# Unseen Passage

For Class 4 to Class 12

# Application of Integrals MCQ Questions Class 12 Mathematics

Please refer to MCQ Questions for Class 12 Mathematics Chapter 8 Application of Integrals with answers below. These multiple-choice questions have been prepared based on the latest NCERT book for Class 12 Mathematics. Students should refer to MCQ Questions for Class 12 Mathematics with Answers to score more marks in Grade 12 Mathematics exams. Students should read the chapter Application of Integrals and then attempt the following objective questions.

## MCQ Questions for Class 12 Mathematics Chapter 8 Application of Integrals with answers

MCQ Questions for Class 12 Mathematics Chapter 8 provided below covers all important topics given in this chapter. These MCQs will help you to properly prepare for exams.

Question. The ratio of the area bounded by the curves y = cos x and y = cos 2 x between x = 0, π / 3 and X-axis, is
(a) √2 : 1
(b) 1: 1
(c) 1: 2
(d) 2 : 1

D

Question. The parabola y2=2x  divides the circle x2+ y2 = 8 in two parts. Then, the ratio of the areas of these parts is
(a) 3π- 2/10π+2
(b) 3π+ 2/9π-2
(c) 6π -3/11π-5
(d) 2π- 9/9π+ 2

B

Question. The area in the first quadrant between x2+y22and y= sin x is
(a) π3-8/4
(b) π3/4
(c) π3-16/4
(d) π3-8/2

A

Question. The area bounded by the curve y = log x, y= log| x|, y =|logx| and y=| log|x|| is
(a) 4 sq units
(b) 6 sq units
(c) 10 sq units
(d) None of these

A

Question. The area of bounded region by the curve y= logex and y = (loge x)2 is
(a) 3 – e
(b) e – 3
(c) 1/2(3(- e)
(d) 1/2(e-3)

A

Question. The area enclosed between the curve y= loge(x+e) and the coordinate axes is
(a) 9 sq units
(b) 27/4 sq units
(c) 1 sq unit
(d) 18 sq units

C

Question. If a curve y a√x +bx =+ passes through the point( 1,2) and the area bounded by the curve, line x = 4 and X-axis is 8 sq units, then
(a) a= 3, b=-1
(b) a=3,b=1
(c) a =-3, b=1
(d) a=-3,b=-1

A

Question. Let y be the function which passes through (1, 2) having slope (2x +1.)The area bounded between the curve and X-axis is
(a) 6 sq units
(b) 5/6 sq unit
(c) 1/6 sq unit
(d) None of these

C

Question. The sine and cosine curves intersects infinitely many times giving bounded regions of equal areas. The area of one of such region is
(a) √2 sq units
(b) 2√2 sq units
(c) 3√2 sq units
(d) 4√2 sq units

B

Question. The area between the curve y= – 2×4-x2, the X-axis and the ordinates of two minima of the curve is
(a) 7/120 sq unit
(b) 9/120 sq unit
(c) 11/120 sq unit
(d) 13/120 sq unit

A

Question. Area bounded by the curve y = x sin X-axis between x = 0 and x = 2π is
(a) 2π  sq units
(b) 3π  sq units
(c) 4π  sq units
(d) 5π  sq units

C

Question. Area bounded by the curve y=(x-1) (x-2) (x-3) and x-axis lying between the ordinates x = 0and x = 3 is equal to
(a) 9/4 sq units
(b) 11/4 sq units
(c) 13/4 sq units
(d) 15/4 sq units

B

Question. If the ordinate x =a  divides the area bounded by X-axis part of the curve y=1+8/x2 and the ordinates x=2,  x=4 into two equal parts, then a is equal to
(a) √2 sq units
(b) 2 √2 sq units
(c) 3 √2 sq units
(d) None of these

B

Question. The area of the region bounded by 1-y2=| x|and |x |+|y| = 1 is
(a) 1/3 sq unit
(b) 2/3 sq unit
(c) 4/3 sq unit
(d) 1 sq unit

B

Question. The area of the region (in sq units), in the first quadrant, bounded by the parabola y = 9x2 and the lines x = 0, y =1and y = 4, is
(a) 12/9
(b) 14/3
(c) 7/3
(d) 14/9

D

Question. The area bounded by the curve y = ln(x ) and the lines y = 0, y = ln(3) and x = 0 is equal to
(a) 3
(b) 3 ln (3) -2
(c) 3 ln (3)+2
(d) 2

D

Question. The area of the region bounded by the curve ay2 = x3 , theY-axis and the lines y =a and y = 2a, is

A

Question. The area of the region bounded by y = | x – 3| and y = 2, is
(a) 4.5 sq units
(b) 6.3 sq units
(c) 3.5 sq units
(d) None of these

D

Question. The area between the curve y = 4 – | x | and X-axis is
(a) 16 sq units
(b) 20 sq units
(c) 12 sq units
(d) 18 sq units

A

Question. The area under the curve y =| cos x – sin x| ,0 ≤ x ≤ π/2 and above X-axis is
(a) 2 √2
(b) 2 √2 – 2
(c) 2 √2 + 2
(d) 0

B

Question. The area of the region enclosed by the curves y = x, x = e, y = 1/x and the positive X-axis is
(a) 1 sq unit
(b) 3/2 sq units
(c) 5/2 sq units
(d) 1/2 sq units

B

Question. The area bounded by y = sin x , X-axis and the lines |x| = π is
(a) 2 sq units
(b) 3 sq units
(c) 4 sq units
(d) None of these

B

Question. For 0 ≤ x ≤ π, the area bounded by y = x and y = x + sin x, is
(a) 2
(b) 4
(c) 2π
(d) 4π

A

Question. The area (in sq unit) of the region described by (x, y ): y2 ≤ 2x and y ≥ 4x – 1} is
(a) 7/32
(b) 5/34
(c) 15/64
(d) 9/32

D

Question. Let f : [-1, 2]→ [0, ∞) be a continuous function such that f (x ) = f (1 – x ) for all x ∈ [-1, 2]. Let R1 =

R2 be the area of region bounded by y = f (x ),x = -1, x = 2 and the X-axis. Then
(a) R1 = 2R2
(b) R1= 3R2
(c) 2R1 = R2
(d) 3R1 = R2

C

Question. If the area bounded by y = ax2 and x = ay2 , a > 0 is 1, then a is equal to
(a) 1
(b) 1/√3
(c) 1/3
(d) None of these

B

Question. The area enclosed the curves y = x3 and y = √x is
(a) 5/3 sq units
(b) 5/4 sq units
(c) 5/12 sq units
(d) 12/5 sq units

C

Question. The curve y = a √x + bx passes through the point (1, 2) and the area enclosed by the curve, the X-axis and the line x = 4 is 8 square units. Then a – b is equal to
(a) 2
(b) – 1
(c) – 2
(d) 4

D

Question. If y = f (x )makes positive intercepts of 2 and 1 unit on x and y-coordinates axes and encloses an area of 3/4 sq unit with the axes, then

(a) 3/2
(b) 1
(c) 5/4
(d) -3/4

D

Question. The area (in sq units) of the region {(x, y ) : y2 ≥ 2x and x2 + y2 ≤ 4x , x ≥ 0 , y ≥ 0 } is
(a) π-4/3
(b) π-8/3
(c) π-4√2/3
(d) π/2 – 2√2/3

B

Question. The area of the region described by
A = {(x,y) : x2 + y2 ≤ 1 and y2 ≤ 1 – x} is
(a) π/2 + 4/3
(b) π/2 – 4/3
(c) π/2 – 2/3
(d) π/2 + 2/3

A

Question. The parabola y2 = 2x divides the circle x2 + y2 = 8 in two parts. Then, the ratio of the areas of these parts is
(a) 3π – 2/10π + 2
(b) 3π + 2/9π – 2
(c) 6π – 3/11π – 5
(d) 2π – 9/9π + 2

B

Question. The area bounded by the curves y = 2 – | 2 – x| and x| y = 3 is
(a) (5 – 4 ln2)/ 3
(b) (2 – ln3)/ 2
(c) (4 – 3 ln3)/ 2
(d) None of these

C

Question. The area bounded by the curves y = | x | -1 and y = – | x | + 1, is
(a) 1 sq unit
(b) 2 sq units
(c) 2 √2 sq units
(d) 4 sq units

B

Question. If the ordinate x = a divides the area bounded by the curve y = (1+8/x2) , X-axis and the ordinates x = 2, 4 into two equal parts, then a is equal to
(a) 8
(b) 2 √2
(c) 2
(d) √2

B

Question. The area bounded by the curve y = f (x ), X-axis and the lines x =1, x = b is (√b2 +1 -√2) for all b > 1, then f (x ) equals to
(a) √x2+1
(b) √x+1
(c) x/√x2+1
(d) None of these

C

Question. The area bounded between the parabolas x2 = y/4 and x2 = 9 y and the straight line y = 2, is
(a) 20√2
(b) 10√2/3
(c) 20√2/3
(d) 10√2

C

Question. The area bounded by f (x ) =min(| x| ,| x -1| ,| x +1| ) in [-1,1] and X-axis, is (in sq unit)
(a) 1/5
(b) 1/4
(c) 1/3
(d) 1/2

D

Question. The area (in sq units) of the region {(x, y ) : x ≥ 0, x + y ≤ 3, x2 ≤ 4 y and y ≤ 1 + √x } is
(a) 7/3
(b) 5/2
(c) 59/12
(d) 3/2

B

Question. The area enclosed between the curves y = loge (x +e ), x = log(1/y ) and the X-axis is equal to
(a) 2e
(b) 2
(c) 2/e
(d) None of these

B

Question. Let A1 be the area of the parabola y2 = 4ax lying between the vertex and the latus rectum and A2 be the area between the latus rectum and the double ordinate x = 2a.
Then A1/A2 is
(a) (2 √2 – 1) /7
(b) (2 √2 + 1) /7
(c) (2 √2 + 1)
(d) (2 √2 – 1)

B

Question. The area bounded by the curves y = x2 and y = 2/(1+x2) is
(a) (3π + 2)/3
(b) (3π – 2)/3
(c) (3π – 2)/6
(d) None of these

B

Question. Area of the region bounded by the parabola (y – 2)2 = x -1, the tangent to it at the point with the ordinate 3, and the X-axis is given by
(a) 9/2 sq units
(b) 9 sq units
(c) 18 sq units
(d) None of these

B

Question. Area of the region bounded by curves x =1/2, x = 2, y = loge x and y = 2x is equal to (4 – √2)/log2 + b – c log 2. Then, b + c equals
(a) 2
(b) – 1
(c) 4
(d) None of these

C

Question. Let the straight line x = b divide the area enclosed by y = (1-x2)2 , y = 0 and x = 0 into two parts R1(0 ≤ x ≤ b) and R2 (b ≤ x ≤1) such that R1 – R2 =1/4. Then b equals
(a) 3/4
(b) 1/2
(c) 1/3
(d) 1/4

B

Question. The area of bounded region by the curve y = loge x and y = (loge x)2 , is
(a) 3 – e
(b) e – 3
(c) 1/2(3-e)
(d) 1/2 (e-3)

A

Question. The area bounded by the curve xy2 = 4(2 -x )and Y-axis is
(a) 2π
(b) 4π
(c) 12π
(d) 6π

B

Question.The area of smaller segment cut off from the circle x2 + y2 = 9 by x =1 is
(a) 1/2(9sec-1 3-√8)
(b) 9 sec-1(3) – √8
(c) √8 – 9 sec-1(3)
(d) None of these

B

Question. The area (in sq units) bounded by the curves y = √x , 2y – x + 3 = 0 , X-axis and lying in the first quadrant is
(a) 9
(b) 36
(c) 18
(d) 27/4

A

Question. The area of the region R = {(x, y ) : | x | ≤ | y | and x2 + y2 ≤ 1} is
(a) 3π/8 sq units
(b) 5π/8 sq units
(c) π/2 sq units
(d) π/8 sq units

C

Question. Let f (x ) be a real valued function satisfying the relation f (x /y ) = f(x) – f(y) and 10
bounded by the curve y = f (x ), y-axis and the line y = 3 is equal to
(a) e
(b) 2e
(c) 3e
(d) None of these

C

Question. Let the circle x2 + y2 + = 4 divide the area bounded by tangent and normal at (1, √3) and X-axis in A1 and A2.
Then A1/A2 equals to
(a) π/ (3 √3 – π)
(b) π/ (3 √3 + π)
(c) π/ (3 – π √3)
(d) None of these

A

Question. The area bounded by y = tan x, y = cot x, X-axis in 0 ≤ x ≤ π/2 is
(a) 3log2
(b) log2
(c) 2log2
(d) None of these

B

Question. The area bounded by the lines y = 2, x =1, x = a and the curve y = f (x ), which cuts the last two lines above the first line for all a ≥ 1, is equal to 2/3 [(2a)3/2 – 3a + 3 – 2√2]
Then f (x ) =
(a) 2 √2x, x ≥ 1
(b) √2x, x ≥ 1
(c) 2 √x,x ≥ 1
(d) None of these

A

Question. Let f (x ) = x2 , g(x) = cos x and h(x) = f (g(x )). Then area bounded by the curve y = h(x ), and X-axis between x = x1 and x = x2 , where x1, x2 are roots of the equation 18x2 – 9πx + π2 = 0 , is
(a) π/12
(b) π/6
(c) π/3
(d) None of these

A

Question. The area of the smaller region bounded by the circle x2 + y2 = 1 and the lines | y | = x + 1 is
(a) (π – 2)/4
(b) (π – 2)/2
(c) (π + 2)/2
(d) None of these

B

Question. A point P(x, y )moves in such a way that [| x | ] + [| y| ] =1, [.] = G.I.F. Area of the region representing all possible positions of the point P is equal to
(a) 8
(b) 4
(c) 16
(d) None of these

A

Question. The area bounded by the curve y = cos-1(sin x) – sin-1 (cos x) and the lines y = 0, x = 3π/2 , x = 2π is
(a) π2/4
(b) π2/2
(c) π2
(d) 2π2

A

Question. The area bounded by the curves y2 = 4x and x2 = 4y , is
(a) 0
(b) 32/3
(c) 16/3
(d) 8/3

C

Question. Let f (x ) be continuous function such that f (0) =1, f (x ) – f (x /7) = x /7 ∀ x ∈ R. The area bounded by the curve y = f (x ) and the coordinate axes is
(a) 2
(b) 3
(c) 6
(d) 9

B

Question. The area bounded by the curve y = (x + 1)2, y = (x -1)2 and the line y = 1/4 is
(a) 1/6 sq unit
(b) 2/3 sq unit
(c) 1/4 sq unit
(d) 1/3 sq unit

D

Question. Let An be the area bounded by the curve y = (tan x)n and the lines x = 0 and x = p / 4. Then, for n > 2

C

Question. If area bounded by the curves y = x – bx2 and by = x2 is maximum, then b is equal to
(a) 1
(b) – 1
(c) b = ± 1
(d) None of these