Class 12 Mathematics Previous Year Questions, Chapter-wise (2022-2026)

A relation R on set A = {1, 2, 3} defined as R = {(1, 1), (2, 2), (1, 2)} is

1. Reflexive only
2. Reflexive and Transitive
3. Symmetric and Transitive
4. Transitive only

A relation R on set A = {1, 2, 3} defined as R = {(1, 2), (2, 1), (2, 2)} is

1. Reflexive only
2. Reflexive and Transitive
3. Symmetric and Transitive
4. Symmetric only

Let f : ℝ → ℝ be a function defined as f(x) = x³, then f is :

1. one-one onto function
2. many-one function
3. one-one but not onto function
4. neither one-one nor onto function

A relation R on set A = {1, 2, 3} is defined as R = {(1, 3), (3, 3), (1, 1), (2, 2), (3, 1)} is

1. only reflexive and symmetric
2. reflexive only
3. only reflexive and transitive
4. reflexive, symmetric and transitive

Assertion (A): A function f : N → N given by f(x) = x³ + 2, x ∈ N is one-one but not onto. Reason (R): Since ∀ y ∈ N (Codomain), there does not exist x = (y − 2)^(1/3) in N (Domain) such that f(x) = x³ + 2 = y.

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 points (2, 3), (0, 4) and (p, 2) are collinear, then the value of p is:

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

If the area of triangle ABC with vertices A(3, 1), B(–2, 1) and C(0, k) is 5 sq. units, then values of k are:

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

Assertion (A): Let ℤ be the set of integers. A function f: ℤ → ℤ defined as f(x) = 3x - 5, ∀x ∈ ℤ is a bijective. Reason (R): A function is a bijective if it is both surjective and injective.

1. Both Assertion (A) and Reason (R) are true and the 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 points A(3, -2), B(k, 2) and C(8, 8) are collinear, then the value of k is

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

A function f : R+ → R (where R+ is the set of all non-negative real numbers) defined by f(x) = 4x + 3 is:

1. one-one but not onto
2. onto but not one-one
3. both one-one and onto
4. neither one-one nor onto

A relation R defined on set A = {x : x ∈ Z and 0 ≤ x ≤ 10} as R = {(x, y) : x = y} is given to be an equivalence relation. The number of equivalence classes is:

1. 1
2. 2
3. 10
4. 11

A relation R defined on a set of human beings as R = {(x, y) : x is 5 cm shorter than y} is:

1. reflexive only
2. reflexive and transitive
3. symmetric and transitive
4. neither transitive, nor symmetric, nor reflexive

Let f : R+ → [-5, ∞) be defined as f(x) = 9x² + 6x - 5, where R+ is the set of all non-negative real numbers. Then, f is:

1. one-one
2. onto
3. bijective
4. neither one-one nor onto

Which of the following statements is not true about equivalence classes A_i (i = 1, 2, .... n) formed by an equivalence relation R defined on a set A?

1. ∪ from i=1 to n of A_i = A
2. A_i ∩ A_j ≠ ∅, i ≠ j
3. x ∈ A_i and x ∈ A_j ⟹ A_i = A_j
4. All elements of A_i are related to each other, for all i

Let R+ denote the set of all non-negative real numbers. Then the function f : R+ → R+ defined as f(x) = x² + 1 is :

1. one-one but not onto
2. onto but not one-one
3. both one-one and onto
4. neither one-one nor onto

A function f : R → A defined as f(x) = x^2 + 1 is onto, if A is:

1. (–∞, ∞)
2. (1, ∞)
3. [1, ∞)
4. [–1, ∞)

Let Z denote the set of integers, then function f: Z → Z defined as f(x) = x³ – 1 is:

1. both one-one and onto
2. one-one but not onto
3. onto but not one-one
4. neither one-one nor onto

Assertion (A): The relation R = {(x, y): (x + y) is a prime number and x, y ∈ N} is not a reflexive relation. Reason (R): The number '2n' is composite for all natural numbers n.

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

Check whether f : ℝ – {3} → ℝ defined as f(x) = (x² - 2)/(x - 3) is onto or not.

Check whether f: ℝ – {3} → ℝ defined as f(x) = (x – 2)/(x – 3) is onto or not.

A relation R on A = {1, 2, 3} is defined as R = {(1, 1) (3, 3), (1, 2)}. Is R a symmetric relation? Justify. Write the smallest relation set R1 such that R ⊆ R1 becomes an equivalence relation on the set {1, 2, 3}.

(a) A function f : A → B defined as f(x) = 2x is both one-one and onto. If A = {1, 2, 3, 4}, then find the set B. OR (b) Evaluate: sin⁻¹(sin 4π/3) + cos⁻¹(cos 4π/3) + tan⁻¹(1)

(a) A function f : A → B defined as f(x) = 2x is both one-one and onto. If A = {1, 2, 3, 4}, then find the set B. OR (b) Evaluate: sin⁻¹(√3/2) + cos⁻¹(√3/2) + tan⁻¹(1)

A function f : A → B defined as f(x) = 2x is both one-one and onto. If A = {1, 2, 3, 4}, then find the set B.

(a) Let A = ℝ − {3} and B = ℝ − {1}. A function f : A → B is defined by f(x) = (x − 2)/(x − 3). Find whether f is one-one and onto.

A relation R on set A = {1, 2, 3, 4, 5} is defined as R = {(x, y) : |x² – y²| < 8}. Check whether the relation R is reflexive, symmetric and transitive.

A relation R is defined on Z, the set of integers, as R = {(x, y) : |x – y| is divisible by a prime number 'p', x, y ∈ Z}. Check whether R is an equivalence relation or not.

Show that f: R+ → [− 5, ∞) given by f(x) = 4x² + 4x − 5 is both one-one and onto where R+ = [0, ∞). Also, find p ∈ R+ such that f(p) = 3.

Show that a function f : R+ → A ⊆ N, defined as f(x) = 4x² + 12x + 15 is one-one. Find set A so that f is onto where R+ = [0, ∞). Also, find if there exists a ∈ R+ such that f(a) = 7. Justify.

Let A = {x | x ∈ Z, 0 ≤ x ≤ 12}. Show that the relation R = {(a, b) : a, b ∈ A, (a - b) is divisible by 4} is an equivalence relation. Find the set of elements related to 2.

(a) Show that a function f : R → R defined by f(x) = 2x/(1+x²) is neither one-one nor onto. Further, find set A so that the given function f : R → A becomes an onto function.

(a) Show that a function f : R → R defined by f(x) = 2x²/(1 + x²) is neither one-one nor onto. Further, find set A so that the given function f : R → A becomes an onto function.

Show that a function f : R → R defined as f(x) = x² + x + 1 is neither one-one nor onto. Also, find all the values of x for which f(x) = 3.

A relation R on set A = {–4, –3, –2, –1, 0, 1, 2, 3, 4} be defined as R = {(x, y) : x + y is an integer divisible by 2}. Show that R is an equivalence relation. Also, write the equivalence class [2].

A relation R on set A = {x : –10 ≤ x ≤ 10, x ∈ Z} is defined as R = {(x, y) : (x – y) is divisible by 5}. Show that R is an equivalence relation. Also, write the equivalence class [5].

Let A = R - {5} and B = R - {1}. Consider the function f: A → B, defined by f(x) = (x - 3)/(x - 5). Show that f is one-one and onto.

A relation R is defined on a set of real numbers ℝ as R = {(x, y) : x · y is an irrational number}. Check whether R is reflexive, symmetric and transitive or not.

A relation R is defined on a set of real numbers as R = {(x, y) : x · y is an irrational number}. Check whether R is reflexive, symmetric and transitive or not.

A function f: [-4, 4] → [0, 4] is given by f(x) = √(16 - x²). Show that f is an onto function but not a one-one function. Further, find all possible values of 'a' for which f(a) = 7.

Prove that a function f: [0, ∞) → [-5, ∞) defined as f(x) = 4x² + 4x - 5 is both one-one and onto.

Check whether a function f: [-1/2, 1/2] defined as f(x) = x/(1 + 2x) is one-one and onto or not.

If N denotes the set of all natural numbers and R is the relation on N × N defined by (a, b) R (c, d), if ad(b + c) = bc(a + d). Show that R is an equivalence relation.

If 2 cos⁻¹x = y, then

1. 0 ≤ y ≤ π
2. –π ≤ y ≤ π
3. 0 ≤ y ≤ 2π
4. –π ≤ y ≤ 0

If tan⁻¹ x = 3y, then

1. -π/2 < y < π/2
2. -3π/2 < y < 3π/2
3. -π/6 < y < π/6
4. -π/6 ≤ y ≤ π/6

If sin⁻¹x + π/2 = y, then

1. -π/2 ≤ y ≤ π/2
2. -3π/2 ≤ y ≤ -π/2
3. π/2 ≤ y ≤ 3π/2
4. 0 ≤ y ≤ π

If y = tan⁻¹[(1 – cos x)/sin x], then dy/dx is equal to :

1. 1
2. 2
3. 1/2
4. –1/2

The domain of f(x) = cos⁻¹ (2x – 5) is :

1. [– 1, 1]
2. [4, 6]
3. [– 7, – 3]
4. [2, 3]

The domain of sin⁻¹ (1 − 2x) is:

1. [−1, 1]
2. [−1, 3]
3. [− 2, 2]
4. [0, 1]

The domain of cos⁻¹ (4x + 1) is:

1. [−1, 1]
2. [− 3, 5]
3. [− 4, 4]
4. [−1/2, 0]

The principal value of sec⁻¹(√2) + 2 cosec⁻¹(–√2) is:

1. –π/2
2. –π/4
3. π/4
4. π/2

For the inverse trigonometric functions, which of the following Principal Value Branch is not correctly defined ?

1. tan⁻¹ : R → (– π/2, π/2)
2. sec⁻¹ : R – (–1, 1) → [0, π] – {π/2}
3. cot⁻¹ : R → (0, π)
4. cosec⁻¹ : R – (–1, 1) → [–π/2, π/2]

Derivative of cos⁻¹ (sin x + cos x)/√2, – π/4 < x < π/4 with respect to x is :

1. –1
2. 1
3. π/4
4. – π/4

The derivative of tan⁻¹(x²) w.r.t. x is:

1. x/(1+x⁴)
2. 2x/(1+x⁴)
3. -2x/(1+x⁴)
4. 1/(1+x⁴)

If y = cos⁻¹(eˣ), then dy/dx is:

1. 1/(eˣ√(1-e²ˣ))
2. -1/(eˣ√(1-e²ˣ))
3. 1/√(1-e²ˣ)
4. -1/√(1-e²ˣ)

Assertion (A): The range of the function f(x) = 2 sin⁻¹(x) + 2π/3, where x ∈ [-1, 1], is [2π/5, 2π]. Reason (R): The range of the principal value branch of sin⁻¹(x) is [0, π].

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 and Reason (R) is false.
4. Assertion (A) is false and Reason (R) is true.

Assertion (A) : All trigonometric functions have their inverses over their respective domains. Reason (R) : The inverse of tan1 x exists for some x .

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 and Reason (R) is false.
4. Assertion (A) is false and Reason (R) is true.

Assertion (A): All trigonometric functions have their inverses over their respective domains. Reason (R): The inverse of tan⁻¹ x exists for some x ∈ ℝ.

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 and Reason (R) is false.
4. Assertion (A) is false and Reason (R) is true.

Simplify: tan⁻¹((cos 2x + sin 2x)/(cos 2x – sin 2x)), 0 < x < π/4.

Simplify: cot⁻¹((1 + cos 2x)/(1 - cos 2x)), x ∈ [0, π/2].

Simplify: sin⁻¹(√(1 + cos 2x)/2), 0 < x < π/2

If x = e^(sin⁻¹ t), y = e^(cos⁻¹ t), find dy/dx at t = 1/2

Find the value of sin [cot⁻¹ (√2 (cos (tan⁻¹ 1)))].

Evaluate: tan⁻¹((√3 − 1)/(√3 + 1)) + cot⁻¹(1/√3) + tan⁻¹(sin(–π/2)) + tan⁻¹(2tan(3π/8))

Evaluate: tan⁻¹(1−√3)/√3 + cot⁻¹(1/√3) + tan⁻¹(sin(−π/2)) + tan⁻¹(2tan(3π/8))

Evaluate: tan⁻¹(1 − √3/1 + √3) + cot⁻¹(1/√3) + tan⁻¹(sin(−π/2)) + tan⁻¹(2tan(3π/8))

Evaluate sin⁻¹(3tan(tan⁻¹(1/4))).

Evaluate sin[1 – 7cos²θ cos θ – sin θ]/(sin 6θ)

Evaluate tan(tan⁻¹((1 – 3cosθ)/(sinθ + √3cosθ))).

Evaluate: tan⁻¹(2 sin⁻¹(2/√3) - cos⁻¹(2/√3))

Find the value of tan⁻¹(1) + cos⁻¹(-1/2) + sin⁻¹(-1/2).

Differentiate cot⁻¹((1 + x + x²)/x) w.r.t. x.

(a) Find the value of tan⁻¹(-1/√3) + cot⁻¹(1/√3) + tan⁻¹(sin(-π/2)).

Find the value of sin⁻¹(-1/2) + cos⁻¹(-√3/2) + cot⁻¹(tan(4π/3)).

Evaluate: sec²(tan⁻¹(1/2)) + cosec²(cot⁻¹(1/3))

Find the value of {sin⁻¹(cos(tan⁻¹(sec⁻¹(3)))) + sin⁻¹(3/5)}.

Express tan⁻¹(cos x - sin x)/(cos x), where -π/2 < x < π/2 in the simplest form.

Evaluate sin⁻¹(sin 3π/4) + cos⁻¹(cos π) + tan⁻¹(1).

Evaluate: sin⁻¹(1/2) + cos⁻¹(√3/2) + cos⁻¹(0).

Evaluate: sin⁻¹(sin(3π/11)) + cos⁻¹(cos(2·3π/11))

Evaluate: sin⁻¹(cos(sin⁻¹(√3/2))) + cos⁻¹(cos(sin⁻¹(√3/2)))

Find the domain of y = sin⁻¹(x² - 4).

Find the value of sin⁻¹(3/5) + tan⁻¹(5/6) + cos⁻¹(13/6).

Evaluate: ∫₀¹ tan⁻¹ x dx

Find the domain of g(x) = cos⁻¹ (x² - 1). Hence, find the value of x for which g(x) = π/3. Also, write the range of cos⁻¹ x other than its principal branch.

Find the domain of p(x) = sin⁻¹(1 – 2x²). Hence, find the value of x for which p(x) = π/6. Also, write the range of 2p(x) + π/2.

Find the domain of q(x) = cos⁻¹ (4x² – 3). Hence, find the value of x for which q(x) = 0. Also, write the range of 3q(x) – π.

Which of the following cannot be the order of a row-matrix?

1. 2 × 1
2. 1 × 2
3. 1 × 1
4. 1 × n

Which of the following properties is/are true for two matrices of suitable orders? (i) (A + B)' = A' + B' (ii) (A – B)' = B' – A' (iii) (AB)' = AB (iv) (kAB)' = kBA (k is a scalar)

1. (i) only
2. (i), (ii) and (iii)
3. (i) and (ii)
4. (i) and (iv)

If A and B are skew symmetric matrices of same order, then which of the following matrices is also skew symmetric?

1. AB
2. AB + BA
3. (A + B)²
4. A – B

Which of the following cannot be an order of a column-matrix?

1. 1 × 2
2. 2 × 1
3. 1 × 1
4. m × 1, where m ∈ ℕ

If A and B are symmetric matrices of same order, then which of the following matrices is a skew-symmetric matrix?

1. A² + B²
2. A² – B²
3. AB + BA
4. AB – BA

If A = [aij]₃ₓ₃ is a scalar matrix then which of the following must be true?

1. A must be a symmetric matrix
2. A must be a skew-symmetric matrix
3. A must be an identity matrix
4. A must be a null matrix

Which of the following properties is/are true for two matrices of suitable orders? (i) (A + B)ᵀ = Aᵀ + Bᵀ (ii) (A – B)ᵀ = Bᵀ – Aᵀ (iii) (AB)ᵀ = ABᵀ (iv) (kAB)ᵀ = kBAᵀ (k is a scalar)

1. (i) only
2. (i), (ii) and (iii)
3. (i) and (ii)
4. (i) and (iv)

If A and B are symmetric matrices of same order, then (AB – BA) is a

1. Zero matrix
2. Identity matrix
3. Symmetric matrix
4. Skew symmetric matrix

If A and B are square matrices of same order, then which of the following statements is/are always true? (i) (A + B) (A – B) = A² – B² (ii) AB = BA (iii) (A + B)² = A² + AB + BA + B² (iv) AB = 0 ⟹ A = 0 or B = 0

1. Only (i) and (iii)
2. Only (ii) and (iii)
3. Only (iii)
4. Only (iii) and (iv)

If A = [[1, a, b], [-1, 2, c], [0, 3, 5]] is a symmetric matrix, then the value of 3a + b + c is

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

If A = [[cos x, sin x], [-sin x, cos x]] and A + A' = I, then the value of x ∈ [0, 2π] is

1. 0
2. π/4
3. π/3
4. π/2

For a square matrix A, (3A)⁻¹ =

1. 3A⁻¹
2. 9A⁻¹
3. (1/3)A⁻¹
4. (1/9)A⁻¹

If A = [cos x sin x; –sin x cos x] and A + A' = I, then value of x ∈ [–π/2, π/2] is

1. π/2
2. π/3
3. 0
4. –π/2

The number of all possible matrices of order 2 × 3 with each entry 1 or 2 is :

1. 16
2. 81
3. 64
4. 32

If A = [1 a b; -1 2 c; 0 5 3] is a symmetric matrix, then the value of 3a + b + c is

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

If A = [tan x -cot x; cot x tan x] and A + A' = 2I, then value of x in (π/2, 0) is

1. 0
2. π/3
3. π/4
4. π/2

If A and B are skew-symmetric matrices of same order, then AB + BA is a/an :

1. symmetric matrix
2. skew-symmetric matrix
3. null matrix
4. identity matrix

If [4 1 3] A = [– 4 8 4; – 1 2 1; – 3 6 3], then order of A must be :

1. 3 × 1
2. 1 × 3
3. 1 × 1
4. 3 × 3

If A² = 4A + 3I and A–1 = xA + yI, then the value of (x + y) is:

1. – 1
2. 1
3. 5/3
4. 7

If a matrix B is such that B[1 2 3] = [1 2 3; 0 1 1; 2 0 1], then the order of matrix B is:

1. 1 × 3
2. 3 × 1
3. 3 × 3
4. 1 × 1

If a square matrix A is such that A² = A and (I − A)³ = xA + I, then value of x must be:

1. 7
2. 5
3. – 7
4. – 1

If A = [aij] is a 2 × 2 matrix whose elements are given by aij = (i – 3j)/2, then A is:

1. [(-5/2, 1/2), (-1, 2)]
2. [(-1/2, 1/2), (5/2, 2)]
3. [(-5/2, -1/2), (1, 1/2)]
4. [(-1/2, -2/2), (5/2, 1/2)]

Assertion (A): If A and B are two square matrices such that AB and BA are defined, then it is not necessary that AB = BA. Reason (R): Product of two diagonal matrices of same order is commutative.

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.

Let A = [aᵢⱼ] be a 2 × 2 matrix whose elements are given by aᵢⱼ = 2(2i – j)/3. Then A is:

1. [1/3, -3/4; 4/3, 0]
2. [1/3, 0; 4/3, -3]
3. [4/3, -3/3; 1/3, 0]
4. [4/3, 0; 1/3, -3]

Let A = [aᵢⱼ] be a 2 × 2 matrix whose elements are given by aᵢⱼ = (2(i + 2j))/3. Then A is given by:

1. [25/12, 3; 16/3, 33]
2. [16/12, 3; 25/3, 33]
3. [25/3, 3; 16/12, 3]
4. [16/3, 3; 25/12, 3]

If sin cos A = cos -sin and A + A = I, then is equal to:

1. 0
2. π/6
3. π/3
4. π/2

If A is a symmetric matrix, then for any matrix B of order same as A, BAB is a/an:

1. Skew symmetric matrix
2. Identity matrix
3. Symmetric matrix
4. Null matrix

Let A = [0 3 4; -2 0 1; -4 -1 0] and B = [0 -0 1; 2 4 -3]. If A + B + C = O, then matrix C is:

1. [-2 -3 5; 3 4 -3]
2. [-3 -3 -5; 3 4 2]
3. [3 3 -5; -3 -4 2]
4. [-5 -3 -3; 2 3 4]

For any square matrix A with real entries, if A + A' is a symmetric matrix then:

1. (A – A') cannot be a skew symmetric matrix
2. (A – A') is a skew symmetric matrix
3. A is always a symmetric matrix
4. A is always a skew symmetric matrix

A matrix B = [bij]m × m is said to be a diagonal matrix, if:

1. bij = 0, when i = j
2. bij = 1, when i = j
3. bij = 1, when i ≠ j
4. bij = 0, when i ≠ j

Let A = [1 2 -1; 0 4 -1; 3 2 -1], B = [2; 5; 7], C = [9 8 7] which of the following is defined?

1. Only AB
2. Only AC
3. Only BA
4. All AB, AC and BA

If A and B are invertible matrices, then which of the following is not correct?

1. (A + B)⁻¹ = B⁻¹ + A⁻¹
2. (AB)⁻¹ = B⁻¹A⁻¹
3. adj(A) = |A| A⁻¹
4. |A|⁻¹ = |A⁻¹|

If A = [sin x, -cos x; cos x, sin x], then the value of x, for which A is an identity matrix, is

1. π/2
2. π
3. 0
4. 3π/2

If the matrix A = [0, a, -7; -3, 0, -5; b, 3, 0] is a skew-symmetric matrix, then the values of 'a' and 'b' are

1. a = 5, b = 3
2. a = 5, b = -7
3. a = -5, b = -7
4. a = -5, b = 7

If [8, 14; 9, 7] = [1, 2; 3, 1] X, then matrix X is

1. [3, 2; 0, 7]
2. [2, 7; 0, 3]
3. [2, 0; 3, 7]
4. [2, -0; 3, 7]

If a matrix has 36 elements, the number of possible orders it can have, is:

1. 13
2. 3
3. 5
4. 9

If A and B are two non-zero square matrices of same order such that (A + B)² = A² + B², then:

1. AB = O
2. AB = –BA
3. BA = O
4. AB = BA

If [[x + y, 2], [5, xy]] = [[6, 2], [5, 8]], then the value of (2x + 2y)/(24/x + 24/y) is:

1. 7
2. 6
3. 8
4. 18

If matrices A and B are of order 1 × 3 and 3 × 1 respectively, then the order of A'B' is:

1. 1 × 1
2. 3 × 1
3. 1 × 3
4. 3 × 3

If [x + 2y, 5; 2, x + y] = [6, 5; 2, 8], then the value of (x/24 + y/24) is:

1. 7
2. 6
3. 8
4. 18

If the sum of all the elements of a 3 × 3 scalar matrix is 9, then the product of all its elements is:

1. 0
2. 9
3. 27
4. 729

If A = [aᵢⱼ] be a 3 × 3 matrix, where aᵢⱼ = i - 3j, then which of the following is false?

1. a₁₁ < 0
2. a₁₂ + a₂₁ = -6
3. a₁₃ > a₃₁
4. a₃₁ = 0

If F(x) = |cos x, sin x, 0; -sin x, cos x, 0; 0, 0, 1| and [F(x)]² = F(kx), then the value of k is:

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

Assertion (A): For any symmetric matrix A, B'AB is a skew-symmetric matrix. Reason (R): A square matrix P is skew-symmetric if P' = -P.

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 number of all scalar matrices of order 3, with each entry –1, 0 or 1, is:

1. 1
2. 3
3. 2
4. 39

If for the matrix A = [1, -tan(x); tan(x), 1], A + A' = (2/3)I, then the value of x ∈ [0, π/2] is:

1. 0
2. π/4
3. π/3
4. π/6

If A = [a_ij] is an identity matrix, then which of the following is true ?

1. a_ij = 1 if i = j, 0 if i ≠ j
2. a_ij = 1, for all i, j
3. a_ij = 0, for all i, j
4. a_ij = 0 if i = j, 1 if i ≠ j

If A and B are two skew symmetric matrices, then (AB + BA) is :

1. a skew symmetric matrix
2. a symmetric matrix
3. a null matrix
4. an identity matrix

If A = [2 0 0; 0 3 0; 0 0 5], then A⁻¹ is :

1. [1/2 0 0; 0 1/3 0; 0 0 1/5]
2. 30[1/2 0 0; 0 1/3 0; 0 0 1/5]
3. (1/30)[2 0 0; 0 3 0; 0 0 5]
4. (1/30)[1/2 0 0; 0 1/3 0; 0 0 1/5]

Assertion (A): Every scalar matrix is a diagonal matrix. Reason (R): In a diagonal matrix, all the diagonal elements are 0.

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.

For any square matrix A, (A – A^T) is always:

1. an identity matrix
2. a null matrix
3. a skew symmetric matrix
4. a symmetric matrix

If the matrix [5, 0, 0; 0, d, b; 0, c, a] is a scalar matrix, then the value of a + 2b + 3c + 4d is:

1. 0
2. 5
3. 10
4. 25

Given that A⁻¹ = [2/7, -3/2; 1/7, 2], matrix A is:

1. 7[2, 3; 1, -2]
2. [2, 3; 1, -2]
3. [2, 3; 1, -2] · 1/7
4. [2, 3; 1, -2] · 1/49

If A = [2, -4; -1, 2], then the value of I - A + A² - A³ + ... is:

1. [-3/4, -1; 1, -1]
2. [3, -4; -1, 1]
3. [0, 0; 0, 0]
4. [1, 0; 0, 1]

If A = [[1, 3], [3, 4]] and A² - kA - 5I = O, then the value of k is:

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

If [[a, c, 0], [b, d, 0], [0, 0, 5]] is a scalar matrix, then the value of a + 2b + 3c + 4d is:

1. 0
2. 5
3. 10
4. 25

If A = [[-1, -4], [2, 2]], then the value of I - A + A² - A³ + ... is:

1. [[−1, −4], [1, 3]]
2. [[3, −4], [−1, 1]]
3. [[0, 0], [0, 0]]
4. [[1, 0], [0, 1]]

Given that A⁻¹ = (1/7)[[2, -1], [-3, 2]], matrix A is:

1. 7[[2, 1], [3, -2]]
2. [[2, 1], [3, -2]]
3. (1/7)[[2, 1], [3, -2]]
4. (1/49)[[2, 1], [3, -2]]

If for a square matrix A, A² - 3A + I = O and A⁻¹ = xA + yI, then the value of x + y is:

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

If |A| = 2, where A is a 2×2 matrix, then |4A⁻¹| equals:

1. 4
2. 2
3. 8
4. 32

If A = [2 5; 4 3] and 2A + B is a null matrix, then B is equal to:

1. [-4 -10; -8 -6]
2. [4 10; 8 6]
3. [3 10; 8 5]
4. [3 -10; -8 5]

If A = [0 1; 1 0] and (3I + 4A)(3I - 4A) = x²I, then the value(s) of x is/are:

1. -7
2. 0
3. -5
4. ±5

If A is a 2 × 3 matrix such that AB and AB' both are defined, then order of the matrix B is

1. 2 × 2
2. 2 × 1
3. 3 × 2
4. 3 × 3

If |2 0; 5 4| = P + Q, where P is a symmetric and Q is a skew symmetric matrix, then Q is equal to

1. |2 5/2; 5/2 4|
2. |0 -5/2; 5/2 0|
3. |0 5/2; -5/2 0|
4. |2 -5/2; 5/2 4|

If matrix A is a symmetric matrix, then the value of x + y + z is:

1. 10
2. 6
3. 8
4. 0

For what value of x, where 0 < x < π/2, is A + A^T = I₃, where A = [cos x, sin x; -sin x, cos x]?

1. π/3
2. π/6
3. 0
4. π/2

If A = [0 0; 0 1] and B = [0 0; 1 1], then BA is equal to:

1. [0 0; 0 1]
2. [0 0; 1 0]
3. [1 1; 1 1]
4. [0 0; 0 0]

A and B are skew-symmetric matrices of same order. AB is symmetric, if:

1. AB = O
2. AB = -BA
3. AB = BA
4. BA = O

A and B are square matrices of same order. If (A + B)² = A² + B², then

1. AB = BA
2. AB = -BA
3. AB = O
4. BA = O

For what value of x ∈ [0, 2π], is A + A' = I₃, where A = [cos x sin x; -sin x cos x]?

1. π/3
2. π/6
3. 0
4. π/2

If A is a square matrix and A² = A, then (I + A)² - 3A is equal to:

1. I
2. A
3. 2A
4. 3I

If A is a 3 × 4 matrix and B is a matrix such that AB and A'B are both defined, then the order of the matrix B is:

1. 3 × 4
2. 3 × 3
3. 4 × 4
4. 4 × 3

If 9/4 - 5/2 = (y - 2)/(1 - x), then:

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

For the matrix A = [3 1; 2 1], find the numbers a and b so that A² + aA + bI = O.

If A = [[3, 2, 0], [1, 4, 0], [0, 0, 5]], then compute A² − 7A + 10 I.

If A = [[1, 2, 2], [2, 1, 2], [2, 2, 1]], then compute A² − 4A − 5I.

If A = [2, 0, 1; 2, 1, 3; 1, −1, 0], then compute A² − 5A + 4I.

If A = [0 1 1; 2 1 2; 1 2 0], find A⁻¹ and use it to solve the following system of equations: –2y + z = 7, 2x – y – z = 8, x – 2y = 10

If P = [[−1, 1, 0], [2, 3, 4], [0, 1, 2]] and Q = [[2, −2, 4], [−4, 2, −4], [−2, 1, 5]], find (QP)⁻¹ and hence solve the following system of equations using matrices: x - y = 3, 2x + 3y + 4z = 17, y + 2z = 7

If P = [–1 1 0; 2 3 4; 0 1 2] and Q = [2 2 –4; –4 2 –4; –2 –1 5], find (QP) and hence solve the following system of equations using matrices: x – y = 3, 2x + 3y + 4z = 17, y + 2z = 7

If A = [1 1 1; 2 1 -3; 1 -1 1], find A⁻¹. Using A⁻¹ solve the system of equations: x – y + z = 4; 2x + y – 3z = 0; x + y + z = 2

On the inauguration day of a new showroom, a lucky draw was organized and some vouchers of ₹ 1,000 and ₹ 500 were given to the lucky draw winners. A total of 60 vouchers were given on the day. The number of ₹ 1,000 vouchers added to 3 times the number of ₹ 500 vouchers, gives 100. Express the given information as a system of linear equations in two variables. Hence, find the number of vouchers of each type by matrix method.

A man goes to buy fruits from the market. The shopkeeper informs him that 4 apples, 3 oranges and 2 bananas cost ₹ 60; 2 apples, 4 oranges and 6 bananas cost ₹ 90; whereas 6 apples, 2 oranges and 3 bananas cost ₹ 70. Using matrix method, find the cost of one fruit of each kind.

Three students A, B and C go to a book-store to buy art books, story books and puzzle solving books. A buys one of each type of book for a total of ₹21. B buys 4 art books, 3 story books and 2 puzzle solving books for ₹60. C buys 6 art books, 2 story books and 3 puzzle solving books and pays ₹10 more than B. Use matrix method to find the cost of each type of book.

A school wants to allocate students into three clubs: Sports, Music and Drama, under following conditions: The number of students in Sports club should be equal to the sum of the number of students in Music and Drama club. The number of students in Music club should be 20 more than half the number of students in Sports club. The total number of students to be allocated in all three clubs are 180. Find the number of students allocated to different clubs, using matrix method.

If A = |–1, 2, 0; –1, 1, –2; 0, –2, 1|, find A⁻¹ and use it to solve the following system of equations: x – 2y = 10, 2x – y – z = 8, –2y + z = 7

(a) If A = [5, 0, 4; 2, 3, 2; 1, 2, 1] and B⁻¹ = [1, 3, 1; 1, 4, 3; 1, 3, 4], find (AB)⁻¹. Also, find |(AB)⁻¹|.

Solve the following system of equations, using matrices: 2x + 3y + 10z = 4, 4x - 6y + 5z = 1, 6x + 9y - 20z = 2, where x, y, z ≠ 0

Use the product of matrices [[1, -2, 3], [-2, 3, 2], [0, 1, -1]] and [[4, -7, 7], [-7, -7, 0], [5, -4, 4]] to solve the following system of equations: x + 2y - 3z = 6, 3x + 2y - 2z = 3, 2x - y + z = 2

(a) If A = [1 2 3; 0 2 1; 1 2 2] and B⁻¹ = [2 2 5; 6 1 5; 1 1 3], find (AB)⁻¹. OR (b) Solve the following system of equations by matrix method: x + 2y + 3z = 6, 2x - y + z = 2, 3x + 2y - 2z = 3.

(a) If A = [1 2 2; 1 3 0; 0 1 2] and B⁻¹ = [1 3 1; 1 5 5; -1 -2 -2], find (AB)⁻¹. OR (b) Solve the following system of equations by matrix method: x + 2y + 3z = 6, 2x - y + z = 2, 3x + 2y - 2z = 3

If A = [1, 2, 0; 0, 3, 1; 2, 2, 1] and B = [1, 1, 3; 1, 5, 1; 6, 1, 5], find (AB)⁻¹.

If A = [1, 2, 0; 2, 1, 0; 0, 2, 3], then show that A³ - 6A² + 7A + 2I = O.

If A = [1 0 2; 0 2 1; 2 1 3], then show that A³ - 6A² + 7A + 2I = O.

If A = [1 0 2; 0 2 1; 2 0 3], then show that A³ - 6A² + 7A + 2I = O.

Find the inverse of the matrix A = [1 1 2; 0 2 -3; 3 -2 4]. Using the inverse A⁻¹, solve the system of linear equations x - y + 2z = 1; 2y - 3z = 1; 3x - 2y + 4z = 3.

If A₁ = [1 0 0; 0 2 0; 0 0 3] and A₂ = [0 0 6; 0 0 1; 0 2 0], then

1. |A₁| = 2|A₂|
2. |A₂| = –2|A₁|
3. |A₁| = |A₂|
4. |A₂| = –|A₁|

One of the values of x for which |cos x sin x; sin x cos x| = 1 is

1. 0
2. π/4
3. π/3
4. π/2

If A₁ = [3 0 0; 0 2 0; 0 0 1] and A₂ = [6 0 0; 0 0 1; 0 2 0], then

1. |A₁| = 2|A₂|
2. |A₂| = -2|A₁|
3. |A₁| = |A₂|
4. |A₂| = -|A₁|

One of the values of x for which |sin x cos x; sin x cos x| = 1 is

1. 0
2. π/4
3. π/3
4. π/2

If D₁ = |1 0 0; 0 2 0; 0 0 3| and D₂ = |6 0 0; 0 0 1; 0 2 0|, then

1. D₁ = 2D₂
2. D₂ = -2D₁
3. D₁ = D₂
4. D₂ = -D₁

If |[-1, -2, -2], [-a, 2, 1], [0, 4, 2a]| = -86, then the sum of all possible values of a is

1. 4
2. 5
3. -4
4. 9

For a square matrix A, (3A)⁻¹ =

1. 3A⁻¹
2. 9A⁻¹
3. (1/3)A⁻¹
4. (1/9)A⁻¹

If |–1 –2 5; –2 a –1; 0 4 a 2| = –86, then the sum of all possible values of a is

1. 4
2. 5
3. –4
4. 9

If the area of a triangle is 15 sq. units with vertices (3, –7), (6, 3) and (k, 3), then the value(s) of k is/are :

1. 3
2. 9
3. –9, –3
4. 9, 3

If |−1 −2 5; −2 a 1; 0 4 2a| = -86, then the sum of all possible values of a is

1. 4
2. 5
3. -4
4. 9

If B(adj B) = [1003 1003 1003], then the value of det (B–1) =

1. 1/3
2. 1/9
3. 3
4. 9

If A(adj A) = [2026 0 0; 0 2026 0; 0 0 2026], then the value of |adj A| is equal to:

1. 2026
2. (2026)−1
3. (2026)−2
4. (2026)2

If A(adj A) = [3 0 0; 0 3 0; 0 0 3], then the value of |2A| is:

1. 6
2. 54
3. 12
4. 24

If points (2, 3), (0, 4) and (p, 2) are collinear, then the value of p is:

1. 4/7
2. –3/7
3. 4
4. –4

If A is a non-singular matrix, then which of the following is not true ?

1. adj A is singular
2. (adj A)–1 = (adj A–1)
3. |A| ≠ 0
4. A–1 exists

If the area of △ ABC with vertices A(3, 1), B(– 2, 1) and C(0, k) is 5 sq. units, then values of k are :

1. 3, 1
2. –1, 3
3. –1, 2
4. 0, 2

If the area of triangle ABC with vertices A(3, 1), B(– 2, 1) and C(0, k) is 5 sq. units, then values of k are:

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

If A is a square matrix of order 2 such that det(A) = 4, then det(4 adj A) is equal to:

1. 16
2. 64
3. 256
4. 512

If |x+2, 2x+4; -x, 3x| = |6, -1; 2, 3|, then the value of x is

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

Assertion (A): For matrix A = [1 –cos θ –1; –cos θ 1 cos θ; –1 cos θ 1], where θ ∈ [0, 2π], |A| ∈ [2, 4]. Reason (R): cos θ ∈ [–1, 1], ∀ θ ∈ [0, 2π].

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 aᵢⱼ and Aᵢⱼ represent the (ij)ᵗʰ element and its cofactor of the matrix [[2, -3, 5], [6, 0, 4], [1, 5, -7]] respectively, then the value of a₁₁A₂₁ + a₁₂A₂₂ + a₁₃A₂₃ is:

1. 0
2. -28
3. 114
4. -114

|(x+1, x-1); (x, x+1)| is equal to:

1. 2x³
2. 2
3. 0
4. 2x³ – 2

Assertion (A): For matrix A = |1, –cos θ, 1; 1, cos θ, –cos θ; 1, –cos θ, 1|, where θ ∈ [0, 2π], |A| ∈ [2, 4]. Reason (R): cos θ ∈ [–1, 1], ∀ θ ∈ [0, 2π].

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 |c-b a c; b-a c b; a c-| = kabc, then the value of k is:

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

If |a–b, a, c; b–a, c, b; a, b, –c| = kabc, then the value of k is:

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

If |a-b, a, c; b-a, c, b; a, b, a-c| = kabc, then the value of k is:

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

Let A = [a b; c d] be a square matrix such that adj A = A. Then, (a + b + c + d) is equal to :

1. 2a
2. 2b
3. 2c
4. 0

If |1 3 1; 0 k 0; 0 1 0| = ±6, then the value of k is :

1. 2
2. –2
3. ±2
4. ≠2

If |100 10k 131| = ±6, then the value of k is:

1. 2
2. –2
3. ±2
4. ≠2

If A = [2, 0, 0; 1, 3, 1; 5, -1, -1], then the value of |A(adj. A)| is:

1. 100 I
2. 10 I
3. 10
4. 1000

Given that [1, x] [4, -2; 0, 0] = 0, the value of x is:

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

If A = [aij] = [[-2, 1, 5], [1, 3, 2], [5, 0, 4]] and cij is the cofactor of element aij, then the value of a21 × c11 + a22 × c12 + a23 × c13 is:

1. -57
2. 0
3. 9
4. 57

Let A be a 3×3 matrix such that |adj A| = 64. Then |A| is equal to:

1. 8 only
2. -8 only
3. 64
4. 8 or -8

If 2A⁻¹ = |A|A^(k-1), where A is a 3 × 3 matrix, then the value of k is:

1. 81
2. 8
3. 2
4. 1/2

If [1, x; 3, 2x] is a singular matrix, then the product of all possible values of x is:

1. 6
2. -6
3. 0
4. -7

If (a, b), (c, d) and (e, f) are the vertices of triangle ABC and Δ denotes the area of triangle ABC, then |a c e; b d f; 1 1 1|/2 is equal to

1. 2Δ²
2. 4Δ²
3. 2Δ
4. 4Δ

If |1 2 1; 2 3 1; 3 a 1| is non-singular matrix and a ∈ A, then the set A is

1. ℝ
2. {0}
3. {4}
4. ℝ - {4}

If |A| = |kA|, where A is a square matrix of order 2, then sum of all possible values of k is

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

If A . (adj A) = 3I, then the value of A + adj A is equal to:

1. 12
2. 9
3. 3
4. 27

Let A be the area of a triangle having vertices (x₁, y₁), (x₂, y₂) and (x₃, y₃). Which of the following is correct?

1. A = |1/2 determinant with rows [1, x₁, y₁], [1, x₂, y₂], [1, x₃, y₃]|
2. 2A = |determinant with rows [1, x₁, y₁], [1, x₂, y₂], [1, x₃, y₃]|
3. 2A = determinant with rows [1, x₁, y₁], [1, x₂, y₂], [1, x₃, y₃]
4. A = determinant with rows [x₁, y₁], [x₂, y₂], [x₃, y₃]

Let A be a skew-symmetric matrix of order 3. If A = x, then (2023)^x is equal to:

1. 2023
2. 2023^(-1)
3. (2023)^2
4. 1

If A·(adj A) = [3 0 0; 0 3 0; 0 0 3], then the value of |A| + adj A is equal to:

1. 12
2. 9
3. 3
4. 27

The value of the determinant 1810111172 is:

1. 47
2. -79
3. 49
4. -51

If the area of a triangle with vertices (2, -6), (5, 4) and (k, 4) is 35 sq units, then k is

1. 12
2. -2
3. -12, 2
4. 12, -2

The product [a b; -b a] [a b; -b a] is equal to:

1. [(a² - b²) 0; 0 (a² - b²)]
2. [0 (b² - a²); 0 (b² - a²)]
3. [0 (a² - b²); 0 (a² - b²)]
4. [b 0; 0 a]

If A = [[0, 1, 1], [2, 1, 2], [1, 2, 0]], find A⁻¹ and use it to solve the following system of equations: –2y + z = 7, 2x – y – z = 8, x – 2y = 10.

Obtain the value of Δ = |[1, x, 1, 1], [1, y, 1, 1], [1, 1, z, 1], [1, 1, 1, −3]| in terms of x, y and z. Further, if Δ = 0 and x, y, z are non-zero real numbers, prove that x⁻¹ + y⁻¹ + z⁻¹ = -1.

Find cost (per kg) of each fertilizer A, B and C that the farmer needs to buy, such that 1 kg each of fertilizer A and C added to 2 kg of B costs him ₹ 400. Also, cost of each kg of fertilizer B and C added together is equal to cost of 1 kg of fertilizer A. However, cost of 3 kg of fertilizer B added to ₹ 200 is the same as cost of 1 kg of fertilizer A and C together. Use matrix method to find the solution.

Using matrices, solve the following system of linear equations: 3x + 4y + 2z = 8; 2y - 3z = 3; x - 2y + 6z = -2

If A = [–1 2 0; –1 1 0; –2 2 1], find A⁻¹ and use it to solve the following system of equations: x – 2y = 10, 2x – y – z = 8, –2y + z = 7

If A = [[-1, -2, 0], [-1, 1, -2], [0, -2, 1]], find A⁻¹ and use it to solve the following system of equations: x - 2y = 10, 2x - y - z = 8, -2y + z = 7

(a) Solve the following system of equations, using matrices: 2x + 3y + 10z = 4, 4x - 6y + 5z = 1, 6x + 9y - 20z = 2, where x, y, z ≠ 0

If A = [2, -1, 3; 1, 2, 3; 5, -1, -1], find A⁻¹ and hence solve the following system of equations: 2x + y - 3z = 13, 3x + 2y + z = 4, x + 2y - z = 8.

If A = |(-3 -2 -4); (2 1 2); (2 1 3)| and B = |(1 2 0); (-2 -1 -2); (0 -1 1)|, then find AB and use it to solve the following system of equations: x - 2y = 3, 2x - y - z = 2, -2y + z = 3

If e⁻ˣ + e⁻ʸ = 2, then dy/dx is

1. eˣ⁻ʸ
2. eʸ⁻ˣ
3. -eˣ⁻ʸ
4. -eʸ⁻ˣ

If e^(x + y) = 3x, then dy/dx is

1. y – 3e^(–x)
2. y – e^(–x)
3. (e^y – 1)/(e^x)
4. (e^y – 3)/(e^x)

The value of k for which f(x) = {sin 7x/5x, x ≠ 0; k, x = 0} is continuous at x = 0 is :

1. 5/7
2. 7/5
3. 0
4. 1/5

If x + y = xy, then dy/dx is

1. 1 - x/y
2. 1 - 1/x
3. 1 - (1-y)/x
4. 1 - (y-1)/x

The value of k for which the function f(x) = {x² sin(1/x), x ≠ 0; k(x + 1), x = 0} is a continuous function, is:

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

If 2 cos⁻¹ x = y, then dy/dx is:

1. − 2 sin⁻¹ x
2. 1/sin(y/2) – sin(2y/2)
3. −2/(√(1 – 2x²))
4. − 2 cosec(y/2)

Differential of e^(e^x) with respect to x is:

1. log x
2. e^(e^x)
3. e^x e^(e^x)
4. (e^x)²

If f(x) = {(sin x + cos x)/x, x ≠ 0; k, x = 0} is continuous at x = 0, then the value of k is:

1. 0
2. –√2
3. –1
4. √2

The greatest integer function, f(x) = [x], 0 < x < 3 is not differentiable at how many points?

1. At only one point
2. At only two points
3. At no point
4. At three points

If f(x) = (sin x + cos x)/x, x ≠ 0 and k, x = 0 is continuous at x = 0, then the value of k is:

1. 0
2. −√2
3. −1
4. √2

If f(x) = [x²/(x+1) - 4x/(x-5), x ≠ -1; k, x = -1] is continuous at x = -1, then the value of k is:

1. Any real value
2. 6
3. -1
4. -6

Derivative of cos⁻¹[(sin x + cos x)/√2], -π/4 < x < π/4 with respect to x is:

1. -1
2. 1
3. π/4
4. -π/4

If f(x) = {(k - 4x²)/(x - 5), x ≠ -1; (1)/(x + 1), x = -1} is continuous at x = -1, then the value of k is:

1. Any real value
2. 6
3. -1
4. -6

Which of the following is not a homogeneous function of x and y?

1. y² - xy
2. x - 3y
3. sin(y/x) + y/x
4. tan x - sec y

If f(x) = |x| + |x - 1|, then which of the following is correct?

1. f(x) is both continuous and differentiable, at x = 0 and x = 1.
2. f(x) is differentiable but not continuous, at x = 0 and x = 1.
3. f(x) is continuous but not differentiable, at x = 0 and x = 1.
4. f(x) is neither continuous nor differentiable, at x = 0 and x = 1.

Assertion (A): f(x) = {3x - 8, x ≤ 5; 2k, x > 5} is continuous at x = 5 for k = 5/2. Reason (R): For a function f to be continuous at x = n, lim(x→n⁻) f(x) = lim(x→n⁺) f(x) = f(n).

1. Both Assertion (A) and Reason (R) are true and the 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 f(x) = {(cos x + sin x)/(x + 3/π), if x ≠ -π/3; k, if x = -π/3} is continuous at x = -π/3, is

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

Which of the following statements is true for the function f(x) = 2x + 3, x ≠ 0; 1, x = 0?

1. f(x) is continuous and differentiable ∀ x ∈ ℝ
2. f(x) is continuous ∀ x ∈ ℝ
3. f(x) is continuous and differentiable ∀ x ∈ ℝ – {0}
4. f(x) is discontinuous at infinitely many points

The derivative of sin(x²) w.r.t. x, at x = π is:

1. 1
2. –1
3. –2π
4. 2π

The derivative of sin(x²) with respect to x, at x = √π is:

1. 1
2. -1
3. -2√π
4. 2√π

The number of points, where f(x) = [x], 0 < x < 3 ([·] denotes greatest integer function) is not differentiable is:

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

If x = at, y = t/a, then dy/dx is:

1. t²
2. –t²
3. 1/t²
4. –1/t²

Which of the following statements is true for the function f(x) = {x² + 3, x ≠ 0; 1, x = 0}?

1. f(x) is continuous and differentiable ∀ x ∈ ℝ
2. f(x) is continuous ∀ x ∈ ℝ
3. f(x) is continuous and differentiable ∀ x ∈ ℝ – {0}
4. f(x) is discontinuous at infinitely many points

The number of points of discontinuity of f(x) = {x+3, if x ≤ -3; -2x, if -3 < x < 3; 6x+2, if x ≥ 3} is:

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

d/dx[cos(log x + e^x)] at x = 1 is:

1. –sin e
2. sin e
3. –(1 + e)sin e
4. (1 + e)sin e

A function f(x) = |1 – x + |x|| is :

1. discontinuous at x = 1 only
2. discontinuous at x = 0 only
3. discontinuous at x = 0, 1
4. continuous everywhere

The derivative of 2^x w.r.t. 3^x is :

1. (2^x / 3^x) × (log 2 / log 3)
2. (3^x / 2^x) × (log 3 / log 2)
3. (2^x / 3^x) × (log 3 / log 2)
4. (3^x / 2^x) × (log 2 / log 3)

The derivative of 5^x w.r.t. e^x is:

1. (5^x/e^x) · (1/log 5)
2. (e^x/5^x) · (1/log 5)
3. (5^x/e^x) · log 5
4. (e^x/5^x) · log 5

Derivative of e²ˣ with respect to eˣ is:

1. eˣ
2. 2eˣ
3. 2e²ˣ
4. 2e³ˣ

For what value of k, the function f(x) = {(x² + x - 4)/x, x ≠ 0; k, x = 0} is continuous at x = 0?

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

The value of constant c that makes the function f defined by f(x) = {x² - c², if x < 4; cx + 20, if x ≥ 4} continuous for all real numbers is:

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

For what value of k may the function f(x) = {k(3x² - 5x) for 0 ≤ x, cos x for x > 0} become continuous?

1. 0
2. 1
3. -1/2
4. No value

The function f(x) = |x| - x is:

1. continuous but not differentiable at x = 0
2. continuous and differentiable at x = 0
3. neither continuous nor differentiable at x = 0
4. differentiable but not continuous at x = 0

If tan((x + y)/(x - y)) = k, then dy/dx is equal to

1. -y/x
2. y/x
3. sec²(y/x)
4. -sec²(y/x)

The value of k for which function f(x) = x² for 0 ≤ x, kx for x > 0 is differentiable at x = 0 is:

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

The value of k for which function f(x) = {x² for x ≤ 0; kx for x > 0} is differentiable at x = 0 is:

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

For what value of x ∈ [0, 2π], is A + A^T = I₃, where A = [cos x sin x; -sin x cos x]?

1. π/3
2. π/6
3. 0
4. π/2

The value of k for which function f(x) = {x, 0 ≤ x, kx², x < 0} is differentiable at x = 0 is

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

If y = (cos x - sin x)/(cos x + sin x), then dy/dx is

1. -2 sec²x
2. 2 sec²x
3. -2 sec x log x
4. 2 sec x log x

The function f(x) = x is

1. continuous and differentiable everywhere.
2. continuous and differentiable nowhere.
3. continuous everywhere, but differentiable everywhere except at x = 0.
4. continuous everywhere, but differentiable nowhere.

If y = log (sin ex), then dy/dx is:

1. cot ex
2. cosec ex
3. ex cot ex
4. ex cosec ex

If f(x) = 2x + 3sin x + 6, then the right hand derivative of f(x) at x = 0 is:

1. 6
2. 5
3. 3
4. 2

If y = sin²(x³), then dy/dx is equal to:

1. 2 sin x³ cos x³
2. 3x³ sin x³ cos x³
3. 6x² sin x³ cos x³
4. 2x² sin²(x³)

Check whether function f(x) defined as f(x) = {(x³ + 2(x-3), 3 ≤ x; 6, x = 3; (6x - x³)/3, x < 3} is continuous at x = 3 or not?

Check whether function f(x) defined as f(x) = {x/(3-x), x < 3; (x-6)/(x-3), x ≥ 3} is continuous at x = 3 or not?

Check whether function f(x) defined as f(x) = {x²(3 - 2x) for 0 ≤ x ≤ 3; 6x - x for x > 3} is continuous at x = 3 or not?

If x = a sin³ t, y = b cos³ t, then find dy/dx at t = π/4.

If x = sin t – cos t, y = sin t cos t, find dy/dx at t = π/4.

If x = t + 1/t and y = t – 1/t, find dy/dx at t = 2.

If x = e^(t + 1/t) and y = e^(−t − 1/t), then find dy/dx at t = − 2.

If x = e^(sin⁻¹ t) and y = e^(cos⁻¹ t), then find dy/dx at t = 1/2.

(a) Show that the function f(x) = {(cos x − x/2), x ≠ π/2; 1, x = π/2} is continuous at x = π/2.

(a) Show that the function f(x) = {cos x − 2x², x ≤ π/2; −1, x > π/2} is continuous at x = π/2.

Show that the function f(x) = cos x − x/2, x ≠ π/2 and 1, x = π/2 is continuous at x = π/2.

(a) Differentiate x^x with respect to x log x.

(a) Differentiate 2^(cos 2x) w.r.t. cos 2x.

If (cos x)^y = (cos y)^x, find dy/dx.

Check whether the function f(x) = x² |x| is differentiable at x = 0 or not.

If y = cos(x) + cos(y), prove that dy/dx = -sin(x)/(1 - sin(y)).

Check the differentiability of function f(x) = [x] at x = –3, where [.] denotes greatest integer function.

(a) If f(x) = |tan 2x|, then find the value of f'(x) at x = π/3.

Show that f(x) = e^x – e^(–x) + x – tan⁻¹x is strictly increasing in its domain.

(a) Check the differentiability of f(x) = |cos x| at x = π/2.

If x = e^(x/y), prove that (dy/dx) = (x log x) / (log x - 1)²

If x = e^(x/y), prove that dy/dx = (log x)/(1 + (log x)²).

If y = cos³(sec² 2t), find dy/dt.

If y = (x + √(1-x²))², then show that (x² - 1) dy/dx = 4y².

If y = (x + √(1-x²))², then show that (x² - 1)(d²y/dx²) = 4y².

If y = aˣ + b, prove that y(d²y/dx²) + (dy/dx)² = 0.

If y = (√x)/(1+√x), then find dy/dx at x = 1.

If (x² + y²)² = xy, then find dy/dx.

If xʸ = eˣ⁻ʸ, then show that dy/dx = (y(1-x))/(1-y·x).

If (sin x)^y = y^(cos x), then find dy/dx.

(a) If xy = e^(x − y), then find dy/dx.

If x = a cos θ and y = b sin θ, then prove that d²y/dx² = -(b⁴)/(a²y³).

If 2^(x–1) + 2^(y–1) = a(x – y), prove that dy/dx = 2^(y–1) / 2^(x–1).

If 2^(x-1) + 2^(y-1) = a(x - y), prove that dy/dx = (1 - 2^(x-1))/(1 - 2^(y-1)).

If 2^(x–1) + 2^(y–1) = a(x – y), prove that dy/dx = 2^(x–1)/2^(y–1).

(a) If x = e^(cos 3t) and y = e^(sin 3t), prove that dy/dx = (y log x)/(x log y).

(a) If x³⁰y²⁰ = (x + y)⁵⁰, prove that dy/dx = y/x.

If y = (log x)², prove that x²y'' + xy' = 2.

If x cos(p + y) + cos p sin(p + y) = 0, prove that cos p (dy/dx) = –cos²(p + y), where p is a constant.

Find the value of a and b so that function f defined as: f(x) = {(a + (x–2)/|x–2|), if x < 2; (a + b), if x = 2; (b + (x–2)/|x–2|), if x > 2} is a continuous function.

Given that y = (sin x)ˣ · x^(sin x) + aˣ, find dy/dx.

Given that xʸ + yˣ = aᵇ, where a and b are positive constants, find dy/dx.

If x = cos t, y = cos mt, prove that (1 – x²) d²y/dx² – x dy/dx + m²y = 0.

If x = 3 sin t – sin 3t, y = 3 cos t – cos 3t, find dy/dx and prove that d²y/dx² = –cosec³(2t) · cosec t/3

If x = a(sin t – t cos t) and y = b(cos t + t sin t), then find dy/dx and d²y/dx².

(a) If y^(√(2x + 1)) = log(√(2x + 1) – √x), show that (x² + 1)(dy/dx) + xy + 1 = 0.

(a) If sin⁻¹(√(1 - x²)) + sin⁻¹(√(1 - y²)) = a², then prove that dy/dx = -y/√(1 - x²)/√(1 - y²).

The least value of f(x) = x³ – 12x, x ∈ [0, 3] is

1. –16
2. –9
3. 0
4. 16

The absolute maximum value of f(x) = x² + 1 in [–5, 2] is

1. 26
2. 1
3. 5
4. 2

The least value of f(x) = e⁻ˣ in [0, 3] is

1. e⁻³
2. -1
3. 1
4. -e³

For f(x) = x + 1/x (x ≠ 0)

1. local maximum value is 2
2. local minimum value is -2
3. local maximum value is -2
4. local minimum value < local maximum value

The interval in which y = x³ . e^(-3x) is an increasing function is :

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

Assertion (A): f(x) = e^(-x) is a decreasing function, where x ∈ ℝ. Reason (R): If f'(x) < 0, then f(x) is a decreasing function.

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 rate of change of volume of a sphere with respect to its diameter, when its radius is 5 cm, is :

1. 400 cm3/cm
2. 100 cm3/cm
3. 50 cm3/cm
4. 25 cm3/cm

The surface area of a sphere when its volume changes at the same rate as its radius is:

1. 4π sq. units
2. 1 sq. unit
3. 4 sq. units
4. π sq. units

If the distance travelled by a particle in t seconds is given by S = 72t + 3t² − t³, then time taken by the particle to come to rest is:

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

A cylindrical tank is being filled with sand at a rate of 314 m³/h. If the radius of the tank is 10 m, then the height of sand in the tank increases at the rate of:

1. 1.1 m/h
2. 1 m/h
3. π m/h
4. 2/π m/h

Absolute minimum value of f(x) = (x – 2)2 + 5 in the interval [–3, 2] is :

1. – 3
2. 2
3. 5
4. 30

The absolute maximum value of function f(x) = x³ - 3x + 2 in [0, 2] is:

1. 0
2. 2
3. 4
4. 5

Let f(x) be a continuous function on [a, b] and differentiable on (a, b). Then, this function f(x) is strictly increasing in (a, b) if:

1. f'(x) < 0, ∀ x ∈ (a, b)
2. f'(x) > 0, ∀ x ∈ (a, b)
3. f'(x) = 0, ∀ x ∈ (a, b)
4. f(x) > 0, ∀ x ∈ (a, b)

The function f(x) = x³ - 3x² + 12x - 18 is:

1. strictly decreasing on R
2. strictly increasing on R
3. neither strictly increasing nor strictly decreasing on R
4. strictly decreasing on (-∞, 0)

If the sides of a square are decreasing at the rate of 1·5 cm/s, the rate of decrease of its perimeter is :

1. 1·5 cm/s
2. 6 cm/s
3. 3 cm/s
4. 2·25 cm/s

The point of inflexion of a function f(x) is the point where:

1. f'(x) = 0 and f'(x) changes its sign from positive to negative from left to right of that point.
2. f'(x) = 0 and f'(x) changes its sign from negative to positive from left to right of that point.
3. f'(x) = 0 and f'(x) does not change its sign from left to right of that point.
4. f'(x) ≠ 0.

The rate of change of surface area of a sphere with respect to its radius 'r', when r = 4 cm, is:

1. 64π cm²/cm
2. 48π cm²/cm
3. 32π cm²/cm
4. 16π cm²/cm

The function f(x) = kx - sin x is strictly increasing for

1. k > 1
2. k < 1
3. k > -1
4. k < -1

If f(x) = a(x - cos x), where a ∈ ℝ is strictly decreasing in which of the following?

1. {0}
2. (0, π)
3. (-π, 0)
4. (-π, π)

If f(x) = a(x - cos x) is strictly decreasing in ℝ, then a lies in which of the following?

1. {0}
2. (0, ∞)
3. (-∞, 0)
4. (-∞, ∞)

If f(x) = a(x - cos x) is strictly decreasing in:

1. {0}
2. (0, π)
3. (-π, 0)
4. (-π, π)

If y = (cos x + sin x)/(cos x - sin x), then dy/dx is:

1. 2sec²x/4
2. -2sec²x/4
3. sec x log 4
4. -sec x log 4

If y = (cos x + sin x)/(sin x - cos x), then dy/dx is:

1. sec²(x/4)
2. -sec²(x/4)
3. log|sec(x/4)|
4. -log|sec(x/4)|

A room freshner bottle in the shape of an inverted cone sprays the perfume at regular intervals such that volume of the perfume in the bottle decreases at the steady rate of 1 mm³/min. Find the rate at which level of perfume is dropping at an instant when level of perfume in the bottle is 10 mm, if the semi-vertical angle of conical bottle is π/6.

Find the absolute maximum value of f(x) = cos x + sin² x, x ∈ [0, π].

The volume of a spherical balloon is increasing at the rate of 20 cm³/s. Find the rate of change of its surface area when its radius is 4 cm.

Find the sub-interval(s) of [0, π/2] in which f(x) = tan x – 4x is increasing.

Find the sub-interval of (0, ∞) in which f(x) = x² e^(−x) is increasing.

Find the sub-interval of (0, ∞) in which f(x) = x log x is increasing.

Find the values of x for which f(x) = x^x, x > 0 is increasing.

Find the interval(s) in which the function f(x) = x/log x, where x ∈ (0, 1) ∪ (1, ∞), is increasing.

Find the interval(s) for which the function f(x) = x/3 + 3/x, x > 0 is increasing.

Determine the values of x for which f(x) = (x – 3)/(x + 1), x ≠ – 1 is an increasing function.

Determine the interval(s) in which f(x) = 5x^(3/2) – 3x^(5/2), x > 0 is increasing.

Find the intervals in which function f(x) = x^(5/3) - (5/2)x^(2/3) is (i) increasing (ii) decreasing.

Find the intervals on which the function f(x) = 10 - 6x - 2x² is (a) strictly increasing (b) strictly decreasing.

Show that of all rectangles inscribed in a given circle, the square has the maximum area.

Show that the function f(x) = 4x³ – 18x² + 27x – 7 has neither maxima nor minima.

Find the absolute maximum and minimum values of the function f(x) = 12x^(4/3) - 6x^(1/3), x ∈ [0, 1].

Find local maximum value and local minimum value (whichever exists) for the function f(x) = 4x² + 1/x (x ≠ 0).

If M and m denote the local maximum and local minimum values of the function f(x) = x + 1/x (x ≠ 0) respectively, find the value of (M - m).

Show that f(x) = eˣ - e⁻ˣ + x - tan⁻¹ x is strictly increasing in its domain.

Determine whether the function f(x) = x² - 6x + 3 is increasing or decreasing in [4, 6].

Find the interval in which the function f(x) = x⁴ - 4x³ + 10 is strictly decreasing.

The volume of a cube is increasing at the rate of 6 cm³/s. How fast is the surface area of cube increasing, when the length of an edge is 8 cm?

Show that f(x) = 4 sin x + cos x - x is an increasing function of x in [0, π/2].

Show that the function f(x) = sin⁴(x) cos⁻⁴(x), x ∈ [π/4, 3π/2], is strictly decreasing.

Find the sub-intervals in which f(x) = log(2 + x) - (2x)/(2 + x), x > -2 is increasing or decreasing.

If the product of two positive numbers is 9, find the numbers so that the sum of their squares is minimum.

If t₁ = tan⁻¹(a/x), t₂ = tan⁻¹(1/cot a/y), then show that dy/dx + ydx/dy = 0.

If the circumference of circle is increasing at the constant rate, prove that rate of change of area of circle is directly proportional to its radius.

If f(x) = a(tan x - cot x), where a ≠ 0, then find whether f(x) is increasing or decreasing function in its domain.

If y = x^(1/x), then find dy/dx at x = 1.

There exists at least one value of b for which f(x) = b^x, b ≠ 0 is strictly increasing in ℝ\{0}. State True or False. Justify.

Find the points on the curve 6y = x³ + 2 at which ordinate is changing 8 times as fast as abscissa.

Find the interval in which the function f(x) = 2x³ - 3x is strictly increasing.

Find the point on the curve y² = 8x for which the abscissa and ordinate change at the same rate.

A spherical balloon loses its volume due to escape of air from it in such a way that decrease of volume at any instant is proportional to its surface area. Show that the radius is decreasing at a constant rate.

A thin metallic wire in the shape of a circular ring has its enclosed area increasing at a uniform rate when heated. Show that the rate of change of circumference varies inversely as the radius.

The volume of a wooden block in the shape of a cube increases at a constant rate as the air becomes moist during the rainy season. Show that the rate of change of its surface area varies inversely as the length of edge of the cube.

The side of an equilateral triangle is increasing at the rate of 3 cm/s. At what rate its area increasing when the side of the triangle is 15 cm?

Find the intervals in which the function f(x) = (log x) / x is strictly increasing or strictly decreasing.

Find the sub intervals in which f(x) = cot⁻¹ (sin x + cos x), x ∈ (0, π) is increasing and decreasing.

Find the sub-interval of (0, π) in which f(x) = tan⁻¹(sin x – cos x) is increasing and decreasing.

Of all the closed right circular cylindrical cans of given volume 200 cm³, find the dimensions of the can which has the minimum surface area.

Find the sub-interval of [0, π/2] in which f(x) = log (sin x + cos x) is increasing and decreasing.

Find the absolute maximum and absolute minimum of function f(x) = 2x³ - 15x² + 36x + 1 on [1, 5].

It is given that function f(x) = x⁴ – 62x² + ax + 9 attains local maximum value at x = 1. Find the value of 'a', hence obtain all other points where the given function f(x) attains local maximum or local minimum values.

(a) It is given that function f(x) = x⁴ – 62x² + ax + 9 attains local maximum value at x = 1. Find the value of 'a', hence obtain all other points where the given function f(x) attains local maximum or local minimum values. OR (b) The perimeter of a rectangular metallic sheet is 300 cm. It is rolled along one of its sides to form a cylinder. Find the dimensions of the rectangular sheet so that volume of cylinder so formed is maximum.

If ∫ (3ax)/(b² + c²x²) dx = A log |b² + c²x²| + K, then the value of A is

1. 3a
2. 3a/(2b²)
3. 3a/(b²c²)
4. 3a/(2c²)

The value of ∫₋₁¹ x³/(x² + 2|x| + 1) dx is

1. 0
2. log 2
3. 2 log 2
4. (1/2) log 2

If ∫₀¹ (2a + 1 - x²) dx = 4, then the value of a is

1. 3/4
2. 3/2
3. 3
4. 2√3

∫(1/(cos x)) dx is equal to

1. tan²(x/2) + C
2. 1 – cot²(x/2) + C
3. –cot(2x) + C
4. tan(2x) + C

∫ cos 2x/(sin² x cos² x) dx is equal to :

1. cot x + tan x + C
2. –cot x + sec x + C
3. –cot x + tan x + C
4. –(tan x + cot x) + C

∫ dx/(x² + 6x + 10) is equal to :

1. x tan⁻¹(x + 3) + C
2. tan⁻¹(x + 3) + C
3. (1/3) tan⁻¹(x + 3) + C
4. tan⁻¹ x + C

∫ sec x tan x / (sec x - tan x) dx is equal to

1. log |sec x + tan x| + C
2. log |sec x - tan x| + C
3. log |1 + cos x| + C
4. log |1 + sin x| + C

∫_{-1}^{1} (1 – |x|) dx is equal to :

1. 2 ∫_0^1 (1 + x) dx
2. 2 ∫_{-1}^0 (1 + x) dx
3. 0
4. 2 ∫_0^{-1} (1 – x) dx

∫ e^(−2x)/(e^(−x) − 1) dx is equal to:

1. sin⁻¹ e−x + C
2. log|e−x + √(e^(−2x) − 1)| + C
3. sin⁻¹ ex + C
4. log|e−x − √(e^(−2x) − 1)| + C

∫ cos x / √(sin²x + 1) dx is equal to:

1. tan⁻¹ (sin x) + C
2. sin⁻¹ (sin x) + C
3. log|sin x + √(1 + sin²x)| + C
4. log|sin x − √(sin²x + 1)| + C

∫ dx/(25 – 16x²) is equal to:

1. (1/5) sin⁻¹ 4x + C
2. (1/25) sin⁻¹ 16x + C
3. (1/4) sin⁻¹(4x/5) + C
4. (1/16) sin⁻¹(4x/5) + C

If ∫₀¹ dx/(e^x + e^(–x)) = tan⁻¹ e + k, then the value of k is:

1. e
2. π/4
3. 0
4. –π/4

∫ 2dx/(25 – 16x²) is equal to:

1. (1/5) sin⁻¹ 4x + C
2. (1/25) sin⁻¹ 16x + C
3. (1/4) sin⁻¹ (4x/5) + C
4. (1/16) sin⁻¹ (4x/5) + C

The value of ∫–1–5 x–1 dx is equal to :

1. – log 5
2. x6
3. log (– 5)
4. x– 6

∫(1 - cos 2x)/(1 + cos 2x) dx is equal to:

1. log |sec² x| + C
2. log |cos x| + C
3. log |1 + cos 2x| + C
4. log |sec x| + C

The value of ∫₀¹ 1/(3x - 4) dx is:

1. 1/3 log 4
2. -1/3 log 4
3. log (–4)
4. log 4

∫ (1 + cos x)/(1 – cos x) dx is equal to:

1. 2 log |sin(x/2)| + C
2. (1/2) log |sin 2x| + C
3. log |1 – cos 2x| + C
4. log |1 + cos 2x| + C

If ∫₀¹ (6x² – 4x + k) dx = 0, then the value of k is:

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

∫₀¹ |x|/x dx is equal to:

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

If ∫ (1 - x²)/x² dx = k log x + C, then k is equal to:

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

∫(cos²2x)/(sin x cos x) dx is equal to

1. cot x + tan x + c
2. -cot x + tan x + c
3. cot x - tan x + c
4. -cot x - tan x + c

∫[a to b] f(x) dx is equal to:

1. ∫[a to b] f(a – x) dx
2. ∫[a to b] f(a + b – x) dx
3. ∫[a to b] f(x – (a + b)) dx
4. ∫[a to b] f((a – x) + (b – x)) dx

If ∫₀ᵃ dx/(4 + x²) = π/6, then the value of 'a' is:

1. 2√3
2. √3/2
3. √3
4. 1/√3

If f(x) is an odd function, then ∫_{-π/2}^{3π/2} f(x) cos x dx equals:

1. 2∫_0^{3π/2} f(x) cos x dx
2. 0
3. 2∫_0^{π/2} f(x) dx
4. 2∫_0^{3π/2} cos x dx

∫₀^(π/2) (sin x - cos x)/(1 + sin x cos x) dx is equal to:

1. π
2. Zero (0)
3. ∫₀^(π/2) sin²x/(1 + sin x cos x) dx
4. π²/4

If ∫₀² 2e^(2x) dx = ∫₀ᵃ e^x dx, then the value of 'a' is:

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

Anti-derivative of √(1 + sin²x), x ∈ [0, π/4] is:

1. cos x + sin x
2. -cos x + sin x
3. cos x - sin x
4. -cos x - sin x

∫_{-a}^{a} f(x) dx = 0, if :

1. f(–x) = f(x)
2. f(–x) = –f(x)
3. f(a – x) = f(x)
4. f(a – x) = –f(x)

If g(x) is a continuous function satisfying g(–x) = –g(x), then ∫₀^(2a) g(x) dx is equal to:

1. 0
2. 2∫₀^a g(x) dx
3. ∫₋ₐ^a g(x) dx
4. –∫₋₂ₐ^0 g(x) dx

The value of ∫₀³ √(9 - x²) dx is:

1. 6π
2. 4π
3. 2π
4. 18π

The value of ∫₋₁¹ (1-|x|) dx is:

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

∫₀^(π/6) sec²(6x) dx is equal to:

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

∫ dx/(x tan⁻¹x tan⁻¹(1/x)) is equal to:

1. sec²(x/4) + C
2. -sec²(x/4) + C
3. log sec(x/4) + C
4. -log sec(x/4) + C

The primitive of x²cos⁻¹(1/2) is:

1. sec² x
2. 2 sec² x tan x
3. tan x
4. -cot x

∫₀^(π/6) (sec² x - 6x) dx is equal to:

1. 1/3
2. -1/3
3. √3
4. -√3

If d/dx f(x) = 2x + 3/x and f(1) = 1, then f(x) is

1. x² + 3 log |x| + 1
2. x² + 3 log |x|
3. 2 - 3/x²
4. x² + 3 log |x| - 4

Anti-derivative of (tan x - 1)/(tan x + 1) with respect to x is

1. sec²(π/4 - x) + c
2. -sec²(π/4 - x) + c
3. log|sec(π/4 - x)| + c
4. -log|sec(π/4 - x)| + c

∫(1 - 2cos²x)/(sin²x) dx is equal to:

1. x - 2cos x + C
2. x + 2cos x + C
3. -x - 2cos x + C
4. -x + 2cos x + C

∫dx/(2^(2x)) is equal to:

1. 2^(x+2) + C
2. 2^(x+2) log 2 + C
3. 2^x/(2 log 2) + C
4. -2^(-x)/(2 log 2) + C

∫₀²dx/(x²-4) equals:

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

∫ 2^(2x) dx is equal to

1. 2^(x+2) + C
2. 2^(x+2)/(log 2) + C
3. 2^(2x)/(log 2) + C
4. 2^(2x) log 2 + C

∫ x e^x/(1 + x)² dx is equal to

1. x e^x + C
2. -x e^x + C
3. x e^(2x) + C
4. e^x/(1 + x) + C

The value of ∫₀^(π/2) log(tan x) dx is

1. π/2
2. 0
3. π
4. 1

dx ex log 5 is equal to :

1. 5x5 + C
2. 6x6 + C
3. 5x4 + C
4. 6x5 + C

∫ ex log 5 dx is equal to:

1. 5x⁵ + C
2. 6x⁶ + C
3. 5x⁴ + C
4. 6x⁵ + C

∫₀⁴ (ex)² dx is equal to:

1. 2e - 15/8
2. 2e - 16/8
3. 2/15 - e/8
4. 2/15 - e⁸

If ∫₀ᵃ x³ dx = 8, then a is:

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

Find: ∫ (x³ – 4)e^x/(x – 2) dx

Find: ∫ cosec(3x+1) cot(3x+1) dx

Find: ∫ x/√(1 + x²) dx

Evaluate: ∫₀ᵃ x²/(6+x) dx

Find: ∫(e^x⁴ + 1)/(e^x⁴(1 + e^(-x⁴))) dx

Evaluate: ∫₀^(π/2) sin 2x cos 3x dx

Find: ∫ x/(√(x² - 1)) dx.

Evaluate: ∫₁² (1 - cos x)/(1 - x)² × log((1 + x)/(1 - x)) dx

Find: ∫₋₂² sin(3x)sin(x) dx

Find: ∫ sin3x sin x dx

Evaluate: ∫₀¹ xe^x dx / (1 + x)²

Evaluate: ∫₀¹ e^x √(x²+1) dx

Evaluate: ∫₀¹ x tan⁻¹x dx.

Find ∫ (x + 2)/(x – 2) dx.

If I₁ = ∫₍₋π/₄₎^(π/4) 1/(1 + cos 2x) dx and I₂ = ∫₍₋₁/₂₎^(1/2) |x| dx, then show that I₁ – 4I₂ = 0.

If I₁ = ∫₋π/₄^(π/4) dx/(1 + cos 2x) and I₂ = ∫₋₁/₂^(1/2) |x| dx, then show that I₁ – 4I₂ = 0.

Evaluate: ∫₀¹ x sin⁻¹x dx

Evaluate: ∫₀¹ log(1 + x²) dx

Find: ∫ x²/(x² - 9) dx

Evaluate: ∫₅π/₁₂^π/₁₂ cot⁻¹ x dx

If d/dx(F(x)) = (1 + e^(1/x))/(x²), then find F(x) given that F(0) = log(1/2).

Find: ∫(x + 1)/(x² – 6x + 2) dx

Evaluate: ∫(5π/12 to π/12) cot⁻¹(x) dx

Find: ∫ x/(x² – 6x + 7) dx

Find: ∫ (3x² - 1)/(x⁴ - 4x) dx

Find : ∫ dx/(x^(1/2) + x^(1/3))

Evaluate : ∫_0^π (sin^2026 x)/(sin^2026 x + cos^2026 x) dx

Find : ∫ cos x/((2 + sin x)(4 + sin x)) dx

Evaluate: ∫_{−π/2}^{π/2} x² cos x / (2^(1 + x) + 1) dx

Evaluate: ∫_0^(π/2) sin²x / (1 + sin 2x) dx

Find : ∫ (x sin x)/(1 – cos x) dx

(a) Find: ∫ (sin x + 1 + cos x)/(x + cos x) dx

Find: ∫ x²/((x+1)²(x+3)) dx

Evaluate: ∫₋₆⁶ |x-2| dx

Find: ∫ x / ((x² + 4)(x² + 9)) dx

Find: ∫ x²/((x² + 4)(x² + 9)) dx

(a) Evaluate: ∫ 2x/(x² - 2x + 2) dx

Find: ∫ x² · sin⁻¹(x^(3/2)) dx

(a) Evaluate: ∫ (x² – 2x + 2)/x² dx

Find: ∫ x² log(x² - 1) dx

Find: ∫ (x² + 1) / ((x² + 2)(x² + 4)) dx

Find: ∫ e^x (2 sin² x + 1) / (1 + cos 2x) dx

Find: ∫ (–1)/(x²(log x)² – 5log x + 4) dx

Find: ∫ e^x/(2 + sin²x + 2cos x) dx

Evaluate: ∫₀^(π/4) x/(1 + cos 2x + sin 2x) dx.

Find: ∫ (3x + 5)/(x² + 2x + 4) dx.

Find: ∫ x/(x² + 3x + 3) dx

Evaluate: ∫₀^(2π) [log (sin x) - log (2 cos x)] dx.

Find: ∫ x/[(x+1)(x+2)] dx.

(a) Evaluate: ∫₀^(2π) eˣ sin(x) dx OR (b) Find: ∫ 1/[cos(a-x)cos(b-x)] dx.

(a) Evaluate: ∫₀^(π/2) e^x sin x dx OR (b) Find: ∫ 1/(cos(a-x)cos(b-x)) dx

Evaluate: ∫₀^π |sin x - cos x| dx

Find: ∫ 1/(√x(1-x)(2-x)) dx

Evaluate: ∫₀^π eˣ sin x dx

Evaluate: ∫₀^(π/2) [log(sin x) - log(2 cos x)] dx

Find: ∫ (x dx)/((x² + 4)(x² + 9))

Evaluate ∫(log 2 to log 3) 1/((eˣ + e⁻ˣ)(eˣ - e⁻ˣ)) dx

Evaluate ∫(-1 to 1) |x⁴ - x| dx

Find ∫ eˣ((1 - sin x)/(1 - cos x)) dx

Evaluate: ∫₁³ (4x)/(4x + x⁴) dx.

Evaluate: ∫₁² e (1 + 4x - x² log x)/(x²) dx.

Find: ∫ cos x/(sin³ x) dx.

Find: ∫(e^x)/(5 - 4e^x - e^(2x)) dx

Evaluate: ∫₋π/2^(π/2) (sin¹⁰⁰ x)/(sin¹⁰⁰ x + cos¹⁰⁰ x) dx

Evaluate: ∫₀^(π/4) log(1 + tan x) dx

Find: ∫ 1/(4 + x² - 2x + 1) dx

Find: ∫ (6x - 12)/(x² - 4x + 1) dx

Evaluate: ∫₀^(π/4) e^(2x) sin⁻¹(2x)/(1 - 4x²) cos 2x dx

Find: ∫ (5e^x + 4)/(e^(2x) - e^x) dx

Find: ∫ 1/((1+x)(1-x²)) dx

Evaluate: ∫ x(x^(13/3) + x^(14/3)) dx

Evaluate: ∫ eˣ sin⁻¹(2x)/(√(1-4x²)) - 2x cos(2x) dx

Evaluate: ∫₀¹⁰ tan⁻¹(x) dx

Evaluate: ∫₀¹ tan⁻¹x dx

Find: ∫ cot x · sin x cos x dx

Find: ∫ dx / (1 + e^(-x))

Evaluate: ∫₀¹ x e^x / (x+1)² dx

Find: ∫ 1/(e^x - 1) dx

Find: ∫₋₁¹ 1/(1+e^(1/x)) dx

(a) Find: ∫ x²/((x – 1)(x + 4)) dx

Evaluate: ∫[0 to π/4] (sin(x) + cos(x)) / (9 + 16sin(2x)) dx

Evaluate: ∫₀^(π/2) e^x * (1 + sin x)/(1 + cos x) dx

Find: ∫ (3cos²x – 5sinx + sin²x)/(4 – sinx – cosx) dx

Evaluate: ∫₀¹⁰ n(x)/(x(1+x)) dx

Evaluate: ∫₀^(π/2) (sin x)/(sin x + cos x) dx

Evaluate: ∫₀¹ ln(x/(1-x)) dx

Evaluate: ∫₀^(π/2) (cos log x - sin log x)² dx

Evaluate: ∫₀^(π/2) (9sin²x + 16cos²x) / √(x) dx

Evaluate: ∫ (e^(x/3))/((3-4e^x)³) dx

The area bounded by the curve y = x|x|, x-axis and the ordinates x = –1 and x = 1 is given by

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

Which of the following expressions will give the area of region bounded by the curve y = x² and line y = 16?

1. ∫₀⁴ x² dx
2. 2∫₀⁴ x² dx
3. ∫₀¹⁶ √y dy
4. 2∫₀¹⁶ √y dy

The area of the shaded region of the circle given below is equal to:

1. ∫_1^3 √(9 − y²) dy
2. ∫_1^3 2√(9 − y²) dy
3. ∫_0^3 √(9 − x²) dx
4. ∫_0^3 2√(9 − x²) dx

The area of the region bounded by the curve y = √x and x-axis, between x = 0 and x = 2 is:

1. 2 sq. units
2. 1/2 sq. unit
3. 1 sq. unit
4. 4 sq. units

An ant is observed crawling on a sheet of paper along a straight line given by equation y = 2x – 4. Area of the surface covered by the ant bounded by y-axis, x-axis and x = 1 is :

1. 1 sq. unit
2. 3 sq. units
3. 2 sq. units
4. 4 sq. units

Sketch the graph of y = |x + 3| and find the area of the region enclosed by the curve, x-axis, between x = -6 and x = 0, using integration.

Find the area of the region {(x, y) : x² ≤ y ≤ x + 2}, using integration.

Using integration, find the area of the region {(x, y) : y² ≤ x ≤ y}.

If the area of the region bounded by the line y = mx and the curve x² = y is 32/3 sq. units, then find the positive value of m, using integration.

A rectangle of perimeter 36 cm is revolved around one of its sides to sweep out a cylinder of maximum volume. Find the dimensions of the rectangle.

Find the area of the region bounded by the line y = 3x + 2, the x-axis and the ordinates x = –1 and x = 1.

Using integration, find the area of the region enclosed by the curve y = |x − 6|, the x-axis, and between x = 4 and x = 8.

Using integration, find the area of the region bounded by the curve y = x|x|, x-axis, x = −2 and x = 2.

Using integration, find the area of the region bounded by y = 5x + 4, y = 0, x = −1 and x = 1.

Sketch the curve {(x, y) : 100x2 + 25y2 = 2500} and find the area of the region enclosed by it, using integration.

Sketch the curve described by {(x, y) : 9x² + 16y² = 144} and find the area of the region enclosed by it, using integration.

Sketch the graph defined by {(x, y): x²/125 + y²/25 = 1}. Find the area of the region of minor segment cut off by the line x = 5/2, using integration.

Using integration, find the area of the region bounded by the curve y = x², x = -1, x = 1 and the x-axis.

Using integration, find the area of the ellipse 16x² + 4y² = 1, included between the lines x = –2 and x = 2.

If A₁ denotes the area of region bounded by y² = 4x, x = 1 and x-axis in the first quadrant and A₂ denotes the area of region bounded by y² = 4x, x = 4, find A₁ : A₂.

Using integration, find the area of the region enclosed between the circle x² + y² = 16 and the lines x = –2 and x = 2.

Find the area of the region bounded by the lines x – 2y = 4, x = –1, x = 6 and x-axis, using integration.

Sketch the graph of y = x|x| and hence find the area bounded by this curve, X-axis and the ordinates x = -2 and x = 2, using integration.

Find the area of the region bounded by the curves x² = y, y = x + 2 and x-axis, using integration.

Find the area of the region {(x, y) : x² + y² ≤ 1 ≤ x + y}, using integration.

Find the area of the region bounded by the lines y = 4x + 5, x + y = 5 and 4y = x + 5, using integration.

Using Integration, find the area of triangle whose vertices are (-1, 1), (0, 5) and (3, 2).

The area of the region bounded by the line y = mx (m ≠ 0), the curve x² + y² = 4 and the x-axis in the first quadrant is 2 units. Using integration, find the value of m.

Using integration, find the area of the region bounded by the circle x² + y² = 16, line y = x and y-axis, but lying in the 1st quadrant.

Using integration, find the area of the region bounded by the parabola y² = 4ax and its latus rectum.

Using integration, find the area of the smaller region enclosed by the curve 4x² + 4y² = 9 and the line 2x + 2y = 3.

The integrating factor of differential equation R(dy/dx) + Px = Q where P, Q, R are functions of y is

1. e^(∫Q/P dy)
2. e^(∫P dy)
3. e^(∫R/P dy)
4. e^(∫R/P dx)

The order and degree of the differential equation d/dx(e^y) = 0 respectively are

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

The order and degree of the differential equation: d/dx(sin y) = y² respectively are

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

The order and degree of the differential equation d/dx(y')³ + (y')³ = 1 respectively are, where y' = dy/dx

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

The general solution of the differential equation dy/dx = y/x is

1. log y = log x + C
2. y + x = C
3. y - x = C
4. log y + log x = C

The integrating factor of the differential equation 2x(dy/dx) - y = 3 is

1. x
2. 1/x
3. eˣ
4. e⁻ˣ

The general solution of the differential equation dy/dx = e^(y – x) is :

1. e^x – e^y = C
2. e^(-y) + e^x = C
3. e^(-y) – e^(-x) = C
4. e^(-x) + e^(-y) = C

The integrating factor of (1 + y²) dx = (tan⁻¹ y – x) dy is :

1. tan y
2. tan⁻¹ y
3. e^(tan⁻¹ y)
4. e^(tan⁻¹ x)

The general solution of the differential equation: x²dy + y²dx = 0 is

1. x³ + y³ = k
2. 1/x - 1/y = k
3. 1/x + 1/y = k
4. log y² + log x² = k

dy/dx = F(x, y) will be a homogeneous differential equation for which of the following functions? (i) F(x, y) = 3x + 2y (ii) F(x, y) = sin(y/x) + log y − log x (iii) F(x, y) = e^(y/x) + 1 (iv) F(x, y) = √(x²y²) − y

1. (i) and (ii)
2. (i), (ii) and (iii)
3. (ii), (iii) and (iv)
4. (ii) and (iii)

Product of the order and degree of differential equation 1 + (dy/dx)³ = (d³y/dx³)² is:

1. 5
2. 6
3. 2
4. 3

Which of the following is not a Linear Differential Equation?

1. (1 + x²) dy + 2xy dx = cot x dx
2. y + d/dx(xy) = x (sin x + log x)
3. x (1 + y²) dx − y (1 + x²) dy = 0
4. y dx − (x + 3y²) dy = 0

The sum of the order and the degree of the differential equation d²y/dx² + 4(dy/dx) + 3(1 – y²) = 0 is:

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

The sum of the order and the degree of the differential equation y(d²y/dx²) + (dy/dx)² − 3(dy/dx) − 2 = x is:

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

The order and degree of the differential equation 1 + (dy/dx)3/3 = (d2y/dx2)2 is :

1. Order = 3, Degree = 3
2. Order = 2, Degree = 2
3. Order = 3, Degree = 1
4. Order = 2, Degree = 1

The general solution for the differential equation dy/dx = e3x–y is :

1. 3ey = e3x + C
2. log (3x – y) = C
3. e3x–y = C
4. – ey + 3e3x = C

The order and degree of differential equation d²y/dx² = 1 - (3(dy/dx)²)^(2/3) is:

1. Order = 3, Degree = 3
2. Order = 2, Degree = 2
3. Order = 3, Degree = 2
4. Order = 2, Degree = 1

The order and degree of differential equation (d²y/dx²)² – (dy/dx) = y³ is:

1. Order = 2, Degree = 2
2. Order = 2, Degree = 3
3. Order = 1, Degree = 2
4. Order = 2, Degree = 4

The integrating factor of differential equation (x + 2y³) dy/dx = y² is:

1. e^(y²/2)
2. 1/y
3. 1/(2y)
4. e^(-y²/2)

The solution of the differential equation dy/dx = 1 - x + y - xy is

1. log |1 + y| = x - x²/2 + c
2. log |1 + y| = -x + x²/2 + c
3. e^y = x - x²/2 + c
4. e^(1+y) = -x + x²/2 + c

The degree of the differential equation x(d²y/dx²)³ + y(dy/dx)⁴ + y⁵ = 0 is

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

The integrating factor of the differential equation dy/dx + y tan x = 2x + x² tan x is

1. e^(sec x)
2. sec x + tan x
3. sec x
4. cos x

The order and degree of the differential equation [1 + (dy/dx)²]^(3/2) = d²y/dx² respectively are:

1. 1, 2
2. 2, 3
3. 2, 1
4. 2, 6

The integrating factor of the differential equation x dy/dx - y = x⁴ - 3x is:

1. x
2. -x
3. x⁻¹
4. log(x⁻¹)

The solution of the differential equation dy/dx = 1/log y is:

1. log y = x + c
2. y log y – y = x + c
3. log y – y = x + c
4. y log y + y = x + c

The differential equation dy/dx = F(x, y) will not be a homogeneous differential equation, if F(x, y) is:

1. cos x - sin(y/x)
2. y/x
3. (x² + y²)/(xy)
4. cos²(y/x)

The degree of the differential equation (y'')² + (y')³ = x sin(y') is:

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

The number of solutions of differential equation dx/dy – y = 1, given that y(0) = 1, is:

1. 0
2. 1
3. 2
4. infinitely many

The degree and order of differential equation y'' + log(y') = x⁵ respectively are:

1. not defined, 5
2. not defined, 2
3. 5, not defined
4. 2, 2

x log x (dy/dx) + y = 2 log x is an example of a :

1. variable separable differential equation
2. homogeneous differential equation
3. first order linear differential equation
4. differential equation whose degree is not defined

The general solution of the differential equation x dy + y dx = 0 is:

1. xy = c
2. x + y = c
3. x² + y² = c²
4. log y = log x + c

The integrating factor of the differential equation (x + 2y²) x(dy/dx) = y (y > 0) is:

1. 1/x
2. x
3. y
4. 1/y

The number of arbitrary constants in the particular solution of the differential equation log(dy/dx) = 3x + 4y; y(0) = 0 is/are:

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

The sum of the order and the degree of the differential equation sin(d²y/dx²) = d/dx(dy/dx)² is:

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

The general solution of the differential equation x dy - (1 + x²) dx = dx is:

1. y = 2x + 3x³ + C
2. y = 2 log x + 3x³ + C
3. y = 2x² + C
4. y = 2 log x + 2x² + C

The sum of the order and the degree of the differential equation y sin(d²y/dx²) + (dy/dx)³ is:

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

The sum of the order and the degree of the differential equation y sin(d²y/dx²) = (d³y/dx³)² is:

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

Degree of the differential equation sin x + cos(dy/dx) = y² is

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

The solution of the differential equation x dx + y dy = 0 is:

1. 1/x + 1/y = C
2. log x - log y = C
3. xy = C
4. x + y = C

What is the product of the order and degree of the differential equation d²y/dx² + sin(dy/dx) + y cos y = 0?

1. 3
2. 2
3. 6
4. not defined

What is the product of the order and degree of the differential equation d²y/dx² - sin(dy/dx) + y cos(dy/dx) = y²?

1. 3
2. 2
3. 6
4. not defined

What is the product of the order and degree of the differential equation d²y/dx² + y sin(dy/dx) + y cos(dy/dx) = 0?

1. 3
2. 2
3. 6
4. not defined

The integrating factor for solving the differential equation dy/dx - 2xy = x² is:

1. e^y
2. e^x
3. x
4. 1/x

The number of solutions of the differential equation dy/dx = 1/(1 - y), when y(1) = 2, is:

1. zero
2. one
3. two
4. infinite

The order and degree (if defined) of the differential equation d³y/dx³ + (d²y/dx²)² = sin x · dy/dx respectively are:

1. 2, 2
2. 1, 3
3. 2, 3
4. 2, degree not defined

The integrating factor for solving the differential equation dy/dx + 2x²y = x is:

1. eʸ
2. eˣ
3. x
4. 1/x

(b) If y = A sin 2x + B cos 2x and d²y/dx² – ky = 0, find the value of k.

Find the product of the order and the degree of the differential equation d²y/dx² - xy(dy/dx) = 0.

Find the particular solution of the differential equation 2y² dx/dy, given y = 1 when x = 1.

Find the general solution of the differential equation sec²x · tan y dx + sec²y · tan x dy = 0.

Find the general solution of the following differential equation: x² dy/dx = x² + xy + y².

Find the general solution of the differential equation: y²dx + (x² – xy + y²)dy = 0

Find the general solution of the differential equation (y² – x²)dx = 2xy dy

Solve the following differential equation: x(dy/dx) = y - x sin²(y/x), given that y(1) = π/6.

Solve the differential equation: x(dy/dx) = y – x tan(y/x)

Find the general solution of the differential equation (x² + y²) dy = xy dx.

Find the general solution of the differential equation (x² − y²) dx + 2xy dy = 0.

Solve the differential equation (x + 2y3) dy = y dx.

Solve the differential equation y dx + (x – y³) dy = 0.

Solve the differential equation (x – sin y) dy + tan y dx = 0.

Find the particular solution of the differential equation 2xy dy/dx = x² + 3y², given that y(1) = 0.

Find the particular solution of the differential equation given by x² dy/dx – xy = x² cos²(y/(2x)), given that when x = 1, y = π/2.

Find the general solution of the differential equation dy/dx = (x² + y²)/(2xy).

Solve the following differential equation: (tan⁻¹ y – x) dy = (1 + y²) dx

(a) Find the particular solution of the differential equation given by 2xy + y² - 2x² dy/dx = 0; y = 2, when x = 1.

If x cos(p + y) + cos p sin(p + y) = 0, prove that cos p dy/dx = –cos²(p + y), where p is a constant.

Find the particular solution of the differential equation x(dy/dx) = y cot 2x, given that y(π/4) = 2.

(a) Find the particular solution of the differential equation dy/dx + sec²(x) · y = tan(x) · sec²(x), given that y(0) = 0. OR (b) Solve the differential equation given by x dy - y dx - √(y²+x²) dx = 0.

(a) Find the general solution of the differential equation: dy/dx = (e^(11y/x) - y/x)/(e^(11y/x) + y/x). OR (b) Find the particular solution of the differential equation dy/dx + cot x · y = cos² x, given that when x = π/2, y = 0.

Find the general solution of the differential equation: dy/dx - xy² = (x sin x)/x

Find the general solution of the differential equation: (xy - x²) dy = y² dx.

Find the general solution of the differential equation: (x + y)² dy/dx = (x + y)² - 1.

Find the general solution of the differential equation (x² + y²)dy/dx = 2(1 - x)y

Find the general solution of the differential equation: (x - y)dy/dx = (x + 2y)² - 1

Find the particular solution of the differential equation (y + 3x²) dy/dx = x, given that y = 1, when x = 1.

Find the general solution of the differential equation x dy/dx + y sin(xy) = 0.

Solve the following differential equation: (1 + eʸ/ˣ) dy + eʸ/ˣ(1 - y/x) dx = 0

(a) Solve the differential equation y e^y dx = (y³ + 2x e^y) dy, when y(0) = 1.

Find the particular solution of the differential equation x dy - y dx = x√(1 + (y/x)²) dx, given that y(1) = 0.

Find the particular solution of the differential equation 2x dy/dx - 11y = x√y, given that y(1) = 0.

Find the particular solution of the differential equation (1-x²)(dy/dx) - xy = 0, given that y(1) = 0.

The value of p for which vectors î + 2ĵ + 3k̂ and 2î – pĵ + k̂ are perpendicular to each other is

1. 0
2. 1
3. 5/2
4. –5/2

The value of m for which the points with position vectors –î – ĵ + 2k̂, 2î + mĵ + 5k̂ and 3î + 11ĵ + 6k̂ are collinear, is

1. 8
2. –8
3. 2
4. 5/2

If |a| = 8, |b| = 3 and |a × b| = 12, then the value of |a · b| is

1. 6√3
2. 8√3
3. 12√3
4. √(3·12)

The value of p for which vectors i + 2j + 3k and 2i – pj + k are perpendicular to each other is

1. 0
2. 1
3. 5/2
4. -5/2

If |a| = 5 and a·b = -2 and |a·b| = 1, then the sum of greatest and the smallest value of |a| is

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

Vector of magnitude 3 making equal angles with x and y axes and perpendicular to z axis is

1. î + 2√2ĵ
2. 3k̂
3. (3√2/2)î + (3√2/2)ĵ
4. 3î + 3ĵ + 3k̂

Assertion (A): For two vectors a and b, |a × b|² + (a·b)² = |a|²|b|². Reason (R): |a × b| = (a·b) tan θ, where 0 ≤ θ ≤ π/2.

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 |a| = 5 and –2 ≤ a·b ≤ 1, then the sum of greatest and the smallest value of |b| is

1. –5
2. 5
3. 10
4. 15

Assertion (A): A line can have direction cosines < 1, 1, 1 > Reason (R): cos θ = 1 is possible for θ = 0.

1. Both Assertion (A) and Reason (R) are true and the 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.

For two vectors a and b Assertion (A): |a × b|² + (a · b)² = |a|² |b|² Reason (R): |a × b| = (a · b) tan θ, where 0 ≤ θ ≤ π/2

1. Both Assertion (A) and Reason (R) are true and the 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.

Let a and b be two unit vectors and θ is the angle between them. Then a + 2b is a unit vector if :

1. θ = 2π/3
2. 2θ = 3π
3. 3θ = 4π
4. θ = π

The area (in square units) of a parallelogram, whose diagonals are a = î + ĵ – 2k̂ and b = î – 3ĵ + 4k̂ is :

1. 2√14
2. √14
3. √38
4. √26

If |a| = 5 and -2 ≤ a·b ≤ 1, then the sum of greatest and the smallest value of |a × b| is

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

For two vectors a and b, Assertion (A): |a × b|² + (a·b)² = |a|²|b|². Reason (R): |a × b| = |a||b|sin θ, π/2 ≥ θ ≥ 0

1. Both Assertion (A) and Reason (R) are true and the 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 (→a + →b) · (→a – →b) = 198 and |→a| = 10|→b|, then :

1. |→a| = √2
2. |→b| = √2
3. |→b| = 10√2
4. |→a| = 10√2

If (3î − 2ĵ + 5k̂) × (4î + pĵ + qk̂) = 0, then the values of p and q are:

1. p = −2/3, q = 5/3
2. p = −8/3, q = 20/3
3. p = 20/3, q = −8/3
4. p = 0, q = 0

Three points A(0, 1, 1), B(2, 0, −1) and C(1, 0, 3) form triangle ABC. The ar(ABC) is:

1. √53/2 sq. units
2. √53 sq. units
3. √11/2 sq. units
4. √11 sq. units

If position vector p→ of a point (24, n) is such that |p→| = 25, then the value of n is :

1. 49
2. 5
3. 1
4. 7

If vectors a = 3î + 2ĵ + k̂ and b = 2î – 4ĵ + 5k̂, represent the two strips of the Red Cross sign placed outside a doctor's clinic, then the value of a · b is:

1. 1
2. 5/2
3. 2/5
4. 0

If vector a = 3i + 2j - k and vector b = i - j + k, then which of the following is correct?

1. a || b
2. a ⊥ b
3. |b| > |a|
4. |a| = |b|

Let the vectors a and b be such that |a| = 3 and |b| = (3√2), then a × b is a unit vector, if the angle between a and b is

1. π/3
2. π/4
3. π/6
4. π/2

If a = 2î - 2ĵ + k̂, b = î + 2ĵ - 3k̂ and c = 2î - ĵ + 4k̂, then the projection of (c - b) along a is

1. 15
2. 5
3. 3/2
4. 1

If a, b and c are unit vectors such that a + b + c = 0, then (a·b + b·c + c·a) is equal to

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

Let θ be the angle between two unit vectors â and b̂ such that sin θ = 3/5. Then, â · b̂ is equal to:

1. ± 3/5
2. ± 4/3
3. ± 4/5
4. ± 3/4

The vector with terminal point A (2, –3, 5) and initial point B (3, –4, 7) is:

1. î – ĵ + 2k̂
2. î + ĵ + 2k̂
3. –î – ĵ – 2k̂
4. –î + ĵ – 2k̂

For any two vectors a and b, which of the following statements is always true?

1. a·b ≥ |a||b|
2. a·b = |a||b|
3. a·b ≤ |a||b|
4. a·b < |a||b|

The unit vector perpendicular to both vectors î + k̂ and î - k̂ is:

1. 2ĵ
2. ĵ
3. î - 2k̂
4. î + 2k̂

Assertion (A): For two non-zero vectors a and b, a·b = b·a. Reason (R): For two non-zero vectors a and b, a × b = b × a.

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 unit vector perpendicular to both vectors i + k and i – k is:

1. 2j
2. j
3. –i + 2k
4. i + 2k

If a = 2î – ĵ + k̂ and b = î + ĵ – k̂, then a and b are :

1. collinear vectors which are not parallel
2. parallel vectors
3. perpendicular vectors
4. unit vectors

If |a| = 2 and –3 ≤ k ≤ 2, then |ka| ∈ :

1. [–6, 4]
2. [0, 4]
3. [4, 6]
4. [0, 6]

Assertion (A): Projection of a on b is same as projection of b on a. Reason (R): Angle between a and b is same as angle between b and a numerically.

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 vectors a and b are such that |a| = 1, |b| = 2 and a · b = √3, then the angle between 2a and -b is:

1. π/6
2. π/3
3. 5π/6
4. 11π/6

The vectors a = 2î - ĵ + k̂, b = î - 3ĵ - 5k̂ and c = -3î + 4ĵ + 4k̂ represent the sides of

1. an equilateral triangle
2. an obtuse-angled triangle
3. an isosceles triangle
4. a right-angled triangle

Let a be any vector such that |a| = a. The value of |a × î|² + |a × ĵ|² + |a × k̂|² is:

1. a²
2. 2a²
3. 3a²
4. 0

A vector perpendicular to the line r = î + ĵ - k̂ + λ(3î - ĵ) is:

1. 5î + ĵ + 6k̂
2. î + 3ĵ + 5k̂
3. 2î - 2ĵ
4. 9î - 3ĵ

The vectors a = 2î - ĵ + k̂, b = î - 3ĵ - 5k̂ and c = -3î + 4ĵ + 4k̂ represents the sides of:

1. an equilateral triangle
2. an obtuse-angled triangle
3. an isosceles triangle
4. a right-angled triangle

If a and b are two vectors such that |a| = 1, |b| = 2 and a · b = √3, then the angle between 2a and -b is:

1. π/6
2. π/3
3. 5π/6
4. 11π/6

The value of p for which the vectors 2î + pĵ + k̂ and -4î - 6ĵ + 26k̂ are perpendicular to each other, is:

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

The value of (î × ĵ) · ĵ + (ĵ × î) · k̂ is:

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

If a + b = î and a = 2î - 2ĵ + 2k̂, then |b| equals:

1. √14
2. 3
3. √12
4. √17

If the vector î - bĵ + k̂ is equally inclined to the coordinate axes, then the value of b is:

1. -1
2. 1
3. -√3
4. -1/√3

If a + b = î and a = 2î - 2ĵ + 2k̂, then |b| equals:

1. √14
2. 3
3. √12
4. √17

The value of p for which the vectors 2î + pĵ + k̂ and -4î - 6ĵ + 26k̂ are perpendicular to each other, is:

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

For what value of λ, the projection of vector î + λĵ on vector î - ĵ is 2?

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

Unit vector along PQ, where coordinates of P and Q respectively are (2, 1, -1) and (4, 4, -7), is

1. 2i + 3j - 6k
2. -2i - 3j + 6k
3. (-2i/7) - (3j/7) + (6k/7)
4. (2i/7) + (3j/7) - (6k/7)

If in triangle ABC, BA = 2a and BC = 3b, then AC is

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

If |a × b| = 3 and a · b = -3, then angle between a and b is

1. 2π/3
2. π/6
3. π/3
4. 5π/6

If a vector makes an angle of π/4 with the positive directions of both x-axis and y-axis, then the angle which it makes with positive z-axis is:

1. π/4
2. 3π/4
3. π/2
4. 0

a and b are two non-zero vectors such that the projection of a on b is 0. The angle between a and b is:

1. π/2
2. π
3. π/4
4. 0

The direction cosines of vector BA, where coordinates of A and B are (1, 2, -1) and (3, 4, 0) respectively, are:

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

In triangle ABC, AB = î + ĵ + 2k̂ and AC = 3î - ĵ + 4k̂. If D is mid-point of BC, then vector AD is equal to

1. 4î + 6k̂
2. 2î - 2ĵ + 2k̂
3. î - ĵ + k̂
4. 2î + 3k̂

A unit vector along the vector 4î - 3k̂ is:

1. 1/7(4î - 3k̂)
2. 1/5(4î - 3k̂)
3. -1/7(4î - 3k̂)
4. -1/5(4î - 3k̂)

If θ is the angle between two vectors a and b, then a · b ≥ 0 only when:

1. 0 ≤ θ ≤ π/2
2. 0 ≤ θ < π/2
3. 0 < θ ≤ π
4. 0 < θ ≤ π/2

The sine of the angle between the vectors a = 3î + ĵ + 2k̂ and b = î + ĵ + 2k̂ is:

1. 2/√15
2. -2/√15
3. 2/√13
4. 2/√14

Find the vector of magnitude 14 in the direction of QP, where P and Q are the points (1, 3, 2) and (–1, 0, 8) respectively.

Vectors a = 3î – 2ĵ + 2k̂ and b = î + 2k̂ represent the two adjacent sides of a parallelogram. Find the vectors representing its diagonals and hence find their lengths.

A vector a of magnitude 14 has direction ratios <2, 3, –6>. Find the projection of the vector a on î.

Using vectors, find the area of triangle ABC with vertices A(1, 2, 3), B(2, –1, 4) and C(4, 5, –1).

Vectors a = 3i – 2j + 2k and b = i + 2k represent the two adjacent sides of a parallelogram. Find the vectors representing its diagonals and hence find their lengths.

If AB = ĵ + k̂ and AC = 3î - ĵ + 4k̂ represent the two vectors along the sides AB and AC of triangle ABC, prove that the median is 2AD = AB + AC, where D is midpoint of BC. Hence, find the length of median AD.

If AB = ĵ + k̂ and AC = 3î – ĵ + 4k̂ represent the two vectors along the sides AB and AC of △ABC, prove that the median AD = (1/2)(AB + AC), where D is midpoint of BC. Hence, find the length of median AD.

Find a unit vector perpendicular to both of the vectors AB and BC, where A, B and C are (3, –1, 2), (1, –1, –3) and (4, –3, 1), respectively.

If for two unit vectors →a and →b, |→a + 2→b| = |2→a – →b|, then find the angle between →a and →b.

If the position vectors of three points A, B and C are 3î + ĵ, 5î + 6ĵ − 3k̂ and 4ĵ respectively, then show that they form an isosceles triangle.

(a) Let two rods placed on the ground be represented by vectors 4î − ĵ + 3k̂ and −2î + ĵ − 2k̂. Find a vector representing a flag-post of height 5 m that has to be erected perpendicular to both the rods.

If (2î + 3ĵ + k̂) and (2î + ĵ − k̂) represent the sides AB and AC respectively of triangle ABC, find the vector representing the median through A.

If in a parallelogram PQRS, PQ→ = 2î + 3ĵ + k̂ and QR→ = î − 2ĵ + k̂, then find the unit vectors parallel to the diagonals PR→ and SQ→.

Three honey bees were found flying along the vectors a→ = 2î – 3ĵ + k̂, b→ = 4ĵ – 2k̂ and c→ = 3î + 2k̂ respectively. Find the value of λ such that the path for a→ + λb→ is perpendicular to c→.

Three honey bees were found flying along the vectors a = 2î – 3ĵ + k̂, b = λĵ – 2k̂ and c = 3î + 2k̂ respectively. Find the value of λ such that the path for λa + b is perpendicular to c.

The diagonals of a parallelogram are given by a = 2i - j + k and b = i + j - 3k. Find the area of the parallelogram.

(a) Two friends while flying kites from different locations, find the strings of their kites crossing each other. The strings can be represented by vectors a = 3i + j + 2k and b = i - 2j + 4k. Determine the angle formed between the kite strings.

If a and b are two non-zero vectors such that (a + b) ⊥ a and (2a + b) ⊥ b, then prove that |b| = 2|a|.

If ā and b̄ are two non-zero vectors such that (ā + b̄) ⊥ ā and (2ā + b̄) ⊥ b̄, then prove that |b̄| = 2|ā|.

Find the position vector of point C which divides the line segment joining points A and B having position vectors î + 2ĵ – k̂ and –î + ĵ + k̂ respectively in the ratio 4 : 1 externally. Further, find |AB| : |BC|.

Let a and b be two non-zero vectors. Prove that |a × b| ≤ |a||b|. State the condition under which equality holds, i.e., |a × b| = |a||b|.

If a, b and c are three unit vectors such that a + b – c = 0, find the angle between vectors a and c.

(a) Position vectors of the points A, B and C are a, b and c respectively. If AC = (4/5)AB, express c in terms of a and b. OR (b) Check whether the lines given by equations x = 2 + 2λ, y = 7 + λ, z = -3 - 3λ and x = -μ - 2, y = 2 + 8μ, z = 4 + 5μ are perpendicular to each other or not.

For two non-zero vectors a and b, if |a - b| = |a + b|, then find the angle between a and b.

Find all the vectors of magnitude 3√3 which are collinear to vector î + ĵ + k̂.

If a, b, c are three non-zero unequal vectors such that a · b = a · c, then find the angle between a and b - c.

If r = 3î - 2ĵ + 6k̂, find the value of (r · ĵ) · (r · k̂) - 12.

If r = 3î - 2ĵ + 6k̂, find the value of (r · ĵ) · (r · k̂) - 12.

If a = 4î - ĵ + k̂ and b = 2î - 2ĵ + k̂, then find a unit vector along the vector a × b.

If the projection of the vector î + ĵ + k̂ on the vector pî + ĵ - 2k̂ is 1/3, then find the value(s) of p.

a and b are two unit vectors such that |2a + 3b| = |3a - 2b|. Find the angle between a and b.

If |a × b|² + |a·b|² = 400 and |b| = 5, then find the value of |a|.

If a, b and c are mutually perpendicular vectors of equal magnitudes, show that the vector a + b + c is equally inclined to a, b and c.

Let three toys A, B and C be placed in the same straight line. If the position vectors of A, B and C are 55î – 2ĵ, 5î + 8ĵ and aî – 52ĵ respectively, find the value of 'a'.

The position vectors of vertices of ∆ ABC are A(2î - ĵ + k̂), B(î - 3ĵ - 5k̂) and C(3î - 4ĵ - 4k̂). Find all the angles of ∆ ABC.

If vectors a, b and 2a + 3b are unit vectors, then find the angle between a and b.

ABCD is a parallelogram such that AC = î + ĵ and BD = 2î + ĵ + k̂. Find AB and AD. Also, find the area of the parallelogram ABCD.

A(1, 1, 2), B(2, b, 5) and C(3, 11, 6) are collinear. Also, determine the ratio in which the point B divides the line-segment AC internally.

If a and b are two vectors such that a = î - ĵ + k̂ and b = 2î - ĵ - 3k̂, then find the vector c, given that a × c = b and a · c = 4.

If a, b and c are mutually perpendicular vectors of equal magnitude, then prove that the vector (2a + b + 2c) is equally inclined to both a and c. Also, find the angle between a and (2a + b + 2c).

If a and b are two vectors of equal magnitude and θ is the angle between them, then prove that |a - b|/|a + b| = cot(θ/2).

If a = 2î - ĵ + k̂, b = î + ĵ - 2k̂ and c = î + 3ĵ - k̂ and the projection of vector λc + b on vector a is 6√2, then find the value of λ.

The length of perpendicular drawn from point (2, 5, 7) on line x/1 = y/0 = z/0 is

1. 2
2. 5
3. √74
4. √78

Assertion (A): Lines given by x = py + q, z = ry + s and x = p'y + q', z = r'y + s' are perpendicular to each other when pp' + rr' = 1. Reason (R): Two lines r = a₁ + λb₁ and r = a₂ + μb₂ are perpendicular to each other if b₁ · b₂ = 0.

1. Both Assertion (A) and Reason (R) are true and the 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 and Reason (R) is false.
4. Assertion (A) is false and Reason (R) is true.

The length of perpendicular drawn from the point (1, 2, 3) on line x/0 = y/1 = z/0 is

1. 2
2. 6
3. 10
4. 14

The length of perpendicular drawn from the point (3, 4, 2) on the line x/0 = y/0 = z/1 is

1. 2
2. 9
3. 5
4. √29

The value of m for which the points with position vectors -i - j + 2k, 2i + mj + 5k and 3i + 11j + 6k are collinear, is

1. 8
2. -8
3. 2
4. 5/2

Direction cosines of the line given by equations: (2x - 1)/4 = (y - 1)/3 = (z - 6)/6 are

1. 2, -3, -6
2. -6/7, -3/7, 2/7
3. 6/7, -3/7, 2/7
4. -6/√61, -3/√61, 4/√61

Assertion (A): A line can have direction cosines < 1, 1, 1 >. Reason (R): cos α = 1 is possible for α = 0.

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.

Direction ratios of lines l1 and l2 are <12, – 3, 9> and <4, q, – p> respectively. The values of p and q for which l1 and l2 are parallel are respectively :

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

Direction ratios of lines l1 and l2 respectively are <1, − 2, 3> and <−2, p, − 6>. The value of p for which l1 || l2, is:

1. − 4
2. 4
3. − 10
4. 10

Direction ratios of lines l1 and l2 respectively are <1, 0, 0> and <0, −1, 0>. The direction ratios of the line perpendicular to both l1 and l2 are:

1. <1, 1, 0>
2. <0, 0, −1>
3. <1, 1, 1>
4. <1, 0, −1>

Three points A(0, 1, 1), B(2, 0, −1) and C(1, 0, 3) form triangle ABC. The area of triangle ABC is:

1. (√53)/2 sq. units
2. √53 sq. units
3. (√11)/2 sq. units
4. √11 sq. units

The angle between the lines (x+1)/2 = (y-2)/(-2) = (z+5)/4 and (x-3)/1 = (y+5)/(-2) = (z-7)/3 is

1. π/4
2. π/2
3. π/3
4. π/6

The Cartesian equations of a line are given as 6x - 2 = 3y + 1 = 2z - 2. The direction ratios of the line are

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

If the direction cosines of a line are 3k, 3k, 3k, then the value of k is:

1. ± 1
2. ± 3
3. ± 1/√3
4. ± 1/3

The distance of point P(a, b, c) from y-axis is:

1. b
2. |b|
3. √(a² + c²)
4. a² + c²

Assertion (A): A line in space cannot be drawn perpendicular to x, y and z axes simultaneously. Reason (R): For any line making angles α, β, γ with the positive directions of x, y and z axes respectively, cos²α + cos²β + cos²γ = 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 coordinates of the foot of the perpendicular drawn from the point (0, 1, 2) on the x-axis are given by:

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

Direction ratios of a vector parallel to line (1-x)/2 = -y = (z+1)/6 are:

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

If a line makes an angle of 30° with the positive direction of x-axis, 120° with the positive direction of y-axis, then the angle which it makes with the positive direction of z-axis is:

1. 90°
2. 120°
3. 60°
4. 0°

Direction ratios of a vector parallel to line (1–x)/2 = –y = (2+z)/6 are:

1. 2, –1, 6
2. 2, 1, 6
3. 2, 1, 3
4. 2, –1, 3

Direction ratios of a vector parallel to line (x-1)/2 = -y = (z+2)/6 are:

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

If α, β and γ are the angles which a line makes with positive directions of x, y and z axes respectively, then which of the following is not true ?

1. cos² α + cos² β + cos² γ = 1
2. sin² α + sin² β + sin² γ = 2
3. cos 2α + cos 2β + cos 2γ = –1
4. cos α + cos β + cos γ = 1

If a line makes an angle of π/4 with the positive directions of both x-axis and z-axis, then the angle which it makes with the positive direction of y-axis is :

1. 0
2. π/4
3. π/2
4. π

The vector equation of a line passing through the point (1, -1, 0) and parallel to Y-axis is:

1. r = î - ĵ + λ(î - ĵ)
2. r = î - ĵ + λĵ
3. r = î - ĵ + λk̂
4. r = λĵ

The lines (x-1)/2 = (1-y)/3 = z/1 and (2x-2p-3)/2 = (y-1)/1 = (4-z)/7 are perpendicular to each other for p equal to:

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

The lines (x-1)/2 = (1-y)/3 = (z-1)/1 and (x-2)/(2p-3) = (y-1)/(-1) = (z-4)/7 are perpendicular to each other for p equal to:

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

Direction cosines of the line (x-1)/2 = 3y/1 = (z-2)/(-12) are:

1. √7/2, √7/3, √7/6
2. 15/√72, -15/√73, 15√7/12
3. √7/2, -√7/3, -√7/6
4. 2/√7, 3/√7, -12/√7

Assertion (A): Equation of a line passing through the points (1, 2, 3) and (3, -1, 3) is (x-2)/3 = (y-3)/(-1) = (z-0)/3. Reason (R): Equation of a line passing through points (x₁, y₁, z₁), (x₂, y₂, z₂) is given by (x-x₁)/(x₂-x₁) = (y-y₁)/(y₂-y₁) = (z-z₁)/(z₂-z₁).

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 and Reason (R) is false.
4. Assertion (A) is false and Reason (R) is true.

Direction cosines of a line perpendicular to both x-axis and z-axis are:

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

Assertion (A): Equation of a line passing through the points (1, 2, 3) and (3, -1, 3) is (x-2)/(3-1) = (y-3)/(−1-2) = (z-0)/(3-3). Reason (R): Equation of a line passing through points (x₁, y₁, z₁), (x₂, y₂, z₂) is given by (x-x₁)/(x₂-x₁) = (y-y₁)/(y₂-y₁) = (z-z₁)/(z₂-z₁).

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 and Reason (R) is false.
4. Assertion (A) is false and Reason (R) is true.

The direction ratios of a line parallel to z-axis are:

1. <1, 1, 0>
2. <1, 1, 1>
3. <0, 0, 0>
4. <0, 0, 1>

Equation of line passing through origin and making 30°, 60° and 90° with x, y, z axes respectively is

1. 2x/3 = y/2 = z/0
2. 2x/3 = 2y/1 = z/0
3. 2x/1 = 2y/3 = z/1
4. 2x/3 = 2y/1 = z/1

The value of λ for which the angle between the lines r = î + ĵ + k̂ + p(2î + ĵ + 2k̂) and r = (1 + q)î + (1 + q)ĵ + (1 + q)k̂ is π/2 is:

1. -π/4
2. π/4
3. π/2
4. -π/2

If the point P(a, b, 0) lies on the line (x-1)/2 = (y-2)/3 = (z-4)/3, then (a, b) is:

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

Distance of the point (p, q, r) from y-axis is:

1. q
2. -q
3. q + r
4. √(p² + r²)

If the direction cosines of a line are a, 1/a, 1/a, then:

1. 0 < a < 1
2. a ≥ 2
3. a ≥ 0
4. a ≤ 3

The point (x, y, 0) on the xy-plane divides the line segment joining the points (1, 2, 3) and (3, 2, 1) in the ratio:

1. 1 : 2 internally
2. 2 : 1 internally
3. 3 : 1 internally
4. 3 : 1 externally

Assertion (A): The lines r = a₁ + λb₁ and r = a₂ + μb₂ are perpendicular, when b₁ · b₂ = 0. Reason (R): The angle θ between the lines r = a₁ + λb₁ and r = a₂ + μb₂ is given by cos θ = |b₁·b₂|/(|b₁||b₂|).

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 and Reason (R) is false.
4. Assertion (A) is false and Reason (R) is true.

Find the co-ordinates of the point on the line r = -ĵ + 3k̂ + λ(2î - 2ĵ + k̂) such that the sum of co-ordinates is 3.

Find the co-ordinates of foot of perpendicular drawn from (0, 0, 0) to line (x–1)/1 = (y–1)/1 = (3–z)/2.

Find the image of the point (2, –1, 5) in the line r = (11î – 2ĵ – 8k̂) + λ(10î – 4ĵ – 11k̂).

Find the co-ordinates of the point on line (y - 1/2) = (z - 2/3) = (x - 3) whose y co-ordinate is 3 times the x co-ordinate.

If the lines (x – 3)/1 = (1 – y)/1 = (z + 2)/p and (2 – x)/3 = (y + 1)/5 = (z + 56)/(2p) are perpendicular to each other, then find the value(s) of p.

If the lines (x – 3)/1 = (1 – y)/1 = (z + 2)/p and (2 – x)/3 = (y + 1)/5 = (z + 5)/(2p) are perpendicular to each other, then find the value(s) of p.

Find the angle between the following pair of lines : (x – 6)/–2 = (y + 5)/–3 = (1 – z)/–2 and (x – 2)/2 = (y + 6)/–1 = (7 – z)/1

Find the angle between the following pair of lines: (x – 6)/(-5) = (y + 5)/(-2) = (z – 1)/3 and (x – 2)/(-7) = (y – 6)/(-1) = (z – 2)/2

Find the angle between the following pair of lines: (x + 5)/(-6) = (y + 1)/3 = (z – 1)/2 and (x – 6)/2 = (y – 1)/(-7) = (z)/(-2)

If the equation of a line is x = ay + b, z = cy + d, then find the direction ratios of the line and a point on the line.

If the angle between the lines (x-5)/5 = (y-2)/2 = (z-4)/5 and (x-1)/1 = (y-0)/0 = (z-1)/1 is π/4, find the relation between α and β.

Find the value of p, so that lines (x-1)/2 = (y-3)/4 = (z-p)/3 and (x-4)/p = (y-5)/2 = (z-1)/2 are perpendicular to each other.

Find the direction cosines of the line whose Cartesian equations are 5x - 3 = 15y + 7 = 3 - 10z.

Find the vector equation of the line passing through the point (2, 1, 3) and perpendicular to both the lines (x-1)/2 = (y-2)/3 = (z-3)/1 and 3x - 2y + 5z = 0.

Find the vector equation of the line passing through the point (2, 1, 3) and perpendicular to both the lines (x-1)/1 = (y-2)/2 = (z-3)/3 and (3x - 2y + 5z = 0).

Write the cartesian equation of the line PQ passing through points P(2, 2, 1) and Q(5, 1, -2). Hence, find the y-coordinate of the point on the line PQ whose z-coordinate is -2.

The foot of a perpendicular drawn from the point (2, 1, 3) on a plane is (1, 3, 3). Find the equation of the plane.

(a) Find a point on the line (x − 2)/3 = (1 − y)/2 = (z − 3)/2 at a distance of √2 units from the point (1, 2, 3).

(a) Verify that lines given by r = (1 - λ)i + (2 + λ)j + (3 - 2λ)k and r = (1 + μ)i + (2μ - 1)j + (-2μ + 1)k are skew lines. Hence, find shortest distance between the lines.

Find the equation of the plane passing through points (2, 1, 0), (3, -2, -2) and (1, 1, -7). Also, obtain its distance from the origin.

If a line makes 60° and 45° angles with the positive directions of x-axis and z-axis respectively, then find the angle that it makes with the positive direction of y-axis. Hence, write the direction cosines of the line.

Check whether the lines given by (x – 1)/2 = (y – 2)/3 = (z – 3)/4 and (x – 4)/5 = (y – 1)/2 = z are parallel or not. If parallel, find the distance between them, otherwise find their point of intersection, if the lines are intersecting.

Prove that the line through points A(0, –1, –1) and B(4, 5, 1) intersects the line through points C(3, 9, 4) and D(–4, 4, 4). Hence, write the equation of line passing through the point of intersection of lines AB and CD as well as origin.

Show that line AB passing through points A(0, 4, 1), B(2, 3, –1) and the line CD passing through points C(4, 5, 0), D(2, 6, 2) are parallel. Also, find distance between them.

A line passing through the points A(1, 2, 3) and B(5, 8, 11) intersects the line r = 4î + ĵ + λ(5î + 2ĵ + k̂). Find the co-ordinates of the point of intersection. Hence, write the equation of a line passing through the point of intersection and perpendicular to both the lines.

A line passing through the points A(1, 2, 3) and B(5, 8, 11) intersects the line r = 4î + ĵ + μ(5î + 2ĵ + k̂). Find the co-ordinates of the point of intersection. Hence, write the equation of a line passing through the point of intersection and perpendicular to both the lines.

Find the shortest distance between the lines: r = (î + 2ĵ + 3k̂) + λ(î – 3ĵ + 2k̂) and r = (4î + 5ĵ + 6k̂) + μ(2î + 3ĵ + k̂).

Represent the equations of lines l1 and l2 in vector form and check whether they are intersecting or not. l1 : (x + 3)/(–3) = (y – 1)/1 = (z – 5)/5, l2 : (x + 1)/(–1) = (2 – y)/(–2) = (z – 5)/5

Find the equation of a line (in vector and cartesian form) that passes through the point of intersection of lines (x − 1)/2 = (y − 2)/3 = (z − 3)/4 and (x − 4)/5 = (y − 1)/2 = z and is parallel to the vector 3î + 2ĵ – 8k̂.

Find the vector and cartesian equations of the line passing through the point of intersection of the lines r = (î + ĵ − k̂) + λ(3î − ĵ) and r = (4î − k̂) + μ(2î + 3k̂) and parallel to the line (x − 1)/(−2) = (7 − y)/(−3) = z.

Find the length of the perpendicular drawn from the point P(1, 2, 3) to the line (x − 6)/3 = (y − 7)/2 = (z − 7)/(−2). Also, find the equation of the perpendicular line joining P and the foot of the perpendicular.

Find the foot of the perpendicular from the point (0, 2, 3) on the line (–x – 3)/(–5) = (1 – y)/(–2) = (3z + 12)/9 and hence find the length of the perpendicular.

(a) Find the image A' of the point A(1, 6, 3) in the line (x-1)/1 = (y-2)/2 = (z-3)/3. Also, find the equation of the line joining A and A'.

Write the vector equations of the following lines and hence find the shortest distance between them: (x+1)/2 = (y+1)/1 = (z+1)/1 and (x-3)/1 = (y-5)/(-2) = (z-7)/1

The image of point P(x, y, z) with respect to line x/1 = (1–y)/2 = (z–2)/3 is P'(1, 0, 7). Find the coordinates of point P.

Equations of sides of a parallelogram ABCD are as follows: AB: (x+1)/1 = (y-2)/-2 = (z-1)/2, BC: (x-1)/3 = (y+2)/5 = (z-5)/3, CD: (x-4)/1 = (y+7)/-2 = (z-8)/2, DA: (x-2)/3 = (y+3)/5 = (z-4)/3. Find the equation of diagonal BD.

Find the equation of a line l₂ which is the mirror image of the line l₁ with respect to line l: x/1 = (1–y)/2 = (z–2)/3, given that line l₁ passes through the point P(1, 6, 3) and parallel to line l.

Find the equation of the line which bisects the line segment joining points A(2, 3, 4) and B(4, 5, 8) and is perpendicular to the lines (x-8)/3 = (y+19)/16 = (z-10)/7 and (x-15)/3 = (y-29)/8 = (z+5)/5.

Find the value of p for which the lines r = l i + (2l + 1) j + (3l + 2) k and r = i – 3m j + (pm + 7) k are perpendicular to each other and also intersect. Also, find the point of intersection of the given lines.

The vertices of ∆ ABC are A(1, 1, 0), B(1, 2, 1) and C(-2, 2, -1). Find the equations of the medians through A and B. Use the equations so obtained to find the coordinates of the centroid.

Find the equation of the line passing through the point of intersection of the lines (x/1) = (1–y)/2 = (2–z)/3 and (1–x)/0 = (–y)/3 = (7–z)/2 and perpendicular to these given lines.

Find the distance between the line (x-1)/2 = (6-2y)/4 = (z+1)/(-1) and another line parallel to it passing through the point (4, 0, -5).

(a) Find the vector and the Cartesian equations of a line passing through the point (1, 2, -4) and parallel to the line joining the points A(3, 3, -5) and B(1, 0, -11). Hence, find the distance between the two lines. OR (b) Find the equations of the line passing through the points A(1, 2, 3) and B(3, 5, 9). Hence, find the coordinates of the points on this line which are at a distance of 14 units from point B.

Find the vector and the Cartesian equations of a line passing through the point (1, 2, -4) and parallel to the line joining the points A(3, 3, -5) and B(1, 0, -11). Hence, find the distance between the two lines.

Find the equations of the diagonals of the parallelogram PQRS whose vertices are P(4, 2, -6), Q(5, -3, 1), R(12, 4, 5) and S(11, 9, -2). Use these equations to find the point of intersection of diagonals.

Find the value of b so that the lines (x-1)/2 = (y-3)/(2b) = (z-4)/3 and (x-5)/4 = (y-2)/1 = (z-1)/2 are intersecting lines. Also, find the point of intersection of these given lines.

Find the value of b so that the lines (x-1)/2 = (y-3)/4 = (z-3)/b and (x-4)/5 = (y-2)/1 = (z-1)/2 are intersecting lines. Also, find the point of intersection of these given lines.

Find the value of b so that the lines (x-1)/2 = (y-2)/b = (z-3)/4 and (x-4)/5 = (y-1)/4 = (z-2)/1 are intersecting lines. Also, find the point of intersection of these given lines.

Show that the following lines do not intersect each other: (x-2)/4 = (y-3)/1 = (z-3)/2 and (x-3)/1 = (y-2)/(-1) = (z-5)/5

Find the image of the point (2, -1, 5) in the line (11-x)/10 = (y-2)/4 = (z-8)/11.

Find the distance of the point (-1, -5, -10) from the point of intersection of the line (x-3)/2 = (y-4)/1 = (z-2)/2 and the plane x - y + z = 5.

Find the distance of the point (-1, -5, -10) from the point of intersection of the line (x-2)/3 = (y-4)/(-1) = (z-2)/2 and the plane x - y + z = 5.

Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the zx-plane.

Find the equations of the planes passing through the line of intersection of the planes r·(î + 3ĵ) = 6 and r·(3î - ĵ - 4k̂) = 0, which are at a distance of 1 unit from the origin.

The degree of an objective function of a linear programming problem is

1. 0
2. 1
3. 2
4. Any natural number

In a linear programming problem, the linear function which has to be maximized or minimized is called

1. a feasible function
2. an objective function
3. an optimal function
4. a constraint

The corner points of the feasible region for an LPP are (0, 2), (6, 0), (6, 8) and (0, 5). Let Z = 4x + 6y be the objective function. (Maximum of Z) – (Minimum of Z) is equal to :

1. 60
2. 48
3. 42
4. 18

In an LPP, if the objective function has the same minimum value on two corner points of the feasible region, then the number of points at which the minimum value of objective function occurs is :

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

The region represented by the system of inequations 3x + y ≤ 3, 2x − y ≤ −5, x, y ≥ 0 is:

1. unbounded in 1st quadrant
2. bounded in 1st quadrant
3. unbounded in 2nd quadrant
4. bounded in 2nd quadrant

In the graph, the feasible region representing the Linear Programming Problem for maximising objective function Z = px + qy, p, q > 0 is shaded. If all points on segment AB give max (Z), then which of the following is true?

1. p = 2q
2. p = 3q
3. q = 3p
4. q = 2p

The corner points of the feasible region determined by the system of linear constraints are (0, 0), (0, 40), (20, 40) (60, 20) and (60, 0). If the objective function of an LPP is Z = 4x + 3y, then the maximum value is :

1. 200
2. 300
3. 240
4. 120

The corner points of the feasible region in graphical representation of a L.P.P. are (2, 72), (15, 20) and (40, 15). If Z = 18x + 9y be the objective function, then

1. Z is maximum at (2, 72), minimum at (15, 20)
2. Z is maximum at (15, 20) minimum at (40, 15)
3. Z is maximum at (40, 15), minimum at (15, 20)
4. Z is maximum at (40, 15), minimum at (2, 72)

If the feasible region of a linear programming problem with objective function Z = ax + by, is bounded, then which of the following is correct?

1. It will only have a maximum value.
2. It will only have a minimum value.
3. It will have both maximum and minimum values.
4. It will have neither maximum nor minimum value.

The solution set of the inequation 2x + 3y < 6 is

1. open half-plane not containing origin
2. whole xy-plane except the points lying on the line 2x + 3y = 6
3. open half-plane containing origin
4. half-plane containing the origin and the points lying on the line 2x + 3y = 6

The maximum value of the objective function z = 3x + 5y subject to the constraints x ≥ 0, y ≥ 0 and 4x + 3y ≤ 12 is

1. 15
2. 29
3. 9
4. 20

A linear programming problem deals with the optimization of a/an:

1. logarithmic function
2. linear function
3. quadratic function
4. exponential function

The number of corner points of the feasible region determined by constraints x ≥ 0, y ≥ 0, x + y ≥ 4 is:

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

The common region determined by all the constraints of a linear programming problem is called:

1. an unbounded region
2. an optimal region
3. a bounded region
4. a feasible region

The restrictions imposed on decision variables involved in an objective function of a linear programming problem are called :

1. feasible solutions
2. constraints
3. optimal solutions
4. infeasible solutions

Of the following, which group of constraints represents the feasible region given below ?

1. x + 2y ≤ 76, 2x + y ≥ 104, x, y ≥ 0
2. x + 2y ≤ 76, 2x + y ≤ 104, x, y ≥ 0
3. x + 2y ≥ 76, 2x + y ≤ 104, x, y ≥ 0
4. x + 2y ≥ 76, 2x + y ≥ 104, x, y ≥ 0

The maximum value of Z = 4x + y for a L.P.P. whose feasible region is given below is:

1. 50
2. 110
3. 120
4. 170

The corner points of the feasible region in the graphical representation of a linear programming problem are (2, 72), (15, 20) and (40, 15). If z = 18x + 9y be the objective function, then:

1. z is maximum at (2, 72), minimum at (15, 20)
2. z is maximum at (15, 20), minimum at (40, 15)
3. z is maximum at (40, 15), minimum at (15, 20)
4. z is maximum at (40, 15), minimum at (2, 72)

The number of corner points of the feasible region determined by the constraints x - y ≤ 0, 2y - x + 2 ≥ 0, x ≥ 0, y ≥ 0 is:

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

The objective function Z = ax + by of an LPP has maximum value 42 at (4, 6) and minimum value 19 at (3, 2). Which of the following is true?

1. a = 9, b = 1
2. a = 5, b = 2
3. a = 3, b = 5
4. a = 5, b = 3

The corner points of the feasible region of a linear programming problem are (0, 4), (8, 0) and (20/3, 4/3). If Z = 30x + 24y is the objective function, then (maximum value of Z - minimum value of Z) is equal to

1. 40
2. 96
3. 120
4. 136

The number of feasible solutions of the linear programming problem given as Maximize z = 15x + 30y subject to constraints: 3x + y ≤ 12, x + 2y ≤ 10, x ≥ 0, y ≥ 0 is:

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

Which of the following points satisfies both the inequations 2x + y ≤ 10 and x + 2y ≥ 8?

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

The solution set of the inequation 3x + 5y ≥ 7 is:

1. whole xy-plane except the points lying on the line 3x + 5y = 7
2. whole xy-plane along with the points lying on the line 3x + 5y = 7
3. open half plane containing the origin except the points of line 3x + 5y = 7
4. open half plane not containing the origin

Solve the following linear programming problem graphically: Minimize Z = 13x – 15y subject to constraints x + y ≤ 7, 2x – 3y + 6 ≥ 0, x ≥ 0, y ≥ 0.

Solve the following linear programming problem graphically: Maximize Z = 10500x + 9000y Subject to constraints: x + y ≤ 50, 2x + y ≤ 80, x, y ≥ 0

Solve the following linear programming problem graphically: Maximize Z = 8000x + 12000y Subject to constraints 3x + 4y ≤ 60, x + 3y ≤ 30, x ≥ 0, y ≥ 0

The corner points of the feasible region determined by the system of linear constraints in an LPP are (0, 10), (5, 5), (15, 15) and (0, 20). Let Z = px + qy, p, q > 0, be the objective function, then find the relation between p and q so that the maximum of Z occurs at both the points (15, 15) and (0, 20).

Solve the following linear programming problem graphically: Maximize Z = 4500x + 5000y Subject to constraints x + y ≤ 250, 25x + 40y ≤ 7000, x ≥ 0, y ≥ 0

Solve the following Linear Programming Problem graphically: Maximise Z = 200x + 120y subject to the constraints x + y ≤ 300, 3x + y ≤ 600, x − y ≤ −100, x, y ≥ 0

Solve the following Linear Programming Problem graphically: Minimize Z = 20x + 10y subject to x + 2y ≤ 40, 3x + y ≤ 30, 4x + 3y ≥ 60, x, y ≥ 0

Solve the following Linear Programming Problem graphically: Maximize Z = 20x + 10y subject to constraints x + 2y ≤ 28, 3x + y ≤ 24, x ≥ 2, x, y ≥ 0

Solve the following Linear Programming Problem graphically : Maximize Z = (2x + 3y + 5)/10 subject to constraints 2x + y ≤ 1000, x + y ≤ 800, x, y ≥ 0.

Solve the following linear programming problem graphically: Maximise Z = x + 2y Subject to the constraints: x - y ≥ 0, x - 2y ≥ -2, x ≥ 0, y ≥ 0

The corner points of the feasible region determined by the system of linear constraints are A(0, 40), B(20, 40), C(60, 20) and D(60, 0). The objective function of the L.P.P. is z = 4x + 3y. Find the point of the feasible region at which the value of objective function is maximum and the point at which the value is minimum. Hence, find the maximum and the minimum values.

Solve the following linear programming problem graphically: Maximise z = 500x + 300y, subject to constraints x + 2y ≤ 12, 2x + y ≤ 12, 4x + 5y ≥ 20, x ≥ 0, y ≥ 0

Solve the following linear programming problem graphically: Maximise z = 5x + 4y subject to the constraints x + 2y ≥ 4, 3x + y ≤ 6, x + y ≤ 4, x, y ≥ 0

Solve the following linear programming problem graphically: Minimise z = 5x – 2y subject to the constraints x + 2y ≤ 120, x + y ≥ 60, x – 2y ≥ 0, x, y ≥ 0

Solve the following linear programming problem graphically: Maximise z = 4x + 3y, subject to the constraints x + y ≤ 800, 2x + y ≤ 1000, x ≤ 400, x, y ≥ 0.

Solve the following linear programming problem graphically: Minimize z = 600x + 400y, subject to the constraints x + y ≥ 8, x + 2y ≤ 16, 4x + y ≤ 29, x, y ≥ 0.

Solve the following linear programming problem graphically: Maximize z = x + y subject to constraints 2x + 5y ≤ 100 8x + 5y ≤ 200 x ≥ 0, y ≥ 0

Solve the following linear programming problem graphically: Maximise Z = 2x + 3y subject to the constraints: x + y ≤ 6, x ≥ 2, y ≤ 3, x, y ≥ 0.

Solve the following L.P.P. graphically: Maximise Z = x + 3y subject to the constraints: x + 2y ≤ 200, x + y ≤ 150, y ≤ 75, x, y ≥ 0

Solve graphically the following linear programming problem: Maximise z = 6x + 3y, subject to the constraints 4x + y ≤ 80, 3x + 2y ≤ 150, x + 5y ≤ 115, x ≥ 0, y ≥ 0.

Solve the following linear programming problem graphically: Maximize P = 100x + 5y subject to the constraints x + y ≤ 300, 3x + y ≤ 600, y ≤ x + 200, x, y ≥ 0.

Solve the following linear programming problem graphically: Maximize z = 600x + 400y subject to the constraints: x + 2y ≤ 12, 2x + y ≤ 12, x + 1.25y ≥ 5, x, y ≥ 0

Solve the following linear programming problem graphically: Maximize: Z = x + 2y subject to constraints: x + 2y ≤ 100, 2x - y < 0, 2x + y ≤ 200, x ≥ 0, y ≥ 0.

Determine graphically the minimum value of the following objective function: z = 500x + 400y subject to constraints x + y ≤ 200, x ≥ 20, y ≥ 4x, y ≥ 0.

Solve the following linear programming problem graphically: Maximize z = 3x + 9y subject to the constraints x + y ≤ 10, x + 3y ≤ 60, x ≥ y, x ≥ 0, y ≥ 0.

Solve the following linear programming problem graphically: Minimize z = x + 2y subject to the constraints 2x + y ≥ 3, x + 2y ≥ 6, x ≥ 0, y ≥ 0.

Solve the following linear programming problem graphically: Maximise z = 3x - 5y subject to the constraints 2x + y ≤ 4, x + y ≤ 3, x - 2y ≤ 2, x ≥ 0, y ≥ 0.

Solve the following linear programming problem graphically: Maximise z = 5x + 3y subject to the constraints 3x + 5y ≤ 15, 5x + 2y ≥ 10, x, y ≥ 0.

Solve the following Linear Programming Problem graphically : Maximise Z = 600x + 400y subject to the constraints x + 2y ≥ 12, 4x + 5y ≥ 20, 2x + y ≥ 12, x, y ≥ 0

Solve the following Linear Programming Problem graphically: Maximise Z = 12x + 18y subject to the constraints x + y ≤ 1200 x − 2y ≥ 0 x + 3y ≥ 600 x ≥ 0, y ≥ 0

Solve the following Linear Programming Problem graphically: Maximize Z = 8x + 8y subject to the constraints x − 20 ≥ 0, 2x + 3y ≤ 120, 2x + y ≤ 60, x ≥ 0, y ≥ 0

Assertion (A): In an experiment of throwing an unbiased die, the probability of getting a prime number given that number appearing on the die being odd is 2/3. Reason (R): For any two events A and B, P(A|B) = P(A ∩ B)/P(B)

1. Both Assertion (A) and Reason (R) are true and the 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 and Reason (R) is false.
4. Assertion (A) is false and Reason (R) is true.

Assertion (A): In an experiment of throwing an unbiased die, the probability of getting a prime number given that the number appearing on the die being odd is 2/3. Reason (R): For any two events A and B, P(A|B) = P(A ∩ B)/P(B).

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 and Reason (R) is false
4. Assertion (A) is false and Reason (R) is true

For two events A and B such that P(A) ≠ 0 and P(B) ≠ 1, P(A'|B) =

1. 1 - P(A|B)
2. 1 - P(A'|B)
3. [1 - P(A∩B)]/P(B)
4. [1 - P(A'∩B)]/P(B)

If events A and B are such that P(A|B) = P(B|A), then :

1. A ⊂ B but A ≠ B
2. A = B
3. A ∩ B = ∅
4. P(A) = P(B)

The probability of getting an odd prime number on each die, when a pair of dice is rolled, is :

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

In a family with three children, the probability that the eldest child is a boy, given that the family has at least one boy, is :

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

Assertion (A): For two independent events A and B, P(A|B) = P(A). Reason (R): A and B are independent events, then P(A ∩ B) = P(A) . P(B).

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 E and F are two independent events such that P(E) = 3/10, P(E ∪ F) = 1/2, then P(E|F) – P(F|E) is equal to :

1. 2/7
2. 3/35
3. 1/70
4. 1/7

A box contains 4 red, 5 blue and 1 green marble. A child randomly takes out a marble from the box, notes down the colour and puts it back in the box. If the activity is repeated 3 times, what is the probability that at least one marble is red?

1. 27/125
2. 8/125
3. 2/125
4. 98/125

The probability that it will rain tomorrow in cities A, B and C is 60%, 70% and 80% respectively. The probability that it will rain tomorrow in at least one of the cities is:

1. 3/250
2. 244/250
3. 1
4. 9/10

The probability that a particular item is available in three shops A, B and C is 4/5, 3/4 and 2/3 respectively. If a person visits all the three shops to buy the item, then what is the probability that it will be available in at least one of the shops?

1. 59/60
2. 1
3. 1/60
4. 9/60

If 3P(A) = P(B) = 3/5 and P(A|B) = 2/3, then P(A ∩ B) is :

1. 3/5
2. 1/5
3. 2/15
4. 2/5

If E and F are two independent events such that P(E) = 2/3, P(F) = 3/7, then P(E|F) is equal to:

1. 1/6
2. 1/2
3. 2/3
4. 7/9

The probabilities of A, B and C solving a problem are 1/3, 1/5 and 1/6 respectively. The probability that the problem is solved, is

1. 4/9
2. 5/9
3. 1/90
4. 1/3

If P(A|B) = P(A'|B), then which of the following statements is true?

1. P(A) = P(A')
2. P(A) = 2 P(B)
3. P(A ∩ B) = (1/2) P(B)
4. P(A ∩ B) = 2 P(B)

Let E be an event of a sample space S of an experiment, then P(S|E) =

1. P(S ∩ E)
2. P(E)
3. 1
4. 0

Let E and F be two events such that P(E) = 0·1, P(F) = 0·3, P(E ∪ F) = 0·4, then P(F|E) is :

1. 0·6
2. 0·4
3. 0·5
4. 0

The probability distribution of a random variable X is given. X: 0, 1, 2, 3, 4; P(X): 0.1, k, 2k, k, 0.1. The probability that the random variable X takes the value 2 is:

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

The probability distribution of a random variable X is: X: 0, 1, 2, 3, 4; P(X): 0.1, k, 2k, k, 0.1, where k is some unknown constant. The probability that the random variable X takes the value 2 is:

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

If P(B|A) = 0·3, P(A) = 0·4 and P(B) = 0·8, then P(A|B) is equal to:

1. 0·6
2. 0·3
3. 0·06
4. 0·4

Assertion (A): The number of onto functions from a set P containing 5 elements to a set Q containing 2 elements is 30. Reason (R): Number of onto functions from a set containing m elements to a set containing n elements is n^m.

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 and Reason (R) is false.
4. Assertion (A) is false and Reason (R) is true.

Ashima can hit a target 2 out of 3 times. She tried to hit the target twice. The probability that she missed the target exactly once is:

1. 3/2
2. 3/1
3. 9/4
4. 9/1

If A and B are two events such that P(A/B) = 2 P(B/A) and P(A) + P(B) = 2/3, then P(B) is equal to

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

If P(A ∩ B) = 1/8 and P(A) = 3/4, then P(B|A) is equal to:

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

For any two events A and B, if P(A) = 1/2, P(B) = 2/3 and P(A ∩ B) = 1/4, then P(A|B) equals:

1. 3/8
2. 8/9
3. 1/8
4. 1/4

X and Y are independent events such that P(X ∩ Y) = 2/5 and P(X) = 3/5. Then P(Y) is equal to

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

The probability that A speaks the truth is 54and that of B speaking the truth is 43. The probability that they contradict each other in stating the same fact is :

1. 207
2. 51
3. 203
4. 54

For two events A and B, if P(A) = 0·4, P(B) = 0·8 and P(B/A) = 0·6, then P(A ∩ B) is:

1. 0·24
2. 0·3
3. 0·48
4. 0·96

The events E and F are independent. If P(E) = 0.3 and P(E ∪ F) = 0.5, then P(E/F) - P(F/E) equals:

1. 1/7
2. 2/7
3. 3/35
4. 1/70

A pair of dice is thrown. It is given that the sum of numbers appearing on both dice is an even number. Find the probability that the number appearing on at least one die is 3.

Probabilities of A and B solving a specific problem are 2/3 and 3/5, respectively. If both of them try independently to solve the problem, then find the probability that the problem is solved.

A bag contains 3 red and 4 white balls. Three balls are drawn at random, one-by-one without replacement from the bag. If the first ball drawn is red in colour, then find the probability that the remaining two balls drawn are also red in colour.

Three friends A, B and C got their photograph clicked. Find the probability that B is standing at the central position, given that A is standing at the left corner.

Out of two bags, bag I contains 3 red and 4 white balls and bag II contains 8 red and 6 white balls. A die is thrown. If it shows a number less than 3 then a ball is drawn at random from bag I, otherwise a ball is drawn at random from bag II. Find the probability that the ball drawn from one of the bags is a red ball.

The probability of hitting the target by a trained sniper is three times the probability of not hitting the target on a stormy day due to high wind speed. The sniper fired two shots on the target on a stormy day when wind speed was very high. Find the probability that (i) target is hit (ii) atleast one shot misses the target.

A survey was conducted on the patients who have undergone knee replacement surgeries. It was found that Robotic Knee replacement surgeries have 90% success rate. On a particular day, robotic surgery was performed on three patients, A, B and C, one after the other. Assuming that the success and failure of each surgery is independent of each other, find the probability that : (i) exactly one surgery is successful, (ii) at most two surgeries are successful.

In a school, the probability of holding a debate competition is 1/3 and that of a quiz competition is 2/3. In the two participating teams, A has 4 girls and 6 boys and B has 7 girls and 3 boys. If a debate competition is held, the students are selected from team A and for the quiz competition they are selected from team B. If only two students are to be chosen from the teams, then find the probability that one will be a girl and the other a boy.

A die is rolled. Consider events : A = {1, 2, 5}, B = {3, 5}, C = {2, 3, 4, 5} and hence find : (i) P(A|C) and P(C|A) (ii) P(A ∩ B|C) and P(A ∪ B|C)

(a) The probability distribution for the number of students being absent in a class on a Saturday is as follows: X: 0, 2, 4, 5; P(X): p, 2p, 3p, p. where X is the number of students absent. (i) Calculate p. (ii) Calculate the mean of the number of absent students on Saturday.

A card is randomly drawn from a well-shuffled pack of 52 playing cards. Events A and B are defined as under: A: Getting a card of diamond, B: Getting a queen. Determine whether the events A and B are independent or not.

E and F are two independent events such that P(¬E) = 0.6 and P(E ∩ F) = 0.6. Find P(F) and P(¬E ∩ ¬F).

A pair of dice is thrown simultaneously. If X denotes the absolute difference of the numbers appearing on top of the dice, then find the probability distribution of X.

The chances of P, Q and R getting selected as CEO of a company are in the ratio 4 : 1 : 2 respectively. The probabilities for the company to increase its profits from the previous year under the new CEO, P, Q or R are 0·3, 0·8 and 0·5 respectively. If the company increased the profits from the previous year, find the probability that it is due to the appointment of R as CEO.

A card from a well shuffled deck of 52 playing cards is lost. From the remaining cards of the pack, a card is drawn at random and is found to be a King. Find the probability of the lost card being a King.

(a) The probability distribution of a random variable X is given below: X: 1, 2, 3; P(X): 2k, 3k, 6k. (i) Find the value of k. (ii) Find P(1 ≤ X < 3). (iii) Find E(X), the mean of X. OR (b) A and B are independent events such that P(A ∩ B) = 1/4 and P(A ∪ B) = 1/6. Find P(A) and P(B).

The probability distribution of a random variable X is given below: X: 1, 2, 3; P(X): 2k, 3k, 6k. (i) Find the value of k. (ii) Find P(1 ≤ X < 3). (iii) Find E(X), the mean of X.

Two balls are drawn at random one by one with replacement from an urn containing equal number of red balls and green balls. Find the probability distribution of number of red balls. Also, find the mean of the random variable.

A pair of dice is thrown simultaneously. If X denotes the absolute difference of numbers obtained on the pair of dice, then find the probability distribution of X.

From a lot of 30 bulbs which include 6 defective bulbs, a sample of 2 bulbs is drawn at random one by one with replacement. Find the probability distribution of the number of defective bulbs and hence find the mean number of defective bulbs.

Two fair dice are thrown simultaneously. If X denotes the number of sixes, find the mean of X.

Probabilities of A and B solving a specific problem are 3/2 and 5/3, respectively. If both of them try independently to solve the problem, then find the probability that the problem is solved.

There are two bags. Bag I contains 1 red and 3 white balls, and Bag II contains 3 red and 5 white balls. A bag is selected at random and a ball is drawn from it. Find the probability that the ball so drawn is red in colour.