KCET 2023 Mathematics Question Paper: Download Set D1 Question Paper with Answer Key PDF

If u = sin^-1(2x/(1+x²)) and v = tan^-1((2x)/(1-x²)), then du/dv is:

• (A) 1/2
• (B) 1
• (C) 1
• (D) 2

The function f(x) = cot x is discontinuous on every point of the set:

• (A) x = (2n + 1)(π/2), n ∈ Z
• (B) x = nπ, n ∈ Z
• (C) x = nπ/2, n ∈ Z
• (D) x = 2nπ, n ∈ Z

If the function is f(x) = 1/(x+2), then the point of discontinuity of the composite function y = f(f(x)) is:

• (A) 2/5
• (B) -5/2
• (C) 1/2
• (D) 1/5

If y = a sin x + b cos x, then y² + (dy/dx)² is:

• (A) function of x and y
• (B) function of x
• (C) constant
• (D) function of y

If f(x) = 1 + nx + n(n-1)x²/2! + n(n-1)(n-2)x³/3! + ... + xⁿ, then f''(1) =:

• (A) n(n - 1)2ⁿ
• (B) (n - 1)2ⁿ
• (C) 2^n-1
• (D) n(n - 1)2^n-2

If A = [1 tan(α/2); -tan(α/2) 1] and AB = I, then B is:

• (A) cos²(α/2) * I
• (B) cos²(α/2) * A^T
• (C) sin²(α/2) * A
• (D) sin²(α/2) * I

A circular plate of radius 5 cm is heated. Due to expansion, its radius increases at the rate of 0.05 cm/sec. The rate at which its area is increasing when the radius is 5.2 cm is:

• (A) 5.05π cm²/sec
• (B) 5.2π cm²/sec
• (C) 0.52π cm²/sec
• (D) 27.4π cm²/sec

The distance *s* in meters travelled by a particle in *t* seconds is given by s = (2t³/3) - 18t + (5/3). The acceleration when the particle comes to rest is:

• (A) 12 m/sec²
• (B) 3 m/sec²
• (C) 18 m/sec²
• (D) 10 m/sec²

A particle moves along the curve x²/16 + y²/4 = 1. When the rate of change of abscissa is 4 times that of its ordinate, then the quadrant in which the particle lies is:

• (A) III or IV
• (B) I or III
• (C) II or III
• (D) II or IV

An enemy fighter jet is flying along the curve given by y = x² + 2. A soldier is placed at (3, 2) and wants to shoot down the jet when it is nearest to him. Then the nearest distance is:

• (A) 2 units
• (B) √3 units
• (C) 5 units
• (D) √6 units

Evaluate the integral: ∫₂⁸ (5√(10-x))/(5√x + 5√(10-x)) dx

• (A) 4
• (B) 5
• (C) 3
• (D) 6

Evaluate the integral: ∫ (cosec x - sin x) dx

• (A) 2√(sin x) + C
• (B) √(sin x) + C
• (C) 2/(√(sin x)) + C
• (D) √(sin x)/2 + C

If f(x) and g(x) are two functions with g(x) = x - 1/x and f o g(x) = x³ - 1/x³, then f'(x) =:

• (A) x² - 1/x²
• (B) 3x² + 3/x⁴
• (C) 1 - x/x²
• (D) 3x² + 2/x⁴

Evaluate the integral: ∫ 1 / (1 + 3 sin²x + 8 cos²x) dx

• (A) (1/6)tan^-1((2tan x)/√3) + C
• (B) πtan^-1((2tan x)/3) + C
• (C) 6 tan^-1((2tan x)/3) + C
• (D) tan^-1((2tan x)/3) + C

Evaluate the integral ∫_-2⁰ (x³ + 3x² + 3x + 3 + (x + 1)cos(x + 1)) dx:

• (A) 4
• (B) 0
• (C) 1
• (D) 3

Evaluate the integral ∫₀^π (x tan x)/(sec x - csc x) dx:

• (A) π/2
• (B) π/7
• (C) π/4
• (D) π/3

Evaluate the integral: ∫ √(5 - 2x + x²) dx

• (A) (x-1)/2 √(5 + 2x + x²) + 2 log |x - 1 + √(5 + 2x + x²)| + C
• (B) (x-1)/2 √(5 - 2x + x²) + 2 log |x + 1 + √(x² + 2x + 5)| + C
• (C) (x-1)/2 √(5 - 2x + x²) + 2 log |x - 1 + √(5 - 2x + x²)| + C
• (D) (x-1)/2 √(5 - 2x + x²) + 4 log |x + 1 + √(x² - 2x + 5)| + C

The area of the region bounded by the line y = x + 1, and the lines x = 3 and x = 5 is:

• (A) 16/2 sq. units
• (B) 10 sq. units
• (C) 7 sq. units
• (D) 11/2 sq. units

If a curve passes through the point (1,1) and at any point (x, y) on the curve, the product of the slope of its tangent and the x-coordinate of the point is equal to the y-coordinate of the point, then the curve also passes through the point:

• (A) (-1, 2)
• (B) (2, 2)
• (C) (√3, 0)
• (D) (3, 0)

The degree of the differential equation 1 + (dy/dx)² = √(d²y/dx²) - 1

• (A) 1
• (B) 6
• (C) 2
• (D) 3

If |a + b| = |a - b| then:

• (A) a and b are coincident.
• (B) a and b are perpendicular.
• (C) Inclined to each other at 60°.
• (D) a and b are parallel.

The component of *i* in the direction of the vector *i + j + 2k* is:

• (A) 6√6
• (B) √6/6
• (C) 6/√6
• (D) 6

In the interval (0, π/2), the area lying between the curves y = tan x and y = cot x and the X-axis is:

• (A) 4 log 2 sq. units
• (B) 3 log 2 sq. units
• (C) log 2 sq. units
• (D) 2 log 2 sq. units

If a + 2b + 3c = 0 and (a x b) + (b x c) + (c x a) = λ(b x c) then the value of λ is equal to:

• (A) 4
• (B) 2
• (C) 6
• (D) 3

If a line makes an angle of π/3 with each X and Y axis, then the acute angle made by the Z-axis is:

• (A) π/3
• (B) π/6
• (C) π/2
• (D) π/4

The length of the perpendicular drawn from the point (3, -1, 11) to the line (x/2) = (y-2)/3 = (z-3)/4 is:

• (A) √33
• (B) √66
• (C) √53
• (D) √29

The equation of the plane through the points (2, 1, 0), (3, 2, -2), and (3, 1, 7) is:

• (A) 6x - 3y + 2z - 7 = 0
• (B) 3x - 2y + 6z - 27 = 0
• (C) 7x - 9y - z - 5 = 0
• (D) 2x - 3y + 4z - 27 = 0

The point of intersection of the line (x+1)/3 = (y+3)/3 = (-z+2)/2 with the plane 3x + 4y + 5z = 10 is:

• (A) (2, 6, -4)
• (B) (-2, 6, -4)
• (C) (2, 6, 4)
• (D) (2, -6, -4)

If (2, 3, -1) is the foot of the perpendicular from (4, 2, 1) to a plane, then the equation of the plane is:

• (A) 2x - y + 2z = 0
• (B) 2x - y + 2z + 1 = 0
• (C) 2x + y + 2z - 5 = 0
• (D) 2x + y + 2z - 1 = 0

If |a x b|² + |a . b|² = 144 and |a| = 4 then |b| is equal to:

• (A) 8
• (B) 12
• (C) 4
• (D) 3

If A and B are events such that P(A) = 1/4, P(A/B) = 1/2, P(B/A) = 2/3, then P(B) is:

• (A) 1/3
• (B) 1/8
• (C) 2/3
• (D) 1/4

A bag contains 2n + 1 coins. It is known that *n* of these coins have heads on both sides, whereas the other *n + 1* coins are fair. One coin is selected at random and tossed. If the probability that the toss results in heads is 31/42, then the value of n is:

• (A) 8
• (B) 5
• (C) 10
• (D) 6

Let A = {x, y, z, u} and B = {a, b}. A function f: A -> B is selected randomly. The probability that the function is an onto function is:

• (A) 5/8
• (B) 7/8
• (C) 1/8
• (D) 1/4

The shaded region in the figure given is the solution of which of the inequalities? 

• (A) x + y ≥ 7, 2x - 3y + 6 ≥ 0, x ≥ 0, y ≥ 0
• (B) x + y ≤ 7, 2x - 3y + 6 ≥ 0, x ≥ 0, y ≥ 0
• (C) x + y ≤ 7, 2x - 3y + 6 ≤ 0, x ≥ 0, y ≥ 0
• (D) x + y ≥ 7, 2x - 3y + 6 ≤ 0, x ≥ 0, y ≥ 0

If f(x) = ax + b, where a and b are integers, f(-1) = -5 and f(3) = 3, then a and b are respectively:

• (A) 0, 2
• (B) -3, -1
• (C) 2, 3
• (D) 2, -3

The value of log₁₀ tan 1° + log₁₀ tan 2° + log₁₀ tan 3° + ... + log₁₀ tan 89° is:

• (A) 1/2
• (B) 0
• (C) 1
• (D) 3

The value of | sin² 14° sin² 66° tan 135°| | sin² 66° tan 135° sin² 14° | is: | tan 135° sin² 14° sin² 66° |

• (A) 1
• (B) -1
• (C) 2
• (D) 0

The modulus of the complex number (1+i)²(1+3i) / (2-6i)(2-2i) is:

• (A) 4/√2
• (B) √2/4
• (C) 1/√2
• (D) 2/√2

Given that a, b, and x are real numbers and a < b, x < 0, then:

• (A) a/x < b/x
• (B) a/x > b/x
• (C) a/x ≤ b/x
• (D) a/x ≥ b/x

Ten chairs are numbered 1 to 10. Three women and two men wish to occupy one chair each. First, the women choose the chairs marked 1 to 6, then the men choose the chairs from the remaining. The number of possible ways is:

• (A) 6C3 × 4P2
• (B) 6C3 × 4C2
• (C) 6P3 × 4C2
• (D) 6P3 × 4P2

Which of the following is an empty set?

• (A) {x : x² - 9 = 0, x ∈ R}
• (B) {x : x² - 1 = 0, x ∈ R}
• (C) {x : x² = x + 2, x ∈ R}
• (D) {x : x² + 1 = 0, x ∈ R}

The n^th term of the series 3/1 + 4/7 + 5/7² + 1/7³ + ... is:

• (A) (n+2)/7^n-1
• (B) (n+3)/7ⁿ
• (C) (2n+1)/7ⁿ
• (D) (2n+1)/7^n-1

If p(1/q + 1/r), q(1/r + 1/p), r(1/p + 1/q) are in A.P., then p, q, and r are:

• (A) in A.P.
• (B) not in A.P.
• (C) not in G.P.
• (D) in G.P.

A line passes through (2, 2) and is perpendicular to the line 3x + y = 3. Its y-intercept is:

• (A) 1
• (B) -2/3
• (C) 4/3
• (D) 3

The distance between the foci of a hyperbola is 16 and its eccentricity is √2. Its equation is:

• (A) 2x² - 3y² = 7
• (B) x² - y² = 32
• (C) y² - x² = 32
• (D) x²/32 - y²/32 = 1

If lim_x→0 (sin(2+x) - sin(2-x))/x = A cos B, then the values of A and B respectively are:

• (A) 2, 1
• (B) 2, 2
• (C) 1, 1
• (D) 1, 2

If n is even and the middle term in the expansion of (x² + 1/x)ⁿ is 924 x⁶, then n is equal to:

• (A) 12
• (B) 10
• (C) 8
• (D) 14

The mean of 100 observations is 50 and their standard deviation is 5. Then the sum of squares of all observations is:

• (A) 250000
• (B) 50000
• (C) 255000
• (D) 252500

Let f : R → R and g : [0, ∞) → R be defined by f(x) = x² and g(x) = √x. Which of the following is not true?

• (A) (f o g)(2) = 2
• (B) (g o f)(4) = 4
• (C) (g o f)(-2) = 2
• (D) (f o g)(-4) = 4

Let f : R → R be defined by f(x) = 3x² - 5 and g : R → R by g(x) = x/(x²+1). Then g o f is:

• (A) (3x² - 5)/(x⁴ + 2x² - 4)
• (B) (3x² - 5)/(9x⁴ - 30x² + 26)
• (C) (3x² - 5)/(9x⁴ + 30x² - 2)
• (D) (3x² - 5)/(9x⁴ - 6x² + 26)

Let the relation R be defined in N by aRb if 3a + 2b = 27. Then R is:

• (A) {(1, 12), (3, 9), (5, 6), (7, 3), (9, 0)}
• (B) {(1, 12), (3, 9), (5, 6), (6, 5), (7, 3)}
• (C) {(2, 1), (9, 3), (6, 5), (3, 7)}
• (D) {(9/3, 0), (1, 12), (3, 9), (6, 6), (7, 3)}

Let f(x) = sin 2x + cos 2x and g(x) = x² - 1. Then g(f(x)) is invertible in the domain:

• (A) x ∈ [-π/2, π/2]
• (B) x ∈ [-π/4, π/4]
• (C) x ∈ [0, π/2]
• (D) x ∈ [-π/2, π/4]

The contrapositive of the statement "If two lines do not intersect in the same plane then they are parallel." is:

• (A) If two lines are not parallel, then they do not intersect in the same plane.
• (B) If two lines are not parallel, then they intersect in the same plane.
• (C) If two lines are parallel, then they do not intersect in the same plane.
• (D) If two lines are parallel, then they intersect in the same plane.

The value of cot^-1((√(1 - sin x) + √(1 + sin x)) / (√(1 - sin x) - √(1 + sin x))), where x ∈ (0, π/4), is:

• (A) π/4
• (B) x/2
• (C) π/2
• (D) x - π

If x[3 2] + y[1 -1] = [15 5], then the value of x and y are:

• (A) x = -4, y = -3
• (B) x = 4, y = 3
• (C) x = -4, y = 3
• (D) x = 4, y = -3

If A and B are two matrices such that AB = B and BA = A, then A² + B² =

• (A) AB
• (B) A + B
• (C) 2BA
• (D) 2AB

If A = [2-k  2; 1 3-k] is a singular matrix, then the value of 5k - k² is equal to:

• (A) -4
• (B) 4
• (C) 6
• (D) -6

The area of a triangle with vertices (-3, 0), (3, 0), and (0, k) is 9 sq. units. The value of k is:

• (A) 6
• (B) 9
• (C) 3
• (D) -9

If Δ = |1 a a²; 1 b b²; 1 c c²| and Δ₁ = |1 bc ca; 1 ca ab; a b c|, then:

• (A) Δ₁ ≠ Δ
• (B) Δ₁ = Δ
• (C) Δ₁ = -Δ
• (D) Δ₁ = 3Δ

If sin^-1(2a/(1+a²)) + cos^-1((1-a²)/(1+a²)) = tan^-1(2x/(1-x²)), where a, x ∈ (0, 1), then the value of x is:

• (A) a²
• (B) 0
• (C) a
• (D) 2a