15. If $f ( x - y ) = f ( x ) \cdot g ( y ) - f ( y ) \cdot g ( x )$ and $g ( x - y ) = g ( x ) \cdot g ( y ) + f ( x ) \cdot f ( y )$ for all $x$, $y \hat { I } R$. If right hand derivative at $x = 0$ exists for $f ( x )$. Find derivative of $g ( x )$ at $x = 0$.

48. $\frac { d ^ { 2 } x } { d y ^ { 2 } }$ equals
(A) $\left( \frac { d ^ { 2 } y } { d x ^ { 2 } } \right) ^ { - 1 }$
(B) $- \left( \frac { d ^ { 2 } y } { d x ^ { 2 } } \right) ^ { - 1 } \left( \frac { d y } { d x } \right) ^ { - 3 }$
(C) $\left( \frac { d ^ { 2 } y } { d x ^ { 2 } } \right) \left( \frac { d y } { d x } \right) ^ { - 2 }$
(D) $- \left( \frac { d ^ { 2 } y } { d x ^ { 2 } } \right) \left( \frac { d y } { d x } \right) ^ { - 3 }$

Answer

◯ ◯
(A)
(B)
(C)
(D)

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 $f$ and $g$ be real valued functions defined on interval $( - 1,1 )$ such that $g ^ { \prime \prime } ( x )$ is continuous, $g ( 0 ) \neq 0 , g ^ { \prime } ( 0 ) = 0 , g ^ { \prime \prime } ( 0 ) \neq 0$, and $f ( x ) = g ( x ) \sin x$.
STATEMENT-1 : $\lim _ { x \rightarrow 0 } [ g ( x ) \cot x - g ( 0 ) \operatorname { cosec } x ] = f ^ { \prime \prime } ( 0 )$. and STATEMENT-2 : $\quad f ^ { \prime } ( 0 ) = g ( 0 )$.
(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
(B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
(C) STATEMENT-1 is True, STATEMENT-2 is False
(D) STATEMENT-1 is False, STATEMENT-2 is True

Let
$$L = \lim _ { x \rightarrow 0 } \frac { a - \sqrt { a ^ { 2 } - x ^ { 2 } } - \frac { x ^ { 2 } } { 4 } } { x ^ { 4 } } , \quad a > 0 .$$
If $L$ is finite, then
(A) $\quad a = 2$
(B) $\quad a = 1$
(C) $\quad L = \frac { 1 } { 64 }$
(D) $\quad L = \frac { 1 } { 32 }$

Let $f ( x ) = x \sin \pi x , x > 0$. Then for all natural numbers $n , f ^ { \prime } ( x )$ vanishes at
(A) a unique point in the interval $\left( n , n + \frac { 1 } { 2 } \right)$
(B) a unique point in the interval $\left( n + \frac { 1 } { 2 } , n + 1 \right)$
(C) a unique point in the interval $( n , n + 1 )$
(D) two points in the interval $( n , n + 1 )$

The value of $g'\left(\frac{1}{2}\right)$ is
(A) $\frac{\pi}{2}$
(B) $\pi$
(C) $-\frac{\pi}{2}$
(D) 0

The largest value of the nonnegative integer $a$ for which $$\lim_{x \rightarrow 1} \left\{\frac{-ax + \sin(x-1) + a}{x + \sin(x-1) - 1}\right\}^{\frac{1-x}{1-\sqrt{x}}} = \frac{1}{4}$$ is

Let $m$ and $n$ be two positive integers greater than 1 . If
$$\lim _ { \alpha \rightarrow 0 } \left( \frac { e ^ { \cos \left( \alpha ^ { n } \right) } - e } { \alpha ^ { m } } \right) = - \left( \frac { e } { 2 } \right)$$
then the value of $\frac { m } { n }$ is

Let $F : \mathbb { R } \rightarrow \mathbb { R }$ be a thrice differentiable function. Suppose that $F ( 1 ) = 0 , F ( 3 ) = - 4$ and $F ^ { \prime } ( x ) < 0$ for all $x \in ( 1 / 2,3 )$. Let $f ( x ) = x F ( x )$ for all $x \in \mathbb { R }$. The correct statement(s) is(are)
(A) $f ^ { \prime } ( 1 ) < 0$
(B) $f ( 2 ) < 0$
(C) $f ^ { \prime } ( x ) \neq 0$ for any $x \in ( 1,3 )$
(D) $f ^ { \prime } ( x ) = 0$ for some $x \in ( 1,3 )$

Let $f ( x ) = \lim _ { n \rightarrow \infty } \left( \frac { n ^ { n } ( x + n ) \left( x + \frac { n } { 2 } \right) \cdots \left( x + \frac { n } { n } \right) } { n ! \left( x ^ { 2 } + n ^ { 2 } \right) \left( x ^ { 2 } + \frac { n ^ { 2 } } { 4 } \right) \cdots \left( x ^ { 2 } + \frac { n ^ { 2 } } { n ^ { 2 } } \right) } \right) ^ { \frac { x } { n } }$, for all $x > 0$. Then
(A) $f \left( \frac { 1 } { 2 } \right) \geq f ( 1 )$
(B) $f \left( \frac { 1 } { 3 } \right) \leq f \left( \frac { 2 } { 3 } \right)$
(C) $f ^ { \prime } ( 2 ) \leq 0$
(D) $\frac { f ^ { \prime } ( 3 ) } { f ( 3 ) } \geq \frac { f ^ { \prime } ( 2 ) } { f ( 2 ) }$

Let $f : ( 0 , \pi ) \rightarrow \mathbb { R }$ be a twice differentiable function such that
$$\lim _ { t \rightarrow x } \frac { f ( x ) \sin t - f ( t ) \sin x } { t - x } = \sin ^ { 2 } x \text { for all } x \in ( 0 , \pi )$$
If $f \left( \frac { \pi } { 6 } \right) = - \frac { \pi } { 12 }$, then which of the following statement(s) is (are) TRUE?
(A) $f \left( \frac { \pi } { 4 } \right) = \frac { \pi } { 4 \sqrt { 2 } }$
(B) $f ( x ) < \frac { x ^ { 4 } } { 6 } - x ^ { 2 }$ for all $x \in ( 0 , \pi )$
(C) There exists $\alpha \in ( 0 , \pi )$ such that $f ^ { \prime } ( \alpha ) = 0$
(D) $f ^ { \prime \prime } \left( \frac { \pi } { 2 } \right) + f \left( \frac { \pi } { 2 } \right) = 0$

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be given by $$f ( x ) = \left\{ \begin{aligned} x ^ { 5 } + 5 x ^ { 4 } + 10 x ^ { 3 } + 10 x ^ { 2 } + 3 x + 1 , & x < 0 \\ x ^ { 2 } - x + 1 , & 0 \leq x < 1 \\ \frac { 2 } { 3 } x ^ { 3 } - 4 x ^ { 2 } + 7 x - \frac { 8 } { 3 } , & 1 \leq x < 3 \\ ( x - 2 ) \log _ { e } ( x - 2 ) - x + \frac { 10 } { 3 } , & x \geq 3 \end{aligned} \right.$$ Then which of the following options is/are correct?
(A) $f$ is increasing on $( - \infty , 0 )$
(B) $f ^ { \prime }$ has a local maximum at $x = 1$
(C) $f$ is onto
(D) $f ^ { \prime }$ is NOT differentiable at $x = 1$

Let the function $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined by $f ( x ) = x ^ { 3 } - x ^ { 2 } + ( x - 1 ) \sin x$ and let $g : \mathbb { R } \rightarrow \mathbb { R }$ be an arbitrary function. Let $f g : \mathbb { R } \rightarrow \mathbb { R }$ be the product function defined by $( f g ) ( x ) = f ( x ) g ( x )$. Then which of the following statements is/are TRUE?
(A) If $g$ is continuous at $x = 1$, then $f g$ is differentiable at $x = 1$
(B) If $f g$ is differentiable at $x = 1$, then $g$ is continuous at $x = 1$
(C) If $g$ is differentiable at $x = 1$, then $f g$ is differentiable at $x = 1$
(D) If $f g$ is differentiable at $x = 1$, then $g$ is differentiable at $x = 1$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ be functions satisfying $$f(x + y) = f(x) + f(y) + f(x)f(y) \text{ and } f(x) = xg(x)$$ for all $x, y \in \mathbb{R}$. If $\lim_{x \rightarrow 0} g(x) = 1$, 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 $f: \mathbb{R} \to \mathbb{R}$ 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 $\_\_\_\_$.

If
$$\beta = \lim _ { x \rightarrow 0 } \frac { e ^ { x ^ { 3 } } - \left( 1 - x ^ { 3 } \right) ^ { \frac { 1 } { 3 } } + \left( \left( 1 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } - 1 \right) \sin x } { x \sin ^ { 2 } x }$$
then the value of $6 \beta$ is $\_\_\_\_$ .

Let $k \in \mathbb { R }$. If $\lim _ { x \rightarrow 0 + } ( \sin ( \sin k x ) + \cos x + x ) ^ { \frac { 2 } { x } } = e ^ { 6 }$, then the value of $k$ is
(A) 1
(B) 2
(C) 3
(D) 4

Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be a function defined by
$$f ( x ) = \left\{ \begin{array} { c l } x ^ { 2 } \sin \left( \frac { \pi } { x ^ { 2 } } \right) , & \text { if } x \neq 0 \\ 0 , & \text { if } x = 0 \end{array} \right.$$
Then which of the following statements is TRUE?
(A) $f ( x ) = 0$ has infinitely many solutions in the interval $\left[ \frac { 1 } { 10 ^ { 10 } } , \infty \right)$.
(B) $f ( x ) = 0$ has no solutions in the interval $\left[ \frac { 1 } { \pi } , \infty \right)$.
(C) The set of solutions of $f ( x ) = 0$ in the interval $\left( 0 , \frac { 1 } { 10 ^ { 10 } } \right)$ is finite.
(D) $f ( x ) = 0$ has more than 25 solutions in the interval $\left( \frac { 1 } { \pi ^ { 2 } } , \frac { 1 } { \pi } \right)$.

Let $S$ be the set of all $( \alpha , \beta ) \in \mathbb { R } \times \mathbb { R }$ such that
$$\lim _ { x \rightarrow \infty } \frac { \sin \left( x ^ { 2 } \right) \left( \log _ { e } x \right) ^ { \alpha } \sin \left( \frac { 1 } { x ^ { 2 } } \right) } { x ^ { \alpha \beta } \left( \log _ { e } ( 1 + x ) \right) ^ { \beta } } = 0 .$$
Then which of the following is (are) correct?
(A) $( - 1,3 ) \in S$
(B) $( - 1,1 ) \in S$
(C) $( 1 , - 1 ) \in S$
(D) $( 1 , - 2 ) \in S$

Let the function $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined by
$$f ( x ) = \frac { \sin x } { e ^ { \pi x } } \frac { \left( x ^ { 2023 } + 2024 x + 2025 \right) } { \left( x ^ { 2 } - x + 3 \right) } + \frac { 2 } { e ^ { \pi x } } \frac { \left( x ^ { 2023 } + 2024 x + 2025 \right) } { \left( x ^ { 2 } - x + 3 \right) }$$
Then the number of solutions of $f ( x ) = 0$ in $\mathbb { R }$ is $\_\_\_\_$ .

Let $\mathbb { R }$ denote the set of all real numbers. Define the function $f : \mathbb { R } \rightarrow \mathbb { R }$ by
$$f ( x ) = \left\{ \begin{array} { c c } 2 - 2 x ^ { 2 } - x ^ { 2 } \sin \frac { 1 } { x } & \text { if } x \neq 0 \\ 2 & \text { if } x = 0 \end{array} \right.$$
Then which one of the following statements is TRUE?

(A)	The function $f$ is NOT differentiable at $x = 0$
(B)	There is a positive real number $\delta$, such that $f$ is a decreasing function on the interval ( $0 , \delta$ )
(C)	For any positive real number $\delta$, the function $f$ is NOT an increasing function on the interval ( $- \delta , 0$ )
(D)	$x = 0$ is a point of local minima of $f$

Let $\alpha$ and $\beta$ be the real numbers such that
$$\lim _ { x \rightarrow 0 } \frac { 1 } { x ^ { 3 } } \left( \frac { \alpha } { 2 } \int _ { 0 } ^ { x } \frac { 1 } { 1 - t ^ { 2 } } d t + \beta x \cos x \right) = 2$$
Then the value of $\alpha + \beta$ is $\_\_\_\_$ .