Limits and Derivatives Test

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Limits and Derivatives Test - 33

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Weekly Quiz Competition

Question 1
1 / -0

For $$x>y$$, $$\displaystyle\lim_{x\rightarrow 0}{\left[\left(\sin{x}\right)^{1/x}+\left(\cfrac{1}{x}\right)^{\sin{x}}\right]}$$ is :

A
0

B
-1

C
1

D
2
Question 2
1 / -0

If $$f : R \rightarrow (0, \infty)$$ is an increasing function and if $$\displaystyle\lim_{x \rightarrow 2018} \dfrac{f(3x)}{f(x)} = 1$$, then $$\displaystyle\lim_{x \rightarrow 2018} \dfrac{f(2x)}{f(x)}$$ is equal to

A
$$\dfrac{2}{3}$$

B
$$\dfrac{3}{2}$$

C
$$2$$

D
$$3$$

E
$$1$$

Solution

Given $$f : R \rightarrow (0, \infty)$$ is an increasing function.
And $$\underset{x \rightarrow 2018}{\lim} \dfrac{f(3x)}{f(x)} = 1$$
So, $$\underset{x \rightarrow 2018}{\lim} f(3x) = \underset{x \rightarrow 2018}{\lim} f(x)$$
$$\Rightarrow f(x) = $$ constant.
Therefore,$$ \underset{x \rightarrow 2018}{\lim} \dfrac{f(2x)}{f(x)} = 1$$
Question 3
1 / -0

If $$f$$ is differentiable at $$x = 1$$ and $$\underset{h \rightarrow 0}{\lim} \dfrac{1}{h} f (1 + h) = 5, f'(1) = $$

A
$$0$$

B
$$1$$

C
$$3$$

D
$$4$$

E
$$5$$

Solution

$$f'(1)=\underset{h\rightarrow 0}{lim} \dfrac{f(1+h)-f(1)}{h}$$; function is differentiable.

and $$\underset{h\rightarrow 0}{lim} \dfrac{f(1+h)}{h}=5$$ [Given function is continuous]
$$\Rightarrow f(1)=0$$

Hence, $$f'(1)=\underset{h\rightarrow 0}{lim} \dfrac{f(1+h)}{h}=5$$
Question 4
1 / -0

If $$y=\sqrt{\sin x+\sqrt{\sin x+\sqrt{\sin x+}}}.....\infty$$ then $$\dfrac{dy}{dx}=?$$

A
$$\dfrac{\sin x}{(2y-1)}$$

B
$$\dfrac{\cos x}{(y-1)}$$

C
$$\dfrac{\cos x}{(2y-1)}$$

D
$$none\ of\ these$$

Solution
Question 5
1 / -0

If $$y=(\tan x)^{\cot x}$$ then $$\dfrac{dy}{dx}=?$$

A
$$\cot x.(\tan x)^{\cot x-1}.\sec^2x$$

B
$$-(\tan x)^{\cot x}.\csc^2x$$

C
$$(\tan x)^{\cot x}.\csc^2 x(1-\log \tan x)$$

D
$$none\ of\ these$$

Solution
Question 6
1 / -0

If $$x=a(\cos\theta+\theta\sin\theta)$$ and $$y=a(\sin\theta-\theta\cos\theta)$$ then $$\dfrac{dy}{dx}=?$$

A
$$\cot\theta$$

B
$$\tan\theta$$

C
$$a\cot\theta$$

D
$$a\tan\theta$$

Solution
Question 7
1 / -0

If $$x=a\sec\theta, y=b\tan\theta$$ then $$\dfrac{dy}{dx}=$$

A
$$\dfrac{b}{a}\sec\theta$$

B
$$\dfrac{b}{a} \ cosec\theta$$

C
$$\dfrac{b}{a}\cot\theta$$

D
$$none\ of\ these$$

Solution
Question 8
1 / -0

If $$y=\sqrt{x\sin x}$$ then $$\dfrac{dy}{dx}=?$$

A
$$\dfrac{(x\cos x+\sin x)}{2\sqrt{x\sin x}}$$

B
$$\dfrac{1}{2}(x\cos x+\sin x).\sqrt{x\sin x}$$

C
$$\dfrac{1}{2\sqrt{x\sin x}}$$

D
$$none\ of\ these$$

Solution
Question 9
1 / -0

If $$y=\sqrt{\dfrac{1+\sin x}{1-\sin x}}$$ then $$\dfrac{dy}{dx}=?$$

A
$$\dfrac{1}{2}\sec^2\left(\dfrac{\pi}{4}-\dfrac{x}{2}\right)$$

B
$$\dfrac{1}{2}\csc^2\left(\dfrac{\pi}{4}-\dfrac{x}{2}\right)$$

C
$$\dfrac{1}{2}\csc \left(\dfrac{\pi}{4}-\dfrac{x}{2}\right)\cot \left(\dfrac{\pi}{4}-\dfrac{x}{2}\right)$$

D
$$none\ of\ these$$

Solution

$$y=\left\{\dfrac{1+\cos \left(\dfrac{\pi}{2}-x\right)}{1-\cos \left(\dfrac{\pi}{2}-x\right)}\right\}^{\dfrac{1}{2}}=\left\{\dfrac{2\cos^2\left(\dfrac{\pi}{4}-\dfrac{x}{2}\right)}{2\sin^2\left(\dfrac{\pi}{4}-\dfrac{x}{2}\right)}\right\}^{\dfrac{1}{2}}=\cot\left(\dfrac{\pi}{4}-\dfrac{x}{2}\right)$$.
$$\Rightarrow \dfrac{dy}{dx}=-\csc^2\left({\dfrac{\pi}{4}-\dfrac{x}{2}}\right)\times\dfrac{dy}{dx}\left({\dfrac{\pi}{4}-\dfrac{x}{2}}\right)$$
$$\therefore \dfrac{dy}{dx}=\dfrac12\csc^2\left({\dfrac{\pi}{4}-\dfrac{x}{2}}\right)$$
Question 10
1 / -0

The value of $$ \displaystyle \lim _{x \rightarrow \pi} \dfrac{1+\cos ^{3} x}{\sin ^{2} x} $$ is

A
1/3

B
2/3

C
-1/4

D
3/2

Solution

We have $$ \displaystyle \lim _{x \rightarrow \pi} \dfrac{1+\cos ^{3} x}{\sin ^{2} x} $$
$$=\displaystyle \lim _{x \rightarrow \pi} \dfrac{(1+\cos x)\left(1-\cos x+\cos ^{2} x\right)}{(1-\cos x)(1+\cos x)}$$
$$=\displaystyle \lim _{x \rightarrow \pi} \dfrac{1-\cos x+\cos ^{2} x}{1-\cos x}=\dfrac{1+1+1}{1+1}=\dfrac{3}{2}$$

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Re-Attempt Weekly Quiz Competition

Limits and Derivatives Test - 33

Result

Limits and Derivatives Test - 33

Score

-

Rank

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SHARING IS CARING

Question 1

0

-1

1

2

Question 2

$$\dfrac{2}{3}$$

$$\dfrac{3}{2}$$

$$2$$

$$3$$

$$1$$

Solution

Question 3

$$0$$

$$1$$

$$3$$

$$4$$

$$5$$

Solution

Question 4

$$\dfrac{\sin x}{(2y-1)}$$

$$\dfrac{\cos x}{(y-1)}$$

$$\dfrac{\cos x}{(2y-1)}$$

$$none\ of\ these$$

Solution

Question 5

$$\cot x.(\tan x)^{\cot x-1}.\sec^2x$$

$$-(\tan x)^{\cot x}.\csc^2x$$

$$(\tan x)^{\cot x}.\csc^2 x(1-\log \tan x)$$

$$none\ of\ these$$

Solution

Question 6

$$\cot\theta$$

$$\tan\theta$$

$$a\cot\theta$$

$$a\tan\theta$$

Solution

Question 7

$$\dfrac{b}{a}\sec\theta$$

$$\dfrac{b}{a} \ cosec\theta$$

$$\dfrac{b}{a}\cot\theta$$

$$none\ of\ these$$

Solution

Question 8

$$\dfrac{(x\cos x+\sin x)}{2\sqrt{x\sin x}}$$

$$\dfrac{1}{2}(x\cos x+\sin x).\sqrt{x\sin x}$$

$$\dfrac{1}{2\sqrt{x\sin x}}$$

$$none\ of\ these$$

Solution

Question 9

$$\dfrac{1}{2}\sec^2\left(\dfrac{\pi}{4}-\dfrac{x}{2}\right)$$

$$\dfrac{1}{2}\csc^2\left(\dfrac{\pi}{4}-\dfrac{x}{2}\right)$$

$$\dfrac{1}{2}\csc \left(\dfrac{\pi}{4}-\dfrac{x}{2}\right)\cot \left(\dfrac{\pi}{4}-\dfrac{x}{2}\right)$$

$$none\ of\ these$$

Solution

Question 10

1/3

2/3

-1/4

3/2

Solution

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