Question Set A

CBSE 12th Standard Maths Applications of the Integrals HOT Questions Study Material - QB365

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Questions:

Multiple Choice Question

46 x 1 = 46

The area bounded by the curve \(y=\sqrt{x}\), Y-axis and between the lines y = 0 and y = 3 is

(a)

2\(\sqrt{3}\)

(b)

(c)

(d)

The area of the region bounded by the curve y²= 4x and x = 1 is

(a)

\(\frac{4}{3}\)

(b)

\(\frac{8}{3}\)

(c)

\(\frac{64}{3}\)

(d)

\(\frac{32}{3}\)

Area of the region bounded by the curve y² = 4x and the X-axis between x = 0 and x = 1 is

(a)

\(\frac{2}{3}\)

(b)

\(\frac{8}{3}\)

(c)

(d)

\(\frac{4}{3}\)

Area of the region bounded by the curve y = cos x between x = 0 and x = 2\(\pi\) is

(a)

(b)

(c)

(d)

The area of the region bounded by the parabolas y = x² and y² = x is

(a)

\(\frac{1}{3}\)

(b)

\(\frac{2}{3}\)

(c)

\(\frac{1}{6}\)

(d)

\(\frac{1}{4}\)

Area of the region bounded by the curve y² = 4x, y-axis and the line y = 3 is

(a)

(b)

\(\frac{9}{4}\)

(c)

\(\frac{9}{3}\)

(d)

\(\frac{9}{2}\)

Area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x² + y² = 32 is

(a)

16 \(\pi\)

(b)

4 \(\pi\)

(c)

32 \(\pi\)

(d)

none of these

The area enclosed by the circle x² + y² = 16 is

(a)

(b)

20 \(\pi\)

(c)

16 \(\pi\)

(d)

256 \(\pi\)

The area of the smaller region between the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \text { and the line } \frac{x}{a}+\frac{y}{b}=1\) in first quadrant is

(a)

\(\frac{1}{2} a b\)

(b)

\(\frac{1}{2} \pi a b\)

(c)

\(\pi a b\)

(d)

\(\frac{a b}{4}(\pi-2)\)

10.

The area bounded by the curve x² = 4y and the straight line x = 4y - 2 is

(a)

\(\frac{3}{8}\)

(b)

\(\frac{5}{8}\)

(c)

\(\frac{7}{8}\)

(d)

\(\frac{9}{8}\)

11.

The area enclosed within the curve IxI + IyI = 1 is

(a)

(b)

1.5

(c)

(d)

none of these

12.

If the area bounded by the curves y² = 4ax and y = mx is \(\frac{a^{2}}{3}\) then the value of m is

(a)

(b)

-2

(c)

\(\frac{1}{2}\)

(d)

none of these

13.

The area bounded by the y -axis. y cos x and y = sin x, \(0 \leq x \leq \frac{\pi}{2} \text { is }\)

(a)

\(\sqrt{2}\)

(b)

\(\sqrt{2} +1\)

(c)

\(\sqrt{2}-1\)

(d)

\(2\sqrt{2}-1\)

14.

The area bounded between the curves y² = 6x and x² = 6 is

(a)

6 sq units

(b)

12 sq units

(c)

36 sq units

(d)

24 sq units

15.

The area enclosed by the circle x² + y² = 8 is

(a)

\(16 \pi\) sq units

(b)

\(2 \sqrt{2} \pi\) sq units

(c)

\(8 \pi^{2}\) sq units

(d)

\(8 \pi\) sq units

16.

If a curve \(y=a \sqrt{x}+b x\) 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

17.

Let the straight line x = b divide the area enclosed by \(y=(1-x)^{2}, y=0 \text { and } x=0\) into two parts \(R_{1}(0 \leq x \leq b) \text { and } R_{2}(b \leq x \leq 1)\) such that \(R_{1}-R_{2}=\frac{1}{4}\). Then, b equals

(a)

\(\frac{3}{4}\)

(b)

\(\frac{1}{2}\)

(c)

\(\frac{1}{3}\)

(d)

\(\frac{1}{4}\)

18.

The area bounded by the curve \(y=x|x|\) X-axis and the coordinates x = -1 and x = 1 is given by

(a)

0 sq units

(b)

\(\frac{1}{3} \text { sq units }\)

(c)

\(\frac{2}{3} \text { sq units }\)

(d)

\(\frac{4}{3} \text { sq units }\)

19.

Area of the region bounded by the curve \(y=|x+1|+1, x=-3, x=3 \text { and } y=0\) is

(a)

8 sq units

(b)

16 sq units

(c)

32 sq units

(d)

None of these

20.

The line \(x=\frac{\pi}{4}\) divides the area of the region bounded by y = sinx, y = cosx and x-axis \(\left(0 \leq x \leq \frac{\pi}{2}\right)\) into two regions of areas A₁ and A₂. Then, A₁ : A₂ is equal to

(a)

4:1

(b)

3:1

(c)

2:1

(d)

1:1

21.

The area enclosed between the curves \(y=a x^{2}\).and \(x=a y^{2},(a>0)\) is 1 sq unit. Then the value of a is

(a)

\(\frac{1}{\sqrt{3}}\)

(b)

\(\frac{1}{2}\)

(c)

(d)

\(\frac{1}{3}\)

22.

The area bounded by the lines y = 4x + 5,y = 5 - x and 4y = x+-5 is

(a)

\(\frac{15}{2} \text { sq units }\)

(b)

\(\frac{9}{2} \text { sq units }\)

(c)

\(\frac{13}{2} \text { sq units }\)

(d)

None of these

23.

The area of the region bounded by the curve y = sinx between the ordinates x = 0 \(x=\frac{\pi}{2}\) and the X-axis is.

(a)

2 sq units

(b)

4 sq units

(c)

3 sq units

(d)

1 sq unit

24.

The area of the region bounded by parabola \(y^{2}=x\) and the straight line 2y = x is

(a)

\(\frac{4}{3} \text { sq units }\)

(b)

1 sq unit

(c)

\(\frac{2}{3} \mathrm{squnit}\)

(d)

\(\frac{1}{3} \text { sq unit }\)

25.

The area of the region bounded by the curve \(y=\sin x\) between \(0 \text { and } 2 \pi\)

(a)

2 sq. units

(b)

4 sq. units

(c)

3 sq. units

(d)

1 sq. unit

26.

The area of the region bounded by the curve x = 2y + 3 and the lines y = 1, y =-1 is

(a)

4 sq units

(b)

\(\frac{3}{2} \text { sq units }\)

(c)

6 sq units

(d)

8 sq units

27.

The area of the region bounded by the curve y = x + 1and the lines x = 2, x = 3, is

(a)

\(\frac{7}{2} \mathrm{sq} \text { units }\)

(b)

\(2(\sqrt{2}+1)\)

(c)

\(\frac{11}{2} \text { sq units }\)

(d)

\(\frac{13}{2} \text { squnit }\)

28.

The area bounded by the curve: y = cos²x between x = 0, x = ㅠ and x - axis

(a)

2π sq. units

(b)

π² sq. units

(c)

π/2 sq. units

(d)

π sq. units

29.

Write the shaded region as an integral

(a)

\(-\left| \int _{ a }^{ b }{ f(x)dx } \right| \)

(b)

\(\int _{ a }^{ b }{ \left| f(x)dx \right| } \)

(c)

\(\left| \int _{ a }^{ b }{ f(x)dx } \right| \)

(d)

\(-\int _{ a }^{ b }{ \left| f(x)dx \right| } \)

30.

If the area above x-axis, bounded by the curves y = 2^kx, x = 0 and x = 2 is \(\frac { 3 }{ { log }_{ e }2 } \) then k = ?

(a)

k = 0

(b)

k = 1

(c)

k = 2

(d)

k = -1

31.

Area of the region \(\left\{ (x,y):{ x }^{ 2 }\le y\le |x| \right\} \) is:

(a)

1/3 sq. units

(b)

1/5 sq. units

(c)

1/2 sq. units

(d)

1/4 sq. units

32.

The area of the region bounded between the line x=9 and the parabola y²=16x is

(a)

144 sq units

(b)

27 sq units

(c)

104 sq units

(d)

54 sq units

33.

Area of the region bounded by the curve y² = 2y – x and y-axis is:

(a)

4/3 sq. units

(b)

3/4 sq. units

(c)

4 sq. units

(d)

3 sq. units

34.

Area of the shaded region in the given figure is:

(a)

144/3 sq. units

(b)

142/3 sq. units

(c)

145/3 sq. units

(d)

143/2 sq. units

35.

Area under the circle x² + y² = 16 is

(a)

π sq units

(b)

13π sq units

(c)

16 π sq units

(d)

4 π sq units

36.

If the area of y = f(x) between x = a and x = b is \(\int _{ a }^{ c }{ f(x)dx } +\int _{ c }^{ a }{ f(x)dx } \) then the point c is the point of intersection of the curve with:

(a)

Line x = a

(b)

Y – axis

(c)

X – axis

(d)

Line x = b

37.

The area enclosed between the lines x = 2 and x = 7 is

(a)

Infinite

(b)

7 units

(c)

5 units

(d)

2 units

38.

The area bounded by the y-axis, y = cos x and y = sin x when 0 ≤ x ≤ \(\frac{\pi}{3}\) is

(a)

\(2(\sqrt { 2-1 } )\)

(b)

\(\sqrt { 2-1 } \)

(c)

\(\sqrt { 2-1 } \)

(d)

\(\sqrt { 2 } \)

39.

The area of the circle x² + y² = 16 exterior to the parabola y² = 6x is

(a)

\(\frac { 4 }{ 3 } (4\pi -\sqrt { 3 } )\)

(b)

\(\frac { 4 }{ 3 } (4\pi +\sqrt { 3 } )\)

(c)

\(\frac { 4 }{ 3 } (8\pi -\sqrt { 3 } )\)

(d)

\(\frac { 4 }{ 3 } (8\pi +\sqrt { 3 } )\)

40.

The area bounded by the curve y = x |x| , x-axis and the ordinates x = – 1 and x = 1 is given by

(a)

(b)

\(\frac13\)

(c)

\(\frac23\)

(d)

\(\frac43\)

41.

Area bounded by the curve y = x³, the x-axis and the ordinates x = – 2 and x = 1 is

(a)

-9

(b)

\(\frac{-15}{4}\)

(c)

\(\frac{15}{4}\)

(d)

\(\frac{17}{4}\)

42.

Area lying between the curves y² = 4x and y = 2x is

(a)

\(\frac23\)

(b)

\(\frac13\)

(c)

\(\frac14\)

(d)

\(\frac34\)

43.

Smaller area enclosed by the circle x² + y² = 4 and the line x + y = 2 is

(a)

2 (ㅠ – 2)

(b)

ㅠ - 2

(c)

2ㅠ - 1

(d)

2(ㅠ+2)

44.

Area lying in the first quadrant and bounded by the circle x² + y² = 4 and the lines x = 0 and x = 2 is

(a)

(b)

\(\frac { \pi }{ 2 } \)

(c)

\(\frac { \pi }{ 3 } \)

(d)

\(\frac { \pi }{ 3 } \)

45.

Area of the region bounded by the curve y = \(\sqrt { 49-{ x }^{ 2 } } \) and the x-axis is

(a)

\(\frac { 49 }{ 2 } \pi \) sq units

(b)

98π sq units

(c)

49π sq units

(d)

240π sq units

46.

Area bounded by the curve y = sin x and the x-axis between x = 0 and x = 2π is

(a)

2 sq units

(b)

0 sq units

(c)

3 sq units

(d)

4 sq units

Assertion and reason

7 x 1 = 7

47.

Assertion (A) The area bounded by the line y = x and the curve y = x³ is \(\frac{1}{2}\)sq units.
Reason (R) \(\int_{0}^{1}\)(x - x³) dx = \(\frac{1}{4}\).
(a) Both A and R are correct; R is the correct explanation of A
(b) Both A and R are correct; R is not the correct explanation of A
(c) A is correct; R is incorrect
(d) R is correct; A is incorrect

48.

Assertion (A) The area bounded by the curve y = sin x between x =0 and x = 4\(\pi\) is 4 sq units.
Reason (R) \(\int_0^{{\pi}/{2}}sin x \space dx = 1\)
(a) Both A and R are correct; R is the correct explanation of A
(b) Both A and R are correct; R is not the correct explanation of A
(c) A is correct; R is incorrect
(d) R is correct; A is incorrect

49.

Assertion (A) The area bounded by y² = 4x and y = x is \(\frac{8}{3}\) sq units.
Reason (R) The area bounded by y² = 4ax and y = mx is \(\frac{8a^2}{3m^3}\)sq units.
(a) Both A and R are correct; R is the correct explanation of A
(b) Both A and R are correct; R is not the correct explanation of A
(c) A is correct; R is incorrect
(d) R is correct; A is incorrect

50.

Assertion: The area bounded by the curve y = cos x in I quadrant with the coordinate axes is 1 sq.unit.
Reason: \(\int_{0}^{\pi/2}\)cos xdx = 1
(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
(c) Assertion is correct, Reason is incorrect
(d) Assertion is incorrect, Reason is correct.

51.

Assertion: The area bounded by the circle y = sin x and y = -sin x from 0 to \(\pi\) is 3 sq.unit.
Reason: The area bounded by the curves is symmetric about x-axis.
(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
(c) Assertion is correct, Reason is incorrect
(d) Assertion is incorrect, Reason is correct.

52.

Assertion: The area bounded by the circle x² + y² = a² in the first quadrant is given by \( \int_{0}^{a}\sqrt{a^{2}-x^{2}}dx\)
Reason: The same area can also be found by \( \int_{0}^{a}\sqrt{a^{2}-y^{2}}dy\)
(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
(c) Assertion is correct, Reason is incorrect
(d) Assertion is incorrect, Reason is correct.

53.

Assertion: The area bounded by the curves y²= 4a²(x - 1) and lines x = 1 and y = 4a is ,\(\frac{16a}{3}\) sq.units.
Reason: The area enclosed between the parabola y²= x² - x + 2 and the line y = x +2 is \(\frac{8}{3}\) sq. units.
(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.
(b) Assertion is correct, Reason is correct; Reason is not a correct explanation for Assertion
(c) Assertion is correct, Reason is incorrect
(d) Assertion is incorrect, Reason is correct.

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Answers Key:

Multiple Choice Question

46 x 1 = 46

(b)
\(\frac{8}{3}\)

(b)
\(\frac{8}{3}\)

(a)
4

(a)
\(\frac{1}{3}\)

(b)
\(\frac{9}{4}\)

(b)
4 \(\pi\)

(d)
\(\frac{a b}{4}(\pi-2)\)

10.

(d)
\(\frac{9}{8}\)

11.

12.

(a)
2

13.

(d)
\(2\sqrt{2}-1\)

14.

(b)
12 sq units

15.

(d)
\(8 \pi\) sq units

16.

(a)
a = 3, b = -1

17.

(b)
\(\frac{1}{2}\)

18.

19.

(b)
16 sq units

20.

(d)
1:1

21.

(a)
\(\frac{1}{\sqrt{3}}\)

22.

(a)
\(\frac{15}{2} \text { sq units }\)

23.

(d)
1 sq unit

24.

(a)
\(\frac{4}{3} \text { sq units }\)

25.

(b)
4 sq. units

26.

27.

(a)
\(\frac{7}{2} \mathrm{sq} \text { units }\)

28.

29.

30.

(b)
k = 1

31.

(a)
1/3 sq. units

32.

(a)
144 sq units

33.

(a)
4/3 sq. units

34.

(b)
142/3 sq. units

35.

36.

37.

(a)
Infinite

38.

(b)
\(\sqrt { 2-1 } \)

39.

40.

41.

(d)
\(\frac{17}{4}\)

42.

(b)
\(\frac13\)

43.

(b)
ㅠ - 2

44.

(a)
π

45.

As area is above the x-axis
∴ area = \(2\int _{ 0 }^{ 7 }{ \sqrt { 49-{ x }^{ 2 } } } \)
= \({ \left[ \frac { x }{ 2 } \sqrt { 49-{ x }^{ 2 } } +\frac { 49 }{ 2 } { sin }^{ -1 }\frac { x }{ 7 } \right] }_{ 0 }^{ 7 }\)
= \(2\left[ \left( \frac { 7 }{ 2 } \times 0+\frac { 49 }{ 2 } { sin }^{ -1 }1 \right) -(0) \right] \)
= \(\frac { 49 }{ 2 } \pi \) sq units

46.

As sin x is positive in 1^st and 2^nd quadrant and negative is 3^rd and 4^th quadrant.
Area = \(\int _{ 0 }^{ 2\pi }{ |sinx|dx } \)
\(=\int _{ 0 }^{ \pi }{ sinxdx } +\int _{ \pi }^{ 2\pi }{ (-sinx)dx } \)
= 4 sq units

Assertion and reason

7 x 1 = 7

47.

(a) Both A and R are correct; R is the correct explanation of A

48.

(d) R is correct; A is incorrect

49.

(a) Both A and R are correct; R is the correct explanation of A

50.

(a) Assertion is correct, Reason is correct; Reason is a correct explanation for assertion.

51.

(d) Assertion is incorrect, Reason is correct.