CIE 0606/2019/Winter/Paper 12/Question

1 (i) On the axes below, sketch the graph of $y=2 \cos 3 x-1$ for $-90^{\circ} \leqslant x \leqslant 90^{\circ}$.


(ii) Write down the amplitude of $2 \cos 3 x-1$.[1]

(iii) Write down the period of $2 \cos 3 x-1$.[1]

2 When $\lg y^2$ is plotted against $x,$ a straight line is obtained passing through the points (5, 12) and (3, 20).

Find $y$ in terms of $x$, giving your answer in the form $y = 10^{ ax + b}$ , where $a$ and $b$ are integers.[5]

3 The first three terms in the expansion of $\left(1-\frac{x}{7}\right)^{14}(1-2 x)^{4}$ can be written as $1+a x+b x^{2}$. Find the value of each of the constants $a$ and $b .$

4 (i) On the axes below, sketch the graph of $y=\left|2 x^{2}-9 x-5\right|$ showing the coordinates of the points where the graph meets the axes.[4]


(ii) Find the values of $k$ for which $\left|2 x^{2}-9 x-5\right|=k$ has exactly 2 solutions.$[3]$

5 $\begin{array}[t]{ll}\text { (a) } \text { It is given that } & \mathrm{f}: x \mapsto \sqrt{x} & \text { for } x \geqslant 0,\end{array}$$\mathrm{g}: x \mapsto x+$for $x \geqslant 0 .$

Identify each of the following functions with one of $\mathrm{f}^{-1}, \mathrm{~g}^{-1}, \mathrm{fg}, \mathrm{gf}, \mathrm{f}^{2}, \mathrm{~g}^{2}$

(i) $\sqrt{x+5}$$[1]$

(ii) $x-5$$[1]$

(iii) $x^{2}$[1

(iv) $x+10$[1)

(b) It is given that $h(x)=a+\frac{b}{x^{2}}$ where $a$ and $b$ are constants.

(i) Why is $-2 \leqslant x \leqslant 2$ not a suitable domain for $h(x) ?$$[1]$

(ii) Given that $h(1)=4$ and $h^{\prime}(1)=16$, find the value of $a$ and of $b$.[2]

6 (a) Write $\dfrac{\sqrt{p}\left(\dfrac{q p}{r}\right)^{2}}{p^{-1} \sqrt[3]{q r}}$ in the form $p^{a} q^{b} r^{c}$, where $a, b$ and $c$ are constants.$[3]$

(b) Solve $\log _{7} x+2 \log _{x} 7=3$.$[4]$

7 It is given that $y=\left(1+\mathrm{e}^{x^{2}}\right)(x+5)$.

(i) Find $\dfrac{\mathrm{d} y}{\mathrm{~d} x}$$[3]$

(ii) Find the approximate change in $y$ as $x$ increases from $0.5$ to $0.5+p$, where $p$ is small.[2]

(iii) Given that $y$ is increasing at a rate of 2 units per second when $x=0.5$, find the corresponding rate of change in $x .$

8 (a) Five teams took part in a competition in which each team played each of the other 4 teams. The following table represents the results after all the matches had been played.

$\begin{array}{|c|c|c|c|}\hline \text{Team} & \text{Won} & \text{Drawn} & \text{Lost} \\\hline A & 2 & 1 & 1 \\\hline B & 1 & 3 & 0 \\\hline C & 1 & 1 & 2 \\\hline D & 0 & 1 & 3 \\\hline E & 3 & 0 & 1 \\\hline\end{array}$

Points in the competition were awarded to the teams as follows

4 for each match won, $\quad 2$ for each match drawn, $\quad 0$ for each match lost.

(i) Write down two matrices whose product under matrix multiplication will give the total number of points awarded to each team. $\quad[2]$

(ii) Evaluate the matrix product from part (i) and hence state which team was awarded the most points. $\quad[2]$

(b) It is given that $\quad \mathbf{A}=\left(\begin{array}{rr}1 & -1 \\ 2 & 4\end{array}\right)$ and $\quad \mathbf{B}=\left(\begin{array}{rr}5 & 0 \\ 1 & -2\end{array}\right)$.

(i) Find $\mathbf{A}^{-1}$.

(ii) Hence find the matrix $\mathbf{C}$ such that $\mathbf{A C}=\mathbf{B}$.[3]

9 A solid circular cylinder has a base radius of $r \mathrm{~cm}$ and a height of $h \mathrm{~cm} .$ The cylinder has a volume of $1200 \pi \mathrm{cm}^{3}$ and a total surface area of $S \mathrm{~cm}^{2}$.

(i) Show that $S=2 \pi r^{2}+\dfrac{2400 \pi}{r}$.

(ii) Given that $h$ and $r$ can vary, find the stationary value of $S$ and determine its nature.

A curve is such that $\dfrac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=2(3 x-1)^{-\frac{2}{3}}$. Given that the curve has a gradient of 6 at the point $(3,11)$, find the equation of the curve. $[8]$


10 The diagram shows a circle centre $O$, radius $10 \mathrm{~cm}$. The points $A, B$ and $C$ lie on the circumference of the circle such that $A B=B C=18 \mathrm{~cm}$.

(i) Show that angle $A O B=2.24$ radians correct to 2 decimal places.

(ii) Find the perimeter of the shaded region.

(iii) Find the area of the shaded region.[3]

11 A curve is such that $\dfrac{d^2y}{dx^2}=2(3x-1)^{-\frac 23}.$ Given that the curve has a gradient of 6 at the point (3, 11), find the equation of the curve.[8]

Answers
1(i)

(ii) 2 (iii) 120$^{\circ},\dfrac{2\pi}{3}$
2.  $y=10^{16-2x}$
3.  $a=-10,b=\dfrac{293}{7}$
4. (i) 

(ii)  $k=0,k>\dfrac{121}{8}$
5(a) (i) $fg$, (ii) $g^{-1}$ (iii) $f^{-1}$ (iv) $g^2$
(b) (i) Undefined at $x=0$ (ii) $a=12,b=-8$
6(a) $\displaystyle p^{\frac 72}q^{\frac 53}r^{-\frac 73}$
(b) $x=7,49$
7(a) (i) $\dfrac{dy}{dx}=\left(e^{x^2}+1\right) +2xe^{x^2}(x+5)$
(ii) $9.35p$ (iii) $\dfrac{dx}{dt}=0.214$
8(a)(i) $\left(\begin{array}{cc}2&1\\1&3\\1&1\\0&1\\3&0\end{array}\right)$ $4\choose 2$
(ii)$\left(\begin{array}{c}10\\10\\6\\2\\12\end{array}\right)$
(b)(i) $\dfrac 16\left(\begin{array}{cc}4&1\\-2&1\end{array}\right)$
(ii) $\dfrac 16\left(\begin{array}{cc}21&-2\\-9&-2\end{array}\right)$
9(i) ...(ii) $S_{min}=1340$
10 (i) (ii) 33.7 (iii) 41.6
11 $y=\dfrac 12(3x-1)^{\frac 43}+2x-3 $

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