Given that $f(x), g(x)$ have domain $\mathbf{R}$, and $f(x) + g(2-x) = 5, g(x) - f(x-4) = 7$. If the graph of $y = g(x)$ is symmetric about the line $x = 2$, and $g(2) = 4$, then $\sum_{k=1}^{22} f(k) =$
A. $-21$
B. $-22$
C. $-23$
D. $-24$

12. Let the function $f ( x )$ and its derivative $f ^ { \prime } ( x )$ both have domain $\mathbf { R }$. Let $g ( x ) = f ^ { \prime } ( x )$. If $f \left( \frac { 3 } { 2 } - 2 x \right)$ and $g ( 2 + x )$ are both even functions, then
A. $f ( 0 ) = 0$
B. $g \left( - \frac { 1 } { 2 } \right) = 0$
C. $f ( - 1 ) = f ( 4 )$
D. $g ( - 1 ) = g ( 2 )$

III. Fill-in-the-Blank Questions: This section contains 4 questions, each worth 5 points, for a total of 20 points.

If $f ( x ) = \ln \left| a + \frac { 1 } { 1 - x } \right| + b$ is an odd function, then $a = $ $\_\_\_\_$ . $b = $ $\_\_\_\_$ .

If $y = (x - 1)^{2} + ax + \sin\left(x + \frac{\pi}{2}\right)$ is an even function, then $a =$ $\_\_\_\_$ .

Let $f(x)$ be an even function defined on $\mathbb{R}$ with period $2$. When $2 \leq x \leq 3$, $f(x) = 5 - 2x$. Then $f\left(-\frac{3}{4}\right) =$
A. $-\frac{1}{2}$
B. $-\frac{1}{4}$
C. $\frac{1}{4}$
D. $\frac{1}{2}$

Let $f(x)$ be an even function defined on $\mathbf{R}$ with period 2. When $2 \leq x \leq 3$, $f(x) = 5 - 2x$. Then $f\left(-\frac{3}{4}\right) =$
A. $-\frac{1}{2}$
B. $-\frac{1}{4}$
C. $\frac{1}{4}$
D. $\frac{1}{2}$

113. The curve $y = \sqrt{4-x}$ is translated $k$ units in the horizontal direction and $k-2$ units in the vertical direction such that the new curve intersects its own inverse at a point with width 1. Then the resulting curve is shifted 1 unit in the downward direction. The length of the intersection point of the curve with the $x$-axis is:
(1) $-4$ (2) $-3$ (3) $1$ (4) $2$

8. For the function $f(x) = \sqrt{x - 2\sqrt{mx-1}}$ on its natural domain, the line $y = 12 - x$ intersects the graph at 10 points on the $x$-axis. What is the value of $f(m+4)$?
(1) $\dfrac{1}{2}$ (2) $\dfrac{1}{4}$ (3) $2$ (4) $1$

For every real number $x \neq - 1$, let $f ( x ) = \frac { x } { x + 1 }$. Write $f _ { 1 } ( x ) = f ( x )$ and for $n \geq 2 , f _ { n } ( x ) = f \left( f _ { n - 1 } ( x ) \right)$. Then,
$$f _ { 1 } ( - 2 ) \cdot f _ { 2 } ( - 2 ) \cdots \cdots f _ { n } ( - 2 )$$
must equal
(A) $\frac { 2 ^ { n } } { 1 \cdot 3 \cdot 5 \cdots \cdot ( 2 n - 1 ) }$
(B) 1
(C) $\frac { 1 } { 2 } \binom { 2 n } { n }$
(D) $\binom { 2 n } { n }$.

12. Let $0 < a < \Pi / 2$ be a fixed angle. If $P = ( \cos \theta , \sin \theta )$ and $Q = ( \cos ( a - \theta ) , \sin ( a - \theta ) )$ then Q is obtained from P by
(A) Clockwise rotation around origin through an angle a
(B) Anticlockwise rotation around origin through an angle a
(C) Reflection in the line through origin with slope tan a
(D) Reflection in the line through origin with slope $\tan \mathrm { a } / 2$

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 $( - \infty , \infty )$

Let $f ( x ) = \sin \left( \frac { \pi } { 6 } \sin \left( \frac { \pi } { 2 } \sin x \right) \right)$ for all $x \in \mathbb { R }$ and $g ( x ) = \frac { \pi } { 2 } \sin x$ for all $x \in \mathbb { R }$. Let $( f \circ g ) ( x )$ denote $f ( g ( x ) )$ and $( g \circ f ) ( x )$ denote $g ( f ( x ) )$. Then which of the following is (are) true?
(A) Range of $f$ is $\left[ - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right]$
(B) Range of $f \circ g$ is $\left[ - \frac { 1 } { 2 } , \frac { 1 } { 2 } \right]$
(C) $\lim _ { x \rightarrow 0 } \frac { f ( x ) } { g ( x ) } = \frac { \pi } { 6 }$
(D) There is an $x \in \mathbb { R }$ such that $( g \circ f ) ( x ) = 1$

Let the function $f: [0,1] \rightarrow \mathbb{R}$ be defined by $$f(x) = \frac{4^{x}}{4^{x} + 2}$$ Then the value of $$f\left(\frac{1}{40}\right) + f\left(\frac{2}{40}\right) + f\left(\frac{3}{40}\right) + \cdots + f\left(\frac{39}{40}\right) - f\left(\frac{1}{2}\right)$$ is $\_\_\_\_$

The domain of the function $f(x) = \sin^{-1}\left(\frac{|x| + 5}{x^{2} + 1}\right)$ is $(-\infty, -a] \cup [a, \infty)$, then $a$ is equal to
(1) $\frac{\sqrt{17}}{2}$
(2) $\frac{\sqrt{17} - 1}{2}$
(3) $\frac{1 + \sqrt{17}}{2}$
(4) $\frac{\sqrt{17}}{2} + 1$

The set of all values of $k$ for which $\left( \tan ^ { - 1 } x \right) ^ { 3 } + \left( \cot ^ { - 1 } x \right) ^ { 3 } = \mathrm { k } \pi ^ { 3 } , x \in R$, is the interval
(1) $\left[ \frac { 1 } { 32 } , \frac { 7 } { 8 } \right)$
(2) $\left( \frac { 1 } { 24 } , \frac { 13 } { 16 } \right)$
(3) $\left[ \frac { 1 } { 48 } , \frac { 13 } { 16 } \right]$
(4) $\left[ \frac { 1 } { 32 } , \frac { 9 } { 8 } \right)$

The range of $f(x) = 4 \sin ^ { - 1 } \left( \frac { x ^ { 2 } } { x ^ { 2 } + 1 } \right)$ is
(1) $[ 0,2 \pi ]$
(2) $[ 0 , \pi ]$
(3) $[ 0,2 \pi )$
(4) $[ 0 , \pi )$

Let $a_{1} = 1, a_{2}, a_{3}, a_{4}, \ldots$ be consecutive natural numbers. Then $\tan^{-1}\left(\frac{1}{1 + a_{1}a_{2}}\right) + \tan^{-1}\left(\frac{1}{1 + a_{2}a_{3}}\right) + \ldots + \tan^{-1}\left(\frac{1}{1 + a_{2021}a_{2022}}\right)$ is equal to
(1) $\frac{\pi}{4} - \cot^{-1}(2022)$
(2) $\cot^{-1}(2022) - \frac{\pi}{4}$
(3) $\tan^{-1}(2022) - \frac{\pi}{4}$
(4) $\frac{\pi}{4} - \tan^{-1}(2022)$

Let $D$ be the domain of the function $f ( x ) = \sin ^ { - 1 } \left( \log _ { 3 x } \left( \frac { 6 + 2 \log _ { 3 } x } { - 5 x } \right) \right)$. If the range of the function $g : D \rightarrow \mathbb { R }$ defined by $g ( x ) = x - [ x ]$, ([x] is the greatest integer function), is $( \alpha , \beta )$, then $\alpha ^ { 2 } + \frac { 5 } { \beta }$ is equal to
(1) 135
(2) 45
(3) 46
(4) 136

Let $f ( x ) = \frac { 1 } { 7 - \sin 5 x }$ be a function defined on $\mathbf { R }$. Then the range of the function $f ( x )$ is equal to:
(1) $\left[ \frac { 1 } { 7 } , \frac { 1 } { 6 } \right]$
(2) $\left[ \frac { 1 } { 8 } , \frac { 1 } { 5 } \right]$
(3) $\left[ \frac { 1 } { 7 } , \frac { 1 } { 5 } \right]$
(4) $\left[ \frac { 1 } { 8 } , \frac { 1 } { 6 } \right]$

Consider the function $f : \mathbb { R } \rightarrow \mathbb { R }$ defined by $f ( x ) = \frac { 2 x } { \sqrt { 1 + 9 x ^ { 2 } } }$. If the composition of $f , \underbrace { ( f \circ f \circ f \circ \cdots \circ f ) } _ { 10 \text { times } } ( x ) = \frac { 2 ^ { 10 } x } { \sqrt { 1 + 9 \alpha x ^ { 2 } } }$, then the value of $\sqrt { 3 \alpha + 1 }$ is equal to $\_\_\_\_$

Q71. Let $f ( x ) = \frac { 1 } { 7 - \sin 5 x }$ be a function defined on $\mathbf { R }$. Then the range of the function $f ( x )$ is equal to ;
(1) $\left[ \frac { 1 } { 7 } , \frac { 1 } { 6 } \right]$
(2) $\left[ \frac { 1 } { 8 } , \frac { 1 } { 5 } \right]$
(3) $\left[ \frac { 1 } { 7 } , \frac { 1 } { 5 } \right]$
(4) $\left[ \frac { 1 } { 8 } , \frac { 1 } { 6 } \right]$

Consider the quadratic function
$$y = 3 x ^ { 2 } - 6 .$$
(1) Suppose that the graph obtained by a parallel translation of the graph of $y = 3 x ^ { 2 } - 6$ passes through the two points $( 1,5 )$ and $( 4,14 )$. The quadratic function of this graph is
$$y = \mathbf { A } x ^ { 2 } - \mathbf { B C } x + \mathbf { D E } .$$
This graph is the parallel translation of the graph of $y = 3 x ^ { 2 } - 6$ by $\mathbf{F}$ in the $x$-direction and by $\mathbf { G }$ in the $y$-direction.
(2) The quadratic function having the graph which is symmetric to the graph of $y = 3 x ^ { 2 } - 6$ with respect to the straight line $y = c$ is
$$y = - \mathbf { H } x ^ { 2 } + \mathbf { I } c + \mathbf { J } .$$
When the graphs of the two quadratic functions (1) and (2) have just one common point, it follows that $c = \mathbf { K }$, and the coordinates of the common point are ( $\mathbf { L } , \mathbf { M }$ ).

Consider the quadratic function
$$y = 3 x ^ { 2 } - 6 .$$
(1) Suppose that the graph obtained by a parallel translation of the graph of $y = 3 x ^ { 2 } - 6$ passes through the two points $( 1,5 )$ and $( 4,14 )$. The quadratic function of this graph is
$$y = \mathbf { A } x ^ { 2 } - \mathbf { BC } x + \mathbf { D E } .$$
This graph is the parallel translation of the graph of $y = 3 x ^ { 2 } - 6$ by $\mathbf{F}$ in the $x$-direction and by $\mathbf { G }$ in the $y$-direction.
(2) The quadratic function having the graph which is symmetric to the graph of $y = 3 x ^ { 2 } - 6$ with respect to the straight line $y = c$ is
$$y = - \mathbf { H } x ^ { 2 } + \mathbf { I } c + \mathbf { J } .$$
When the graphs of the two quadratic functions (1) and (2) have just one common point, it follows that $c = \mathbf { K }$, and the coordinates of the common point are ( $\mathbf { L } , \mathbf { M }$ ).

3. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \\ \text { MATHEMATICS \& COMPUTER SCIENCE } \end{array} \right\}$ ONLY.

Computer Science applicants should turn to page 14.
(i) The graph $y = f ( x )$ of a certain function has been plotted below. [Figure]
On the next three pairs of axes (A), (B), (C) are graphs of
$$y = f ( - x ) , \quad f ( x - 1 ) , \quad - f ( x )$$
in some order. Say which axes correspond to which graphs. [Figure]
(A) [Figure]
(B) [Figure]
(C)
(ii) Sketch, on the axes opposite, graphs of both of the following functions
$$y = 2 ^ { - x ^ { 2 } } \quad \text { and } \quad y = 2 ^ { 2 x - x ^ { 2 } }$$
Carefully label any stationary points.
(iii) Let $c$ be a real number and define the following integral
$$I ( c ) = \int _ { 0 } ^ { 1 } 2 ^ { - ( x - c ) ^ { 2 } } \mathrm {~d} x$$
State the value(s) of $c$ for which $I ( c )$ is largest. Briefly explain your reasoning. [Note you are not being asked to calculate this maximum value.] [Figure]

Let $\Gamma$ be the graph formed by points $(x, y)$ satisfying $y = \log x$ on the coordinate plane. Which of the following relationships produce graphs that are completely identical to $\Gamma$?
(1) $y + \frac{1}{2} = \log(5x)$
(2) $2y = \log\left(x^{2}\right)$
(3) $3y = \log\left(x^{3}\right)$
(4) $x = 10^{y}$
(5) $x^{3} = 10^{\left(y^{3}\right)}$