Continuity and Differentiability Test

Result

Continuity and Differentiability Test - 48

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Question 1
1 / -0

$$f(x)=\begin{cases} x\left( \dfrac { a{ e }^{ \frac { 1 }{ \left| x \right| } }+3{ e }^{ \frac { -1 }{ x } } }{ \left( a+2 \right) { e }^{ \frac { 1 }{ \left| x \right| } }-{ e }^{ \frac { -1 }{ x } } } \right),\ \ x\neq 0 \\ 0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x=0 \end{cases}$$ is differentiable at $$x=0$$ then $$[a]=......$$ ($$[\ ]$$ denotes greatest integer function)

A
$$0$$

B
$$-1$$

C
$$-2$$

D
$$None\ of\ these$$
Question 2
1 / -0

If $$y = {\cot ^{ - 1}}\left( {\frac{{\sin x}}{{1 + \cos x}}} \right)$$ then $$\frac{{dy}}{{dx}}$$ is

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

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

C
$$0$$

D
$$1$$

Solution
Question 3
1 / -0

let f be the differeniable at x=0 and f'(0)=1 then $$\underset { b\rightarrow 0 }{ lim } \frac { f(h)-f(-2h) }{ h } $$

A
3

B
2

C
1

D
-1
Question 4
1 / -0

If $$f$$ is twice differentiable such that $$f ^ { \prime \prime } ( x ) = - f ( x )$$ and $$f ^ { \prime } ( x ) = g ( x ).$$ If $$h ( x )$$ is a twice differentiable function that $$h ^ { \prime } ( x ) = ( f ( x ) ) ^ { 2 } + ( g ( x ) ) ^ { 2 } .$$ If $$h ( 0 ) = 2 , h ( 1 ) = 4 ,$$ then the equation $$y = h ( x )$$ represents :

A
a curve of degree $$2$$

B
a curve passing through the origin

C
a straight line with slope $$2$$

D
a straight line with $$y$$ intercept equal to $$2$$

Solution
Question 5
1 / -0

Number of points in $$[0,\pi]$$, where $$f(x)=\left[\dfrac{x\tan{x}}{\sin{x}+\cos{x}}\right]$$ is non-differentiable is/are (where $$[.]$$ denotes the greatest integer function)

A
$$0$$

B
$$4$$

C
$$9$$

D
$$None\ of\ these$$

Solution
Question 6
1 / -0

Solve : $$\dfrac { d } { d x } \left[ \tan ^ { - 1 } \sqrt { 1 + x ^ { 2 } } - \cot ^ { - 1 } \left( - \sqrt { 1 + x ^ { 2 } } \right) \right] =?$$

A
$$\pi$$

B
$$1$$

C
$$0$$

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

Solution
Question 7
1 / -0

if the function $$f(x)=\begin{cases} a+{\sin}^{-1}(x+b),\,x\geq 1 \\ x,\,x<1 \end{cases}$$ is differentiable at $$x=1$$, then $$\displaystyle \frac{a}{b}$$ is equal to

A
$$1$$

B
$$0$$

C
$$-1$$

D
$$2$$
Question 8
1 / -0

If $$y=e^{\sin^{-1}(t^{2}-1)}$$ & $$x=e^{\sec^{-1}\left (\frac{1}{t^{2}-1}\right)}$$, then $$\dfrac{dy}{dx}$$ is equal to

A
$$\frac{x}{y}$$

B
$$\dfrac{-y}{x}$$

C
$$\dfrac{y}{x}$$

D
$$\dfrac{-x}{y}$$

Solution
Question 9
1 / -0

The function $$f(x) = \dfrac {x}{1 + |x|}$$ is differentiable at which of the following?

A
Every where

B
Everywhere except at $$x = 1$$

C
Everywhere except at $$x = 0$$

D
Everywhere except at $$x = 0$$ or $$1$$

Solution
Question 10
1 / -0

Let F(x) = $$\left( f\left( x \right) \right) ^{ 2 }+\left( f\left( x \right) \right) ^{ 2 },F\left( 0 \right) -6$$ where f(x) is a differential function such that $$\left| f\left( x \right) \right| \le 1\forall x\notin \left[ -1,1 \right] $$ then choose the correct statement (s)

A
There is atleast one point in each of the intervals (-1, 0) and (0, 1) where $$\left| f'(x) \right| \le 2$$

B
there is atleast one point in a each of the interval (-1, 0) and (0, 1) where f(x)$$\le S$$

C
there is no point of local maximum of F(x) in (-1, 1)

D
For some $$c\in \left( -1,1 \right) $$, $$F'\left( c \right) \ge 6,F"\left( c \right) \le 0$$

Solution