For a positive integer $n$ and a real number $a$, consider the function
$$f_n(a) = \int_0^{\pi} (\cos x + a\sin 2nx)^2\, dx$$
(1) When we transform $f_n(a)$ into
$$f_n(a) = \int_0^{\pi} \left\{\frac{1 + \cos \mathbf{L}\, x}{2} + a^2 \frac{1 - \cos \mathbf{M}\, nx}{2} + a(\sin(2n+1)x + \sin(2n-1)x)\right\} dx$$
and calculate the definite integral on the right side, we obtain
$$f_n(a) = \frac{\pi}{\mathbf{N}}\, a^2 + \frac{\mathbf{O}\, n}{\mathbf{P}\, n^2 - \mathbf{Q}}\, a + \frac{\pi}{\mathbf{R}}.$$
(2) Let $a_n$ denote the value of $a$ at which $f_n(a)$ is minimalized, and set $S_N = \sum_{n=1}^{N} \frac{a_n}{n}$.
Then
$$\begin{aligned} S_N &= -\frac{\mathbf{S}}{\pi} \sum_{n=1}^{N} \left(\frac{1}{2n - \mathbf{T}} - \frac{1}{2n + \mathbf{U}}\right) \\ &= -\frac{\mathbf{S}}{\pi} \left(\mathbf{V} - \frac{1}{\mathbf{W}N + \mathbf{U}}\right) \end{aligned}$$
Hence we obtain
$$\sum_{n=1}^{\infty} \frac{a_n}{n} = \lim_{N \to \infty} S_N = -\frac{\mathbf{Y}}{\pi}.$$

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. Let
$$f ( x ) = \left\{ \begin{array} { c c } x + 1 & \text { for } 0 \leqslant x \leqslant 1 ; \\ 2 x ^ { 2 } - 6 x + 6 & \text { for } 1 \leqslant x \leqslant 2 . \end{array} \right.$$
(i) On the axes provided below, sketch a graph of $y = f ( x )$ for $0 \leqslant x \leqslant 2$, labelling any turning points and the values attained at $x = 0,1,2$.
(ii) For $1 \leqslant t \leqslant 2$, define
$$g ( t ) = \int _ { t - 1 } ^ { t } f ( x ) \mathrm { d } x$$
Express $g ( t )$ as a cubic in $t$.
(iii) Calculate and factorize $g ^ { \prime } ( t )$.
(iv) What are the minimum and maximum values of $g ( t )$ for $t$ in the range $1 \leqslant t \leqslant 2$ ? [Figure]

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. Let
$$f ( x ) = \left\{ \begin{array} { c c } x + 1 & \text { for } 0 \leqslant x \leqslant 1 ; \\ 2 x ^ { 2 } - 6 x + 6 & \text { for } 1 \leqslant x \leqslant 2 . \end{array} \right.$$
(i) On the axes provided below, sketch a graph of $y = f ( x )$ for $0 \leqslant x \leqslant 2$, labelling any turning points and the values attained at $x = 0,1,2$.
(ii) For $1 \leqslant t \leqslant 2$, define
$$g ( t ) = \int _ { t - 1 } ^ { t } f ( x ) \mathrm { d } x$$
Express $g ( t )$ as a cubic in $t$.
(iii) Calculate and factorize $g ^ { \prime } ( t )$.
(iv) What are the minimum and maximum values of $g ( t )$ for $t$ in the range $1 \leqslant t \leqslant 2$ ? [Figure]

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. Let
$$I ( c ) = \int _ { 0 } ^ { 1 } \left( ( x - c ) ^ { 2 } + c ^ { 2 } \right) \mathrm { d } x$$
where $c$ is a real number.
(i) Sketch $y = ( x - 1 ) ^ { 2 } + 1$ for the values $- 1 \leqslant x \leqslant 3$ on the axes below and show on your graph the area represented by the integral $I$ (1).
(ii) Without explicitly calculating $I ( c )$, explain why $I ( c ) \geqslant 0$ for any value of $c$.
(iii) Calculate $I ( c )$.
(iv) What is the minimum value of $I ( c )$ (as $c$ varies)?
(v) What is the maximum value of $I ( \sin \theta )$ as $\theta$ varies? [Figure]

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. For a positive whole number $n$, the function $f _ { n } ( x )$ is defined by
$$f _ { n } ( x ) = \left( x ^ { 2 n - 1 } - 1 \right) ^ { 2 } .$$
(i) On the axes provided opposite, sketch the graph of $y = f _ { 2 } ( x )$ labelling where the graph meets the axes.
(ii) On the same axes sketch the graph of $y = f _ { n } ( x )$ where $n$ is a large positive integer.
(iii) Determine
$$\int _ { 0 } ^ { 1 } f _ { n } ( x ) \mathrm { d } x$$
(iv) The positive constants $A$ and $B$ are such that
$$\int _ { 0 } ^ { 1 } f _ { n } ( x ) \mathrm { d } x \leqslant 1 - \frac { A } { n + B } \text { for all } n \geqslant 1 .$$
Show that
$$( 3 n - 1 ) ( n + B ) \geqslant A ( 4 n - 1 ) n ,$$
and explain why $A \leqslant 3 / 4$.
(v) When $A = 3 / 4$, what is the smallest possible value of $B$ ? [Figure]

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 and Computer Science \& Philosophy applicants should turn to page 14.
In this question we fix a real number $\alpha$ which will be the same throughout. We say that a function $f$ is bilateral if
$$f ( x ) = f ( 2 \alpha - x )$$
for all $x$.
(i) Show that if $f ( x ) = ( x - \alpha ) ^ { 2 }$ for all $x$ then the function $f$ is bilateral.
(ii) On the other hand show that if $f ( x ) = x - \alpha$ for all $x$ then the function $f$ is not bilateral.
(iii) Show that if $n$ is a non-negative integer and $a$ and $b$ are any real numbers then
$$\int _ { a } ^ { b } x ^ { n } \mathrm {~d} x = - \int _ { b } ^ { a } x ^ { n } \mathrm {~d} x$$
(iv) Hence show that if $f$ is a polynomial (and $a$ and $b$ are any reals) then
$$\int _ { a } ^ { b } f ( x ) \mathrm { d } x = - \int _ { b } ^ { a } f ( x ) \mathrm { d } x$$
(v) Suppose that $f$ is any bilateral function. By considering the area under the graph of $y = f ( x )$ explain why for any $t \geqslant \alpha$ we have
$$\int _ { \alpha } ^ { t } f ( x ) \mathrm { d } x = \int _ { 2 \alpha - t } ^ { \alpha } f ( x ) \mathrm { d } x$$
If $f$ is a function then we write $G$ for the function defined by
$$G ( t ) = \int _ { \alpha } ^ { t } f ( x ) \mathrm { d } x$$
for all $t$.
(vi) Suppose now that $f$ is any bilateral polynomial. Show that
$$G ( t ) = - G ( 2 \alpha - t )$$
for all $t$.
(vii) Suppose $f$ is a bilateral polynomial such that $G$ is also bilateral. Show that $G ( x ) = 0$ for all $x$.
If you require additional space please use the pages at the end of the booklet

Let the function $f ( x ) = \left\{ \begin{array} { l l l } x ^ { 3 } e ^ { - 1 / x ^ { 2 } } & \text { if } & x \neq 0 \\ 0 & \text { if } & x = 0 \end{array} \right.$ a) (1 point) Study the continuity and differentiability of $f ( x )$ at $x = 0$. b) ( 0.5 points) Study whether $f ( x )$ presents any type of even or odd symmetry. c) (1 point) Calculate the following integral: $\int _ { 1 } ^ { 2 } \frac { f ( x ) } { x ^ { 6 } } d x$

Let the function
$$f ( x ) = \begin{cases} x & \text { if } x \leq 0 \\ x \ln ( x ) & \text { if } x > 0 \end{cases}$$
a) ( 0.5 points) Study the continuity and differentiability of $f ( x )$ at $x = 0$. b) (1 point) Study the intervals of increase and decrease of $\mathrm { f } ( \mathrm { x } )$, as well as the relative maxima and minima. c) (1 point) Calculate $\int _ { 1 } ^ { 2 } f ( x ) d x$.

Let $F(x)$ and $f(x)$ both be polynomial functions with real coefficients. Given that $F'(x) = f(x)$, select the correct options.
(1) If $a \geq 0$, then $F(a) - F(0) = \int_{0}^{a} f(t)\, dt$
(2) If $F(x)$ divided by $x$ has quotient $Q(x)$, then $Q(0) = f(0)$
(3) If $f(x)$ is divisible by $x + 1$, then $F(x) - F(0)$ is divisible by $(x+1)^{2}$
(4) If $F(x) \geq \frac{x^{2}}{2}$ holds for all real numbers $x$, then $f(x) \geq x$ also holds for all real numbers $x$
(5) If $f(x) \geq x$ holds for all $x > 0$, then $F(x) \geq \frac{x^{2}}{2}$ also holds for all $x > 0$

What is the limit
$$\lim _ { n \rightarrow \infty } \frac { 3 } { n ^ { 2 } } \left( \sqrt { 4 n ^ { 2 } + 9 \times 1 ^ { 2 } } + \sqrt { 4 n ^ { 2 } + 9 \times 2 ^ { 2 } } + \cdots + \sqrt { 4 n ^ { 2 } + 9 \times ( n - 1 ) ^ { 2 } } \right)$$
which of the following definite integrals can represent?
(1) $\int _ { 0 } ^ { 3 } \sqrt { 1 + x ^ { 2 } } d x$
(2) $\int _ { 0 } ^ { 3 } \sqrt { 1 + 9 x ^ { 2 } } d x$
(3) $\int _ { 0 } ^ { 3 } \sqrt { 4 + x ^ { 2 } } d x$
(4) $\int _ { 0 } ^ { 3 } \sqrt { 4 + 9 x ^ { 2 } } d x$
(5) $\int _ { 0 } ^ { 3 } \sqrt { 4 x ^ { 2 } + 9 } d x$

Let $f ( x )$ be a real-coefficient cubic polynomial. The function $y = f ( x )$ has a local minimum at $x = - 3$ and a local maximum at $x = 1$. Given that the slope of the tangent line at the inflection point of the graph of $y = f ( x )$ is 4, so that $f ^ { \prime } ( x )$ is as found in question 14.
Find the value of $\int _ { - 3 } ^ { 1 } f ^ { \prime } ( x ) \, d x$.

15. The smallest possible value of $\int _ { 0 } ^ { 1 } ( x - a ) ^ { 2 } d x$ as $a$ varies is
A $\frac { 1 } { 12 }$
B $\frac { 1 } { 3 }$
C $\frac { 1 } { 2 }$
D $\frac { 7 } { 12 }$
E 2

Find the value of
$$\int _ { 1 } ^ { 2 } \left( x ^ { 2 } - \frac { 4 } { x ^ { 2 } } \right) ^ { 2 } d x$$

Consider this statement about a function $f ( x )$ :
$\left( ^ { * } \right)$ If $( f ( x ) ) ^ { 2 } \leq 1$ for all $- 1 \leq x \leq 1$ then $\int _ { - 1 } ^ { 1 } ( f ( x ) ) ^ { 2 } \mathrm {~d} x \leq \int _ { - 1 } ^ { 1 } f ( x ) \mathrm { d } x$
Which one of the following functions provides a counterexample to (*)?

$f ( x )$ is a function defined for all real values of $x$.
Which one of the following is a sufficient condition for $\int _ { 1 } ^ { 3 } f ( x ) d x = 0$ ?
A $f ( 2 ) = 0$
B $f ( 1 ) = f ( 3 ) = 0$
C $f ( - x ) = - f ( x )$ for all $x$
D $f ( x + 2 ) = - f ( 2 - x )$ for all $x$
E $\quad f ( x - 2 ) = - f ( 2 - x )$ for all $x$

The polynomial function $f ( x )$ is such that $f ( x ) > 0$ for all values of $x$.
Given $\int _ { 2 } ^ { 4 } f ( x ) d x = A$, which one of the following statements must be correct?
A $\int _ { 0 } ^ { 2 } [ f ( x + 2 ) + 1 ] d x = A + 1$
B $\quad \int _ { 0 } ^ { 2 } [ f ( x + 2 ) + 1 ] d x = A + 2$
C $\int _ { 2 } ^ { 4 } [ f ( x + 2 ) + 1 ] d x = A + 1$
D $\int _ { 2 } ^ { 4 } [ f ( x + 2 ) + 1 ] d x = A + 2$
E $\quad \int _ { 4 } ^ { 6 } [ f ( x + 2 ) + 1 ] d x = A + 1$
F $\quad \int _ { 4 } ^ { 6 } [ f ( x + 2 ) + 1 ] d x = A + 2$

The two functions $F ( n )$ and $G ( n )$ are defined as follows for positive integers $n$ :
$$\begin{aligned} & F ( n ) = \frac { 1 } { n } \int _ { 0 } ^ { n } ( n - x ) d x \\ & G ( n ) = \sum _ { r = 1 } ^ { n } F ( r ) \end{aligned}$$
What is the smallest positive integer $n$ such that $G ( n ) > 150$ ?
A 22
B 23
C 24
D 25
E 26

Evaluate
$$\int_{-1}^{3} |x|(1-x) \, dx$$

Given that
$$2\int_0^1 f(x) \, dx + 5\int_1^2 f(x) \, dx = 14$$
and
$$\int_0^1 f(x+1) \, dx = 6$$
find the value of
$$\int_0^2 f(x) \, dx$$

The function $\mathrm { f } ( x )$ is defined for all real values of $x$. Which of the following conditions on $\mathrm { f } ( x )$ is/are necessary to ensure that
$$\int _ { - 5 } ^ { 0 } \mathrm { f } ( x ) \mathrm { d } x = \int _ { 0 } ^ { 5 } \mathrm { f } ( x ) \mathrm { d } x$$
Condition I: $\quad \mathrm { f } ( x ) = \mathrm { f } ( - x ) $ for $- 5 \leq x \leq 5$ Condition II: $\mathrm { f } ( x ) = c$ for $- 5 \leq x \leq 5$, where $c$ is a constant Condition III: $\mathrm { f } ( x ) = - \mathrm { f } ( - x ) $ for $- 5 \leq x \leq 5$
A none of them
B I only
C II only
D III only
E I and II only F I and III only G II and III only H I, II and III

$\mathrm { f } ( x )$ is a function for which
$$\int _ { 0 } ^ { 3 } ( \mathrm { f } ( x ) ) ^ { 2 } \mathrm {~d} x + \int _ { 0 } ^ { 3 } \mathrm { f } ( x ) \mathrm { d } x = \int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x$$
Which of the following claims about $\mathrm { f } ( x )$ is/are necessarily true? I $\mathrm { f } ( x ) \leq 0$ for some $x$ with $1 \leq x \leq 3$ II $\int _ { 0 } ^ { 3 } \mathrm { f } ( x ) \mathrm { d } x \leq \int _ { 0 } ^ { 1 } \mathrm { f } ( x ) \mathrm { d } x$
A neither of them
B I only
C II only
D I and II

Find the value of
$$\int _ { 1 } ^ { 4 } \left( 3 \sqrt { x } + \frac { 4 } { x ^ { 2 } } \right) \mathrm { d } x$$
A - 0.75
B 7.125
C 11
D 17
E 18 F 21.875 G 34.5

The curve $y = x ^ { 3 } - 6 x + 3$ has turning points at $x = \alpha$ and $x = \beta$, where $\beta > \alpha$. Find
$$\int _ { \alpha } ^ { \beta } x ^ { 3 } - 6 x + 3 \mathrm {~d} x$$
A $- 8 \sqrt { 2 }$ B - 10 C $- 10 + 6 \sqrt { 2 }$ D 0 E $\quad 12 - 8 \sqrt { 2 }$ F $\quad 6 \sqrt { 2 }$ G 12

The function f is such that $\mathrm { f } ( 0 ) = 0$, and $x \mathrm { f } ( x ) > 0$ for all non-zero values of $x$. It is given that
$$\int _ { - 2 } ^ { 2 } \mathrm { f } ( x ) \mathrm { d } x = 4$$
and
$$\int _ { - 2 } ^ { 2 } | \mathrm { f } ( x ) | \mathrm { d } x = 8$$
Evaluate
$$\int _ { - 2 } ^ { 0 } \mathrm { f } ( | x | ) \mathrm { d } x$$
A - 8 B - 6 C - 4 D - 2 E 2 F $\quad 4$ G 6 H 8

Which of the following statements about polynomials $f$ and $g$ is/are true?
I If $\mathrm { f } ( x ) \geq \mathrm { g } ( x )$ for all $x \geq 0$, then $\int _ { 0 } ^ { x } \mathrm { f } ( t ) \mathrm { d } t \geq \int _ { 0 } ^ { x } \mathrm {~g} ( t ) \mathrm { d } t$ for all $x \geq 0$.
II If $\mathrm { f } ( x ) \geq \mathrm { g } ( x )$ for all $x \geq 0$, then $\mathrm { f } ^ { \prime } ( x ) \geq \mathrm { g } ^ { \prime } ( x )$ for all $x \geq 0$.
III If $\mathrm { f } ^ { \prime } ( x ) \geq \mathrm { g } ^ { \prime } ( x )$ for all $x \geq 0$, then $\mathrm { f } ( x ) \geq \mathrm { g } ( x )$ for all $x \geq 0$.
A none of them
B I only
C II only
D III only
E I and II only F I and III only G II and III only H I, II and III