CBSE Class 12 Mathematics Application of Integrals MCQs Set 03

Question. The area enclosed by the circle \( x^2 + y^2 = 2 \) is equal to
(a) \( 4\pi \) sq units
(b) \( 2\sqrt{2}\pi \) sq units
(c) \( 4\pi^2 \) sq units
(d) \( 2\pi \) sq units
Answer: (d) \( 2\pi \) sq units

Question. The area enclosed by the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) is equal to
(a) \( \pi^2 ab \) sq units
(b) \( \pi ab \) sq units
(c) \( \pi a^2 b \) sq units
(d) \( \pi ab^2 \) sq units
Answer: (b) \( \pi ab \) sq units

Question. The area of the region bounded by the \( y \)-axis, \( y = \cos x \) and \( y = \sin x \), \( 0 \leq x \leq \frac{\pi}{2} \) is 
(a) \( \sqrt{2} \) sq units
(b) \( (\sqrt{2} + 1) \) sq units
(c) \( (\sqrt{2} - 1) \) sq unit
(d) \( (2\sqrt{2} - 1) \) sq units
Answer: (c) \( (\sqrt{2} - 1) \) sq unit

Question. The area of the region bounded by the curve \( x^2 = 4y \) and the straight line \( x = 4y - 2 \) is
(a) \( \frac{3}{8} \) sq unit
(b) \( \frac{5}{8} \) sq unit
(c) \( \frac{7}{8} \) sq unit
(d) \( \frac{9}{8} \) sq unit
Answer: (d) \( \frac{9}{8} \) sq unit

Question. The area of the region bounded by the curve \( y = \sqrt{16 - x^2} \) and \( x \)-axis is 
(a) \( 8 \pi \) sq units
(b) \( 20\pi \) sq units
(c) \( 16\pi \) sq units
(d) \( 256\pi \) sq units
Answer: (a) \( 8 \pi \) sq units

Question. Area of the region in the first quadrant enclosed by the \( x \)-axis, the line \( y = x \) and the circle \( x^2 + y^2 = 32 \) is 
(a) \( 16\pi \) sq units
(b) \( 4\pi \) sq units
(c) \( 32\pi \) sq units
(d) 24 sq units
Answer: (b) \( 4\pi \) sq units

Question. Area of the region bounded by the curve \( y = \cos x \) between \( x = 0 \) and \( x = \pi \) is 
(a) 2 sq units
(b) 4 sq units
(c) 3 sq units
(d) 1 sq unit
Answer: (a) 2 sq units

Question. The area of the region bounded by the curve \( x = 2y + 3 \) and the lines \( y = 1 \) and \( y = -1 \) is 
(a) 4 sq units
(b) \( \frac{3}{2} \) sq unit
(c) 6 sq units
(d) 8 sq units
Answer: (c) 6 sq units

Question. The area of the region bounded by the curve \( y = x^2 \) and the line \( y = 16 \) is
(a) \( \frac{37}{3} \) sq units
(b) \( \frac{256}{3} \) sq units
(c) \( \frac{64}{3} \) sq units
(d) \( \frac{128}{3} \) sq units
Answer: (b) \( \frac{256}{3} \) sq units

Question. The area of the region bounded by the curve \( y^2 = 9x \), \( y = 3x \) is
(a) 1 sq unit
(b) \( \frac{1}{2} \) sq unit
(c) 4 sq units
(d) 14 sq units
Answer: (b) \( \frac{1}{2} \) sq unit

Question. The area of the curve \( y = \sin x \) between 0 and \( \pi \) is
(a) 2 sq units
(b) 4 sq units
(c) 12 sq units
(d) 14 sq units
Answer: (a) 2 sq units

Question. The area of the region bounded by the curve \( ay^2 = x^3 \), the \( y \)-axis and the lines \( y = a \) and \( y = 2a \) is
(a) 3 sq units
(b) \( \frac{3}{5} a^2 |2 \times 2^{2/3} - 1| \) sq units
(c) \( \frac{3}{5} a |2^{3/2} - 1| \) sq units
(d) 1 sq unit
Answer: (b) \( \frac{3}{5} a^2 |2 \times 2^{2/3} - 1| \) sq units

Question. The area enclosed by the curve \( x = 3 \cos t, y = 2 \sin t \) is
(a) \( 4\pi \) sq units
(b) \( 6\pi \) sq units
(c) \( 14\pi \) sq units
(d) \( 7\pi \) sq units
Answer: (b) \( 6\pi \) sq units

Question. The area of the region bounded by the curves \( x = at^2 \) and \( y = 2at \) between the ordinate corresponding to \( t = 1 \) and \( t = 2 \) is
(a) \( \frac{56}{3} a^2 \) sq units
(b) \( \frac{40}{3} a^2 \) sq units
(c) \( 5\pi \) sq units
(d) None of the options
Answer: (a) \( \frac{56}{3} a^2 \) sq units

Question. The area of a minor segment of the circle \( x^2 + y^2 = a^2 \) cut off by the line \( x = \frac{a}{2} \) is
(a) \( \frac{a^2}{12}(4\pi - 3\sqrt{3}) \) sq units
(b) \( \frac{a^2}{4}(4\pi - 3) \) sq units
(c) \( \frac{a^2}{12}(3\pi - 4) \) sq units
(d) None of the options
Answer: (a) \( \frac{a^2}{12}(4\pi - 3\sqrt{3}) \) sq units

Question. The area of the region bounded by the curve \( y = x^3 \) and \( y = x + 6 \) and \( x = 0 \) is
(a) 7 sq units
(b) 6 sq units
(c) 10 sq units
(d) 14 sq units
Answer: (c) 10 sq units

Question. The area under the curve \( y = 2\sqrt{x} \) included between the lines \( x = 0 \) and \( x = 1 \) is
(a) 4 sq units
(b) 3 sq units
(c) \( \frac{4}{3} \) sq units
(d) None of the options
Answer: (c) \( \frac{4}{3} \) sq units

Question. The area under the curve \( y = \sqrt{a^2 - x^2} \) included between the lines \( x = 0 \) and \( x = a \) is
(a) \( \frac{\pi a^2}{4} \) sq units
(b) \( \frac{a^2}{4} \) sq units
(c) \( \pi a^2 \) sq units
(d) \( 4\pi \) sq units
Answer: (a) \( \frac{\pi a^2}{4} \) sq units

Question. The area of the region bounded by the triangle whose vertices are (-1, 1), (0, 5) and (3, 2) is
(a) \( \frac{15}{2} \) sq units
(b) 15 sq units
(c) 4 sq units
(d) 10 sq units
Answer: (a) \( \frac{15}{2} \) sq units

Question. The area of the region bounded by the line \( y - 1 = x \), the \( x \)-axis and the ordinates \( x = -2 \) and \( x = 3 \) is
(a) \( \frac{4}{3} \) sq units
(b) \( \frac{7}{2} \) sq units
(c) \( \frac{17}{2} \) sq units
(d) \( \frac{16}{3} \) sq units
Answer: (c) \( \frac{17}{2} \) sq units

Assertion-Reason Questions 

The following questions consist of two statements—Assertion(A) and Reason(R). Answer these questions selecting the appropriate option given below:
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.

Question. Assertion (A) : The area of region bounded by the parabolas \( y^2 = 4x \) and \( x^2 = 4y \) is \( \frac{32}{3} \) sq. units.
Reason (R) : The area of region bounded by the parabolas \( y^2 = 4ax \) and \( x^2 = 4by \) is \( \frac{16}{3} ab \).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (d) A is false but R is true.

Question. Assertion (A) : The area of the curve \( y = \sin^2 x \) from 0 to \( \pi \) will be more than that of the curve \( y = \sin x \) from 0 to \( \pi \).
Reason (R) : \( x^2 > x \), if \( x > 1 \)
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (d) A is false but R is true.

Question. Assertion (A) : The area of the ellipse \( 2x^2 + 3y^2 = 6 \) will be more than the area of the circle \( x^2 + y^2 - 2x + 4y + 4 = 0 \).
Reason (R) : The length of the semi-major axis of ellipse \( 2x^2 + 3y^2 = 6 \) is more than the radius of the circle \( x^2 + y^2 - 2x + 4y + 4 = 0 \).
(a) Both A and R are true and R is the correct explanation for A.
(b) Both A and R are true and R is not the correct explanation for A.
(c) A is true but R is false.
(d) A is false but R is true.
Answer: (b) Both A and R are true and R is not the correct explanation for A.