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

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 $2y = \cot^{-1}\left(\frac{\sqrt{3}\cos x + \sin x}{\cos x - \sqrt{3}\sin x}\right)$, $\forall x \in \left(0, \frac{\pi}{2}\right)$, then $\frac{dy}{dx}$ is equal to
(1) $\frac{\pi}{6} - x$
(2) $2x - \frac{\pi}{3}$
(3) $x - \frac{\pi}{6}$
(4) None of these

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 a function $f(x)$ defined by $$f(x) = \begin{cases} ae^{x} + be^{-x}, & -1 \leq x < 1 \\ cx^{2}, & 1 \leq x \leq 3 \\ ax^{2} + 2cx, & 3 < x \leq 4 \end{cases}$$ be continuous for some $a, b, c \in R$ and $f'(0) + f'(2) = e$, then the value of $a$ is
(1) $\frac{1}{e^{2} - 3e + 13}$
(2) $\frac{e}{e^{2} - 3e - 13}$
(3) $\frac{e}{e^{2} + 3e + 13}$
(4) $\frac{e}{e^{2} - 3e + 13}$

The derivative of $\tan^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$ with respect to $\tan^{-1}\left(\frac{2x\sqrt{1-x^2}}{1-2x^2}\right)$ at $x = \frac{1}{2}$ is:
(1) $\frac{2\sqrt{3}}{5}$
(2) $\frac{\sqrt{3}}{12}$
(3) $\frac{2\sqrt{3}}{3}$
(4) $\frac{\sqrt{3}}{10}$

If the function $f ( x ) = \left\{ \begin{array} { c c } k _ { 1 } ( x - \pi ) ^ { 2 } - 1 , & x \leq \pi \\ k _ { 2 } \cos x , & x > \pi \end{array} \right.$ is twice differentiable, then the ordered pair $\left( k _ { 1 } , k _ { 2 } \right)$ is equal to:
(1) $\left( \frac { 1 } { 2 } , 1 \right)$
(2) $( 1,0 )$
(3) $\left( \frac { 1 } { 2 } , - 1 \right)$
(4) $( 1,1 )$

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

If the function $f ( x ) = \left\{ \begin{array} { l l } \frac { \log _ { e } \left( 1 - x + x ^ { 2 } \right) + \log _ { e } \left( 1 + x + x ^ { 2 } \right) } { \sec x - \cos x } , & x \in \left( \frac { - \pi } { 2 } , \frac { \pi } { 2 } \right) - \{ 0 \} \\ k & , x = 0 \end{array} \right.$ is continuous at $x = 0$, then $k$ is equal to:
(1) 1
(2) $- 1$
(3) $e$
(4) 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) $4$

If $a = \lim _ { n \rightarrow \infty } \sum _ { k = 1 } ^ { n } \frac { 2 n } { n ^ { 2 } + k ^ { 2 } }$ and $f ( x ) = \sqrt { \frac { 1 - \cos x } { 1 + \cos x } } , x \in ( 0,1 )$, then:
(1) $2 \sqrt { 2 } f \left( \frac { a } { 2 } \right) = f ^ { \prime } \left( \frac { a } { 2 } \right)$
(2) $f \left( \frac { a } { 2 } \right) f ^ { \prime } \left( \frac { a } { 2 } \right) = \sqrt { 2 }$
(3) $\sqrt { 2 } f \left( \frac { a } { 2 } \right) = f ^ { \prime } \left( \frac { a } { 2 } \right)$
(4) $f \left( \frac { a } { 2 } \right) = \sqrt { 2 } f ^ { \prime } \left( \frac { a } { 2 } \right)$

If $y ( x ) = \left( x ^ { x } \right) ^ { x } , x > 0$ then $\frac { d ^ { 2 } x } { d y ^ { 2 } } + 20$ at $x = 1$ is equal to

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 $f$ and $g$ be twice differentiable functions on $R$ such that $f ^ { \prime \prime } ( x ) = g ^ { \prime \prime } ( x ) + 6 x$ $f ^ { \prime } ( 1 ) = 4 g ^ { \prime } ( 1 ) - 3 = 9$ $f ( 2 ) = 3 g ( 2 ) = 12$ Then which of the following is NOT true ? (1) $g ( - 2 ) - f ( - 2 ) = 20$ (2) If $- 1 < x < 2$, then $| f ( x ) - g ( x ) | < 8$ (3) $\left| f ^ { \prime } ( x ) - g ^ { \prime } ( x ) \right| < 6 \Rightarrow - 1 < x < 1$ (4) There exists $x _ { 0 } \in \left( 1 , \frac { 3 } { 2 } \right)$ such that $f \left( x _ { 0 } \right) = g \left( x _ { 0 } \right)$

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 the total maximum value of the function $f ( x ) = \left( \frac { \sqrt { 3 e } } { 2 \sin x } \right) ^ { \sin ^ { 2 } x } , x \in \left( 0 , \frac { \pi } { 2 } \right)$, is $\frac { k } { e }$, then $\left( \frac { k } { e } \right) ^ { 8 } + \frac { k ^ { 8 } } { e ^ { 5 } } + k ^ { 8 }$ is equal to
(1) $e ^ { 3 } + e ^ { 6 } + e ^ { 11 }$
(2) $e ^ { 5 } + e ^ { 6 } + e ^ { 11 }$
(3) $e ^ { 3 } + e ^ { 6 } + e ^ { 10 }$
(4) $e ^ { 3 } + e ^ { 5 } + e ^ { 11 }$

Let $y = \log _ { e } \left( \frac { 1 - x ^ { 2 } } { 1 + x ^ { 2 } } \right) , - 1 < x < 1$. Then at $x = \frac { 1 } { 2 }$, the value of $225 \left( y ^ { \prime } - y ^ { \prime \prime } \right)$ is equal to
(1) 732
(2) 746
(3) 742
(4) 736

If $f ( x ) = \left\{ \begin{array} { l } x ^ { 3 } \sin \left( \frac { 1 } { x } \right) , x \neq 0 \\ 0 \quad , x = 0 \end{array} \right.$ then
(1) $f ^ { \prime \prime } \left( \frac { 2 } { \pi } \right) = \frac { 24 - \pi ^ { 2 } } { 2 \pi }$
(2) $f ^ { \prime \prime } \left( \frac { 2 } { \pi } \right) = \frac { 12 - \pi ^ { 2 } } { 2 \pi }$
(3) $f ^ { \prime \prime } ( 0 ) = 1$
(4) $f ^ { \prime \prime } ( 0 ) = 0$

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) $9 e ^ { \pi / 6 }$

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) $<$