Limits and Derivatives Test

Result

Limits and Derivatives Test - 57

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Weekly Quiz Competition

Question 1
1 / -0

If $$\alpha$$ and $$\beta$$ be the roots of the equation $$ax^{2} + bx + c = 0$$ then $$\displaystyle \lim_{x\rightarrow \dfrac {1}{\alpha}} \sqrt {\dfrac {1 - \cos^{2} (cx^{2} + bx + a)}{4(1 - \alpha x)^{2}}}$$

A
Does not exist

B
Equals $$\left |\dfrac {c}{2\alpha} \left (\dfrac {1}{\alpha} +\dfrac {1}{\beta}\right )\right |$$

C
Equals $$\left |\dfrac {c}{2\alpha} \left (\dfrac {1}{\alpha} - \dfrac {1}{\beta}\right )\right |$$

D
Equals $$\left |\dfrac {c}{2} \left (\dfrac {1}{\alpha} +\dfrac {1}{\beta}\right )\right |$$

Solution
Question 2
1 / -0

The value of $$\begin{matrix} lim \\ x\rightarrow \frac { 1 }{ \sqrt { 2 } } \end{matrix}\dfrac { x-cos\left( { sin }^{ -1 }x \right) }{ 1-tan\left( { sin }^{ -1 }x \right) } is$$

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

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

C
$$\sqrt { 2 } $$

D
$$-\sqrt { 2 } $$
Question 3
1 / -0

If $$x = 3\cos \theta - 2\cos^{3} \theta$$ and $$y = 3\sin \theta - 2\sin^{3}\theta$$, then $$\dfrac {dy}{dx} =$$

A
$$\sin \theta$$

B
$$\cos \theta$$

C
$$\tan \theta$$

D
$$\cot \theta$$
Question 4
1 / -0

the value of $$\underset { x\longrightarrow \infty }{ lim } \frac { { X }^{ 4 }sin\left( \frac { 1 }{ x } \right) +{ x }^{ 3 } }{ 1+\left| x \right| ^{ 3 } } $$

A
1

B
-1

C
2

D
does not exist
Question 5
1 / -0

$$\underset { x\rightarrow 1 }{ lim } { \left[ cosec { \dfrac { \pi x }{ 2 } } \right] }^{ { 1 }/{ \left( 1-x \right) } }$$ (where $$[.]$$ represents the greatest integer function) is equal to

A
$$0$$

B
$$1$$

C
$$\infty$$

D
$$Does \ not \ exist$$

Solution
Question 6
1 / -0

$$\displaystyle \lim_{x\rightarrow 0} \dfrac {\sin 2x + 3x}{2x + \sin 3x}$$ is equal to

A
$$1$$

B
$$\dfrac {1}{5}$$

C
$$2$$

D
Does not exist

Solution
Question 7
1 / -0

The value of $$\underset{x\rightarrow 1}{lim}(2-x)^{tan\left(\dfrac{\pi x}{2}\right)}$$ is

A
$$e^{-2/pi}$$

B
$$e^{1/pi}$$

C
$$e^{2/pi}$$

D
$$e^{-1/pi}$$

Solution
Question 8
1 / -0

$$\underset{x \rightarrow 1} {lim}\dfrac{x^2-1}{\sin^2x+\cos x\cos (x+2)-\cos^2(x+1)}$$ is-

A
$$0$$

B
$$\dfrac{1}{\cos 1}$$

C
$$\dfrac{2}{\sin 2}$$

D
$$\dfrac{1}{2\cos 1}$$

Solution
Question 9
1 / -0

$$\lim_{x\rightarrow 1}\frac{1-x^{-2/3}}{1-x^{-1/3}}$$

A
2

B
1

C
0

D
does not exist
Question 10
1 / -0

$$\overset {lim}{x \rightarrow \pi/2} \dfrac{\sin(x \ cos x)}{cos(x\, \ sin x)}$$ is equal to

A
$$1$$

B
$$\dfrac{\pi}{2}$$

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

D
does not exist

Solution