Let $f(x) = x|x|^n$ for $n \geq 1$ a positive integer. Which of the following is true?
(A) $f$ is differentiable everywhere except at $x = 0$
(B) $f$ is continuous but not differentiable at $x = 0$
(C) $f$ is differentiable everywhere
(D) None of the above

The limit $$\lim _ { x \rightarrow 0 } \frac { \left( e ^ { x } - 1 \right) \tan ^ { 2 } x } { x ^ { 3 } }$$ (A) does not exist
(B) exists and equals 0
(C) exists and equals $2/3$
(D) exists and equals 1

If the function $$f ( x ) = \begin{cases} \frac { x ^ { 2 } - 2 x + A } { \sin x } & \text { if } x \neq 0 \\ B & \text { if } x = 0 \end{cases}$$ is continuous at $x = 0$, then
(A) $A = 0 , B = 0$
(B) $A = 0 , B = - 2$
(C) $A = 1 , B = 1$
(D) $A = 1 , B = 0$

$\lim _ { x \rightarrow 0 } \frac { \left( e ^ { x } - 1 \right) \tan ^ { 2 } x } { x ^ { 3 } }$
(a) does not exist
(b) exists and equals 0
(c) exists and equals $\frac { 2 } { 3 }$
(d) exists and equals 1.

$\lim _ { x \rightarrow 0 } \frac { \left( e ^ { x } - 1 \right) \tan ^ { 2 } x } { x ^ { 3 } }$
(a) does not exist
(b) exists and equals 0
(c) exists and equals $\frac { 2 } { 3 }$
(d) exists and equals 1.

If the function $$f ( x ) = \begin{cases} \frac { x ^ { 2 } - 2 x + A } { \sin x } & \text { if } x \neq 0 \\ B & \text { if } x = 0 \end{cases}$$ is continuous at $x = 0$, then
(A) $A = 0 , B = 0$
(B) $A = 0 , B = - 2$
(C) $A = 1 , B = 1$
(D) $A = 1 , B = 0$

If the function $$f ( x ) = \begin{cases} \frac { x ^ { 2 } - 2 x + A } { \sin x } & \text { if } x \neq 0 \\ B & \text { if } x = 0 \end{cases}$$ is continuous at $x = 0$, then
(A) $A = 0 , B = 0$
(B) $A = 0 , B = - 2$
(C) $A = 1 , B = 1$
(D) $A = 1 , B = 0$

Which of the following graphs represents the function $$f ( x ) = \int _ { 0 } ^ { \sqrt { x } } e ^ { - u ^ { 2 } / x } d u , \quad \text { for } \quad x > 0 \quad \text { and } \quad f ( 0 ) = 0 ?$$ (A), (B), (C), (D) as given by the respective graphs.

Which of the following graphs represents the function $$f ( x ) = \int _ { 0 } ^ { \sqrt { x } } e ^ { - u ^ { 2 } / x } d u , \quad \text { for } \quad x > 0 \quad \text { and } \quad f ( 0 ) = 0 ?$$ (A), (B), (C), (D) as shown in the graphs.

Consider the function $f : \mathbb{R} \rightarrow \mathbb{R}$ defined as $$f(x) = \begin{cases} \frac{x}{e^x - 1} & \text{if } x \neq 0 \\ 1 & \text{if } x = 0 \end{cases}$$ Then which one of the following statements is correct?
(A) $f$ is not continuous at $x = 0$.
(B) $f$ is continuous but not differentiable at $x = 0$.
(C) $f$ is differentiable at $x = 0$ and $f'(0) = -\frac{1}{2}$.
(D) $f$ is differentiable at $x = 0$ and $f'(0) = \frac{1}{2}$.

For the function on the real line $\mathbb { R }$ given by $f ( x ) = | x | + | x + 1 | + e ^ { x }$, which of the following is true ?
(A) It is differentiable everywhere.
(B) It is differentiable everywhere except at $x = 0$ and $x = - 1$.
(C) It is differentiable everywhere except at $x = 1 / 2$.
(D) It is differentiable everywhere except at $x = - 1 / 2$.

Define $f : \mathbb { R } \rightarrow \mathbb { R }$ by $$f ( x ) = \begin{cases} ( 1 - \cos x ) \sin \left( \frac { 1 } { x } \right) , & x \neq 0 \\ 0 , & x = 0 \end{cases}$$ Then,
(A) $f$ is discontinuous.
(B) $f$ is continuous but not differentiable.
(C) $f$ is differentiable and its derivative is discontinuous.
(D) $f$ is differentiable and its derivative is continuous.

If $f ( x ) = e ^ { x } \sin x$, then $\left. \frac { d ^ { 10 } } { d x ^ { 10 } } f ( x ) \right| _ { x = 0 }$ equals
(a) 1 .
(B) - 1 .
(C) 10 .
(D) 32 .

Let $g ( x ) = \frac { ( x - 1 ) ^ { n } } { \log \cos ^ { m } ( x - 1 ) } ; 0 < x < 2 , m$ and $n$ are integers, $m \neq 0 , n > 0$, and let $p$ be the left hand derivative of $| x - 1 |$ at $x = 1$.
If $\lim _ { x \rightarrow 1 + } g ( x ) = p$, then
(A) $n = 1 , m = 1$
(B) $n = 1 , m = - 1$
(C) $n = 2 , m = 2$
(D) $n > 2 , m = n$

Let $g ( x ) = \log f ( x )$ where $f ( x )$ is a twice differentiable positive function on $( 0 , \infty )$ such that $f ( x + 1 ) = x f ( x )$. Then, for $N = 1,2,3 , \ldots$,
$$g ^ { \prime \prime } \left( N + \frac { 1 } { 2 } \right) - g ^ { \prime \prime } \left( \frac { 1 } { 2 } \right) =$$
(A) $- 4 \left\{ 1 + \frac { 1 } { 9 } + \frac { 1 } { 25 } + \cdots + \frac { 1 } { ( 2 N - 1 ) ^ { 2 } } \right\}$
(B) $4 \left\{ 1 + \frac { 1 } { 9 } + \frac { 1 } { 25 } + \cdots + \frac { 1 } { ( 2 N - 1 ) ^ { 2 } } \right\}$
(C) $- 4 \left\{ 1 + \frac { 1 } { 9 } + \frac { 1 } { 25 } + \cdots + \frac { 1 } { ( 2 N + 1 ) ^ { 2 } } \right\}$
(D) $4 \left\{ 1 + \frac { 1 } { 9 } + \frac { 1 } { 25 } + \cdots + \frac { 1 } { ( 2 N + 1 ) ^ { 2 } } \right\}$

For the function $$f(x)=x\cos\frac{1}{x},\quad x\geq1,$$ (A) for at least one $x$ in the interval $[1,\infty),f(x+2)-f(x)<2$
(B) $\lim_{x\rightarrow\infty}f^{\prime}(x)=1$
(C) for all $x$ in the interval $[1,\infty),f(x+2)-f(x)>2$
(D) $f^{\prime}(x)$ is strictly decreasing in the interval $[1,\infty)$

Let $f$ be a real-valued function defined on the interval $( 0 , \infty )$ by $\mathrm { f } ( \mathrm { x } ) = \ell n \mathrm { x } + \int _ { 0 } ^ { \mathrm { x } } \sqrt { 1 + \sin \mathrm { t } } \mathrm { dt }$. Then which of the following statement(s) is (are) true?
A) $\mathrm { f } ^ { \prime \prime } ( \mathrm { x } )$ exists for all $\mathrm { x } \in ( 0 , \infty )$
B) $f ^ { \prime } ( x )$ exists for all $x \in ( 0 , \infty )$ and $f ^ { \prime }$ is continuous on $( 0 , \infty )$, but not differentiable on $( 0 , \infty )$
C) there exists $\alpha > 1$ such that $\left| \mathrm { f } ^ { \prime } ( \mathrm { x } ) \right| < | \mathrm { f } ( \mathrm { x } ) |$ for all $\mathrm { x } \in ( \alpha , \infty )$
D) there exists $\beta > 0$ such that $| f ( x ) | + \left| f ^ { \prime } ( x ) \right| \leq \beta$ for all $x \in ( 0 , \infty )$

Let $f : (0, \infty) \rightarrow \mathbb{R}$ be given by $$f(x) = \int_{\frac{1}{x}}^{x} e^{-\left(t + \frac{1}{t}\right)} \frac{dt}{t}$$ Then
(A) $f(x)$ is monotonically increasing on $[1, \infty)$
(B) $f(x)$ is monotonically decreasing on $(0,1)$
(C) $f(x) + f\left(\frac{1}{x}\right) = 0$, for all $x \in (0, \infty)$
(D) $f\left(2^x\right)$ is an odd function of $x$ on $\mathbb{R}$

List I P. Let $y(x) = \cos\left(3\cos^{-1}x\right)$, $x \in [-1,1]$, $x \neq \pm\frac{\sqrt{3}}{2}$. Then $\frac{1}{y(x)}\left\{\left(x^2-1\right)\frac{d^2y(x)}{dx^2} + x\frac{dy(x)}{dx}\right\}$ equals Q. Let $A_1, A_2, \ldots, A_n$ $(n > 2)$ be the vertices of a regular polygon of $n$ sides with its centre at the origin. Let $\overrightarrow{a_k}$ be the position vector of the point $A_k$, $k = 1,2,\ldots,n$. If $\left|\sum_{k=1}^{n-1}\left(\overrightarrow{a_k} \times \overrightarrow{a_{k+1}}\right)\right| = \left|\sum_{k=1}^{n-1}\left(\overrightarrow{a_k} \cdot \overrightarrow{a_{k+1}}\right)\right|$, then the minimum value of $n$ is R. If the normal from the point $P(h,1)$ on the ellipse $\frac{x^2}{6} + \frac{y^2}{3} = 1$ is perpendicular to the line $x + y = 8$, then the value of $h$ is S. Number of positive solutions satisfying the equation $\tan^{-1}\left(\frac{1}{2x+1}\right) + \tan^{-1}\left(\frac{1}{4x+1}\right) = \tan^{-1}\left(\frac{2}{x^2}\right)$ is
List II
1. 1
2. 2
3. 3
4. 4
P Q R S
(A) 4321
(B) 2431
(C) 4312
(D) 2413

Let $f _ { 1 } : \mathbb { R } \rightarrow \mathbb { R } , f _ { 2 } : \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right) \rightarrow \mathbb { R } , f _ { 3 } : \left( - 1 , e ^ { \frac { \pi } { 2 } } - 2 \right) \rightarrow \mathbb { R }$ and $f _ { 4 } : \mathbb { R } \rightarrow \mathbb { R }$ be functions defined by
(i) $\quad f _ { 1 } ( x ) = \sin \left( \sqrt { 1 - e ^ { - x ^ { 2 } } } \right)$,
(ii) $\quad f _ { 2 } ( x ) = \left\{ \begin{array} { c c } \frac { | \sin x | } { \tan ^ { - 1 } x } & \text { if } x \neq 0 \\ 1 & \text { if } x = 0 \end{array} \right.$, where the inverse trigonometric function $\tan ^ { - 1 } x$ assumes values in $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$,
(iii) $\quad f _ { 3 } ( x ) = \left[ \sin \left( \log _ { e } ( x + 2 ) \right) \right]$, where, for $t \in \mathbb { R } , [ t ]$ denotes the greatest integer less than or equal to $t$,
(iv) $\quad f _ { 4 } ( x ) = \left\{ \begin{array} { c c } x ^ { 2 } \sin \left( \frac { 1 } { x } \right) & \text { if } x \neq 0 \\ 0 & \text { if } x = 0 \end{array} \right.$.
LIST-I P. The function $f _ { 1 }$ is Q. The function $f _ { 2 }$ is R. The function $f _ { 3 }$ is S. The function $f _ { 4 }$ is
LIST-II

1. NOT continuous at $x = 0$
2. continuous at $x = 0$ and NOT differentiable at $x = 0$
3. differentiable at $x = 0$ and its derivative is NOT continuous at $x = 0$
4. differentiable at $x = 0$ and its derivative is continuous at $x = 0$

The correct option is:
(A) $\mathbf { P } \rightarrow \mathbf { 2 ; } \mathbf { Q } \rightarrow \mathbf { 3 ; } \mathbf { R } \rightarrow \mathbf { 1 ; } \mathbf { S } \rightarrow \mathbf { 4 }$
(B) $\mathbf { P } \rightarrow \mathbf { 4 } ; \mathbf { Q } \rightarrow \mathbf { 1 } ; \mathbf { R } \rightarrow \mathbf { 2 } ; \mathbf { S } \rightarrow \mathbf { 3 }$
(C) $\mathbf { P } \rightarrow \mathbf { 4 } ; \mathbf { Q } \rightarrow \mathbf { 2 } ; \mathbf { R } \rightarrow \mathbf { 1 } ; \mathbf { S } \rightarrow \mathbf { 3 }$
(D) $\mathbf { P } \rightarrow \mathbf { 2 } ; \mathbf { Q } \rightarrow \mathbf { 1 } ; \mathbf { R } \rightarrow \mathbf { 4 } ; \mathbf { S } \rightarrow \mathbf { 3 }$

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $$f(x) = \begin{cases} x^5 \sin\left(\frac{1}{x}\right) + 5x^2, & x < 0 \\ 0, & x = 0 \\ x^5 \cos\left(\frac{1}{x}\right) + \lambda x^2, & x > 0 \end{cases}$$ The value of $\lambda$ for which $f''(0)$ exists is ____.
(A) 0
(B) 1
(C) $-1$
(D) $\frac{1}{2}$

Let $f: \mathbb{R} \to \mathbb{R}$ and $g: \mathbb{R} \to \mathbb{R}$ be functions satisfying $$f(x+y) = f(x) + f(y) + f(x)f(y) \quad \text{and} \quad f(x) = xg(x)$$ for all $x, y \in \mathbb{R}$. If $\lim_{x \to 0} g(x) = 1$, then which of the following statements is(are) TRUE?
(A) $f$ is differentiable at every $x \in \mathbb{R}$
(B) If $g(0) = 1$, then $g$ is differentiable at every $x \in \mathbb{R}$
(C) The derivative $f'(1)$ is equal to 1
(D) The derivative $f'(0)$ is equal to 1

Let $\alpha$ be a positive real number. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ and $g : ( \alpha , \infty ) \rightarrow \mathbb { R }$ be the functions defined by
$$f ( x ) = \sin \left( \frac { \pi x } { 12 } \right) \quad \text { and } \quad g ( x ) = \frac { 2 \log _ { \mathrm { e } } ( \sqrt { x } - \sqrt { \alpha } ) } { \log _ { \mathrm { e } } \left( e ^ { \sqrt { x } } - e ^ { \sqrt { \alpha } } \right) }$$
Then the value of $\lim _ { x \rightarrow \alpha ^ { + } } f ( g ( x ) )$ is $\_\_\_\_$.

Let $x _ { 0 }$ be the real number such that $e ^ { x _ { 0 } } + x _ { 0 } = 0$. For a given real number $\alpha$, define
$$g ( x ) = \frac { 3 x e ^ { x } + 3 x - \alpha e ^ { x } - \alpha x } { 3 \left( e ^ { x } + 1 \right) }$$
for all real numbers $x$.
Then which one of the following statements is TRUE?

(A)	For $\alpha = 2 , \lim _ { x \rightarrow x _ { 0 } } \left| \frac { g ( x ) + e ^ { x _ { 0 } } } { x - x _ { 0 } } \right| = 0$
(B)	For $\alpha = 2 , \lim _ { x \rightarrow x _ { 0 } } \left| \frac { g ( x ) + e ^ { x _ { 0 } } } { x - x _ { 0 } } \right| = 1$
(C)	For $\alpha = 3 , ~ \lim _ { x \rightarrow x _ { 0 } } \left| \frac { g ( x ) + e ^ { x _ { 0 } } } { x - x _ { 0 } } \right| = 0$
(D)	For $\alpha = 3 , ~ \lim _ { x \rightarrow x _ { 0 } } \left| \frac { g ( x ) + e ^ { x _ { 0 } } } { x - x _ { 0 } } \right| = \frac { 2 } { 3 }$

For $x \in \mathbb{R}$, $f(x) = |\log 2 - \sin x|$ and $g(x) = f(f(x))$, then:
(1) $g$ is not differentiable at $x = 0$
(2) $g'(0) = \cos(\log 2)$
(3) $g'(0) = -\cos(\log 2)$
(4) $g$ is differentiable at $x = 0$ and $g'(0) = -\sin(\log 2)$