Let $f(x) = \sin(\pi\cos x)$ and $g(x) = \cos(2\pi\sin x)$ be two functions defined for $x > 0$. Define the following sets whose elements are written in the increasing order: $$\begin{array}{ll} X = \{x : f(x) = 0\}, & Y = \{x : f'(x) = 0\} \\ Z = \{x : g(x) = 0\}, & W = \{x : g'(x) = 0\} \end{array}$$
List-I contains the sets $X$, $Y$, $Z$ and $W$. List-II contains some information regarding these sets.
List-I: (I) $X$ (II) $Y$ (III) $Z$ (IV) $W$
List-II: (P) $\supseteq \left\{\frac{\pi}{2}, \frac{3\pi}{2}, 4\pi, 7\pi\right\}$ (Q) an arithmetic progression (R) NOT an arithmetic progression (S) $\supseteq \left\{\frac{\pi}{6}, \frac{7\pi}{6}, \frac{13\pi}{6}\right\}$ (T) $\supseteq \left\{\frac{\pi}{3}, \frac{2\pi}{3}, \pi\right\}$ (U) $\supseteq \left\{\frac{\pi}{6}, \frac{3\pi}{4}\right\}$
Which of the following is the only CORRECT combination?
(A) (III), (R), (U)
(B) (IV), (P), (R), (S)
(C) (III), (P), (Q), (U)
(D) (IV), (Q), (T)

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

If $y = \left( \frac { x } { x + 1 } \right) ^ { x } + x ^ { \left( \frac { x } { x + 1 } \right) }$, find $\frac { d y } { d x }$ at $x = 1$.
(1) $\frac { 1 } { 2 } + \ln 2$
(2) $1 + \frac { 1 } { 2 } \ln 2$
(3) $1 - \frac { 1 } { 2 } \ln 2$
(4) $\frac { 1 } { 2 } - \ln 2$

If for $x \in \left(0, \dfrac{1}{4}\right)$, the derivative of $\tan^{-1}\left(\dfrac{6x\sqrt{x}}{1 - 9x^3}\right)$ is $\sqrt{x} \cdot g(x)$, then $g(x)$ equals:
(1) $\dfrac{9}{1 + 9x^3}$
(2) $\dfrac{3x\sqrt{x}}{1 - 9x^3}$
(3) $\dfrac{3x}{1 - 9x^3}$
(4) $\dfrac{3}{1 + 9x^3}$

If $y = \left[ x + \sqrt { x ^ { 2 } - 1 } \right] ^ { 15 } + \left[ x - \sqrt { x ^ { 2 } - 1 } \right] ^ { 15 }$, then $\left( x ^ { 2 } - 1 \right) \frac { d ^ { 2 } y } { d x ^ { 2 } } + x \frac { d y } { d x }$ is equal to
(1) $224 y ^ { 2 }$
(2) $125 y$
(3) $225 y$
(4) $225 y ^ { 2 }$

If $y(\alpha) = \sqrt { 2 \left( \frac { \tan \alpha + \cot \alpha } { 1 + \tan ^ { 2 } \alpha } \right) + \frac { 1 } { \sin ^ { 2 } \alpha } }$, $\alpha \in \left( \frac { 3 \pi } { 4 } , \pi \right)$, then $\frac { d y } { d \alpha }$ at $\alpha = \frac { 5 \pi } { 6 }$ is
(1) 4
(2) $\frac { 4 } { 3 }$
(3) $-4$
(4) $- \frac { 1 } { 4 }$

If $y ^ { 2 } + \log _ { e } \left( \cos ^ { 2 } x \right) = y , \quad x \in \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$ then:
(1) $y \prime \prime ( 0 ) = 0$
(2) $| y \prime ( 0 ) | + | y \prime \prime ( 0 ) | = 1$
(3) $| y \prime \prime ( 0 ) | = 2$
(4) $| y \prime ( 0 ) | + | y \prime \prime ( 0 ) | = 3$

Let $f : S \rightarrow S$ where $S = ( 0 , \infty )$ be a twice differentiable function such that $f ( x + 1 ) = x f ( x )$. If $g : S \rightarrow R$ be defined as $g ( x ) = \log _ { \mathrm { e } } f ( x )$, then the value of $\left| g ^ { \prime \prime } ( 5 ) - g ^ { \prime \prime } ( 1 ) \right|$ is equal to :
(1) $\frac { 205 } { 144 }$
(2) $\frac { 197 } { 144 }$
(3) $\frac { 187 } { 144 }$
(4) 1

If $y ( x ) = \cot ^ { - 1 } \left( \frac { \sqrt { 1 + \sin x } + \sqrt { 1 - \sin x } } { \sqrt { 1 + \sin x } - \sqrt { 1 - \sin x } } \right) , x \in \left( \frac { \pi } { 2 } , \pi \right)$, then $\frac { d y } { d x }$ at $x = \frac { 5 \pi } { 6 }$ is: (1) 0 (2) - 1 (3) $\frac { - 1 } { 2 }$ (4) $\frac { 1 } { 2 }$

If $f ( x ) = \left\{ \begin{array} { l l } \int _ { 0 } ^ { x } ( 5 + | 1 - t | ) d t , & x > 2 \\ 5 x + 1 , & x \leq 2 \end{array} \right.$, then
(1) $f ( x )$ is not continuous at $x = 2$
(2) $f ( x )$ is everywhere differentiable
(3) $f ( x )$ is continuous but not differentiable at $x = 2$
(4) $f ( x )$ is not differentiable at $x = 1$

If $y = \tan ^ { - 1 } \left( \sec x ^ { 3 } - \tan x ^ { 3 } \right) , \frac { \pi } { 2 } < x ^ { 3 } < \frac { 3 \pi } { 2 }$, then
(1) $x y ^ { \prime \prime } + 2 y ^ { \prime } = 0$
(2) $x ^ { 2 } y ^ { \prime \prime } - 6 y + \frac { 3 \pi } { 2 } = 0$
(3) $x ^ { 2 } y ^ { \prime \prime } - 6 y + 3 \pi = 0$
(4) $x y ^ { \prime \prime } - 4 y ^ { \prime } = 0$

The value of $\log _ { e } 2 \cdot \frac { \mathrm { d } } { \mathrm { d } x } \log _ { \cos x } \operatorname { cosec } x$ at $x = \frac { \pi } { 4 }$ is
(1) $- 2 \sqrt { 2 }$
(2) $2 \sqrt { 2 }$
(3) $-4$
(4) $4$

Let $f(x) = 2x + \tan^{-1} x$ and $g(x) = \log_e\left(\sqrt{1 + x^2} + x\right)$, $x \in [0, 3]$. Then
(1) There exists $x \in (0, 3)$ such that $f'(x) < g'(x)$
(2) $\max f(x) > \max g(x)$
(3) There exist $0 < x_1 < x_2 < 3$ such that $f(x) < g(x)$, $\forall x \in (x_1, x_2)$
(4) $\min f'(x) = 1 + \max g'(x)$

Let $y = f(x) = \sin^3\left(\frac{\pi}{3}\cos\left(\frac{\pi}{3\sqrt{2}}\left(-4x^3 + 5x^2 + 1\right)^{3/2}\right)\right)$. Then, at $x = 1$,
(1) $2y' + \sqrt{3}\pi^2 y = 0$
(2) $2y' + 3\pi^2 y = 0$
(3) $\sqrt{2}y' - 3\pi^2 y = 0$
(4) $y' + 3\pi^2 y = 0$

If the function $f ( x ) = \left\{ \begin{array} { c l } ( 1 + | \cos x | ) \frac { \lambda } { | \cos x | } , & 0 < x < \frac { \pi } { 2 } \\ \mu , & x = \frac { \pi } { 2 } \\ e ^ { \frac { \cot 6 x } { \cot 4 x } } , & \frac { \pi } { 2 } < x < \pi \end{array} \right.$ is continuous at $x = \frac { \pi } { 2 }$, then $9 \lambda + 6 \log _ { e } \mu + \mu ^ { 6 } - e ^ { 6 \lambda }$ is equal to
(1) 11
(2) 8
(3) $2 e ^ { 4 } + 8$
(4) 10

Let $f ( x ) = \frac { \sin x + \cos x - \sqrt { 2 } } { \sin x - \cos x } , x \in [ 0 , \pi ] - \left\{ \frac { \pi } { 4 } \right\}$, then $f \left( \frac { 7 \pi } { 12 } \right) f ^ { \prime \prime } \left( \frac { 7 \pi } { 12 } \right)$ is equal to
(1) $\frac { 2 } { 9 }$
(2) $\frac { - 2 } { 3 }$
(3) $\frac { - 1 } { 3 \sqrt { 3 } }$
(4) $\frac { 2 } { 3 \sqrt { 3 } }$

If $\lim _ { x \rightarrow 0 } \frac { 3 + \alpha \sin x + \beta \cos x + \log _ { e } ( 1 - x ) } { 3 \tan ^ { 2 } x } = \frac { 1 } { 3 }$, then $2 \alpha - \beta$ is equal to :
(1) 2
(2) 7
(3) 5
(4) 1

If $y ( \theta ) = \frac { 2 \cos \theta + \cos 2 \theta } { \cos 3 \theta + 4 \cos 2 \theta + 5 \cos \theta + 2 }$, then at $\theta = \frac { \pi } { 2 } , y ^ { \prime \prime } + y ^ { \prime } + y$ is equal to :
(1) $\frac { 1 } { 2 }$
(2) 1
(3) 2
(4) $\frac { 3 } { 2 }$

If $\log _ { e } y = 3 \sin ^ { - 1 } x$, then $\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - x y ^ { \prime }$ at $x = \frac { 1 } { 2 }$ is equal to
(1) $3 e ^ { \pi / 6 }$
(2) $9 e ^ { \pi / 2 }$
(3) $3 e ^ { \pi / 2 }$
(4) $9 e ^ { \pi / 6 }$

Q73. If the function $f ( x ) = \left\{ \begin{array} { l l } \frac { 72 ^ { x } - 9 ^ { x } - 8 ^ { x } + 1 } { \sqrt { 2 } - \sqrt { 1 + \cos x } } , & x \neq 0 \\ a \log _ { e } 2 \log _ { e } 3 & , x = 0 \end{array} \right.$ is continuous at $x = 0$, then the value of $a ^ { 2 }$ is equal to
(1) 968
(2) 1152
(3) 746
(4) 1250

Q73. If $y ( \theta ) = \frac { 2 \cos \theta + \cos 2 \theta } { \cos 3 \theta + 4 \cos 2 \theta + 5 \cos \theta + 2 }$, then at $\theta = \frac { \pi } { 2 } , y ^ { \prime \prime } + y ^ { \prime } + y$ is equal to :
(1) $\frac { 1 } { 2 }$
(2) 1
(3) 2
(4) $\frac { 3 } { 2 }$

Q73. If $\log _ { e } y = 3 \sin ^ { - 1 } x$, then $\left( 1 - x ^ { 2 } \right) y ^ { \prime \prime } - x y ^ { \prime }$ at $x = \frac { 1 } { 2 }$ is equal to
(1) $3 e ^ { \pi / 6 }$
(2) $9 e ^ { \pi / 2 }$
(3) $3 e ^ { \pi / 2 }$
(4) $9 e ^ { \pi / 6 }$

Q1 Let $f(x)=\log(4x-\log x)$, where $\log$ is the natural logarithm. We are to find a local extremum of $f(x)$ by using $f''(x)$.
For $\mathbf{K}$ and $\mathbf{L}$, choose the most appropriate answer from among the choices (0)$\sim$(6) below.
First of all, we have
$$\begin{aligned} f'(x) &= \frac{\mathbf{A}-\frac{\mathbf{B}}{x}}{4x-\log x} \\ f''(x) &= \frac{\frac{1}{x^{\mathbf{C}}}(4x-\log x)-\left(\mathbf{A}-\frac{\square}{x}\right)^{\mathbf{D}}}{(4x-\log x)^2} \end{aligned}$$
which give
$$\begin{aligned} f'\left(\frac{\mathbf{E}}{\mathbf{F}}\right) &= 0 \\ f''\left(\frac{\mathbf{E}}{\mathbf{F}}\right) &= \frac{\mathbf{GH}}{\mathbf{I}+\log\mathbf{J}}. \end{aligned}$$
Since
$$f''\left(\frac{\mathbf{E}}{\mathbf{F}}\right) \mathbf{K} \, 0,$$
$f(x)$ has a $\mathbf{L}$ at $x=\frac{\mathbf{E}}{\mathbf{F}}$, and this value is $\log(\mathbf{M}+\log\mathbf{N})$.
(0) $=$ (1) $>$ (2) $\geqq$ (3) $<$ (4) $\leqq$ (5) local maximum (6) local minimum

Let $a$ be a constant. Assume that the function
$$f(x) = 2\sin^3 x + a\sin 2x + \frac{9}{2}\cos 2x - 9\cos x - 2ax + 6$$
takes a local extremum at $x = \frac{\pi}{3}$. We consider about the maximum and minimum values of $f(x)$ over the interval $0 \leqq x \leqq \frac{\pi}{2}$.
(1) Since $f(x)$ takes a local extremum at $x = \frac{\pi}{3}$, it follows that $a = \frac{\mathbf{A}}{\mathbf{B}}$.
Hence the derivative $f'(x)$ of $f(x)$ can be expressed as
$$f'(x) = \mathbf{C}\cos x(\mathbf{D}\cos x - 1)(\sin x - \mathbf{E}).$$
(2) It can be seen from the result of (1) that $f(x)$ over $0 \leqq x \leqq \frac{\pi}{2}$ takes the maximum value at $x = \mathbf{F}$ and the minimum value at $x = \mathbf{G}$, where $\mathbf{F}$ and $\mathbf{G}$ are the appropriate expressions from among (0) $\sim$ (4) below. (0) $0$
(1) $\frac{\pi}{6}$
(2) $\frac{\pi}{4}$
(3) $\frac{\pi}{3}$
(4) $\frac{\pi}{2}$

For each of $\mathbf{A} \sim \mathbf{I}$ in the following sentences, choose the appropriate answer from among (0) $\sim$ (9) at the bottom of this page.
We are to compare the magnitudes of $a ^ { a + 1 }$ and $( a + 1 ) ^ { a }$ by using the properties of the function $f ( x ) = \dfrac { \log x } { x }$, where $a > 0$.
(1) Since the derivative of $f ( x )$ is
$$f ^ { \prime } ( x ) = \frac { \mathbf { A } - \log x } { x^{\mathbf{B}} } ,$$
the interval on $x$ in which $f ( x )$ monotonically increases is
$$\mathbf { C } < x \leqq \mathbf { D }$$
and the interval on $x$ in which $f ( x )$ monotonically decreases is
$$\mathbf { E } \leq x .$$
(2) When we set $p = a ^ { a + 1 }$, $q = ( a + 1 ) ^ { a }$, we have
$$\log p - \log q = \left( a ^ { \mathbf { F } } + a \right) \{ f ( a ) - f ( a + \mathbf { G } ) \} .$$
Hence we see that
$$\text { if } \quad 0 < a < \tfrac{3}{2} \quad \text { then } \quad p \quad \mathbf { H } \quad q ,$$
and
$$\text { if } \quad 3 < a \quad \text { then } \quad p \quad \mathbf { I } \quad q .$$
(0) 0 (1) 1 (2) 2 (3) 3 (4) $e$ (5) $e + 1$ (6) $\dfrac{1}{e}$ (7) $>$ (8) $=$ (9) $<$