# Integrals VBQs Class 12 Mathematics

VBQs Integrals Class 12 Mathematics with solutions has been provided below for standard students. We have provided chapter wise VBQ for Class 12 Mathematics with solutions. The following Integrals Class 12 Mathematics value based questions with answers will come in your exams. Students should understand the concepts and learn the solved cased based VBQs provided below. This will help you to get better marks in class 12 examinations.

## Integrals VBQs Class 12 Mathematics

Question. The integral (image 78) is equal to :
(a) 35/6 – 32/3
(b) 34/3 – 31/3
(c) 37/6 – 35/6
(d) 35/3 – 31/3

C

Question. If (image 31) where k is a constant of integration, then A + B + C equals
(a) 16/5
(b) 27/10
(c) 7/10
(d) 21/5

A

Question. The integral of (x2 – x) / (x3 – x2 + x – 1) w.r.t. x is
(a) (1/2)log (x2 + 1) + C
(b) (1/2)log |x2 – 1| + C
(c) log (x2 + 1) + C
(d) log |x2 + 1| + C

A

Question. If ƒ(x) = (2 − x cos x) / (2 + x cos x) and g(x) = logex, (x > 0) then the value of the integral (image 82)
(a) loge3
(b) logee
(c) loge2
(d) loge1

D

Question. Let (image 83) where g is a non- ero even function. If ƒ(x + 5) = g(x), then (image 83)
(image 83)

A

Question. The integral (image 63) is equal to :
(a) 7/18
(b) – 1/9
(c) – 1/18
(d) 9/2

C

Question. Let {x} and [x] denote the fractional part of x and the greatest integer ≤ x respectively of a real number x. If (image 64) and 10(n2 – n), (n ∈N, n > 1) are three consecutive terms of a G.P., then n is equal to _______________.

21

Question. (image 44)
(a) log tan(x/2 + π/12) + c
(b) log tan(x/2 − π/12) + c
(c) (1/2)log tan(x/2 + π/12) + c
(d) (1/2)log tan(x/2 − π/12) + c

C

Question. ƒ(x) and g(x) are two differentiable functions on [0, 2] such that ƒ”(x) – g”(x) = 0, ƒ'(1) = 2g'(1) = 4ƒ(2) = 3g(2) = 9 then ƒ(x) – g(x) at x = 3/2 is
(a) 0
(b) 2
(c) 10
(d) 5

D

Question. If ∫ x5e-x2dx = g(x)e-x2 + c , where c is a constant of integration, then g(–1) is equal to :
(a) –1
(b) 1
(c) – 5/2
(d) – 1/2

C

Question. (image 53)
(a) −(x4+1)1/4 + c
(b) −((x4+1) / (x4))1/4 + c
(c) ((x4+1) / (x4))1/4 + c
(d) (x4+1)1/4 + c

B

Question. The value of the integral (image 98) is
(a) (3/16)π
(b) 0
(c) (3/8)π
(d) (3/4)π

C

Question. If (image 127) = sin x − x cos x − x2/2, for all x ∈ R − {0}, then the value of ƒ(π/6) is
(a) 1/2
(b) 1
(c) 0
(d) – 1/2

D

Question. If f (a + b – x) = f (x) then (image 148) is equal to
(image 148)

C

Question. Let [.] denote the greatest integer function then the value of (image 128)
(a) 0
(b) 3/2
(c) 3/4
(d) 5/4

C

Question. (image 131) where [ . ] denotes the greatest integer function, is equal to :
(a) 1
(b) –1
(c) – π/2
(d) π/2

C

Question. (image 111)
(a) 4√3 − 4
(b) 4√3 − 4 − π/3
(c) π − 4
(d) 2π/3 − 4 − 4√3

B

Question. (image 151)
(a) π2/4
(b) π2
(c) zero
(d) π/2

B

Question. Let function F be defined as (image 112) then the value of the integral (image 112) where a > 0, is:
(a) ea [F(x) − F(1+a)]
(b) e−a [F(x+a) − F(a)]
(c) ea [F(x+a) − F(1+a)]
(d) e−a [F(x) − F(1+a)]

D

Question. Let (image 132) Then which one of the following is true?
(a) I > 2/3 and J > 2
(b) I < 2/3 and J < 2
(c) I < 2/3 and J > 2
(d) I > 2/3 and J < 2

B

Question. Let a function ƒ : [0, 5] → R be continuous, ƒ(1) = 3 and F be defined as:
(image 155)
Then for the function F, the point x = 1 is :
(a) a point of local minima.
(b) not a critical point.
(c) a point of local maxima.
(d) a point of inflection.

A

Question. The value of the integral (image 125) where [.] denotes the greatest integer function is
(a) 0.9
(b) 1.8
(c) – 0.9
(d) 0

D

Question. The solution for x of the equation (image 133)
(a) √3/2
(b) 2√2
(c) 2
(d) None of these

D

Question. If for a continuous function f(x), (image 113) π2 –t2, for all t ≥ – π, then f(− π/3) is equal to:
(a) π
(b) π/2
(c) π/3
(d) π/6

A

Question. (image 159)
(a) 9/e2
(b) 3 log 3 – 2
(c) 9/e4
(d) 9/e2

D

Question. If [ ] denotes the greatest integer function, then the integral (image 114) is equal to:
(a) π/2
(b) 0
(c) –1
(d) – π/2

D

Question. The value of (image 135) a > 1 where [x] denotes the greatest integer not exceeding x is
(a) af (a) – { f (1) + f (2) +………….. f ([a])}
(b) [a] f (a) – { f (1) + f (2) +………….. f ([a])}
(c) [a] f ([a]) – { f (1) + f (2) +………….. f (a)}
(d) af ([a]) – { f (1) + f (2) +………….. f (a)}

B

Question. (image 140)
(a) I2 > I1
(b) I1 > I2
(c) I3 = I4
(d) I3 > I4

B

Question. (imgae 136)
(a) π4/32
(b) π4/32 + π/2
(c) π/2
(d) π/4 − 1

C

Question. (image 137)
(image 137)

D

Question. The integral (image 99) is equal to :
(a) –1
(b) –2
(c) 2
(d) 4

C

Question. (image 54)
(image 54)

B

Question. If (image 57) then A(x) is equal to :
(a) – x
(b) x
(c) √(1 – x)
(d) √(1 + x)

B

Question. (image 36)
(a) (1/2)sin2 2x + c
(b) –(1/2)sin2 2x + c
(c) –(1/2)sin2 x + c
(d) –sin2 x + c

B

Question. If ∫ƒ(x)dx = ψ(x), then ∫x5ƒ(x3)dx is equal to
(image 37)

C

Question. If (image 60) such that I2 = αI1 then α equals to :
(a) 5049/5050
(b) 5050/5049
(c) 5050/5051
(d) 5051/5050

C

Question. If (image 9) where c is a constant of integration, then λƒ(π/3) is equal to:
(a) – 9/8
(b) 2
(c) 9/8
(d) –2

D

Question. The integral (image 10) is equal to : (Here C is a constant of integration)
(image 10)

C

Question. (image 61)
(a) π/4
(b) π
(c) π/2
(d) 3π/2

C

Question. Let ƒ(x) =| x – 2 | and g(x) = ƒ( ƒ(x)), x∈[0, 4]. Then (image 62) is equal to :
(a) 1
(b) 0
(c) 1/2
(d) 3/2

A

Question. The integral (image 67) is equal to ________.

1.50

Question. Let [t] denote the greatest integer less than or equal to t. Then the value of (image 68) is ___________.

A

Question. If (image 101) then k is equal to :
(a) 1
(b) 2
(c) 3
(d) 4

A

Question. The integral (image 102) equals :
(a) 15/128
(b) 15/64
(c) 13/32
(d) 15/256

A

Question. If for all real triplets (a, b, c), f(x) = a + bx + cx2; then (image 69) is equal to:
(image 69)

D

Question. If ƒ(a + b + 1 – x) = ƒ(x), for all x, where α and b are fixed positive real numbers, then (image 72) is equal to:
(image 72)

C

Question. If(image 93) (k > 0) then the value of k is :
(a) 4
(b) 1/2
(c) 1
(d) 2

D

Question. The value of (image 94)
(a) π/2
(b) 4π
(c) π/4
(d) π/8

C

Question. The value of α for which (image 73)
(a) loge2
(b) loge(3/2)
(c) loge√2
(d) loge(4/3)

A

Question. For x ∈R, x ≠ 0, if y(x) is a differentiable function such that (image 104) then y(x) equals :
(where C is a constant)
(image 104)

D

Question. The value of the integral (image 105) where [x] denotes the greatest integer less than or equal to x, is :
(a) 1/3
(b) 6
(c) 7
(d) 3

D

Question. If (image 76) then m.n is equal to :
(a) 1/2
(b) 1
(c) 1/2
(d) –1

D

Question. If (image 3)where C is a constant of integration, then B(θ)/A can be
(a) (2sinθ + 1) / (sinθ + 3)
(b) (2sinθ + 1) / 5(sinθ + 3)
(c) 5(sinθ + 3) / (2sinθ + 1)
(d) 5(2sinθ + 1) / (sinθ + 3)

D

Question. The integral (image 4) is equal to (where C is a constant of integration) :
(image 4)

A

Question. If f : R → R is a differentiable function and f (2) = 6, then (image 81)
(a) 24 f'(2)
(b) 2 f'(2)
(c) 0
(d) 12 f'(2)

D

Question. If m is a non-zero number and (image 55) then f(x) is:
(image 55)

B

Question. (image 56)
(image 56)

B

Question. Let ƒ and g be continuous functions on [0, a] such that ƒ(x) = ƒ(a – x) and g(x) + g(a – x) = 4, then (image 84) is equal to:
(image 84)

C

Question. Let (image 88) If I is minimum then the ordered pair (a, b) is:
(a) (0, √2)
(b) (- √2, 0)
(c) ( √2, -√2)
(d) (-√2, √2)

D

Question. Let f : (–1, 1) → R be a continuous function. If (image 109) is equal to :
(a) 1/2
(b) √3/2
(c) √(3/2)
(d) √3

D

Question. For x > 0, let (image 110) Then ƒ(x) + ƒ(1/x) is equal to:
(a) 1/4 (log x)2
(b) log x
(c) 1/2 (log x)2
(d) (1/4)log x2

C

Question. The value of (image 91)
(a) 0
(b) 4/3
(c) 2/3
(d) 4/3

B

Question. Let f be a differentiable function from R to R such that |ƒ(x) – ƒ(y)| ≤ 2|x – y|3/2, for all x, y, ∈ R. If ƒ(0) = 1 then (image 92) is equal to :
(a) 1
(b) 2
(c) 1/2
(d) 0

A

Question. (image 97)
(a) I2 > I3 > I1
(b) I3 > I1 > I2
(c) I2 > I1 > I3
(d) I3 > I2 > I1

D

Question. Let In = ∫ tann x dx, (n > 1) . I4 + I6 = a tan5x + bx5 + C, where C is constant of integration, then the ordered pair (a, b) is equal to :
(a) (– 1/5, 0)
(b) (– 1/5, 1)
(c) (1/5, 0)
(d) (1/5, –1)

C

Question. The integral (image 103) is equal to :
(image 103)

D

Question. The value of integral, (image 138)
(a) 1/2
(b) 3/2
(c) 2
(d) 1

B

Question. Let f : R → R be a differentiable function having f(2) = 6, f'(2) = (1/48). Then (image 141)
(a) 24
(b) 36
(c) 12
(d) 18

D

Question. The value of (image 144)
(a) 3
(b) 1
(c) 2
(d) 0

C

Question. Let p(x) be a function defined on R such that p'(x) = p'(1 – x), for all x ∈ [0, 1], p (0) = 1 and p (1) = 41. Then
(image)
(a) 21
(b) 41
(c) 42
(d) √41

A

Question. The value of (image 145)
(a) 1/3
(b) 14/3
(c) 7/3
(d) 28/3

D

Question. (image 163)
(a) 1/5
(b) 1/30
(c) Zero
(d) 1/4

A

Question. (image 164)
(a) 1/(p+1)
(b) 1/(1 − p)
(c) 1/p − 1/(p − 1)
(d) 1/(p+2)

A

Question. The value of the integral (image 146)
(a) 1/(n+1) + 1/(n+2)
(b) 1/(n+1)
(c) 1/(n+2)
(d) 1/(n+1) − 1/(n+2)

D

Question. Let f(x) be a function satisfying f ‘(x) = f(x) with f(0)=1 and g(x) be a function that satisfies f(x) + g(x) = x2 . Then the value of the integral (image 147)
(a) e + e2/2 + 5/2
(b) e − e2/2 − 5/2
(c) e + e2/2 − 3/2
(d) e − e2/2 − 3/2

D

Question. (image 129)
(a) π/8 log 2
(b) π/2 log 2
(c) log 2
(d) π log 2

D

Question. (image 149)
(a) 0
(b) 3
(c) 2
(d) 1

D

Question. If f(y) = ey , g(y) = y; y > 0 y and (image 150)
(a) F(t) = te−t
(b) F(t) = 1 − te−t (1+t)
(c) F(t) = et − (1+t)
(d) F(t) = tet

C

Question. (image 152)
(a) 2 – √2
(b) 2 + √2
(c) √2 – 1
(d) -√2 – √3 + 5

D

Question. (image 153)
(a) 1/2
(b) 1
(c) ∞
(d) zero

B

Question. (image 154)
(a) 20
(b) 8
(c) 10
(d) 18

A

Question. (image 156)
(image 156)

A

Question. (image 157) is equal to :
(a) π/4
(b) tan–1(3)
(c) π/2
(d) tan–1(2)

D

Question. If (image 158) for some positive real number a, then a is equal to :
(a) 7
(b) 8
(c) 15/2
(d) 17/2

A

Question. (image 139)
(a) aπ
(b) π/2
(c) π/a
(d) 2π

B

Question. (image 162)
(a) e + 1
(b) e – 1
(c) 1 – e
(d) e

B

Question. (image 87)

B

Question. (image 106)
(a) π/2 + log2
(b) log2
(c) π/2 − log4
(d) log4

B

Question. (image 70)
(a) 2π
(b) 2π2
(c) π2
(d) 4π

C

Question. (image 71)
(a) 1/8 < I2 < 1/4
(b) 1/9 < I2 < 1/8
(c) 1/16 < I2 < 1/9
(d) 1/6 < I2 < 1/2

B

Question. The integral ∫12 ex · xx(2 + logex)dx equals :
(a) e(4e +1)
(b) 4e2 – 1
(c) e(4e–1)
(d) e(2e – 1)

C

Question. A value of α such that (image 49)
(a) –2
(b) 1/2
(c) – 1/2
(d) 2

A

Question. (image 65)
(a) √2π2
(b) 2π2
(c) π2
(d) π2/2

C

Question. If the value of the integral (image 66) is k/6 then k is equal to :
(a) 2√3 – π
(b) 2√3 + π
(c) 3√2 + π
(d) 3√2 – π

A

Question. Let n ≥ 2 be a natural number and 0 < θ < π/2 Then (image 22) is equal to:
(where C is a constant of integration)
(image 22)

A

Question. For x2 ≠ nπ + 1, n∈N (the set of natural numbers), the integral (image 23) is equal to:
(image 23)

C,D

Question. (image 85)
(a) 1/2 – e – 1/e2
(b) – 1/2 + e – 1/2e2
(c) 3/2 – 1/e – 1/2e2
(d) 3/2 – e – 1/2e2