Continuity and Differentiability Test

Result

Continuity and Differentiability Test - 43

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Question 1
1 / -0

If $$y=\sec^{-1}\left(\dfrac {\sqrt {x}+1}{\sqrt {x}-1}\right)+\sin^{-1}\left(\dfrac {\sqrt {x}-1}{\sqrt {x}+1}\right)$$, then $$\dfrac {dy}{dx}$$ is equal to

A
$$0$$

B
$$\dfrac {1}{\sqrt {x}+1}$$

C
$$1$$

D
$$None\ of\ these$$

Solution
Question 2
1 / -0

Define $$f\left( x \right) = \left\{ \begin{array}{l}{x^2} + bx + c\,\,\,\,\,,x < 1\\x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x \ge 1\end{array} \right.$$. If $$f(x)$$ is differentiable at $$x=1$$ then $$(b-c)=$$

A
$$-2$$

B
$$0$$

C
$$1$$

D
$$2$$

Solution
Question 3
1 / -0

If $$f(x)$$ is a non constant polynomial function $$f:R\rightarrow R$$ such that $$7\dfrac{d}{dx}(xf(x))=3f(x)+4f(x+1),\ f(-1)+f(0)=2$$, then number of such function is

A
$$0$$

B
$$1$$

C
$$2$$

D
$$3$$

Solution
Question 4
1 / -0

If $$y=\tan^{-1}\left(\dfrac{\ell n\dfrac{e}{x^{2}}}{\ell nx^{2}}\right)+\tan^{-1}\dfrac{3+2\ell nx }{1-6\ell nx}$$ then $$\dfrac{d^{2}y}{dx^{2}}=$$

A
$$2$$

B
$$1$$

C
$$0$$

D
$$-1$$

Solution
Question 5
1 / -0

If $$y=\left[x+\sqrt{x^2-1}\right]^{15}+\left[x-\sqrt{x^2-1}\right]^{15}$$, then $$(x^2-1)\dfrac{d^2y}{dx^2}+x\dfrac{dy}{dx}$$ is equal to?

A
$$125y$$

B
$$224$$ $$y^2$$

C
$$225y^2$$

D
$$225y$$

Solution
Question 6
1 / -0

If $$f(x)=p\left|\sin x\right|+qe^{\left|x\right|}+r\left|x\right|^{3}$$ and $$f(x)$$ is differentiable at $$x=0$$, then

A
$$q+r=0;p$$ is any real number

B
$$p+q=0;r$$ is any real number

C
$$q=0,r=0;p$$ is any real number

D
$$r=0,0\p=o\0;q$$ is any real number

Solution
Question 7
1 / -0

The derivative of $$f (\tan^{-1}x)$$, where $$f(x)=\tan x$$ is

A
$$1$$

B
$$\dfrac{1}{1+x^{2}}$$

C
$$2$$

D
$$\dfrac{-1}{1+x^{2}}$$

Solution
Question 8
1 / -0

The value of $$\displaystyle \int _{ 0 }^{ \pi /2 }{ \frac { \sec ^{ 2 }{ x } dx }{ \left( \sec { x } +\tan { x } \right) ^{ n } } , } n>1$$ is equal to

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

B
$$\dfrac{1}{{n}^{2}+1}$$

C
$$\dfrac{n}{{n}^{2}-1}$$

D
$$\dfrac{n}{{n}^{2}+1}$$
Question 9
1 / -0

If $$f(x) = \sqrt{x+2\sqrt{2x-4}} + \sqrt{x-2\sqrt{2x-4}}$$, then f(x) is differentiable on

A
$$(-\infty, \infty)$$

B
$$[2, \infty) - $$ { $$4$$}

C
$$[2, \infty)$$

D
$$[0, \infty)$$

Solution
Question 10
1 / -0

The number of points at which $$g(x)= \dfrac { 1 }{ 1+\dfrac { 2 }{ f(x) } } $$ is not differentiable where $$f(x)= \dfrac { 1 }{ 1+\dfrac { 1 }{ x } }$$ is

A
$$1$$

B
$$2$$

C
$$3$$

D
$$4$$

Solution