KCET 2024 Mathematics Question Paper: Download Question Paper with Solution PDF

Two finite sets have *m* and *n* elements respectively. The total number of subsets of the first set is 56 more than the total number of subsets of the second set. The values of *m* and *n* respectively are:

• (A) 7, 6
• (B) 5, 1
• (C) 6, 3
• (D) 8, 7

If [x]² - 5[x] + 6 = 0, where [x] denotes the greatest integer function, then:

• (A) x ∈ [3, 4)
• (B) x ∈ [2, 4)
• (C) x ∈ [2, 3)
• (D) x ∈ [2, 3]

If in two circles, arcs of the same length subtend angles 30° and 78° at the center, then the ratio of their radii is:

• (A) 5/13
• (B) 13/5
• (C) 13/4
• (D) 4/13

If ∆ABC is right-angled at C, then the value of tan A + tan B is:

• (A) a + b
• (B) c / ac
• (C) c² / ab
• (D) c² / ac

The real value of α for which (1 - i sin α)/(1 + 2i sin α) is purely real is:

• (A) (n + 1)π/2, n ∈ N
• (B) (2n + 1)π/2, n ∈ N
• (C) nπ, n ∈ N
• (D) (2n - 1)π/2, n ∈ N

The length of a rectangle is five times the breadth. If the minimum perimeter of the rectangle is 180 cm, then:

• (A) Breadth < 15 cm
• (B) Breadth ≥ 15 cm
• (C) Length < 15 cm
• (D) Length = 15 cm

The value of ⁴⁹C₃ + ⁴⁸C₃ + ⁴⁷C₃ + ⁴⁶C₃ + ⁴⁵C₃ + ⁴⁵C₄ is:

• (A) ⁵⁰C₄
• (B) ⁵⁰C₃
• (C) ⁵⁰C₂
• (D) ⁵⁰C₁

In the expansion of (1 + x)ⁿ, ⁿC₁ + 2 * ⁿC₂ + 3 * ⁿC₃ + ... + n * ⁿC_n is equal to:

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

If Sn stands for the sum to n terms of a G.P. with a as the first term and r as the common ratio, then S1/S2 is:

• (A) rⁿ + 1
• (B) 1/(rⁿ + 1)
• (C) rⁿ – 1
• (D) -1

If A.M. and G.M. of roots of a quadratic equation are 5 and 4 respectively, then the quadratic equation is:

• (A) x² – 10x – 16 = 0
• (B) x² + 10x + 16 = 0
• (C) x² + 10x – 16 = 0
• (D) x² – 10x + 16 = 0

The angle between the line x + y = 3 and the line joining the points (1, 1) and (-3, 4) is:

• (A) tan^-1(7)
• (B) tan^-1(1/7)
• (C) tan^-1(3/7)
• (D) tan^-1(4/7)

The equation of the parabola whose focus is (6,0) and directrix is x = -6 is:

• (A) y² = 24x
• (B) y² = -24x
• (C) x² = 24y
• (D) x² = -24y

lim_x→π/4 (√2 cos x - 1)/(cot x - 1) is equal to:

• (A) 2
• (B) √2
• (C) 1/2
• (D) 1/√2

The negation of the statement “For every real number x, x² + 5 is positive” is:

• (A) For every real number x, x² + 5 is not positive
• (B) For every real number x, x² + 5 is negative
• (C) There exists at least one real number x such that x² + 5 is not positive
• (D) There exists at least one real number x such that x² + 5 is positive

Let *a*, *b*, *c*, and *d* be the observations with mean *m* and standard deviation *S*. The standard deviation of the observations *a* + *k*, *b* + *k*, *c* + *k*, *d* + *k* is:

• (A) *kS*
• (B) *S* + *k*
• (C) *S/k*
• (D) *S*

Let f : R → R be given by f(x) = tan x. Then f^-1(1) is:

• (A) π/4
• (B) nπ + π/4, n ∈ Z
• (C) 3π/4
• (D) nπ + 5π/4, n ∈ Z

Let f : R → R be defined by f(x) = x² + 1. Then the pre-images of 17 and -3 respectively are:

• (A) Φ, {4, -4}
• (B) {3, -3}, φ
• (C) {4, -4}, ф
• (D) {4, -4}, {2, -2}

Let (g o f)(x) = sin x and (f o g)(x) = (sin √x)². Then:

• (A) f(x) = sin²x, g(x) = x
• (B) f(x) = sin √x, g(x) = √x
• (C) f(x) = sin²x, g(x) = √x
• (D) f(x) = sin √x, g(x) = x²

Let A = {2, 3, 4, 5, . . ., 16, 17, 18}. Let R be the relation on the set A of ordered pairs of positive integers defined by (a, b)R(c, d) if and only if ad = bc for all (a, b), (c, d) ∈ A × A. Then the number of ordered pairs of the equivalence class of (3, 2) is:

• (A) 4
• (B) 5
• (C) 6
• (D) 7

If cos^-1x + cos^-1y + cos^-1z = 3π, then x(y + z) + y(z + x) + z(x + y) equals to:

• (A) 0
• (B) 1
• (C) 6
• (D) 12

If 2 sin^-1x - 3 cos^-1x = 4x, x ∈ [-1,1], then 2 sin^-1x + 3 cos^-1x is equal to:

• (A) 4 - 6π
• (B) 6π - 4
• (C) 3π/2
• (D) 0

If A is a square matrix such that A² = A, then (I + A)³ is equal to:

• (A) 7A - I
• (B) I + 7A
• (C) 7A + I
• (D) I - 7A

If A = [1 1] [1 1] then A¹⁰ is equal to:

• (A) 2⁸A
• (B) 2⁹A
• (C) 2¹⁰A
• (D) 2¹¹A

If then f(1) * f(3) * f(5) + f(5) * f(1) is:

• (A) 1
• (B) 0
• (C) 2
• (D) None of these

If is the adjoint of a 3 x 3 matrix A and |A| = 4, then α is equal to:

• (A) 4
• (B) 5
• (C) 11
• (D) 0

If A = [x 1] [1 x] and B = [x 1 1] [1 x 1] [1 1 x] then dB/dx is:

• (A) 3A
• (B) -3B
• (C) 3B + 1
• (D) 1 - 3A

Let f(x) = [cos x x 1 ] [2sin x x 2x] [sin x x x ] Then lim _x→0 f(x) / x² is:

• (A) -1
• (B) 0
• (C) 3
• (D) 2

Which one of the following observations is correct for the features of the logarithm function to any base b > 1?

• (A) The domain of the logarithm function is R, the set of real numbers.
• (B) The range of the logarithm function is R+, the set of all positive real numbers.
• (C) The point (1,0) is always on the graph of the logarithm function.
• (D) The graph of the logarithm function is decreasing as we move from left to right.

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

• (A) Everywhere continuous and differentiable.
• (B) Everywhere continuous but not differentiable at odd multiples of π/2.
• (C) Neither continuous nor differentiable at 2n + 1, n ∈ Z.
• (D) Not differentiable everywhere.

If y = 2^x3^x, then dy/dx at x = 1 is:

• (A) 2
• (B) 6
• (C) 3
• (D) 1

Let the function satisfy the equation f(x + y) = f(x)f(y) for all x, y ∈ R, where f(0) ≠ 0. If f(5) = 3 and f′(0) = 2, then f′(5) is:

• (A) 6
• (B) 0
• (C) 3
• (D) -6

The value of C in (0,2) satisfying the mean value theorem for the function f(x) = x(x - 1)², x ∈ [0, 2] is equal to:

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

d/dx cos²[cot^-1 √ (3-x) / (2+x) ] is:

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

For the function f(x) = x³ – 6x² + 12x – 3, x = 2 is:

• (A) A point of minimum
• (B) A point of inflection
• (C) Not a critical point
• (D) A point of maximum

The function x^x, x > 0 is strictly increasing at:

• (A) ∀x ∈ R
• (B) x < 1/e
• (C) x > 1/e
• (D) x < 0

The maximum volume of the right circular cone with slant height 6 units is:

• (A) 4√3π cubic units
• (B) 16√3π cubic units
• (C) 3√3π cubic units
• (D) 6√3π cubic units

If f(x) = xe^x(1-x), then f(x) is:

• (A) Increasing in R
• (B) Decreasing in R
• (C) Decreasing in [-1/2, 1]
• (D) Increasing in [-1/2, 1]

∫ sin x / (3 + 4 cos² x) dx

• (A) -1/2√3 tan^-1 (2 cos x / √3) + C
• (B) 1/4 tan^-1 (cos x / √3) + C
• (C) 1/2√3 tan^-1 (cos x / √3) + C
• (D) -1/3 tan^-1 (2 cos x / √3) + C

∫^π_-π (1 - x²) sin x cos² x dx =

• (A) π/2
• (B) 2π - π²
• (C) π³ / 2
• (D) 0

∫ dx / (x (6(log x)² + 7log x + 2)) =

• (A) 1/2 log |(2 log x + 1) / (3 log x + 2)| + C
• (B) log |(2 log x + 1) / (3 log x + 2)| + C
• (C) log |(3 log x + 2) / (2 log x + 1)| + C
• (D) 1/2 log |(3 log x + 2) / (2 log x + 1)| + C

∫ sin 5x / sin x dx

• (A) 2x + sin x + 2 sin 2x + C
• (B) x + 2 sin x - 2 sin 2x + C
• (C) x + 2 sin x + sin 2x + C
• (D) 2x + sin x - sin 2x + C

∫⁵₁ (|x − 3| + |1 – x|) dx

• (A) 12
• (B) 9/2
• (C) 21
• (D) 10

lim_n→∞ ( 1/(n²+1²) + 1/(n²+2²) + ... + 1/(n²+n²)) =

• (A) π/2
• (B) tan^-13
• (C) tan^-12
• (D) π/4

The area of the region bounded by the line y = 3x and the curve y = x³ in sq. units is:

• (A) 10
• (B) 9/2
• (C) 9
• (D) 5

The area of the region bounded by the line y = x and the curve y = x³ is:

• (A) 0.2 sq. units
• (B) 0.3 sq. units
• (C) 0.4 sq. units
• (D) 0.5 sq. units

The solution of e^dy/dx = x + 1, y(0) = 3 is:

• (A) y − 2 = x log x
• (B) y − x − 3 = x log x
• (C) y − x − 3 = (x + 1) log(x + 1)
• (D) y + x − 3 = (x + 1) log(x + 1)

The family of curves whose *x* and *y* intercepts of a tangent at any point are respectively double the *x* and *y* coordinates of that point is:

• (A) xy = C
• (B) x² + y² = C
• (C) x² − y² = C
• (D) x/y = C

The vectors AB = 3i + 4k and AC = 5i – 2j + 4k are the sides of a △ABC. The length of the median through A is:

• (A) √18
• (B) √72
• (C) √33
• (D) √288

The volume of the parallelepiped whose co-terminous edges are i + j, i + k, i + j is:

• (A) 6 cu. units
• (B) 2 cu. units
• (C) 4 cu. units
• (D) 3 cu. units

Let a and b be two unit vectors and θ is the angle between them. Then a + b is a unit vector if:

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

If a, b, c are three non-coplanar vectors and p, q, r are vectors defined by: p = (b x c)/[a b c], q = (c x a)/[a b c], r = (a x b)/[a b c] then (a + b) · p + (b + c) · q + (c + a) · r is:

• (A) 0
• (B) 1
• (C) 2
• (D) 3

If lines (x-1)/(-3) = (y-2)/(2k) = (z-3)/2 and (x-1)/(3k) = (y-5)/2 = (z-6)/(-5) are mutually perpendicular, then k is equal to:

• (A) -10/7
• (B) 7/10
• (C) -10
• (D) -7

The distance between the two planes 2x + 3y + 4z = 4 and 4x + 6y + 8z = 12 is:

• (A) 2 units
• (B) 8 units
• (C) 2/√29 units
• (D) 4 units

The sine of the angle between the straight line (x-2)/3 = (y-3)/4 = (z-4)/-5 and the plane 2x – 2y + z = 5 is:

• (A) 1/√2
• (B) 2/√2
• (C) 3/5√2
• (D) 4/5√2

The equation xy = 0 in three-dimensional space represents:

• (A) A pair of straight lines
• (B) A plane
• (C) A pair of planes at right angles
• (D) A pair of parallel planes

The plane containing the point (3, 2, 0) and the line (x-2)/1 = (y-6)/5 = (z-4)/4 is:

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

Corner points of the feasible region for an LPP are (0, 2), (3, 0), (6, 0), (6, 8) and (0, 5). Let z = 4x + 6y be the objective function. The minimum value of z occurs at:

• (A) Only (0, 2)
• (B) Only (3, 0)
• (C) The mid-point of the line segment joining the points (0, 2) and (3, 0)
• (D) Any point on the line segment joining the points (0, 2) and (3, 0)

A die is thrown 10 times. The probability that an odd number will come up at least once is:

• (A) 11/1024
• (B) 1013/1024
• (C) 1023/1024
• (D) 1/1024

A random variable X has the following probability distribution:

If the mean of the random variable X is 1/3, then the variance is:

• (A) 1
• (B) 5/18
• (C) 11/36
• (D) 1/2

If a random variable X follows the binomial distribution with parameters n = 5, p, and P(X = 2) = 9 P(X = 3), then p is equal to:

• (A) 10
• (B) 1/10
• (C) 5
• (D) 1/5