KCET 2023 Mathematics Question Paper: Download Set C4 Question Paper with Answer Key PDF

Given that |a| = |b| = |a - b|, then:

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

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

• (A) √6/₆
• (B) √6
• (C) 6
• (D) ⁴/₆

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

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

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

• (A) 7 sq. units
• (B) 10 sq. units
• (C) ¹¹/₂ sq. units
• (D) ⁷/₂ 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 x-coordinate of the point is equal to the y-coordinate of the point, then the curve also passes through the point:

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

The degree of the differential equation

• (A) 2
• (B) 6
• (C) 3
• (D) 1

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

• (A) 7x - 9y – 2z - 5 = 0
• (B) 3x - 2y + 6z - 27 = 0
• (C) 2x – 3y + 4z - 27 = 0
• (D) 6x – 3y + 2z-7=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 - 5 = 0
• (B) 2x - y + 2x + 1 = 0
• (C) 2x + y + 2z − 1 = 0
• (D) 2x - y + 2z = 0

If |a × b|² + |a – b|² = 144 and |a| = 4, then |b| is equal to:

• (A) 4
• (B) 12
• (C) 3
• (D) 8

If â+2b+3ĉ = 0 and (axb)+(bxê)+(ĉxâ) = λ(b×ĉ), then the value of 入 is equal to:

• (A) 6
• (B) 2
• (C) 3
• (D) 4

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/₄
• (C) ^π/₆
• (D) ^π/₅

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

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

The shaded region in the figure 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 A and B are events such that P(A) = ¹/₂, P(A/B) = ¹/₃, P(B/A) = ²/₃, then P(B) is:

• (A) ¹/₆
• (B) ¹/₃
• (C) ¹/₂
• (D) ²/₃

A bag contains 2n + 1 coins. It is known that n of these coins have heads on both sides where the other n+1 coins are fair. One coin is selected at random and tossed. If the outcome is heads, what is the probability that the selected coin is the one with heads on both sides?

• (A) ¹/₄
• (B) ¹/₃
• (C) ²/₃
• (D) 1

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

• (A) ³/₅
• (B) ⁷/₈
• (C) ¹/₂
• (D) ⁵/₈

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

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

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 as 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) 5P3 × 6C2
• (B) 6C3 × 4C2
• (C) 5P3 × 4P2
• (D) 6C3 × 4P2

Which of the following is an empty set?

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

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

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

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

• (A) 1
• (B) 0
• (C) 3
• (D) -1

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

• (A) 2
• (B) -1
• (C) 0
• (D) 1

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

• (A) y² – x² = 32
• (B) x² - y² = 32
• (C) (x²/₃) – (y²/₄) = 1
• (D) 2x² – 3y² = 7

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

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

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) 8
• (B) 10
• (C) 14
• (D) 12

The nth term of the series: 1 + ⁵/₇ + ⁵/_7² + ⁵/_7³ + ...

• (A) ²ⁿ⁺¹/_7n
• (B) ^2n-1/_7n
• (C) ²ⁿ⁺¹/_7n-1
• (D) ^2n-1/_7n-1

If p, q, r are in arithmetic progression and if: ^{(1/q + 1/r)}/_p, ^{(1/r + 1/p)}/_q, ^{(1/p + 1/q)}/_r are in A.P., then p, q, r are:

• (A) Not in G.P.
• (B) Not in A.P.
• (C) In G.P.
• (D) In A.P.

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

• (A) ⁴/₃
• (B) ¹/₃
• (C) ²/₃
• (D) 1

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

• (A) {(2, 1), (9, 3), (6, 5), (3, 7)}
• (B){(1, 12), (3, 9), (5, 6), (7, 3)}
• (C){(0, ²⁷/₂), (1, 12), (3, 9), (5, 6), (7, 3)}
• (D){(1, 12), (3, 9), (5, 6), (7, 3), (9,0)}

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

• (A) x ∈ [0,1]
• (B) x ∈ [-π/₄,π/₄]
• (C) x ∈ [-π/₈,π/₈]
• (D) x ∈ [-π/₃,π/₃]

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

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

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

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

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

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

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

If A = |(2-k) k| |( 1 ) (3-k)| is a singular matrix, then the value of 5k – k² is equal to:

• (A) 6
• (B) 4
• (C) -6
• (D) -4

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

• (A) 3
• (B) -9
• (C) -9
• (D) 6

If A = | 1 a²| and Δ1 = |bc ab| | b b²| |ca ab| then:

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

If sin^-1((2α)/(1+α²)) + cos^-1((1-α²)/(1+α²)) = tan^-1((2x)/(1-x²)) where α, x ∈ (0,1), then the value of x is:

• (A) (2α)/(1+α²)
• (B) 0
• (C) α/2
• (D) (1+α²)/(2α)

The value of coth^-1(√(1 + sin x)/√(1 - sin x) + √(1 - sin x)/√(1 + sin x)) where x ∈ (0, π/2) is:

• (A) x/3
• (B) x/2
• (C) π/3
• (D) π - x/3

If |x 2| - |-1 -1| = |-1 5| then the value of x and y are: |y 1| |1 5| = |-1 5|

• (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 is:

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

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

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

If f(x) = 1 + nx + n(n-1)/2 x² + n(n-1)(n-2)/6 x³ + ... + xⁿ, then f"(1) is:

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

If A = | 1 tan α/2| and AB =I then B = |-tan α/2 1 |

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

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

• (A) 1
• (B) 1/2
• (C) 2
• (D) 3/4

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

• (A) x = π/n, n ∈ Z
• (B) x = nπ, n ∈ Z
• (C) x = 2nπ, n ∈ Z
• (D) x = (2n + 1)π/2, 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) 1/4
• (B) -5/2
• (C) 3/4
• (D) 5/4

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

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

• (A) 3
• (B) 5
• (C) 6
• (D) 4

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

• (A) 2√(sin x) + C
• (B) √(sin x) + C
• (C) 4√(sin x) + C
• (D) 2√(cos x) + C

If f(x) and g(x) are two functions with g(x) = x - 1/x and f o g(x) = x³ - 3/x, then f'(x) is:

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

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) 0.52π cm²/sec
• (B) 5.2π cm²/sec
• (C) 27.4π cm²/sec
• (D) 5.05π 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) 18 m/sec²
• (B) 3 m/sec²
• (C) 10 m/sec²
• (D) 12 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) II or III
• (B) I or III
• (C) II or IV
• (D) III or IV