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Mathematics Weekly Test 11/10/2020

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The value of the integral $\int ^{1}_{0}\sqrt{\dfrac{1-x}{1+x}}dx$ is equal to

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Let the straight line x = b divides the area enclosed by y = (1 - x)2, y = 0, and x = 0 into two parts R1 (0 ≤ x ≤ b) and R2 (b ≤ x ≤ 1) such that R1 - R2 = 1/4. The the value of 'b' will be

3 / 15

Let f : [1/2, 1] → R (Set of all real numbers) be a positive, non-constant and differentiable function such that f'(x) < 2f(x) and f(1/2) = 1. Then the value of $\int ^{1}_{1/2}f\left( x\right) dx$ lies in the interval

4 / 15

Let f : [0, 2] → R be a function which is continuous on [0, 2] and is differentiable on (0, 2) with f(0) = 1. Let F(x) = $\int ^{x^{2}}_{0}f\left( \sqrt{t}\right) dt$ for x ∈ [0, 2]. If F'(x) = f'(x) for all x ∈ (0, 2), then F(2) equals.

5 / 15

The value of $\lim _{x\rightarrow 0}\dfrac{1}{x^{3}}\left( \int ^{x}_{0}\left( \dfrac{t\ln \left( 1+t\right) }{t^{4}+4}\right) \right) dt$ is

6 / 15

The value of $\int ^{0}_{-2}\left( x^{3}+3x^{2}+3x+3+\left( x+1\right) \cos \left( x+1\right) \right) dx$ is equal to

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Let f(x) = $\int ^{x}_{1}\sqrt{\left( 2-t^{2}\right) }dt$. Then the real roots of the equation x2 - f'(x) = 0 are

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If f(x) = $\int ^{x^{2}+1}_{x^{2}}e^{-t^{2}}dt$, then f(x) increases in

9 / 15

Let 'f' be a non-negative function defined on the interval [0, 1]. If $\int ^{x}_{0}\sqrt{1-\left( f'\left( t\right) \right) ^{2}}dt=\int ^{x}_{0}f\left( t\right) dt$, 0 ≤ x ≤ 1, and f(0)=0, then

10 / 15

The value of $\int ^{\pi /2}_{-\pi /2}\left( \dfrac{x^{2}\cos x}{1+e^{x}}\right) dx$ is eual to

11 / 15

The value of the integral $\int ^{\sqrt{ln3}}_{\sqrt{ln2}}\left( \dfrac{x\sin x^{2}}{\sin x^{2}+\sin \left( \ln 6-x^{2}\right) }\right) dx$ is

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Let T > 0 be a real number. Suppose 'f' is a continuous function such that for all x ∈ R (Set of all real numbers) f(x + T) = f(x). If L = $\int ^{T}_{0}f\left( x\right) dx$, then the value of $\int ^{3+3T}_{3}f\left( 2x\right) dx$ is

13 / 15

If $\int ^{1}_{\sin x}t^{2}\left( f\left( t\right) \right) dt=\left( 1-\sin x\right)$ ∀ x ∈ (0, π/2), then the value of f(1/√3) is

14 / 15

The value of the integral $\int ^{\pi /2}_{-\pi /2}\left( x^{2}+\ln \left( \dfrac{\pi +x}{\pi -x}\right) \right) \left( \cos x\right) dx$ is

15 / 15

If f(x) is a differentiable function and $\int ^{t^{2}}_{0}xf\left( x\right) dx=\dfrac{2t^{5}}{5}$, the the value of f(4/25) will be