, then verify that cos (A + B) = cos A cos B – sin A sin B.

If sec A = 2 and tan B = 3, then find the value of 2 sin A cos B.

(a) Evaluate: sin 30º/5 – tan 345º sec² 30º + sin 22/7π (b) For A = 60º and B = 30º, verify that tan(A – B) = (tan A – tan B)/(1 + tan A tan B).

If 7 sin2 A + 3 cos2 A = 4, then find the value of tan A.

(a) Find the values of A and B (0° ≤ A < 90°, 0° ≤ B < 90°), if tan (A + B) = 1 and tan (A - B) = 1/√3. OR (b) Prove that tan 45° = 1 geometrically.

Find the values of A and B (0° ≤ A < 90°, 0° ≤ B < 90°), if tan (A + B) = 1 and tan (A – B) = 1/√3.

Evaluate: (sin 30° + cos 30°)² + (sin 60° - cos 60°)²

Evaluate: 2sin²45° - cos²45°

If sin 3A = 1, then find the value of cos 2A - tan² 45°.

(a) If x cos 60° + y cos 0° + sin 30° − cot 45° = 5, then find the value of x + 2y. OR (b) Evaluate: sin² 30° + cos² 45° + tan² 60°

If x cos 60° + y cos 0° + sin 30° − cot 45° = 5, then find the value of x + 2y.

If tan A = 3/4, where A is an acute angle, then find the value of sin A + cos A.

If tan A = 3/4; where A is an acute angle, then find the value of sec A.

If tan A = √3/4; where A is an acute angle, then find the value of cosec A.

If 4k = tan² 60° - 2 cosec² 30° - 2 tan² 30°, then find the value of k.

If tan A + cot A = 6, then find the value of tan²A + cot²A − 4.

If tan θ + sec θ = m, then prove that sec θ = (m² + 1)/(2m).

Evaluate: 2 sin² 30° sec 60° + tan² 60°.

Evaluate: (cos 45° + sin 60°)/(sec 30° + cosec 30°).

Evaluate: (sec² 45° – tan² 45°)/sin² 45°

Evaluate: 5 tan 60°(sin² 60° + cos² 60°) / tan 30°

(a) Evaluate: (5 cos² 60° + 4 sec² 30° – tan² 45°) / (sin² 60° + sin² 30°) OR (b) If sin (A – B) = 1/2, cos (A + B) = 1/2; 0 < A + B ≤ 90°, A > B; find ∠A and ∠B.

(a) Evaluate: (5·2² cos² 60° + 4·sec² 30° – tan² 45°) / (sin² 60° + sin² 30°)

(a) Evaluate: 5 cos² 60° + 4 sec² 30° – tan² 45° / sin² 30° + sin² 60°

Evaluate: (2 tan 30° . sec 60° . tan 45°)² / (1 – sin 60°)

Evaluate: (2 tan 30° × sec 60° × tan 45°)/(1 – sin 60°)²

If 4 cot²45° - sec²60° + sin²60° + p = 4/3, then find the value of p.

Evaluate (5cos²60° + 4sec²30° - tan²45°) / (sin²30° + cos²30°)

(A) Evaluate: 5 + cot²30° + sin²60° – cot²45° + 2sin²90° / 2 OR (B) If θ is an acute angle and sin θ = cos θ, find the value of tan²θ + cot²θ – 2.

Evaluate 2sec² θ + 3cosec² θ – 2sin θ cos θ if θ = 45°.

If a cos θ + b sin θ = m and a sin θ - b cos θ = n, then prove that a² + b² = m² + n².

(a) If a cos θ + b sin θ = m and a sin θ - b cos θ = n, then prove that a² + b² = m² + n². OR (b) Prove that: (1 - sec A)/(tan A) - (tan A)/(1 - sec A) = 2 cosec A

Prove that (cot A – cos A)/(cot A + cos A) = (sec A – tan A)/(sec A + tan A)

Prove that (sec θ + tan θ)² – 1 / (sec θ + tan θ)² + 1 = sin θ.

Prove that (1 – sin A)/(1 + sin A) = (sec A – tan A)².

Prove that: tan² θ + cot² θ + 2 = sec² θ cosec² θ.

Prove that: (1 - cos A)/(1 + cos A) = (cosec A - cot A)².

If cos A + sin A = √2 cos A, prove that cos A – sin A = √2 sin A.

If cos θ + sin θ = √2 cos θ, then prove that cos θ – sin θ = √2 sin θ.

Evaluate: (sin³60° – tan 30°)/cos²45°

Prove that: tan θ/(1 – cot θ) + cot θ/(1 – tan θ) = 1 + tan θ + cot θ

Prove the following trigonometric identity: (1 – cos A)/sin A = sin A/(1 + cos A) = sec A – tan A

Prove the following trigonometric identity: (sin A - cosec A)(cos A - sec A) = 1/(sin A cos A) - sin A cos A

(b) x and y are complementary angles such that x : y = 1 : 2. Express the given information as a system of linear equations in two variables and hence solve it.

Prove the following trigonometric identity: (sin θ + cos θ)² + (sin θ - cos θ)² = 2

Prove the following trigonometric identity: = cot A

(a) Prove that: tan θ + (1/tan θ) = 1 + sec θ cosec θ OR (b) Prove that: (sin θ/(1 − cos θ)) + (sin θ/(1 + cos θ)) = 2 cosec θ

Prove that: (sin A − cos A + 1)/(sin A + cos A − 1) = 1/(sec A − tan A) or equivalently prove that cot A + cosec A − 1 = cot A(cosec A − 1).

Prove that: (sin θ)/(1 − cos θ) + (1 − cos θ)/(sin θ) = 1 + sec θ cosec θ

Prove that: (sin θ + cos θ)² + (sin θ - cos θ)² = 2

(a) If tan θ + sin θ = m and tan θ - sin θ = n, then prove that m² - n² = 4√(mn).

(a) Prove that: sin²θ/(1 - sin²θ) = tan²θ + cot²θ

Prove that: sin A/(1 + cos A) + (1 + cos A)/sin A = 2 cosec A

(a) Prove that: sin A/(1 − cos A) + tan A/(sin A) = 2 cosec A OR (b) Prove that: (sin θ − cos θ + 1)/(sin θ + cos θ − 1) = (1 + sin θ)/cos θ

Prove that: (1 – tan θ)/cot θ + (1 – cot θ)/tan θ = 1 + tan θ + cot θ

Prove that (1 + sec θ – tan θ)/(1 + sec θ + tan θ) = (1 – sin θ)/cos θ.

Prove that sin⁶ θ + cos⁶ θ = 1 – 3 sin² θ cos² θ.

Prove that (cosec² θ – sec² θ)/(cosec² θ + sec² θ) = 3/4, if tan θ = 1/√7.

Prove that (sin θ – 2 sin³ θ)/(2 cos³ θ – cos θ) = tan θ.

Prove that (sin A + cos A)/(sin A – cos A) + (sin A – cos A)/(sin A + cos A) = 2/(2sin²A – 1)

Prove that: (cosec θ – sin θ) (sec θ – cos θ) (tan θ + cot θ) = 1

Prove that (sin θ – cos θ + 1) / (sin θ + cos θ – 1) = 1 / (sec θ – tan θ)

If sin θ + cos θ = p and sec θ + cosec θ = q, then prove that q (p² – 1) = 2p.

Prove that sec² θ + cosec² θ = tan θ + cot θ.

Prove that: (tan A/(1 - cot A)) + (cot A/(1 - tan A)) = 1 + sec A cosec A

Prove that: (sin θ + 1 + cos θ)/(1 + cos θ + sin θ) = 2 cosec θ

Prove that: (tan θ – cot θ) / (sin θ cos θ) = sec² θ – cosec² θ

Prove that (tan θ - cot θ)/(tan θ + cot θ) = 1 + sec θ cosec θ

Prove that: (cos θ)/(cos θ - 1) - (sin θ)/(sin θ - 1) = cot θ - tan θ.

Prove that: cos θ/(1 - sin θ) + sin θ/(cos θ - 1) = 2 cosec θ

Prove that: tan θ/(1 - cos²θ) + cos θ/sin θ · sin³θ = 1 + sin θ/cos θ

Prove that (sin³A - 2sinA) / (2cosA - cos³A) = tanA

(A) Prove that (sin³A – 2sin A) / (2cos A – cos³A) = tanA OR (B) Prove that sec A (1 – sin A) (sec A + tan A) = 1.

Prove that: cot θ/(1 - tan θ) + tan θ/(1 - cot θ) = 1 + sec θ cosec θ

Prove that: (sec A - 1)/(tan A) - (tan A)/(sec A - 1) = 2 cosec A

The length of a pendulum is 70 cm and it describes an arc of length 88 cm when swings. The angle subtended by the arc at the centre is

1. 36°
2. 70°
3. 72°
4. 80°

The angle of elevation from 30° to 60° will:

1. become shorter
2. become longer
3. remain same
4. be doubled

A kite is flying at a height of 150 m from the ground. It is attached to a string inclined at an angle of 30° to the horizontal. The length of the string is:

1. 100 m
2. 300 m
3. 150 m
4. 150√2 m

A ladder 14 m long leans against a wall. If the foot of the ladder is 7 m from the wall, then the angle of elevation of the top of the wall is:

1. 15°
2. 30°
3. 45°
4. 60°

Assertion (A): A ladder leaning against a wall, stands at a horizontal distance of 6 m from the wall. If the height of the wall up to which the ladder reaches is 8 m, then the length of the ladder is 10 m. Reason (R): The ladder makes an angle of 60° with the ground.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

A ladder 14 m long just reaches the top of a vertical wall. If the ladder makes an angle of 60° with the wall, then the height of the wall is:

1. 14√3 m
2. 7 m
3. 14 m
4. 7√3 m

From a point on the ground, which is 30 m away from the foot of a vertical tower, the angle of elevation of the top of the tower is found to be 60°. The height (in metres) of the tower is:

1. 10√3
2. 30√3
3. 60
4. 30

At some time of the day, the length of the shadow of a tower is equal to its height. Then, the Sun's altitude at that time is:

1. 30°
2. 45°
3. 60°
4. 90°

If a vertical pole of length 7·5 m casts a shadow 5 m long on the ground and at the same time, a tower casts a shadow 24 m long, then the height of the tower is:

1. 20 m
2. 40 m
3. 60 m
4. 36 m

The ratio of the length of a pole and its shadow on the ground is 1 : √3. The angle of elevation of the Sun is:

1. 90°
2. 60°
3. 45°
4. 30°

The length of the shadow of a tower on the plane ground is 3 times the height of the tower. The angle of elevation of the Sun is:

1. 30°
2. 45°
3. 60°
4. 90°

A pole 7√3 m high casts a shadow 21 m long on the ground, then the sun's elevation is:

1. 30°
2. 45°
3. 60°
4. 90°

The hour-hand of a clock is 6 cm long. The angle swept by it between 7:20 a.m. and 7:55 a.m. is:

1. 43.5°
2. 23.5°
3. 35°
4. 70°

If a pole 6 m high casts a shadow 2√3 m long on the ground, then sun's elevation is :

1. 60°
2. 45°
3. 30°
4. 90°

Water in a river which is 3 m deep and 40 m wide is flowing at the rate of 2 km/h. How much water will fall into the sea in 2 minutes ?

1. 800 m³
2. 4000 m³
3. 8000 m³
4. 2000 m³

A vertical pole of height 10 m casts a shadow of 15 m on the ground and at the same time, a tower casts a shadow of 45 m on the ground. Find the height of the tower.

The length of the shadow of a tower on the plane ground is 3 times the height of the tower. Find the angle of elevation of the sun.

Chord AB of a circle subtends an angle of 120º at the centre O of the circle. Find the length of arc AB, if radius of the circle is 21 cm.

The angle of elevation of the top of a building from the foot of the tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 50 m high, then find the height of the building.

The tops of two poles of heights 20 m and 28 m are connected with a wire. The wire is inclined to the horizontal at an angle of 30°. Find the length of the wire and the distance between the two poles.

A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle of 30° with it. The height of the breaking point from the ground is 2 m. Find the total height of the tree.

A tower stands vertically on the ground. From a point on the ground, which is 15 m away from the foot of the tower, the angle of elevation of the top of the tower is 60°. Find the height of the tower.

The angle of elevation of the top of a building from the foot of the tower is 30° and the angle of elevation of the top of the tower from the foot of the building is 60°. If the tower is 60 m high, find the height of the building and distance between the building and the tower. (Use √3 = 1.73)

A statue, 2 m tall, stands on the top of a pedestal. From a point on the ground, the angle of elevation of the top of the statue is 60° and from the same point the angle of elevation of the bottom of the statue is 45°. Find the height of the pedestal and its distance from the point of observation on the ground. (use √3 = 1.73)

A straight road leads to foot of a tower whose shadow is found to be 40 m longer when sun's altitude is 30° than when it is 60°. Find the height of the tower and length of shadow in both the situations. (Use √3 = 1.73)

Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60° and 30° respectively. Find the height of the poles and the distance of the point from the poles.

Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60º and 30º respectively. Find the height of the poles and the distance of the point from the poles.

Two poles of equal heights are standing opposite to each other on either side of the road which is 90 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 30° and 60° respectively. Find the height of the poles and the distances of the point from the poles. [Use √3 = 1.732]

The angle of elevation of the top of a tower as observed from a point in a horizontal plane through the foot of the tower is 30°. When the observer moves towards the tower a distance of 100 m, he finds the angle of elevation of the top to be 60°. Find the height of the tower and the distance of first position from the tower.

Two ships are sailing in the sea on either side of a lighthouse. The angles of depression to the two ships as observed from the top of the lighthouse are 60° and 45°, respectively. If the distance between the ships is 100√3 m, then find the height of the lighthouse.

(a) The angle of elevation of an airborne helicopter from a point A on the ground is 45°. After a flight of 15 seconds, the angle of elevation of the helicopter changes to 30°. If the helicopter is flying at a constant height of 2000 m, find the speed of the helicopter. (Take √3 = 1·732)

From the top of a 45 m high light house, the angles of depression of two ships, on the opposite side of it, are observed to be 30° and 60°. If the line joining the ships passes through the foot of the light house, find the distance between the ships. (Use √3 = 1.73)

The angle of elevation of an aircraft from a point A on the ground is 60°. After a flight of 30 seconds, the angle of elevation changes to 30°. The aircraft is flying at a constant height of 3500√3 m at a uniform speed. Find the speed of the aircraft.

A person standing on the bank of a river observes that the angle of elevation of the top of a tower on the opposite bank is 60°. When he moves 30 m away from the bank, he finds the angle of elevation to be 30°. Find the height of the tower and width of the river. (Take √3 = 1.732)

From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a 20 m high building are 45° and 60° respectively. Find the height of the tower.

The angle of elevation of a jet plane from a point A on the ground is 60°. After a flight of 30 seconds, the angle of elevation changes to 30°. If the jet plane is flying at a constant height of 3600√3 m, find the speed of the jet plane.

A man on a cliff observes a boat at an angle of depression of 30° which is approaching the shore to the point immediately beneath the observer with a uniform speed. Six minutes later, the angle of depression of the boat is found to be 60°. Find the time taken by the boat from here to reach the shore.

Two pillars of equal lengths stand on either side of a road which is 100 m wide, exactly opposite to each other. At a point on the road between the pillars, the angles of elevation of the tops of the pillars are 60° and 30°. Find the length of each pillar and distance of the point on the road from the pillars. (Use √3 = 1.732)

The angles of depression of the top and the bottom of a 50 m high building from the top of a tower are 45° and 60°, respectively. Find the height of the tower. (Use √3 = 1·73)

The angles of depression of the top and the bottom of a 8 m tall building from the top of a multi-storeyed building are 30° and 45° respectively. Find the height of the multi-storeyed building and the distance between the two buildings.

From a point on a bridge across the river, the angles of depressions of the banks on opposite sides of the river are 30° and 60° respectively. If the bridge is at a height of 4 m from the banks, find the width of the river.

From a window 15 metres high above the ground in a street, the angles of elevation and depression of the top and the foot of another house on the opposite side of the street are 30° and 45° respectively. Find the height of the opposite house. (Use √3 = 1·732)

From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 45°. Determine the height of the tower. (Use √3 = 1.732)

As observed from the top of a 75 m high lighthouse from the sea-level, the angles of depression of two ships are 30° and 60°. If one ship is exactly behind the other on the same side of the lighthouse, find the distance between the two ships. (Use √3 = 1.73)

(a) The angle of elevation of the top of a tower 30 m high from the foot of another tower in the same plane is 60° and the angle of elevation of the top of the second tower from the foot of the first tower is 30°. Find the distance between the two towers and also the height of the other tower.

A straight highway leads to the foot of a tower. A man standing on the top of the 75 m high tower observes two cars at angles of depression of 30° and 60°, which are approaching the foot of the tower. If one car is exactly behind the other on the same side of the tower, find the distance between the two cars. (use √3 = 1.73)

(A) A straight highway leads to the foot of a tower. A man standing on the top of the 75 m high tower observes two cars at angles of depression of 30° and 60°, which are approaching the foot of the tower. If one car is exactly behind the other on the same side of the tower, find the distance between the two cars. (use √3 = 1.73) OR (B) From the top of a 7 m high building, the angle of elevation of the top of a cable tower is 60° and the angle of depression of its foot is 30°. Determine the height of the tower.

One observer estimates the angle of elevation to the basket of a hot air balloon to be 60°, while another observer 100 m away estimates the angle of elevation to be 30°. Find: (a) The height of the basket from the ground. (b) The distance of the basket from the first observer. (c) The horizontal distance of the second observer from the basket.

The angle of elevation of the top of a vertical tower from a point P on the ground is 60°. From another point Q, 10 m vertically above the first point P, its angle of elevation is 30°. Find: (a) The height of the tower. (b) The distance of the point P from the foot of the tower. (c) The distance of the point P from the top of the tower.

Two pillars are standing on either side of a 80 m wide road. Height of one pillar is 20 m more than the height of the other pillar. From a point on the road between the pillars, the angle of elevation of the higher pillar is 60°, whereas that of the other pillar is 30°. Find the position of the point between the pillars and the height of each pillar. (Use √3 = 1·73)

A straight highway leads to the foot of a tower. A man standing at the top of the tower observes a car at an angle of depression of 30°, which is approaching the foot of the tower with a uniform speed. Ten seconds later, the angle of depression of the car is found to be 60°. Find the time taken by the car to reach the foot of the tower from this point.

The angle of elevation of an aeroplane from a point on the ground is 60°. After a flight of 30 seconds, the angle of elevation from the same point becomes 30°. If the aeroplane is flying at a constant height of 3000√3 m, find the speed of the aeroplane.

The angle of elevation of the top Q of a vertical tower PQ from a point X on the ground is 60°. From a point Y, 40 m vertically above X, the angle of elevation of the top Q of tower PQ is 45°. Find the height of the tower PQ and the distance PX. [Use √3 = 1·73]

PT is tangent to the circle with centre O and radius 5 cm. OP intersects the circle at Q. If PQ = x, then PT² equals:

1. x² + 5x
2. x² + 10x + 50
3. x² + 10x
4. x² – 25

A chord QR subtends an angle of 105° at the centre O of the circle. The measure of ∠RQP is

1. 75°/2
2. 105°/2
3. 75°
4. 15°

PQ is tangent to a circle at a point P on the circle. The number of tangents which can be drawn to the circle parallel to PQ, is

1. 2
2. 1
3. many
4. zero

A chord QR subtends an angle of 105º at the centre O of the circle. The measure of ∠RQP is

1. 75º/2
2. 105º/2
3. 75º
4. 15º

PQ is tangent to the circle with centre O such that OP = 2OQ. m∠OPQ is

1. 15°
2. 60°
3. 45°
4. 30°

Four tangents drawn to a circle are extended from both the sides to form a quadrilateral. Which of these quadrilateral is not possible ?

1. Trapezium
2. Square
3. Rectangle
4. Rhombus

From a point P, tangents PQ and PR are drawn to a circle with centre O and radius 6 cm. If OP = 10 cm, then area of quadrilateral PQOR is :

1. 48 cm2
2. 24 cm2
3. 96 cm2
4. 72 cm2

If two tangents inclined at an angle of 60° are drawn from an external point to a circle of radius 6 cm, then the length of each tangent is:

1. 3√3 cm
2. 6 cm
3. 12 cm
4. 6√3 cm

In two concentric circles with centre O, the radius of outer circle is 25 cm. Chord PQ of the outer circle is tangent to the inner circle at R. If PQ = 14 cm, then the radius of the inner circle is:

1. √429 cm
2. 24 cm
3. √674 cm
4. 20 cm

If the distance of a tangent to a circle from its centre is 4 cm, then the length of diameter of the circle is :

1. 2 cm
2. 4 cm
3. 8 cm
4. 16 cm

In two concentric circles, a tangent to the smaller circle will intersect the larger circle at:

1. zero point
2. one point
3. two points
4. three points

The tangents drawn at the extremities of the diameter of a circle are always:

1. parallel
2. perpendicular
3. equal
4. intersecting

If the length of a chord of a circle is equal to its radius, then the angle subtended by chord at the centre is:

1. 60°
2. 30°
3. 120°
4. 90°

The coordinates of the end points of a diameter of a circle are (5, -2) and (5, 2). The length of the radius of the circle is:

1. ±2
2. ±4
3. 4
4. 2

If tangents PA and PB drawn from an external point P to the circle with centre O are inclined to each other at an angle of 80°, then the measure of angle POA is:

1. 40°
2. 50°
3. 60°
4. 80°

Assertion (A): If two tangents are drawn to a circle from an external point, then they subtend equal angles at the centre of the circle. Reason (R): A parallelogram circumscribing a circle is a rhombus.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

If tangents PA and PB drawn from an external point P to the circle with centre O are inclined to each other at an angle of 80°, then the measure of ∠POA is:

1. 40°
2. 50°
3. 60°
4. 80°

If two tangents inclined at an angle of 60° are drawn to a circle of radius 5 cm, then the length of each tangent is:

1. 5√3/2 cm
2. 10 cm
3. 3/5 cm
4. 5√3 cm

AB and CD are two chords of a circle intersecting at P. Choose the correct statement from the following:

1. △ADP ~ △CBA
2. △ADP ~ △BPC
3. △ADP ~ △BCP
4. △ADP ~ △CBP

Maximum number of common tangents that can be drawn to two circles intersecting at two distinct points is:

1. 4
2. 3
3. 2
4. 1

If an arc subtends an angle of 90° at the centre of a circle, then the ratio of its length to the circumference of the circle is:

1. 2 : 3
2. 1 : 4
3. 4 : 1
4. 1 : 3

The area of the square inscribed in a circle of radius 2√5 cm is:

1. 50 cm²
2. 100 cm²
3. 25 cm²
4. 200 cm²

The length of an arc of a circle with radius 12 cm is 10π cm. The angle subtended by the arc at the centre of the circle, is:

1. 120°
2. 6°
3. 75°
4. 150°

A chord of a circle of radius 10 cm subtends a right angle at its centre. The length of the chord (in cm) is:

1. 2√5
2. 2√10
3. √5
4. 5

A line intersecting a circle in two distinct points is called a

1. secant
2. chord
3. diameter
4. tangent

Assertion (A): A tangent to a circle is perpendicular to the radius through the point of contact. Reason (R): The lengths of tangents drawn from an external point to a circle are equal.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

Assertion (A): If PA and PB are tangents drawn from an external point P to a circle with centre O, then the quadrilateral AOBP is cyclic. Reason (R): The angle between two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the centre.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

The length of tangent drawn to a circle of radius 9 cm from a point 41 cm from the centre is:

1. 40 cm
2. 9 cm
3. 41 cm
4. 50 cm

A circle is inscribed in a right triangle ABC, right angled at B. If the lengths of the two sides containing the right angle are 8 cm and 15 cm, find the radius of the incircle.

Two concentric circles are of radii 6 cm and 10 cm. Find the length of the chord of the larger circle which touches the smaller circle.

Prove that the tangent drawn at any point of the circle is perpendicular to the radius through the point of contact.

A person is standing at P outside a circular ground at a distance of 26 m from the centre of the ground. He found that his distances from the points A and B on the ground are 10 m (PA and PB are tangents to the circle). Find the radius of the circular ground.

At point A on the diameter AB of a circle of radius 10 cm, tangent XAY is drawn to the circle. Find the length of the chord CD parallel to XY at a distance of 16 cm from A.

If a hexagon ABCDEF circumscribes a circle, prove that AB + CD + EF = BC + DE + FA.

(a) If two tangents inclined at an angle of 60° are drawn to a circle of radius 3 cm, then find the length of each tangent.

Prove that the tangents drawn at the ends of a diameter of a circle are parallel to each other.

Prove that the tangents drawn at the ends of a diameter of a circle are parallel.

Prove that a parallelogram circumscribing a circle, is a rhombus.

Two tangents TP and TQ are drawn to a circle with centre O, from an external point T. Prove that ∠PTQ = 2∠OPQ.

Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

Chord AB of a circle subtends an angle of 120° at the centre O of the circle. Find the length of arc AB, if radius of the circle is 21 cm.

A circle touches the side BC of a triangle ABC at P and also touches the sides AB and AC produced at Q and R respectively. Prove that AR = 1/2 (perimeter of ABC).

A circle is inscribed in a right-angled triangle ABC, right-angled at B. If BC = 7 cm and AB = 24 cm, find the radius of the circle.

Prove that the tangents drawn at the end points of a chord of a circle makes equal angles with the chord.

From an external point P, two tangents PA and PB are drawn to the circle with centre O. Prove that OP is the perpendicular bisector of chord AB.

Prove that the parallelogram circumscribing a circle is a rhombus.

(a) In two concentric circles, a chord of length 24 cm of larger circle touches the smaller circle, whose radius is 5 cm. Find the radius of the larger circle.

Prove that the angle between the two tangents drawn from an external point to a circle is supplementary to the angle subtended by the line-segment joining the points of contact at the centre.

Two concentric circles are of radii 5 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.

From an external point, two tangents are drawn to a circle. Prove that the line joining the external point to the centre of the circle bisects the angle between the two tangents.

Two concentric circles are of radii 4 cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.

Draw a circle of radius 3 cm. Take two points P and Q on one of its extended diameter each at a distance of 7 cm from its centre. Construct tangents to the circle from these two points P and Q.

Construct a pair of tangents to a circle of radius 5 cm, which are inclined to each other at an angle of 60°.

Draw a circle of radius 3 cm. From a point P lying outside the circle at a distance of 6 cm from its centre, construct two tangents PA and PB to the circle.

Draw a circle of radius 4 cm. Construct a pair of tangents to the circle from a point 6 cm away from its centre.

Draw a circle of radius 3 cm. Take a point P at a distance of 8 cm from the centre of the circle. Construct a pair of tangents from the point P to the circle.

Draw two concentric circles of radii 5 cm and 2 cm. From a point P on the outer circle, construct a pair of tangents to the inner circle.

Draw a circle of radius 2.5 cm. From a point P lying outside the circle at a distance of 6 cm from the centre of the circle, construct tangents PA and PB to the circle.

PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP = 13 cm, then find the length AB and PA.

Two circles with centres O and O' of radii 6 cm and 8 cm, respectively intersect at two points P and Q such that OP and O'P are tangents to the two circles. Find the length of the common chord PQ.

A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC are of lengths 10 cm and 8 cm respectively. Find the lengths of the sides AB and AC, if it is given that the area of triangle ABC is 90 cm².

(a) A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC are of lengths 10 cm and 8 cm respectively. Find the lengths of the sides AB and AC, if the area of triangle ABC is 90 cm². OR (b) Two circles with centres O and O' of radii 6 cm and 8 cm, respectively intersect at two points P and Q such that OP and O'P are tangents to the two circles. Find the length of the common chord PQ.

Chord AB subtends an angle of 40° at the centre of the circle of radius 9 cm. The length of arc AB is:

1. 22/7 cm
2. 198/7 cm
3. 44/7 cm
4. 54/7 cm

An arc of length 11 cm subtends an angle of 105° at the centre of the circle. The radius of the circle is:

1. 8 cm
2. 4√3 cm
3. 6 cm
4. 7 cm

The perimeter of sector of a circle of radius 21 cm and central angle 60, is

1. 22 cm
2. 44 cm
3. 64 cm
4. 273 cm

Assertion (A) : In a circle of radius 21 cm, an arc of length 22 cm subtends an angle of 60° at the centre. Reason (R) : The length of arc of a sector of a circle of radius r and central angle θ is 2πrθ/360.

1. Both, Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).
2. Both, Assertion (A) and Reason (R) are true, but Reason (R) is not correct explanation for Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

The area (in cm²) of a sector of a circle of radius 14 cm cut off by an arc of length 22 cm is:

1. 77
2. 308
3. 154
4. 462

The minute hand of a clock is 21 cm long. The distance covered by the tip of minute hand from 2:10 pm to 2:25 pm is:

1. 346.5 cm
2. 33 cm
3. 66 cm
4. 16.5 cm

An arc of length 2.2 cm subtends an angle θ at the centre of the circle with radius 2.8 cm. The value of θ is

1. 50°
2. 60°
3. 45°
4. 30°

The ratio of the area of a quadrant of a circle to the area of the same circle is:

1. 1 : 2
2. 2 : 1
3. 1 : 4
4. 4 : 1

The perimeter of a quadrant of a circle of radius r is:

1. πr²
2. πr² + 2r
3. πr/2 + 2r
4. πr + 2r

The ratio of the area of a quadrant of a circle to the area of a circle is:

1. 2 : 1
2. 1 : 2
3. 1 : 4
4. 4 : 1

The angle of the sector of a circle whose area is one-eighth of the area of the circle is:

1. 30°
2. 45°
3. 60°
4. 90°

The perimeter of a quadrant of a circle of circumference 22 cm is:

1. 29 cm
2. 22 cm
3. 12.5 cm
4. 5.5 cm

If a sector of a circle has an area of 40 sq. units and a central angle of 72°, the radius of the circle is:

1. 200 units
2. 100 units
3. 20 units
4. 10√2 units

A piece of wire 20 cm long is bent into the form of an arc of a circle of radius √(60/π) cm. The angle subtended by the arc at the centre of the circle is:

1. 30°
2. 60°
3. 90°
4. 50°

A piece of wire 20 cm long is bent into the form of an arc of a circle of radius 60 cm. The angle subtended by the arc at the centre of the circle is:

1. 30°
2. 60°
3. 90°
4. 50°

An arc of a circle is of length 5 cm and the sector it bounds has an area of 20 cm². Its radius is:

1. 10 cm
2. 1 cm
3. 5 cm
4. 8 cm

If a large circular pizza is divided into 5 equal sectors, then the central angle of each sector will be:

1. 60°
2. 90°
3. 45°
4. 72°

If an arc of a circle of diameter 10 cm subtends an angle of 144° at the centre of the circle, then the length of the arc is:

1. 2 cm
2. 4 cm
3. 5 cm
4. 6 cm

If the area of a sector of circle of radius 36 cm is 54π cm², then the length of the corresponding arc of the sector is:

1. 8 cm
2. 6 cm
3. 4 cm
4. 3 cm

The diameter of a wheel is 63 cm. The distance travelled by the wheel in 100 revolutions is:

1. 99 m
2. 198 m
3. 63 m
4. 136 m

If the area of a sector is one-twelfth that of a complete circle, then the angle of the sector is:

1. 36°
2. 30°
3. 60°
4. 45°

The minute hand of a clock is 21 cm long. The area swept by it in 10 minutes is:

1. 121 cm²
2. 131 cm²
3. 231 cm²
4. 172·5 cm²

Perimeter of a sector of a circle whose central angle is 90° and radius 7 cm is:

1. 35 cm
2. 11 cm
3. 22 cm
4. 25 cm

If the area of a sector of a circle is 7/20 of the area of the circle, then the angle at the centre is equal to:

1. 110°
2. 130°
3. 100°
4. 126°

If an arc subtends an angle of 90° at the centre of a circle, then the ratio of its length to the circumference of the circle is:

1. 2 : 3
2. 1 : 4
3. 4 : 1
4. 1 : 3

The area of the sector of a circle of radius 12 cm is 60π cm². The central angle of this sector is:

1. 120°
2. 6°
3. 75°
4. 150°

Assertion (A): If the circumference of a circle is 176 cm, then its radius is 28 cm. Reason (R): Circumference = 2π × radius of a circle.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

The perimeter of the sector of a circle of radius 21 cm which subtends an angle of 60° at the centre of circle, is:

1. 22 cm
2. 43 cm
3. 64 cm
4. 462 cm

The length of an arc of a circle with radius 12 cm is 10π cm. The angle subtended by the arc at the centre of the circle, is:

1. 120°
2. 6°
3. 75°
4. 150°

The difference of the areas of a minor sector of angle 120° and its corresponding major sector of a circle of radius 21 cm, is

1. 231 cm²
2. 462 cm²
3. 346.5 cm²
4. 693 cm²

What is the length of the arc of the sector of a circle with radius 14 cm and of central angle 90°?

1. 22 cm
2. 44 cm
3. 88 cm
4. 11 cm

What is the area of a semi-circle of diameter 'd' ?

1. πd²/16
2. πd²/4
3. πd²/8
4. πd²/2

A chord AB of a circle of radius 14 cm makes a right angle at the centre of the circle. Find the area of the minor segment.

A chord of a circle of diameter 20 cm subtends an angle of 60° at the centre of the circle. Find the area of the corresponding minor segment of the circle. (Use π = 3.14 and √3 = 1.73)

From a circular sheet of radius 70 cm, a quadrant is cut. Find the area of the remaining sheet.

(a) The area of a smaller circle is equal to the area of a sector of a larger circle with central angle 120°. The radii of the smaller and larger circles are r and R respectively. Find the ratio r : R.

(a) The area of a smaller circle is equal to the area of a sector of a larger circle with central angle 120°. The radii of the smaller and larger circles are r and R respectively. Find the relationship between r and R.

From a circular sheet of radius 10 cm, a quadrant is cut. Find the perimeter of the remaining sheet.

A horse, a cow and a goat are tied, each by ropes of length 14 m, at the corners A, B and C respectively, of a grassy triangular field ABC with sides of lengths 35 m, 40 m and 50 m. Find the total area of grass field that can be grazed by them.

(a) The minute hand of a clock is 14 cm long. Find the area on the face of the clock described by the minute hand in 5 minutes. OR (b) Find the length of the arc of a circle which subtends an angle of 60° at the centre of the circle of radius 42 cm.

(a) The minute hand of a clock is 14 cm long. Find the area on the face of the clock described by the minute hand in 5 minutes.

A chord of a circle of radius 14 cm subtends a right angle at the centre. Find the area of the corresponding (i) minor segment (ii) major segment.

A circle of radius 12 cm is divided into 11 identical sectors. Find the total area of 7 such sectors.

Chord AB of a circle subtends an angle of 120° at the centre O of the circle. Find the length of arc AB, if radius of the circle is 21 cm.

A chord PQ of a circle of diameter 28 cm subtends an angle of 90° at the centre O. Find the area of the sector OPCQ, where C is a point on minor arc PQ. Also, find the area of segment PCQ.

A circle of diameter 20 cm is equally divided into five sectors. Find the area and perimeter of one of the sectors.

The length of the hour hand of a clock is 10 cm. Find the area of the minor sector swept by the hour hand of the clock between 5 a.m. to 8 a.m. Also, find the area of the major sector.

A chord of a circle of radius 10 cm subtends a right angle at the centre of the circle. Find the area of the corresponding minor segment. [Use π = 3·14]

An arc of a circle of radius 10 cm subtends a right angle at the centre of the circle. Find the area of the corresponding major sector. (Use π = 3·14)

A sector is cut from a circle of radius 21 cm. The central angle of the sector is 150°. Find the length of the arc of this sector and the area of the sector.

To warn ships for underwater rocks, a light house spreads a red coloured light over a sector of angle 80° to a distance of 16.5 km. Find the area of the sea over which the ships are warned. (Use π = 3.14)

A car has two wipers which do not overlap. Each wiper has a blade of length 21 cm sweeping through an angle of 120°. Find the area cleaned at each sweep of the blades.

A car has two wipers which do not overlap. Each wiper has a blade of length 21 cm sweeping through an angle of 120°. Find the total area cleaned at each sweep of the two blades.

A chord of a circle of radius 14 cm subtends an angle of 60° at the centre. Find the area of the corresponding minor segment of the circle. (Use π = 3·14 and √3 = 1·73)

The perimeter of a certain sector of a circle of radius 5.6 m is 20.0 m. Find the area of the sector.

A horse is tied to a peg at one corner of a square shaped grass field of side 15 m by means of a 5 m long rope. Find the area of that part of the field in which the horse can graze. Also, find the increase in grazing area if length of rope is increased to 10 m. (Use π = 3.14)

A cone with slant height 10 cm and radius 6 cm is surmounted on a hemisphere of same radius. The height of the toy is:

1. 12 cm
2. 16 cm
3. 14 cm
4. 10 cm

A cylinder of radius r is surmounted on a hemisphere of same radius. If total height of the object is 13 cm, then its inner surface area is

1. 2πr(r + 13)
2. 13πr
3. 2π(13 + r)²
4. 26πr

The total surface area of a solid cone of radius 7 cm and slant height 25 cm, is

1. 724 cm²
2. 704 cm²
3. 550 cm²
4. 616 cm²

The number of conical bottles of radius 2 cm and height 1.2 cm that can be filled from a cylindrical bottle of radius 6 cm and height 8 cm, full of liquid, is

1. 60
2. 130
3. 180
4. 18

Volumes of two spheres are in the ratio 64 : 27. The ratio of their surface areas is:

1. 4 : 3
2. 3 : 4
3. 16 : 9
4. 9 : 16

A conical cavity of maximum volume is carved out from a wooden solid hemisphere of radius 10 cm. Curved surface area of the cavity carved out is (use π = 3.14)

1. 314√2 cm²
2. 314 cm²
3. 3140/3 cm²
4. 3140√2 cm²

A camping tent in hemispherical shape of radius 1.4 m, has a door opening of area 0.50 m². Outer surface area of the tent is

1. 11.78 m²
2. 12.32 m²
3. 11.82 m²
4. 12.86 m²

For which of the following solids is the lateral/curved surface area and total surface area the same?

1. Cube
2. Cuboid
3. Hemisphere
4. Sphere

A cone and a cylinder have the same base radius and volume. The (height of cone : height of cylinder) is:

1. 1 : 1
2. 3 : 1
3. 1 : 3
4. 3 : 2

The total surface area of the entire solid will be:

1. πrh + πr²
2. πr + πr²
3. πr + 2πr²
4. πr + 3πr²

A cone and cylinder have same height and same radius. The volume of the cone and the volume of the cylinder are in the ratio:

1. 1 : 1
2. 1 : 3
3. 3 : 1
4. 1 : 2

Assertion (A): If we join two hemispheres of same radius along their bases, then we get a sphere. Reason (R): Total Surface Area of a sphere of radius r is 3πr².

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

Assertion (A): If we join two hemispheres of same radius along their bases, then we get a sphere. Reason (R): Total Surface Area of a sphere of radius r is 3πr². Select the correct answer from the codes (A), (B), (C) and (D) as given below. (A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A). (B) Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A). (C) Assertion (A) is true, but Reason (R) is false. (D) Assertion (A) is false, but Reason (R) is true.

1. Both A and R are true, R is the correct explanation
2. Both A and R are true, R is not the correct explanation
3. A is true, but R is false
4. A is false, but R is true

If the volumes of two cubes are in the ratio 8 : 125, then the ratio of their surface areas is:

1. 8 : 125
2. 4 : 25
3. 2 : 5
4. 16 : 25

A cone and a cylinder have the same base radius and same height. If their volumes are also same, then the height of the cone is:

1. r
2. 2r
3. 3r
4. 4r

Assertion (A): The area of canvas cloth required to just cover a heap of rice in the form of a cone of diameter 14 m and height 24 m is 175π sq.m. Reason (R): The curved surface area of a cone of radius r and slant height l is πrl.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

Assertion (A): Two cubes each of edge length 10 cm are joined together. The total surface area of newly formed cuboid is 1200 cm². Reason (R): Area of each surface of a cube of side 10 cm is 100 cm².

1. Both Assertion (A) and Reason (R) are true. Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true. Reason (R) does not give correct explanation of (A).
3. Assertion (A) is true but Reason (R) is not true.
4. Assertion (A) is not true but Reason (R) is true.

Assertion (A): Two cubes each of edge length 10 cm are joined together. The total surface area of newly formed cuboid is 1200 cm². Reason (R): Area of each surface of a cube of side 10 cm is 100 cm². Select the correct option.

1. Both Assertion (A) and Reason (R) are true. Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true. Reason (R) does not give correct explanation of (A).
3. Assertion (A) is true but Reason (R) is not true.
4. Assertion (A) is not true but Reason (R) is true.

The ratio of total surface area of a solid hemisphere to the square of its radius is:

1. 2π : 1
2. 4π : 1
3. 3π : 1
4. 1 : 4π

Two identical solid cubes of side 'a' are joined end-to-end. The total surface area of the resulting cuboid is:

1. 6a²
2. 10a²
3. 5a²
4. 4a²

The total surface area of a solid hemisphere of radius 7 cm is:

1. 98π cm²
2. 147π cm²
3. 196π cm²
4. 228(3/2)π cm²

If the area of the base of a cone is 51 cm² and its volume is 85 cm³, then the vertical height of the cone is given as :

1. 65 cm
2. 35 cm
3. 25 cm
4. 5 cm

Water in a river which is 3 m deep and 40 m wide is flowing at the rate of 2 km/h. How much water will fall into the sea in 2 minutes?

1. 800 m³
2. 4000 m³
3. 8000 m³
4. 2000 m³

A cone of height 28 cm and radius of base 7 cm is made up of modelling clay. A child reshapes it in the form of a sphere. Find the radius of the sphere.

A solid hemisphere of radius 8 cm is melted and recast into 4 identical right circular solid cones of base radius 4 cm. Find the height of the cone.

A room is in the form of a cylinder surmounted by a hemispherical dome. The base radius of the hemisphere is half of the height of the cylindrical part. If the room contains m m³ of air, find the height of the cylindrical part. (Use π = 22/7).

If the radii of the bases of a cylinder and a cone are in the ratio 3 : 4 and their heights are in the ratio 2 : 3, find the ratio of their volumes.

A room is in the form of a cylinder surmounted by a hemispherical dome. The base radius of the hemisphere is half of the height of the cylindrical part. If the room contains m³ of air, find the height of the cylindrical part. (Use π = 22/7)

A solid metallic sphere of radius 10·5 cm is melted and recast into a number of smaller cones, each of radius 3·5 cm and height 3 cm. Find the number of cones so formed.

150 spherical marbles, each of diameter 1.4 cm, are dropped in a cylindrical vessel of diameter 7 cm containing some water, and are completely immersed in water. Find the rise in the level of water in the cylindrical vessel.

A solid metallic sphere of radius 3 cm is melted and recast into the shape of a solid cylinder of radius 2 cm. Find the height of the cylinder.

From a solid cylinder whose height is 2.8 cm and radius 2.1 cm, a conical cavity of the same height and same radius is hollowed out. Find the volume and the total surface area of the remaining solid.

(a) A toy is in the form of a cone surmounted on a hemisphere. The cone and hemisphere have the same radii. The height of the conical part of the toy is equal to the diameter of its base. If the radius of the conical part is 5 cm, find the volume of the toy. OR (b) A cubical block is surmounted by a hemisphere of radius 3.5 cm. What is the smallest possible length of the edge of the cube so that the hemisphere can totally lie on the cube? Find the total surface area of the solid so formed.

A toy is in the form of a cone surmounted on a hemisphere. The cone and hemisphere have the same radii. The height of the conical part of the toy is equal to the diameter of its base. If the radius of the conical part is 5 cm, find the volume of the toy.

A hemispherical depression is scooped out from the top face of a wooden cubical block of side 14 cm. If the diameter of the hemisphere is equal to the side of the cube, find the total surface area of the remaining solid block. (Use π = 22/7)

A spherical glass vessel has a cylindrical neck which is 7 cm long and 2 cm in diameter. The diameter of the spherical part is 14 cm. Find the capacity of the entire glass vessel. (Use π = 22/7)

A perfume bottle is in the form of a cylinder but the bottom of the bottle has a hemispherical raised portion to reduce the capacity of the bottle. The inner diameter of the bottle is 5 cm and the height of the bottle is 10 cm. Find the capacity of the perfume bottle in mL. (Use π = 3.14 and 1 cm³ = 1 mL)

From a solid wooden cylinder of height 10 cm and radius 14 cm, a cylinder of radius 7 cm and height 5 cm is scooped out to form a cavity inside the solid cylinder. Find the total surface area of the remaining solid.

A necklace is made up of wooden beads. Each bead is in the form of a sphere of diameter 4·2 mm. A cylinder is hollowed out from each bead. If the radius of the cylinder is 1 mm, find the volume of wood left in each bead.

A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top, which is open, is 5 cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius 0·5 cm, are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.

A solid toy is in the form of a hemisphere surmounted by a right circular cone. The height of the cone is 2 cm and the diameter of the base is 4 cm. Determine the volume of the toy. Also, find the surface area of the toy. (Take π = 3·14)

The interior of a building is in the form of a cylinder of base radius 12 m and height 3·5 m surmounted by a cone of equal base and slant height 14 m. Find the internal curved surface area of the building.

A solid iron pole consists of a solid cylinder of height 200 cm and base diameter 28 cm, which is surmounted by another cylinder of height 50 cm and radius 7 cm. Find the mass of the pole, given that 1 cm³ of iron has approximately 8 g mass.

A tent is in the shape of a cylinder, surmounted by a conical top. If the height and diameter of the cylindrical part are 3·5 m and 6 m, and slant height of the top is 4·2 m, find the area of canvas used for making the tent. Also, find the cost of canvas of the tent at the rate of ₹500 per m².

(a) A textile industry runs in a shed. This shed is in the shape of a cuboid surmounted by a half cylinder. If the base of the industry is of dimensions 14 m × 20 m and the height of the cuboidal portion is 7 m, find the volume of air that the industry can hold. Further, suppose the machinery in the industry occupies a total space of 400 m³. Then, how much space is left in the industry?

A student was asked to make a model shaped like a cylinder with two cones attached to its ends by using a thin aluminium sheet. The diameter of the model is 3 cm and its total length is 12 cm. If each cone has a height of 2 cm, find the volume of air contained in the model.

From a solid cylinder of height 20 cm and diameter 12 cm, a conical cavity of height 8 cm and radius 6 cm is hallowed out. Find the total surface area of the remaining solid.

A wooden article was made by scooping out a hemisphere from each end of a solid cylinder. If the height of the cylinder is 10 cm and its base is of radius 3.5 cm, find the total surface area of the article.

From a solid cylinder of height 30 cm and radius 7 cm, a conical cavity of height 24 cm and same radius is hollowed out. Find the total surface area of the remaining solid.

A spherical glass vessel has a cylindrical neck 8 cm long and 1 cm in radius. The radius of the spherical part is 9 cm. Find the amount of water (in litres) it can hold, when filled completely.

While calculating mean of a grouped frequency distribution using step deviation method x̄ = a + (h/f)Σfu it was found that x̄ = 62, a = 47.5, h = 5. The value of ū is:

1. 3
2. 14.5
3. 2.9
4. 3.1

While calculating mean of a grouped frequency distribution using step deviation method x̄ = a + (h/Σf) × Σ(u × f), it was found that x̄ = 62, a = 47.5, h = 5. The value of ū is:

1. 3
2. 14.5
3. 2.9
4. 3.1

While calculating mean of a grouped frequency distribution using step deviation method x̄ = a + (h/n)Σfu, it was found that x̄ = 62, a = 47.5, h = 5. The value of ū is:

1. 3
2. 14.5
3. 2.9
4. 3.1

Assertion (A): Median of a data is the value of N/2, where N represents sum of all frequencies. Reason (R): Median divides the whole distribution in two equal parts.

1. Both, Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A)
2. Both, Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A)
3. Assertion (A) is true, but Reason (R) is false
4. Assertion (A) is false, but Reason (R) is true

If the mean of first n natural numbers is 6n/11, then n is

1. 11
2. 6
3. 12
4. 22

Assertion (A) : If the Mode and Mean of a data are 12k and 15k, then Median of the data is 14k. Reason (R) : The relation between the Mean, Mode and Median of a data is : Mean = 3 Median – 2 Mode.

1. Both, Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).
2. Both, Assertion (A) and Reason (R) are true, but Reason (R) is not correct explanation for Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

Assertion (A): If the value of mode and mean for a distribution is 50 and 56 respectively, then the value of median is 54. Reason (R): Median = 1/3 (Mode – 2 Mean)

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not correct explanation of Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

While calculating mean of a grouped frequency distribution, step deviation method was used where (x – a)/h = u. It was found that x̄ = 64, h = 5 and a = 62.5. The value of ū is

1. 0.5
2. 1.5
3. 0.3
4. 7.5

The class mark of the median class of the following data is: Class Interval: 10-25, 25-40, 40-55, 55-70, 70-85, 85-100; Frequency: 2, 3, 7, 6, 6, 6

1. 40
2. 55
3. 47.5
4. 62.5

The following distribution shows the number of runs scored by some batsmen in test matches: Runs Scored: 3000-4000, 4000-5000, 5000-6000, 6000-7000; Number of Batsmen: 5, 10, 9, 8. The lower limit of the modal class is:

1. 3000
2. 4000
3. 5000
4. 6000

In the formula of mode given by mode = l + (f - f1)/(2f - f1 - f2) × h, f1 denotes the:

1. frequency of the modal class
2. frequency of class preceding modal class
3. frequency of class succeeding modal class
4. cumulative frequency of modal class

For a distribution, if mean = median = a, then its mode is:

1. 3a
2. 2a
3. a
4. 0

For a distribution, if mean = 15 and mode = 12, then its median is:

1. 12
2. 13
3. 14
4. 15

Mode and Mean of a data are 15x and 18x, respectively. Then the median of the data is:

1. x
2. 11x
3. 17x
4. 34x

What is the mode of a data if median and mean of the same data are 9·6 and 10·5, respectively?

1. 7·8
2. 12·3
3. 8·4
4. 7

If the mean of 2, 9, x+6, 2x+3, 5, 10, 5 is 7, then the value of x is:

1. 9
2. 6
3. 5
4. 3

The mean of seven observations is 17. If the mean of the first four observations is 15 and that of the last four observations is 18, then the fourth observation is:

1. 14
2. 13
3. 12
4. 10

For the above distribution, the modal class is:

Marks: Below 10, Below 20, Below 30, Below 40, Below 50; Number of Students: 3, 12, 27, 57, 75

1. 10–20
2. 20–30
3. 30–40
4. 40–50

If value of each observation in a data is increased by 2, then median of the new data

1. increases by 2
2. increases by 2n
3. remains same
4. decreases by 2

After an examination, a teacher wants to know the marks obtained by maximum number of the students in her class. She requires to calculate _____ of marks.

1. median
2. mode
3. mean
4. range

The mean of five observations is 15. If the mean of first three observations is 14 and that of the last three observations is 17, then the third observation is

1. 20
2. 19
3. 18
4. 17

The mean of five numbers is 15. If we include one more number, the mean of six numbers becomes 17. The included number is:

1. 27
2. 37
3. 17
4. 25

If the difference of mode and median of a data is 24, then the difference of its median and mean is:

1. 12
2. 24
3. 8
4. 36

If the mean of 6, 7, p, 8, q, 14 is 9, then:

1. p – q = 19
2. p + q = 19
3. p – q = 21
4. p + q = 21

For the data 2, 9, x + 6, 2x + 3, 5, 10, 5; if the mean is 7, then the value of x is:

1. 9
2. 6
3. 5
4. 3

If the mean and mode of a data are 24 and 12 respectively, then its median is:

1. 25
2. 18
3. 20
4. 22

The empirical relation between the mode, median and mean of a distribution is :

1. Mode = 3 Median – 2 Mean
2. Mode = 3 Mean – 2 Median
3. Mode = 2 Median – 3 Mean
4. Mode = 2 Mean – 3 Median

For the following distribution: Class: 0-5, 5-10, 10-15, 15-20, 20-25 Frequency: 10, 15, 12, 20, 9 The sum of lower limits of median class and modal class is:

1. 15
2. 25
3. 30
4. 35

For the following distribution: Marks Below 10 20 30 40 50 60, Number of Students 3 12 27 57 75 80. The modal class is :

1. 10–20
2. 20–30
3. 30–40
4. 50–60

If the mean and the median of a data are 12 and 15 respectively, then its mode is :

1. 13.5
2. 21
3. 6
4. 14

If every term of the statistical data consisting of n terms is decreased by 2, then the mean of the data :

1. decreases by 2
2. remains unchanged
3. decreases by 2n
4. decreases by 1

If the mean and the mode of a distribution are 15 and 18 respectively, then the median of the distribution is :

1. 17
2. 15
3. 16
4. 18

Find the mean of the following distribution: Class 30–40 40–50 50–60 60–70 70–80; Frequency 6 13 8 12 11

Calculate the mean of the following data: Class: 4–6, 7–9, 10–12, 13–15; Frequency: 5, 4, 9, 10

In a test, the marks obtained by 100 students (out of 50) are given below: Marks obtained: 0–10, 10–20, 20–30, 30–40, 40–50 Number of students: 12, 23, 34, 25, 6 Find the mean marks of the students.

Find the mean of the following frequency distribution: Classes: 25-30, 30-35, 35-40, 40-45, 45-50, 50-55, 55-60; Frequency: 14, 22, 16, 6, 5, 3, 4

Find the mean of the following distribution: Classes: 0-15, 15-30, 30-45, 45-60, 60-75, 75-90 with Frequencies: 17, 20, 18, 21, 15, 9

Find the mode of the following frequency distribution: Class: 10-20, 20-30, 30-40, 40-50, 50-60 Frequency: 15, 10, 12, 17, 4

For what value of x, is the median of the following frequency distribution 34·5? Class: 0-10, 10-20, 20-30, 30-40, 40-50, 50-60, 60-70 Frequency: 3, 5, 11, 10, x, 3, 2

Following is the daily expenditure on lunch by 30 employees of a company: Daily Expenditure (in Rupees): 100-120, 120-140, 140-160, 160-180, 180-200 Number of Employees: 8, 3, 8, 6, 5 Find the mean daily expenditure of the employees.

If the median of the following data is 33, then find the value of m: Class: 0-15, 15-30, 30-45, 45-60, 60-75, 75-90 Frequency: 17, 35, 40, 18, m, 2

Find the mode of the given frequency distribution: Class: 15-25, Frequency: 6 Class: 25-35, Frequency: 11 Class: 35-45, Frequency: 22 Class: 45-55, Frequency: 23 Class: 55-65, Frequency: 14 Class: 65-75, Frequency: 5

The weights (in kg) of 50 wild animals of a National Park were recorded and the following data was obtained: Weight (in kg): 100-110, Number of animals: 4 Weight (in kg): 110-120, Number of animals: 12 Weight (in kg): 120-130, Number of animals: 23 Weight (in kg): 130-140, Number of animals: 8 Weight (in kg): 140-150, Number of animals: 3 Find the mean weight (in kg) of animals, using assumed mean method.

For the following frequency distribution, find the median: Class: 1400-1550, Frequency: 6 Class: 1550-1700, Frequency: 13 Class: 1700-1850, Frequency: 25 Class: 1850-2000, Frequency: 10

The percentage of marks obtained by 100 students in an examination are given below: Percentage of Marks: 30-35, Number of Students: 16 Percentage of Marks: 35-40, Number of Students: 14 Percentage of Marks: 40-45, Number of Students: 18 Percentage of Marks: 45-50, Number of Students: 20 Percentage of Marks: 50-55, Number of Students: 18 Percentage of Marks: 55-60, Number of Students: 12 Percentage of Marks: 60-65, Number of Students: 2 Determine the median percentage of marks.

Find the mean of the following frequency distribution: Class: 0-10, Frequency: 12 Class: 10-20, Frequency: 18 Class: 20-30, Frequency: 27 Class: 30-40, Frequency: 20 Class: 40-50, Frequency: 17 Class: 50-60, Frequency: 6

Find the mode of the following frequency distribution: Class: 0-20, 20-40, 40-60, 60-80, 80-100; Frequency: 8, 7, 12, 5, 3

Determine the median marks for the following data: Marks: 0-10, 10-20, 20-30, 30-40, 40-50; Number of Students: 3, 8, 15, 10, 8

Find mode of the following frequency distribution: Class: 100-110, 110-120, 120-130, 130-140, 140-150 Frequency: 5, 9, 8, 11, 7

Determine median of the following frequency distribution: Class: 15-20, 20-25, 25-30, 30-35, 35-40, 40-45 Frequency: 8, 13, 21, 12, 5, 4

Find mean and mode of the following data: Class 10-20 20-30 30-40 40-50 50-60 60-70 70-80, Frequency 5 4 10 13 12 10 6

Find mean and mode of the following data: Class 0-15, 15-30, 30-45, 45-60, 60-75, 75-90, 90-105 with Frequency 4, 6, 8, 10, 12, 7, 3

Find the missing frequencies p and q in the following frequency distribution, when sum of frequencies is 40 and mean is 19: Class 0-5: 2, 5-10: 5, 10-15: 6, 15-20: p, 20-25: 10, 25-30: q, 30-35: 4

Find the mean and the mode for the following frequency distribution: Class: 30–35, 35–40, 40–45, 45–50, 50–55; Frequency: 3, 9, 7, 3, 2

The mean of the following frequency distribution is 28. If sum of all frequencies is 100, then find the values of p and q: Class Interval: 0–10, 10–20, 20–30, 30–40, 40–50, 50–60; Frequency: 12, p, 27, 20, q, 6

The following data gives the information on the observed lifetime (in hours) of 200 electrical components: Lifetime (in hours): 0-20, 20-40, 40-60, 60-80, 80-100, 100-120; Number of electrical components: 10, 35, 50, 60, 30, 15. Find the mean lifetime (in hours) of the electrical components.

The following frequency distribution gives the monthly consumption of electricity of 68 consumers of a locality. Find the monthly mean consumption from the data. Monthly Consumption (in units): 50–100, 100–150, 150–200, 200–250, 250–300, 300–350, 350–400 Number of Consumers: 4, 5, 13, 20, 14, 8, 4

A life insurance agent found the following data for the distribution of 100 policy holders on the basis of their ages. Age (in years) 15-20, 20-25, 25-30, 30-35, 35-40, 40-45, 45-50, 50-55, 55-60; Number of policy holders 2, 4, 18, 21, 33, 11, 3, 6, 2. Find the median age of the policy holders.

(a) The lengths of 40 leaves of a plant are measured, correct to the nearest millimetre and data obtained is represented in the following table: Length in (mm) | 100-120 | 120-140 | 140-160 | 160-180 | 180-200; Number of leaves | 8 | 9 | 12 | 5 | 6. Find the median length (in mm) of the leaves. OR (b) A class teacher has the following absentees record of 30 students of a class. Number of days | 0-4 | 4-8 | 8-12 | 12-16 | 16-20 | 20-24; Number of Absent students | 1 | 8 | x | 6 | 5 | y. If the mean number of days a student was absent is 12, find the values of x and y.

The weights of 30 students of a class are given in the following distribution table: Weight (in kg): 40-45, 45-50, 50-55, 55-60, 60-65, 65-70. Number of students: 2, 5, 8, 6, 6, 3. Find the median weight of the students.

The following distribution shows the weekly pocket allowance (in ₹) of some children of a locality. The mean pocket allowance is ₹180. Weekly Pocket Allowance (in ₹): 110-130 (7), 130-150 (6), 150-170 (9), 170-190 (13), 190-210 (f), 210-230 (5), 230-250 (4). Find the value of f. Hence find the mode of given data.

The following frequency distribution gives the monthly consumption of electricity of 68 consumers of a locality. Find the mean and mode of the data: Monthly Consumption (in units): 65−85 (4), 85−105 (5), 105−125 (13), 125−145 (20), 145−165 (14), 165−185 (8), 185−205 (4)

Find the Mean and Mode of the following frequency distribution: Class: 0-10, 10-20, 20-30, 30-40, 40-50, 50-60, 60-70 Frequency: 8, 7, 15, 20, 12, 8, 10

Find the mean and median for the following data: Classes: 5-15, 15-25, 25-35, 35-45, 45-55, 55-65, 65-75 Frequency: 2, 3, 5, 7, 4, 2, 2

Find the Mean and Mode of the following data: Class: 4-8, 8-12, 12-16, 16-20, 20-24, 24-28, 28-32, 32-36; Frequency: 2, 12, 15, 25, 18, 12, 13, 3

The following distribution shows the daily pocket allowance of children of a locality. The mean daily pocket allowance is ₹36.10. Find the missing frequency, f. Daily pocket allowance (in ₹): 20–25, 25–30, 30–35, 35–40, 40–45, 45–50, 50–55. Number of children: 7, 6, 9, 13, f, 5, 4.

The mode of the following frequency distribution is 55. Find the missing frequencies.

Class Interval: 0-15, 15-30, 30-45, 45-60, 60-75, 75-90 Frequency: 6, 7, a, 15, 10, b Total Frequency: 51

The monthly expenditure on milk in 200 families of a Housing Society is given below: Monthly Expenditure (in ₹): 1000-1500, 1500-2000, 2000-2500, 2500-3000, 3000-3500, 3500-4000, 4000-4500, 4500-5000 Number of families: 24, 40, 33, x, 30, 22, 16, 7 Find the value of x and also, find the median and mean expenditure on milk.

Probability of getting an irrational number at random from the numbers 3, 4, ∛9, ∛8, √5, 0, 42/3 is:

1. 0
2. 4/7
3. 3/7
4. 1

A card is drawn from a well shuffled deck of 52 cards. The probability of getting an ace or a ten is:

1. 1/13
2. 2/13
3. 1/26
4. 4/13

Assertion (A): The probability of an event can not be 1/0.9. Reason (R): 0 ≤ P(E) ≤ 1 for an event E.

1. Both, Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A)
2. Both, Assertion (A) and Reason (R) are true, but Reason (R) is not correct explanation for Assertion (A)
3. Assertion (A) is true, but Reason (R) is false
4. Assertion (A) is false, but Reason (R) is true

A card is drawn from a well shuffled deck of 52 cards. The probability that it is not a diamond card is:

1. 1/4
2. 0
3. 1/2
4. 3/4

The probability of getting sum greater than 10, when two dice are rolled together, is

1. 1/9
2. 1/18
3. 1/12
4. 1

A bag contains some red and some white balls. A ball is drawn at random from the bag. If the probability of getting a red ball is 2/7, then the probability of getting a white ball is

1. 1/14
2. 5/7
3. 1/7
4. 2/7

A card is drawn from a well-shuffled deck of 52 playing cards. The probability of getting a queen of spade is

1. 1/26
2. 1/52
3. 0
4. 1/4

Three coins are tossed together. The probability of getting exactly two tails is

1. 2/8
2. 1/2
3. 3/8
4. 1

One ticket is drawn at random from a bag containing 50 tickets numbered 1 to 50. The probability that the drawn ticket has a number which is a multiple of 7 is :

1. 1/5
2. 7/50
3. 3/25
4. 4/25

Two dice are thrown simultaneously. The probability of getting a sum of 7 is :

1. 2/9
2. 1/9
3. 5/36
4. 1/6

A card is drawn from a packet of 50 identical cards numbered from 1 to 50. The probability of drawing a number which is a perfect square, is:

1. 7/50
2. 8/50
3. 6/50
4. 10/50

The probability of getting a sum of 7, when two dice are thrown simultaneously, is:

1. 1/9
2. 1/12
3. 1/6
4. 5/36

Assertion (A): The probability that the date of birth of a man is in the month of June is 1/12. Reason (R): There are 12 months in a year.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true, but Reason (R) is not correct explanation of Assertion (A).
3. Assertion (A) is true, but Reason (R) is false.
4. Assertion (A) is false, but Reason (R) is true.

Which of the following can not be the probability of an event?

1. 39/100
2. 0.001/20
3. 10/0.2
4. 10%

A card is drawn at random from a well shuffled deck of 52 playing cards. The probability that it is either a ten or a king is

1. 1/26
2. 2/13
3. 1/13
4. 8/26

In a random experiment of throwing a die, which of the following is a sure event?

1. Getting a number between 1 and 6
2. Getting an odd number < 7
3. Getting an even number < 7
4. Getting a natural number < 7

A bag contains 3 red, 4 white and 7 green balls. A ball is drawn at random. The probability that the ball drawn is not of red colour is :

1. 3/14
2. 4/14
3. 11/14
4. 7/14

The total number of outcomes in the experiment of simultaneous throw of three dice is:

1. 6
2. 18
3. 36
4. 216

The sum of outcomes in favour of event E is:

1. 4
2. 6
3. 10
4. 26

outcomes in favour of event E is:

1. 4
2. 6
3. 10
4. 26

The number of outcomes in favour of event E is:

1. 4
2. 6
3. 10
4. 26