The function $x ( \alpha - x )$ is strictly increasing on the interval $0 < x < 1$ if and only if
(a) $\alpha \geq 2$
(b) $\alpha < 2$
(c) $\alpha < - 1$
(d) $\alpha > 2$.

The function $x ( \alpha - x )$ is strictly increasing on the interval $0 < x < 1$ if and only if
(A) $\alpha \geq 2$
(B) $\alpha < 2$
(C) $\alpha < - 1$
(D) $\alpha > 2$

If $f(x) = \cos(x) - 1 + \frac{x^2}{2}$, then
(A) $f(x)$ is an increasing function on the real line
(B) $f(x)$ is a decreasing function on the real line
(C) $f(x)$ is increasing on $-\infty < x \leq 0$ and decreasing on $0 \leq x < \infty$
(D) $f(x)$ is decreasing on $-\infty < x \leq 0$ and increasing on $0 \leq x < \infty$

The function $x ( \alpha - x )$ is strictly increasing on the interval $0 < x < 1$ if and only if
(A) $\alpha \geq 2$
(B) $\alpha < 2$
(C) $\alpha < - 1$
(D) $\alpha > 2$

If $f ( x ) = \cos ( x ) - 1 + \frac { x ^ { 2 } } { 2 }$, then
(A) $f ( x )$ is an increasing function on the real line
(B) $f ( x )$ is a decreasing function on the real line
(C) $f ( x )$ is increasing on $- \infty < x \leq 0$ and decreasing on $0 \leq x < \infty$
(D) $f ( x )$ is decreasing on $- \infty < x \leq 0$ and increasing on $0 \leq x < \infty$

Let $f ( x ) = \sin x + \alpha x , x \in \mathbb { R }$, where $\alpha$ is a fixed real number. The function $f$ is one-to-one if and only if
(A) $\alpha > 1$ or $\alpha < - 1$.
(B) $\alpha \geq 1$ or $\alpha \leq - 1$.
(C) $\alpha \geq 1$ or $\alpha < - 1$.
(D) $\alpha > 1$ or $\alpha \leq - 1$.

Let $f(x) = x^x$ for $x > 0$. Then $f$ is
(A) increasing on $(0, \infty)$
(B) decreasing on $(0, \infty)$
(C) increasing on $(0, 1/e)$ and decreasing on $(1/e, \infty)$
(D) decreasing on $(0, 1/e)$ and increasing on $(1/e, \infty)$

Let $f(x) = \frac{x}{\sqrt{a^2+x^2}} - \frac{d-x}{\sqrt{b^2+(d-x)^2}}$, where $a$, $b$, and $d$ are positive constants. Then
(A) $f$ is an increasing function of $x$
(B) $f$ is a decreasing function of $x$
(C) $f$ is neither increasing nor decreasing function of $x$
(D) $f'$ is not a monotonic function of $x$

Let the function $g : ( - \infty , \infty ) \rightarrow \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$ be given by $g ( u ) = 2 \tan ^ { - 1 } \left( e ^ { u } \right) - \frac { \pi } { 2 }$. Then, $g$ is
(A) even and is strictly increasing in $(0 , \infty)$
(B) odd and is strictly decreasing in $( - \infty , \infty )$
(C) odd and is strictly increasing in $( - \infty , \infty )$
(D) neither even nor odd, but is strictly increasing in $( - \infty , \infty )$

Consider the polynomial
$$f ( x ) = 1 + 2 x + 3 x ^ { 2 } + 4 x ^ { 3 }$$
Let s be the sum of all distinct real roots of $\mathrm { f } ( \mathrm { x } )$ and let $\mathrm { t } = | \mathrm { s } |$.
The function $f ^ { \prime } ( x )$ is
A) increasing in $\left( - t , - \frac { 1 } { 4 } \right)$ and decreasing in $\left( - \frac { 1 } { 4 } , t \right)$
B) decreasing in $\left( - t , - \frac { 1 } { 4 } \right)$ and increasing in $\left( - \frac { 1 } { 4 } , t \right)$
C) increasing in (-t, t)
D) decreasing in (-t, t)

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 $f(x) = x + \log_e x - x\log_e x$, $x \in (0, \infty)$.
- Column 1 contains information about zeros of $f(x)$, $f'(x)$ and $f''(x)$. - Column 2 contains information about the limiting behavior of $f(x)$, $f'(x)$ and $f''(x)$ at infinity. - Column 3 contains information about increasing/decreasing nature of $f(x)$ and $f'(x)$.

Column 1	Column 2	Column 3
(I) $f(x) = 0$ for some $x \in (1, e^2)$	(i) $\lim_{x\to\infty} f(x) = 0$	(P) $f$ is increasing in $(0,1)$
(II) $f'(x) = 0$ for some $x \in (1, e)$	(ii) $\lim_{x\to\infty} f(x) = -\infty$	(Q) $f$ is decreasing in $(e, e^2)$
(III) $f'(x) = 0$ for some $x \in (0,1)$	(iii) $\lim_{x\to\infty} f'(x) = -\infty$	(R) $f'$ is increasing in $(0,1)$
(IV) $f''(x) = 0$ for some $x \in (1, e)$	(iv) $\lim_{x\to\infty} f''(x) = 0$	(S) $f'$ is decreasing in $(e, e^2)$

Which of the following options is the only CORRECT combination?
[A] (I) (i) (P)
[B] (II) (ii) (Q)
[C] (III) (iii) (R)
[D] (IV) (iv) (S)

Let $f(x) = x + \log_e x - x\log_e x$, $x \in (0, \infty)$.
- Column 1 contains information about zeros of $f(x)$, $f'(x)$ and $f''(x)$. - Column 2 contains information about the limiting behavior of $f(x)$, $f'(x)$ and $f''(x)$ at infinity. - Column 3 contains information about increasing/decreasing nature of $f(x)$ and $f'(x)$.

Column 1	Column 2	Column 3
(I) $f(x) = 0$ for some $x \in (1, e^2)$	(i) $\lim_{x\to\infty} f(x) = 0$	(P) $f$ is increasing in $(0,1)$
(II) $f'(x) = 0$ for some $x \in (1, e)$	(ii) $\lim_{x\to\infty} f(x) = -\infty$	(Q) $f$ is decreasing in $(e, e^2)$
(III) $f'(x) = 0$ for some $x \in (0,1)$	(iii) $\lim_{x\to\infty} f'(x) = -\infty$	(R) $f'$ is increasing in $(0,1)$
(IV) $f''(x) = 0$ for some $x \in (1, e)$	(iv) $\lim_{x\to\infty} f''(x) = 0$	(S) $f'$ is decreasing in $(e, e^2)$

Which of the following options is the only CORRECT combination?
[A] (I) (ii) (R)
[B] (II) (iii) (S)
[C] (III) (iv) (P)
[D] (IV) (i) (S)

The function $f ( x ) = \tan ^ { - 1 } ( \sin x + \cos x )$ is an increasing function in
(1) $\left( \frac { \pi } { 4 } , \frac { \pi } { 2 } \right)$
(2) $\left( - \frac { \pi } { 2 } , \frac { \pi } { 4 } \right)$
(3) $\left( 0 , \frac { \pi } { 2 } \right)$
(4) $\left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$

Let $f ( x ) = \sin ^ { 4 } x + \cos ^ { 4 } x$. Then, $f$ is an increasing function in the interval:
(1) $]\frac { 5 \pi } { 8 } , \frac { 3 \pi } { 4 } [$
(2) $]\frac { \pi } { 2 } , \frac { 5 \pi } { 8 } [$
(3) $]\frac { \pi } { 4 } , \frac { \pi } { 2 } [$
(4) $]0 , \frac { \pi } { 4 } [$

The function, $f ( x ) = ( 3 x - 7 ) x ^ { \frac { 2 } { 3 } } , x \in \mathrm { R }$, is increasing for all $x$ lying in
(1) $( - \infty , 0 ) \cup \left( \frac { 14 } { 15 } , \infty \right)$
(2) $( - \infty , 0 ) \cup \left( \frac { 3 } { 7 } , \infty \right)$
(3) $\left( - \infty , \frac { 14 } { 15 } \right)$
(4) $\left( - \infty , - \frac { 14 } { 15 } \right) \cup ( 0 , \infty )$

Let $f$ be a real valued function, defined on $R - \{ - 1,1 \}$ and given by $f ( x ) = 3 \log _ { \mathrm { e } } \left| \frac { x - 1 } { x + 1 } \right| - \frac { 2 } { x - 1 }$. Then in which of the following intervals, function $f ( x )$ is increasing?
(1) $( - \infty , - 1 ) \cup \left( \left[ \frac { 1 } { 2 } , \infty \right) - \{ 1 \} \right)$
(2) $( - \infty , \infty ) - \{ - 1,1 \}$
(3) $\left( - 1 , \frac { 1 } { 2 } \right]$
(4) $\left( - \infty , \frac { 1 } { 2 } \right] - \{ - 1 \}$

If $R$ is the least value of $a$ such that the function $f ( x ) = x ^ { 2 } + \mathrm { a } x + 1$ is increasing on $[ 1,2 ]$ and $S$ is the greatest value of $a$ such that the function $f ( x ) = x ^ { 2 } + a x + 1$ is decreasing on $[ 1,2 ]$, then the value of $| R - S |$ is

Let $\lambda ^ { * }$ be the largest value of $\lambda$ for which the function $f _ { \lambda } ( x ) = 4 \lambda x ^ { 3 } - 36 \lambda x ^ { 2 } + 36 x + 48$ is increasing for all $x \in \mathbb { R }$. Then $f _ { \lambda ^ { * } } ( 1 ) + f _ { \lambda ^ { * } } ( - 1 )$ is equal to:
(1) 36
(2) 48
(3) 64
(4) 72

For the function $f ( x ) = 4 \log _ { e } ( x - 1 ) - 2 x ^ { 2 } + 4 x + 5 , x > 1$, which one of the following is NOT correct?
(1) $f ( x )$ is increasing in $( 1,2 )$ and decreasing in $( 2 , \infty )$
(2) $f ( x ) = - 1$ has exactly two solutions
(3) $f ^ { \prime } ( \mathrm { e } ) - f ^ { \prime \prime } ( 2 ) < 0$
(4) $f ( x ) = 0$ has a root in the interval $( e , e + 1 )$

If the maximum value of $a$, for which the function $f _ { a } ( x ) = \tan ^ { - 1 } 2 x - 3 a x + 7$ is non-decreasing in $\left[ - \frac { \pi } { 6 } , \frac { \pi } { 6 } \right]$, is $\bar { a }$, then $f _ { \bar { a } } \left( \frac { \pi } { 8 } \right)$ is equal to
(1) $8 - \frac { 9 \pi } { 49 + \pi ^ { 2 } }$
(2) $8 - \frac { 4 \pi } { 94 + \pi ^ { 2 } }$
(3) $8 \frac { 1 + \pi ^ { 2 } } { 9 + \pi ^ { 2 } }$
(4) $8 - \frac { \pi } { 4 }$

The function $f ( x ) = x e ^ { x ( 1 - x ) } , x \in R$, is
(1) increasing in $\left( - \frac { 1 } { 2 } , 1 \right)$
(2) decreasing in $\left( \frac { 1 } { 2 } , 2 \right)$
(3) increasing in $\left( - 1 , - \frac { 1 } { 2 } \right)$
(4) decreasing in $\left( - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right)$

Let $f : ( 0,1 ) \rightarrow \mathbb { R }$ be a function defined by $f ( x ) = \frac { 1 } { 1 - e ^ { - x } }$, and $g ( x ) = ( f ( - x ) - f ( x ) )$. Consider two statements (I) $g$ is an increasing function in $( 0,1 )$ (II) $g$ is one-one in $( 0,1 )$ Then,
(1) Only (I) is true
(2) Only (II) is true
(3) Neither (I) nor (II) is true
(4) Both (I) and (II) are true

The function $f ( x ) = \frac { x } { x ^ { 2 } - 6 x - 16 } , x \in \mathbb { R } - \{ - 2,8 \}$
(1) decreases in $( - 2,8 )$ and increases in $( - \infty , - 2 ) \cup ( 8 , \infty )$
(2) decreases in $( - \infty , - 2 ) \cup ( - 2,8 ) \cup ( 8 , \infty )$
(3) decreases in $( - \infty , - 2 )$ and increases in $( 8 , \infty )$
(4) increases in $( - \infty , - 2 ) \cup ( - 2,8 ) \cup ( 8 , \infty )$