Limits and Derivatives Test

Result

Limits and Derivatives Test - 39

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Weekly Quiz Competition

Question 1
1 / -0

$$y=sin(x^2): \dfrac{dy}{dx}=$$

A
$$cos\, (x^2)$$

B
$$cos\, (2x)$$

C
$$2x\,cos\, (x^2)$$

D
$$2\,sin\, (x^2)$$

Solution
Question 2
1 / -0

The solution of $$\cos y +(x \sin y-1)\dfrac{dy}{dx}=0$$ is

A
$$\tan y-\sec y=cx$$

B
$$\tan y+\sec y=cx$$

C
$$x \sec y+\tan y=c$$

D
$$x \sec y=\tan y+c$$

Solution
Question 3
1 / -0

If $$\mathop {\lim }\limits_{x \to \infty } \left( {\frac{{{x^2} + x + 1}}{{x + 1}} - ax - b} \right)\, = 4$$,then

A
$$a=1,b=4$$

B
$$a=1,b=-4$$

C
$$a=2,b=-3$$

D
$$a=2,b=3$$

Solution
Question 4
1 / -0

Find derivative of given function w.r.t. the respective independent variable $$y= \frac{(sinx+cosx)}{cosx}$$

A
$$cos^2x$$

B
$$sin^2x$$

C
$$sec^2x$$

D
$$tan^2x$$

Solution
Question 5
1 / -0

$$If {A_i} = \frac{{x - {a_i}}}{{\left| {x - {a_i}} \right|}}, \,i = 1,2,3,.....n$$ and $${a_1}< {a_2}< {a_3}....< {a_{n,}} \, then$$
$$\mathop {\lim }\limits_{x \to {a_m}} \left( {{A_1}{A_2}......{A_n}} \right), 1 \le m \le n$$

A
is equal to $${\left( { - 1} \right)^m}$$

B
is equal to $${\left( { - 1} \right)^{m + 1}}$$

C
is equal to $${\left( { - 1} \right)^{n-m}}$$

D
is equal to $${\left( { - 1} \right)^{n-m-1}}$$

Solution
Question 6
1 / -0

$$\displaystyle \mathop {\lim }\limits_{x \to \frac{\pi }{2}} \frac{{\cot x - \cos x}}{{{{\left( {\frac{\pi }{2} - x} \right)}^3}}}$$

A
$$\dfrac{-1}{2}$$

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

C
$$2$$

D
$$1$$

Solution
Question 7
1 / -0

Let p= $$\lim_{x\rightarrow 0+}(1+tan^{2}\sqrt{x})^{\frac{1}{2x}}$$ then log p is equal to :

A
1

B
$$\frac{1}{2}$$

C
$$\frac{1}{4}$$

D
2

Solution
Question 8
1 / -0

Find that $$\frac { d } { d x } \left[ \frac { 2 } { \pi } \sin x ^ { 0 } \right] = ?$$

A
$$\frac { \pi } { 180 } \cos x ^ { 0 }$$

B
$$\frac { 1 } { 90 } \cos x ^ { 0 }$$

C
$$\frac { \pi } { 90 } \cos x ^ { 0 }$$

D
$$\frac { 2 } { 90 } \cos x ^ { 0 }$$

Solution
Question 9
1 / -0

Consider $$A=\begin{bmatrix} \cos { \theta } & \sin { \theta } \\ -\sin { \theta } & \cos { \theta } \end{bmatrix}$$, then the value of $$\lim_{n \rightarrow \infty} \dfrac{A^{n}}{n}$$ (where $$\theta \in R$$) is equal to

A
$$10$$

B
zero matrix

C
symmetric matrix

D
$$4$$

Solution
Question 10
1 / -0

$$\dfrac{d}{dx}\left\{\tan^{-1}\dfrac{\sqrt{x}+\sqrt{a}}{1-\sqrt{ax}}\right\}=$$

A
$$\dfrac{1}{2\sqrt{x}\left(1+x\right)}$$

B
$$\dfrac{-1}{2\sqrt{x}\left(1+x\right)}$$

C
$$\dfrac{2}{\sqrt{x}\left(1+x\right)}$$

D
$$\dfrac{-2}{\sqrt{x}\left(1+x\right)}$$

Solution