Let $f _ { 1 } : ( 0 , \infty ) \rightarrow \mathbb { R }$ and $f _ { 2 } : ( 0 , \infty ) \rightarrow \mathbb { R }$ be defined by $$f _ { 1 } ( x ) = \int _ { 0 } ^ { x } \prod _ { j = 1 } ^ { 21 } ( t - j ) ^ { j } d t , \quad x > 0$$ and $$f _ { 2 } ( x ) = 98 ( x - 1 ) ^ { 50 } - 600 ( x - 1 ) ^ { 49 } + 2450 , \quad x > 0$$ where, for any positive integer $n$ and real numbers $a _ { 1 } , a _ { 2 } , \ldots , a _ { n } , \prod _ { i = 1 } ^ { n } a _ { i }$ denotes the product of $a _ { 1 } , a _ { 2 } , \ldots , a _ { n }$. Let $m _ { i }$ and $n _ { i }$, respectively, denote the number of points of local minima and the number of points of local maxima of function $f _ { i } , i = 1,2$, in the interval $( 0 , \infty )$. The value of $6 m _ { 2 } + 4 n _ { 2 } + 8 m _ { 2 } n _ { 2 }$ is $\_\_\_\_$.

Let $\mathbb { R }$ denote the set of all real numbers. Let $f : \mathbb { R } \rightarrow \mathbb { R }$ be defined by
$$f ( x ) = \begin{cases} \frac { 6 x + \sin x } { 2 x + \sin x } & \text { if } x \neq 0 \\ \frac { 7 } { 3 } & \text { if } x = 0 \end{cases}$$
Then which of the following statements is (are) TRUE?

(A)	The point $x = 0$ is a point of local maxima of $f$
(B)	The point $x = 0$ is a point of local minima of $f$
(C)	Number of points of local maxima of $f$ in the interval $[ \pi , 6 \pi ]$ is 3
(D)	Number of points of local minima of $f$ in the interval $[ 2 \pi , 4 \pi ]$ is 1

Let $f ( x ) = x ^ { 2 } + \frac { 1 } { x ^ { 2 } }$ and $g ( x ) = x - \frac { 1 } { x } , x \in R - \{ - 1,0,1 \}$. If $h ( x ) = \frac { f ( x ) } { g ( x ) }$, then the local minimum value of $h ( x )$ is:
(1) $2 \sqrt { 2 }$
(2) 3
(3) - 3
(4) $- 2 \sqrt { 2 }$

If $S_1$ and $S_2$ are respectively the sets of local minimum and local maximum points of the function, $f(x) = 9x^4 + 12x^3 - 36x^2 + 25$, $x \in R$, then
(1) $S_1 = \{-2\}$; $S_2 = \{0, 1\}$
(2) $S_1 = \{-1\}$; $S_2 = \{0, 2\}$
(3) $S_1 = \{-2, 0\}$; $S_2 = \{1\}$
(4) $S_1 = \{-2, 1\}$; $S_2 = \{0\}$

If $f ( x )$ is a non-zero polynomial of degree four, having local extreme points at $x = - 1,0,1$; then the set $S = \{ x \in R : f ( x ) = f ( 0 ) \}$ contains exactly
(1) Two irrational and two rational numbers
(2) Four rational numbers
(3) Two irrational and one rational number
(4) Four irrational numbers

Let $f ( x )$ be a polynomial of degree 5 such that $x = \pm 1$ are its critical points. If $\lim _ { x \rightarrow 0 } \left( 2 + \frac { f ( x ) } { x ^ { 3 } } \right) = 4$, then which one of the following is not true?
(1) $f$ is an odd function
(2) $f ( 1 ) - 4 f ( - 1 ) = 4$
(3) $x = 1$ is a point of local minimum and $x = - 1$ is a point of local maximum
(4) $x = 1$ is a point of local maxima of $f$

Let a function $f : [ 0,5 ] \rightarrow R$ be continuous, $f ( 1 ) = 3$ and $F$ be defined as: $F ( x ) = \int _ { 1 } ^ { x } t ^ { 2 } g ( t ) d t$, where $g ( t ) = \int _ { 1 } ^ { t } f ( u ) d u$. Then for the function $F ( x )$, the point $x = 1$ is:
(1) a point of local minima
(2) not a critical point
(3) a point of local maxima
(4) a point of inflection

If $x = 1$ is a critical point of the function $f(x) = (3x^2 + ax - 2 - a)e^x$, then
(1) $x = 1$ and $x = -\frac{2}{3}$ are local minima of $f$
(2) $x = 1$ and $x = -\frac{2}{3}$ is a local maxima of $f$
(3) $x = 1$ is a local maxima and $x = -\frac{2}{3}$ is a local minima of $f$
(4) $x = 1$ is a local minima and $x = -\frac{2}{3}$ are local maxima of $f$

Let $a$ be a real number such that the function $f ( x ) = a x ^ { 2 } + 6 x - 15 , x \in R$ is increasing in $( - \infty , \frac { 3 } { 4 } )$ and decreasing in $\left( \frac { 3 } { 4 } , \infty \right)$. Then the function $g ( x ) = a x ^ { 2 } - 6 x + 15 , x \in R$ has a
(1) local maximum at $x = - \frac { 3 } { 4 }$
(2) local minimum at $x = - \frac { 3 } { 4 }$
(3) local maximum at $x = \frac { 3 } { 4 }$
(4) local minimum at $x = \frac { 3 } { 4 }$

The sum of all the local minimum values of the twice differentiable function $f : R \rightarrow R$ defined by $f ( x ) = x ^ { 3 } - 3 x ^ { 2 } - \frac { 3 f ^ { \prime \prime } ( 2 ) } { 2 } x + f ^ { \prime \prime } ( 1 )$ is:
(1) - 22
(2) 5
(3) - 27
(4) 0

If $m$ and $n$ respectively are the number of local maximum and local minimum points of the function $f ( x ) = \int _ { 0 } ^ { x ^ { 2 } } \frac { t ^ { 2 } - 5 t + 4 } { 2 + e ^ { t } } d t$, then the ordered pair $( m , n )$ is equal to
(1) $( 2,3 )$
(2) $( 3,2 )$
(3) $( 2,2 )$
(4) $( 3,4 )$

The curve $y(x) = ax ^ { 3 } + bx ^ { 2 } + cx + 5$ touches the $x$-axis at the point $P(-2,0)$ and cuts the $y$-axis at the point $Q$, where $y'$ is equal to 3. Then the local maximum value of $y(x)$ is
(1) $\frac { 27 } { 4 }$
(2) $\frac { 29 } { 4 }$
(3) $\frac { 37 } { 4 }$
(4) $\frac { 9 } { 2 }$

Let $f(x) = \int_0^x t(t-1)(t-2)\,dt$, $x > 0$. Then the number of points in the interval $(0, 3)$ at which $f(x)$ has a local maximum is $\_\_\_\_$.

Let $x = 2$ be a local minima of the function $f ( x ) = 2 x ^ { 4 } - 18 x ^ { 2 } + 8 x + 12 , x \in ( - 4,4 )$. If $M$ is local maximum value of the function $f$ in $( - 4,4 )$, then $M =$
(1) $12 \sqrt { 6 } - \frac { 33 } { 2 }$
(2) $12 \sqrt { 6 } - \frac { 31 } { 2 }$
(3) $18 \sqrt { 6 } - \frac { 33 } { 2 }$
(4) $18 \sqrt { 6 } - \frac { 31 } { 2 }$

The function $f ( x ) = 2 x + 3 x ^ { \frac { 2 } { 3 } } , x \in R$, has
(1) exactly one point of local minima and no point of local maxima
(2) exactly one point of local maxima and no point of local minima
(3) exactly one point of local maxima and exactly one point of local minima
(4) exactly two points of local maxima and exactly one point of local minima

Let $f ( x ) = 4 \cos ^ { 3 } x + 3 \sqrt { 3 } \cos ^ { 2 } x - 10$. The number of points of local maxima of $f$ in interval $( 0,2 \pi )$ is
(1) 3
(2) 4
(3) 1
(4) 2

The number of critical points of the function $f ( x ) = ( x - 2 ) ^ { 2 / 3 } ( 2 x + 1 )$ is
(1) 1
(2) 2
(3) 0
(4) 3

Let $f ( x ) = \int _ { 0 } ^ { x ^ { 2 } } \frac { \mathrm { t } ^ { 2 } - 8 \mathrm { t } + 15 } { \mathrm { e } ^ { t } } \mathrm { dt } , x \in \mathbf { R }$. Then the numbers of local maximum and local minimum points of $f$, respectively, are :
(1) 2 and 3
(2) 2 and 2
(3) 3 and 2
(4) 1 and 3

$\lim_{x \rightarrow 0} \operatorname{cosec} x \left(\sqrt{2\cos^2 x + 3\cos x} - \sqrt{\cos^2 x + \sin x + 4}\right)$ is:
(1) 0
(2) $\frac{1}{\sqrt{15}}$
(3) $\frac{1}{2\sqrt{5}}$
(4) $-\frac{1}{2\sqrt{5}}$

Q71. Let $f ( x ) = 4 \cos ^ { 3 } x + 3 \sqrt { 3 } \cos ^ { 2 } x - 10$. The number of points of local maxima of $f$ in interval $( 0,2 \pi )$ is
(1) 3
(2) 4
(3) 1
(4) 2

Q72. The number of critical points of the function $f ( x ) = ( x - 2 ) ^ { 2 / 3 } ( 2 x + 1 )$ is
(1) 1
(2) 2
(3) 0
(4) 3

Let $a$ be a positive real number. We are to investigate local extrema of the function
$$f(x) = x^2 - 5 + 4a\log(2x + a + 8) \quad \left(-\frac{a}{2} - 4 < x < -2\right).$$
(1) When we differentiate the function $f(x)$ with respect to $x$, we obtain
$$f'(x) = \frac{\mathbf{A}(\mathbf{B}\, x + a)(x + \mathbf{C})}{\mathbf{D}\, x + a + \mathbf{E}}.$$
(2) Since a condition of $a$ is that $a > 0$ and the domain of $f(x)$ is $-\frac{a}{2} - 4 < x < -2$, the range of values of $a$ such that $f(x)$ has both a local maximum and a local minimum is
$$\mathbf{F} < a < \mathbf{G}.$$
In such a case, the sum of the local maximum and the local minimum is
$$\frac{a^2}{\mathbf{H}} + \mathbf{I} + \mathbf{IJ}\, a\log\mathbf{K}\, a.$$

3. Let
$$f ( x ) = \left( c - \frac { 1 } { c } - x \right) \left( 4 - 3 x ^ { 2 } \right)$$
where $c$ is a positive constant and $x$ varies over the real numbers.
(i) Show that $f ( x )$ has one maximum and one minimum.
(ii) Show that the difference between the values of $f ( x )$ at its turning points is
$$\frac { 4 } { 9 } \left( c + \frac { 1 } { c } \right) ^ { 3 }$$
(iii) What is the least value that the difference in (ii) can have for $c > 0$ ?

1. The absolute value $| x |$ of a real number $x$ is defined by

$$| x | = \begin{cases} x & \text { if } x \geqslant 0 \\ - x & \text { if } x < 0 \end{cases}$$
(So, for example, $| 3 | = 3$ and $| - 5 | = 5$.)
(i) Sketch the locus of points ( $x , y$ ) in the first quadrant ( $x \geqslant 0 , y \geqslant 0$ ) of the $x - y$ plane that satisfy the equation $x + y = 1$.
(ii) Hence, sketch the set of points $( x , y )$ in the whole $x - y$ plane that satisfy the equation $| x | + | y | = 1$.
(iii) Give the equations of the straight lines that contain the sides of the figure that you have drawn in (ii).
(iv) If $| x | + | y | = 1$, what are the maximum and minimum values of $\sqrt { x ^ { 2 } + y ^ { 2 } }$ ? At which points $( x , y )$ are they attained?
[Figure]
(i)
[Figure]
(ii)

1. The game of Hexaglide is played on a board with $3 \times 3$ squares; White and Black each have 3 pieces, and they begin as shown in the first diagram. White moves first, and the players take turns to move one of their pieces forwards or backwards to an adjacent empty square. (Pieces never move sideways or diagonally, and they are never captured or removed from the board.) [Figure] [Figure] [Figure]
   (i) In a practice game, White plays without any Black pieces on the board, and, from his usual starting position, reaches the position shown in the second diagram. How many different sequences of moves end with this position if White makes (a) 6 moves, (b) 7 moves, (c) 8 moves?

For parts (ii) and (iii) of this question, give 'yes' or 'no' answers in the grids opposite; for both parts, you need not show your working or explain your answers. In the real game, White and Black both play, and a player wins if he can trap his opponent's pieces so that they cannot move: the third diagram shows a win for White.
(ii) (a) Is it possible to reach the position shown in the third diagram?
(b) Is it possible to reach a position where Black has won?
(c) Can White play so as to ensure that either he wins or the game goes on forever?
(d) Can Black play so as to ensure that either he wins or the game goes on forever?
(iii) In an advanced version of the game, the board has $4 \times 4$ squares, and each player has 4 pieces. What would be the answers to the four questions in part (ii) in this case?
[Figure]
(ii)
[Figure]
(iii)

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.
Let $f ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + c$, where the coefficients $a , b$ and $c$ are real numbers. The figure below shows a section of the graph of $y = f ( x )$. The curve has two distinct turning points; these are located at $A$ and $B$, as shown. (Note that the axes have been omitted deliberately.) [Figure]
(i) Find a condition on the coefficients $a , b , c$ such that the curve has two distinct turning points if, and only if, this condition is satisfied.
It may be assumed from now on that the condition on the coefficients in (i) is satisfied.
(ii) Let $x _ { 1 }$ and $x _ { 2 }$ denote the $x$ coordinates of $A$ and $B$, respectively. Show that
$$x _ { 2 } - x _ { 1 } = \frac { 2 } { 3 } \sqrt { a ^ { 2 } - 3 b }$$
(iii) Suppose now that the graph of $y = f ( x )$ is translated so that the turning point at $A$ now lies at the origin. Let $g ( x )$ be the cubic function such that $y = g ( x )$ has the translated graph. Show that
$$g ( x ) = x ^ { 2 } \left( x - \sqrt { a ^ { 2 } - 3 b } \right)$$
(iv) Let $R$ be the area of the region enclosed by the $x$-axis and the graph $y = g ( x )$. Show that if $a$ and $b$ are rational then $R$ is also rational.
(v) Is it possible for $R$ to be a non-zero rational number when $a$ and $b$ are both irrational? Justify your answer.

4. For APPLICANTS IN $\left\{ \begin{array} { l } \text { MATHEMATICS } \\ \text { MATHEMATICS \& STATISTICS } \\ \text { MATHEMATICS \& PHILOSOPHY } \end{array} \right\}$ ONLY. Mathematics \& Computer Science, Computer Science and Computer Science \& Philosophy applicants should turn to page 20.
(i) Sketch the graph of $y = \sqrt { x } - \frac { x } { 4 }$ for $x \geqslant 0$, and find the coordinates of the turning point.
(ii) Describe in words how the graph of $y = \sqrt { 4 x + 1 } - x - 1$ for $x \geqslant - \frac { 1 } { 4 }$ is related to the graph that you sketched in part (i). Write down the coordinates of the turning point of this new graph.
Point $A$ is at $( - 1,0 )$ and point $B$ is at $( 1,0 )$. Curve $C$ is defined to be all points $P$ that satisfy the equation $| A P | \times | B P | = 1$, that is; the distance from $P$ to $A$, multiplied by the distance from $P$ to $B$, is 1 .
(iii) Find all points that lie on both the $x$-axis and also on the curve $C$.
(iv) Find an equation in the form $y = f ( x )$ for the part of the curve $C$ in the region where $x > 0$ and $y > 0$. You should explicitly determine the function $f ( x )$.
(v) Use part (ii) to determine the coordinates of any turning points of the curve $C$ in the region where $x > 0$ and $y > 0$.
(vi) Sketch the curve $C$, including any parts of the curve with $x < 0$ or $y < 0$ or both.
This page has been intentionally left blank