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 $( - \infty , - 2 ) \cup ( - 2,8 ) \cup ( 8 , \infty )$

The interval in which the function $f ( x ) = x ^ { x } , x > 0$, is strictly increasing is
(1) $\left( 0 , \frac { 1 } { e } \right]$
(2) $( 0 , \infty )$
(3) $\left[ \frac { 1 } { e } , \infty \right)$
(4) $\left[ \frac { 1 } { e ^ { 2 } } , 1 \right)$

Let $\mathrm { g } ( \mathrm { x } ) = 3 \mathrm { f } ^ { \mathrm { x } } + \mathrm { f } ( 3 - \mathrm { x } )$ and $\mathrm { f } ^ { \prime \prime } ( \mathrm { x } ) > 0$ for all $\mathrm { x } \in ( 0,3 )$. If g is decreasing in ( $0 , \alpha$ ) and increasing in $( \alpha , 3 )$, then $8 \alpha$ is
(1) 24
(2) 0
(3) 18
(4) 20

For the function $f ( x ) = ( \cos x ) - x + 1 , x \in \mathbb { R }$, between the following two statements (S1) $f ( x ) = 0$ for only one value of $x$ in $[ 0 , \pi ]$. (S2) $f ( x )$ is decreasing in $\left[ 0 , \frac { \pi } { 2 } \right]$ and increasing in $\left[ \frac { \pi } { 2 } , \pi \right]$.
(1) Both (S1) and (S2) are correct.
(2) Both (S1) and (S2) are incorrect.
(3) Only (S2) is correct.
(4) Only (S1) is correct.

Let the set of all values of $p$, for which $f ( x ) = \left( p ^ { 2 } - 6 p + 8 \right) \left( \sin ^ { 2 } 2 x - \cos ^ { 2 } 2 x \right) + 2 ( 2 - p ) x + 7$ does not have any critical point, be the interval $( a , b )$. Then $16 a b$ is equal to $\_\_\_\_$

Let $(2, 3)$ be the largest open interval in which the function $f(x) = 2\log_{\mathrm{e}}(x - 2) - x^{2} + ax + 1$ is strictly increasing and $(\mathrm{b}, \mathrm{c})$ be the largest open interval, in which the function $\mathrm{g}(x) = (x - 1)^{3}(x + 2 - \mathrm{a})^{2}$ is strictly decreasing. Then $100(a + b - c)$ is equal to:
(1) 420
(2) 360
(3) 160
(4) 280

Q74. For the function $f ( x ) = \sin x + 3 x - \frac { 2 } { \pi } \left( x ^ { 2 } + x \right)$, where $x \in \left[ 0 , \frac { \pi } { 2 } \right]$, consider the following two statements : (I) f is increasing in ( $0 , \frac { \pi } { 2 }$ ). (II) $f ^ { \prime }$ is decreasing in ( $0 , \frac { \pi } { 2 }$ ).
Between the above two statements,
(1) only (II) is true.
(2) only (I) is true.
(3) neither (I) nor (II) is true.
(4) both (I) and (II) are true

Q74. The interval in which the function $f ( x ) = x ^ { x } , x > 0$, is strictly increasing is
(1) $\left( 0 , \frac { 1 } { e } \right]$
(2) $( 0 , \infty )$
(3) $\left. \left[ \frac { 1 } { e } , \infty \right) \right] _ { V }$
(4) $\left[ \frac { 1 } { e ^ { 2 } } , 1 \right)$

Consider vectors $\vec{a}$ and $\vec{b}$ on the coordinate plane satisfying $|\vec{a}| + |\vec{b}| = 9$ and $|\vec{a} - \vec{b}| = 7$. Let $|\vec{a}| = x$, where $1 < x < 8$, and let the angle between $\vec{a}$ and $\vec{b}$ be $\theta$. Using the triangle formed by vectors $\vec{a}$, $\vec{b}$, and $\vec{a} - \vec{b}$, we can express $\cos\theta$ in terms of $x$ as $f(x) = \frac{c}{9x - x^2} + d$, where $c$ and $d$ are constants with $c > 0$, with domain $\{x \mid 1 < x < 8\}$. Explain where $f(x)$ is increasing and decreasing in its domain. Determine the value of $x$ for which the angle $\theta$ between $\vec{a}$ and $\vec{b}$ is maximum. (Non-multiple choice question, 4 points)

The function $\frac { 1 - x } { \sqrt [ 3 ] { x ^ { 2 } } }$ is defined for all $x \neq 0$. The complete set of values of $x$ for which the function is decreasing is
A $x \leq - 2 , x > 0$ B $- 2 \leq x < 0$ C $x \leq 1 , x \neq 0$ D $x \geq 1$ E $- 2 \leq x \leq 1 , \quad x \neq 0$ F $x \leq - 2 , x \geq 1$

The functions $f$ and $g$ are given by $f ( x ) = 3 x ^ { 2 } + 12 x + 4$ and $g ( x ) = x ^ { 3 } + 6 x ^ { 2 } + 9 x - 8$.
What is the complete set of values of $x$ for which one of the functions is increasing and the other decreasing?
A $x \geq - 1$
B $x \leq - 1$
C $\quad - 3 \leq x \leq - 2 , x \geq - 1$
D $x \leq - 2 , x \geq - 1$
E $\quad x \leq - 3 , - 2 \leq x \leq - 1$
F $x \leq - 3 , x \geq - 2$
G $\quad - 2 \leq x \leq - 1$

The function f is given by
$$\mathrm { f } ( x ) = x ^ { \frac { 1 } { 7 } } \left( x ^ { 2 } - x + 1 \right)$$
Find the fraction of the interval $0 < x < 1$ for which $\mathrm { f } ( x )$ is decreasing.
A $\frac { 2 } { 15 }$
B $\frac { 1 } { 5 }$
C $\frac { 1 } { 3 }$
D $\frac { 1 } { 2 }$
E $\frac { 2 } { 3 }$
F $\frac { 4 } { 5 }$
G $\frac { 13 } { 15 }$

3

For real numbers $t$ satisfying $-1 \leq t \leq 1$, let $$x(t) = (1+t)\sqrt{1+t}, \quad y(t) = 3(1+t)\sqrt{1-t}$$ Consider the point $\mathrm{P}(x(t),\ y(t))$ in the coordinate plane.

(1) Show that the function $\dfrac{y(t)}{x(t)}$ of $t$ on $-1 < t \leq 1$ is strictly decreasing.

(2) Let $f(t)$ be the distance from the origin to $\mathrm{P}$. Investigate the monotonicity of the function $f(t)$ of $t$ on $-1 \leq t \leq 1$, and find its maximum value.

(3) Let $C$ be the locus of $\mathrm{P}$ as $t$ ranges over $-1 \leq t \leq 1$, and let $D$ be the region enclosed by $C$ and the $x$-axis. When $D$ is rotated $90°$ clockwise about the origin, find the area of the region swept out by $D$.
%% Page 4

Let $a$ and $b$ be real numbers. The function $f$ defined on the set of real numbers as
$$f(x) = x^{3} + 9x^{2} + ax + b$$
takes positive values on positive real numbers and negative values on negative real numbers.
What is the smallest integer value that $a$ can take?
A) 9 B) 13 C) 17 D) 21 E) 25

Let $a$ and $b$ be real numbers. The function $f$ defined as
$$f(x) = ax^{3} + bx^{2} + x + 7$$
is always increasing.
If $f(-1) = 0$, what is the sum of the different integer values that $b$ can take?
A) 11 B) 13 C) 15 D) 17 E) 19