Limits and Derivatives Test

Result

Limits and Derivatives Test - 52

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Weekly Quiz Competition

Question 1
1 / -0

$$\displaystyle \lim _{ x\rightarrow \dfrac { \pi }{ 2 } }{ \dfrac { \sin { x } }{ \cos ^{ -1 }{ \left[ \dfrac { 1 }{ 4 } \left( 3\sin { x } -\sin { 3x } \right) \right] } } } $$, where [.] denotes greatest integer function is :

A
$$\dfrac{2}{\pi}$$

B
$$1$$

C
$$\dfrac{4}{\pi}$$

D
$$does\ not\ exist$$

Solution
Question 2
1 / -0

$$Im _{ }{ \left( \dfrac { 1 }{ 1-\cos { \theta } +i\sin { \theta } } \right) } $$ is equal to

A
$$\dfrac{1}{2}\tan\dfrac{\theta}{2}$$

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

C
$$-\dfrac{1}{2}\tan\dfrac{\theta}{2}$$`

D
$$-\dfrac{1}{2}\cot\dfrac{\theta}{2}$$

Solution
Question 3
1 / -0

$$\lim _ { x \rightarrow 0 } \frac { \sqrt [ 3 ] { 1 + \sin x } - \sqrt [ 3 ] { 1 - \sin x } } { x } =$$

A
0

B
1

C
3$$/ 2$$

D
2$$/ 3$$

Solution
Question 4
1 / -0

$$\displaystyle\underset{x\rightarrow \dfrac{\pi}{4}}{Lt}\dfrac{\sqrt{2}-\cos x-\sin x}{(4x-\pi)^2}=?$$

A
$$\dfrac{1}{16\sqrt{2}}$$

B
$$\dfrac{1}{32\sqrt{2}}$$

C
$$\dfrac{1}{16}$$

D
$$\dfrac{1}{8}$$

Solution
Question 5
1 / -0

$$\underset { \theta \longrightarrow 0 }{ Lt } \dfrac { 3tan\theta -tan3\theta }{ { 2\theta }^{ 3 } } =$$

A
1/4

B
3/4

C
4

D
-4

Solution
Question 6
1 / -0

$$\lim _ { x \rightarrow 0 } \frac { \ln ( \sin 3 x ) } { \ln ( \sin x ) }$$ is equal to

A
0

B
1

C
2

D
none of these

Solution
Question 7
1 / -0

If $$y = \frac{{\sin x}}{{1 + \cos x}},$$ then $$\frac{{dy}}{{dx}}$$ is equal to

A
$$\frac{1}{{1 + \cos x}}$$

B
$$\frac{1}{{1 + \sin x}}$$

C
$$\frac{cos x}{{1 + \sin x}}$$

D
$$1 + \cos x$$

Solution
Question 8
1 / -0

$$\underset { x\rightarrow \infty }{ Lt } { 5 }^{ x }sin\left( \cfrac { a }{ { 5 }^{ x } } \right) =$$

A
0

B
5

C
log5

D
None of these.

Solution
Question 9
1 / -0

$$L\underset { x\rightarrow 0 }{ im } \frac { \sec { 4x-\sec { 2x } } }{ \sec { 3x-\sec { x } } }=$$

A
3/2

B
2/3

C
1/3

D
3/4

Solution
Question 10
1 / -0

$$\cfrac { d }{ dx } \left( \sin ^{ 5 }{ x } \sin { 5x } \right) =$$

A
$$ \left( \sin ^{ 4 }{ x } \sin { 5x } \right)$$

B
$$5 \left( \sin ^{ 4 }{ x } \sin { 5x } \right)$$

C
$$5 \left( \sin ^{ 4 }{ x } \sin { 6x } \right)$$

D
$$-5 \left( \sin ^{ 4 }{ x } \sin { 6x } \right)$$

Solution