Application of Integrals Test

Result

Application of Integrals Test - 63

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Weekly Quiz Competition

Question 1
1 / -0

The area bounded by the curves $$y=xe^{x},y=xe^{-x}$$ and the line $$x=1$$, is

A
$$\dfrac {2}{e}$$

B
$$1-\dfrac {2}{e}$$

C
$$\dfrac {1}{e}$$

D
$$1-\dfrac {1}{e}$$

Solution
Question 2
1 / -0

The ratio of the areas of two regions of the curve $$C_1 : 4x^2 + \pi^2y^2 = 4\pi^2$$ divided by the curve $$C_2 : y = -sgn \left(x - \dfrac{\pi}{2}\right) \cos x$$ (where sgn(x) denotes signum function) is -

A
$$\dfrac{\pi^2 +4}{\pi^2-2\sqrt{2}}$$

B
$$\dfrac{\pi^2-2}{\pi^2+2}$$

C
$$\dfrac{\pi^2+6}{\pi^2+3\sqrt{3}}$$

D
$$\dfrac{\pi^2+1}{\pi^2-\sqrt{2}}$$

Solution
Question 3
1 / -0

If $$z$$ is not purely real then area bounded by curves $$lm\left(z+\dfrac{1}{z}\right) = 0$$ and $$|z-1| = 2$$ is (in square units)-

A
$$4\pi$$

B
$$3\pi$$

C
$$2\pi$$

D
$$\pi$$

Solution
Question 4
1 / -0

Area bounded by the curves y= $$[\frac{x^{2}}{64}+2],y=x-1$$ and x=0 above x-axis is where $$[.]$$ denotes greatest $$x$$.

A
2 sq. unit

B
3 sq. unit

C
4 sq. unit

D
none of these

Solution
Question 5
1 / -0

The graphs of $$f(x)=x^{2}$$ and $$g(x)=cx^{3}(c>0)$$ intersect at the points $$(0, 0)$$ and $$(\dfrac{1}{c}, \dfrac{1}{c^{2}})$$. If the region which lies between these graphs and over the interval $$[0, \dfrac{1}{c}]$$ has the area equal to $$(\dfrac{2}{3})sq.\ units$$, then the value of $$c$$ is :

A
$$\dfrac{1}{3}$$

B
$$\dfrac{1}{2}$$

C
$$1$$

D
$$2$$

Solution
Question 6
1 / -0

Let function $$f_n$$ be the number of way in which a positive integer n can be written as an ordered sum of several positive integers. For example, for $$n=3$$, $${f_3} = 3,since, 3 = 3, 3 = 2 + 1$$ and $$ 3 = 1+1+1$$. Then$${f_5} =$$

A
$$4$$

B
$$5$$

C
$$6$$

D
$$7$$
Question 7
1 / -0

The area (in square units) bounded by the curve $$y=\sqrt{x},2y=x+3=0$$, x-axis, and lying in the first quadrant is

A
$$9$$

B
$$36$$

C
$$18$$

D
none

Solution

Finding intersection of $$y=\sqrt{x}$$ and $$2y=x+3$$,
$$2\sqrt{x}=x+3$$
$$\Rightarrow 4x=x^2+9+6x$$
$$\Rightarrow x^2+2x+9$$
$$D=4-36 < 0$$
$$\Rightarrow$$ No intersection
Area bounded by these curves is not bounded, hence it years the infinite.
Question 8
1 / -0

The area bounded by circles $$x^2+y^2=r^2$$, $$r=1, 2$$ and rays given by $$2x^2-3xy-2y^2=0$$($$y > 0$$) is?

A
$$\pi$$

B
$$\dfrac{3\pi}{4}$$

C
$$\dfrac{\pi}{2}$$

D
$$\dfrac{\pi}{4}$$

Solution
Question 9
1 / -0

In a system of three curves $$C_{1}, C_{2}$$ and $$C_{3}, C_{1}$$ is a circle whose equation is $$x^{2}+y^{2}=4$$. $$C_{2}$$ is the locus of orthogonal tangents drawn on $$C_{1}. C_{3}$$ is the intersection of perpendicular tangents drawn on $$C_{2}$$. Area enclosed between the curve $$C_{2}$$ and $$C_{3}$$ is-

A
$$8\pi\ sq.\ units$$

B
$$16\pi\ sq.\ units$$

C
$$32\pi\ sq.\ units$$

D
$$None\ of\ these$$

Solution
Question 10
1 / -0

The area bounded by the curve $$y = f(x)$$ the x-axis & the ordinates x=1 & x=b is $$\left( {b - 1} \right)\,\sin \left( {3b + 4} \right).\,then\,f\left( x \right)is:$$

A
$$\left( {x - 1} \right)\cos \left( {3x + 4} \right)$$

B
$$\sin (3x + 4)$$

C
$$\sin (3x + 4) + 3\left( {x - 1} \right).cos\left( {3x + 4} \right)$$

D
none

Solution