CIE Graph (Additional Mathematics -2018)

$\def\D{\displaystyle}$
1 (CIE 2012, s, paper 11, question 1)
(i) Sketch the graph of $\D y = |2x - 5|,$ showing the coordinates of the points where the graph meets the coordinate axes. [2]
(ii) Solve $\D |2x - 5| = 3 .$ [2]

2 (CIE 2012, s, paper 12, question 7)
(i) Sketch the graph of $\D y = |x^2 - x - 6|,$ showing the coordinates of the points where the curve meets the coordinate axes. [3]
(ii) Solve $|x^2 - x - 6| = 6.$ [3]

3 (CIE 2012, s, paper 21, question 3)
The diagram shows a sketch of the curve $\D y = a\sin(bx) + c$ for $\D 0^{\circ}\le  x \le  180^{\circ}.$ Find the
values of $\D a, b$ and $\D c.$ [3]
(b) Given that $\D f(x) = 5\cos3x + 1,$ for all $\D x,$ state
(i) the period of $\D f,$ [1]
(ii) the amplitude of $\D f.$ [1]

4 (CIE 2012, w, paper 11, question 1)
(i) Sketch the graph of $\D y = |3 + 5x|,$ showing the coordinates of the points where your graph meets the coordinate axes. [2]
(ii) Solve the equation $\D |3 + 5x| = 2.$ [2]

5 (CIE 2012, w, paper 12, question 9)
(a) (i) Using the axes below, sketch for $\D 0\le   x \le  \pi,$ the graphs of $\D y = \sin 2x$ and $\D y = 1 + \cos 2x.$ [4]
(ii) Write down the solutions of the equation $\D \sin 2x - \cos 2x = 1,$ for $\D 0 \le x \le  \pi.$ [2]
(b) (i) Write down the amplitude and period of $\D 5 \cos 4x - 3.$ [2]
(ii) Write down the period of $\D 4 \tan 3x.$ [1]

6 (CIE 2012, w, paper 13, question 4)
(i) On the axes below sketch, for $\D 0\le   x \le  \pi,$ the graphs of $\D y = \tan x$ and $\D y = 1 + 3\sin 2x.$ [3]
Write down
(ii) the coordinates of the stationary points on the curve $\D y = 1 + 3\sin 2x$ for $\D 0 \le x \le \pi,$ [2]
(iii) the number of solutions of the equation $\D \tan x = 1 + 3\sin 2x$ for $\D 0 \le x \le \pi.$ [1]

7 (CIE 2012, w, paper 21, question 3)
(i) On the grid below draw, for $\D 0^{\circ} \le  x \le  360^{\circ},$ the graphs of $\D y = 3 \sin 2x$ and $\D y = 2 + \cos x.$ [4]
(ii) State the number of values of $\D x$ for which $\D 3 \sin 2x = 2 + \cos x$ in the interval $\D 0^{\circ} \le  x \le  360^{\circ}. $ [1]

8 (CIE 2013, s, paper 11, question 1)
On the axes below sketch, for $\D 0 \le  x \le  2\pi,$ the graph of
(i) $\D y = \cos x - 1,$ [2]
(ii) $\D y = \sin 2x.$ [2]


(iii) State the number of solutions of the equation $\D \cos x - \sin 2x = 1,$ for $\D 0 \le x \le  2\pi.$ [1]

9 (CIE 2013, s, paper 21, question 2)
The velocity-time graph represents the motion of a particle moving in a straight line.
(i) Find the acceleration during the first 5 seconds. [1]
(ii) Find the length of time for which the particle is travelling with constant velocity. [1]
(iii) Find the total distance travelled by the particle. [3]

10 (CIE 2013, s, paper 21, question 4)
(i) Sketch the graph of $\D y = |4x - 2|$, showing the coordinates of the points where the graph meets the axes. [3]
(ii) Solve the equation $\D |4x - 2| = x.$ [3]

11 (CIE 2013, s, paper 22, question 3)
(i) Write down the letter of each graph which does not represent a function. [2]
(ii) Write down the letter of each graph which represents a function that does not have an inverse. [2]
(b)
The diagram shows the graph of a function $\D y = f(x).$ On the same axes sketch the graph of $\D y = f^{-1}(x).$

12 (CIE 2013, s, paper 22, question 10)
(a) The function $\D f$ is defined, for $\D 0^{\circ} \le x\le  360^{\circ},$ by $\D f(x) = 1 + 3 \cos 2x.$
(i) Sketch the graph of $\D y = f(x)$ on the axes below. [4]
(ii) State the amplitude of $\D f.$ [1]
(iii) State the period of $\D f.$ [1]
(b) Given that $\D \cos x = p ,$ where $\D 270^{\circ} < x < 360^{\circ},$ find  cosec $\D x$ in terms of $\D p.$ [3]

13 (CIE 2013, w, paper 11, question 1)
The diagram shows the graph of $\D y = a \sin(bx) + c$ for $\D 0 \le  x \le  2\pi,$ where $\D a, b$ and $\D c$ are positive integers. State the value of $\D a,$ of $\D b$ and of $\D c.$ [3]

14 (CIE 2013, w, paper 11, question 8)
(i) On the grid below, sketch the graph of $\D y = |(x - 2) (x + 3)|$ for $\D - 5 \le  x \le  4,$ and state the coordinates of the points where the curve meets the coordinate axes. [4]
(ii) Find the coordinates of the stationary point on the curve $\D y = |(x - 2) (x + 3)| .$ [2]
(iii) Given that $\D k$ is a positive constant, state the set of values of $\D k$ for which $\D |(x - 2) (x + 3)| = k$ has 2 solutions only. [1]

15 (CIE 2013, w, paper 23, question 4)
(a) (i) The diagram shows the graph of $\D y = A + C \tan(Bx)$ passing through the points (0, 3) and $\D \left(\frac{\pi}{2},3\right).$ Find the value of $\D A$ and of $\D B.$ [2]
(ii) Given that the point $\D \left(\frac{\pi}{8},7\right)$  also lies on the graph, find the value of $\D C.$ [1]
(b) Given that $\D f (x) = 8 - 5 \cos 3x,$ state the period and the amplitude of $\D f.$ [2]

16 (CIE 2014, s, paper 11, question 9a)
(a) The diagram shows the velocity-time graph of a particle $\D P$ moving in a straight line with velocity $D v$ ms$\D^{-1}$ at time $\D t$ s after leaving a fixed point.
Find the distance travelled by the particle $\D P.$ [2]
(b) The diagram shows the displacement-time graph of a particle $\D Q$ moving in a straight line with displacement $\D s$ m from a fixed point at time $\D t$ s.
On the axes below, plot the corresponding velocity-time graph for the particle $\D Q.$ [3]

(c) The displacement $\D s$ m of a particle $\D R,$ which is moving in a straight line, from a fixed point at time $\D t$ s is given by $\D s = 4t - 16 \ln(t+1)+ 13.$
(i) Find the value of $\D t$ for which the particle $\D R$ is instantaneously at rest. [3]
(ii) Find the value of $\D t$ for which the acceleration of the particle $\D R$ is 0.25ms$\D ^{-1}.$ [2]

17 (CIE 2014, s, paper 11, question 9b)
18 (CIE 2014, s, paper 12, question 3)
(i) Sketch the graph of $\D y = |(2x+1)(x-2)|$ for $\D -2\le  x\le   3,$ showing the coordinates of the points where the curve meets the x- and y-axes. [3]
(ii) Find the non-zero values of $\D k$ for which the equation $\D |(2x+1)(x-2)| = k$ has two solutions only.
[2]

19 (CIE 2014, s, paper 21, question 3)
(i) On the axes below, sketch the graph of $\D y = |(x-4)(x+2)|$ showing the coordinates of the points where the curve meets the x-axis. [2]
(ii) Find the set of values of k for which ^x - 4h^x + 2h = k has four solutions. [3]

20 (CIE 2014, w, paper 11, question 2)
(a) On the axes below, sketch the curve $\D y = 3 \cos 2x - 1$ for  $\D 0^{\circ}\le  x \le  180^{\circ}.$ [3]
(b) (i) State the amplitude of $\D 1 - 4 \sin 2x.$ [1]
(ii) State the period of $\D 5 \tan 3x + 1.$ [1]

21 (CIE 2014, w, paper 13, question 1)
The diagram shows the graph of $\D y = a \cos bx + c$ for $\D 0^{\circ} \le x \le  360^{\circ},$ where $\D a, b$ and $\D c$ are
positive integers. State the value of each of $\D a, b$ and $\D c.$ [3]
22 (CIE 2014, w, paper 13, question 2)
The line $\D 4y = x + 8$ cuts the curve $\D xy = 4 + 2x$ at the points $\D A$ and $\D B.$ Find the exact length of $\D AB.$ [5]


$23(\mathrm{CIE} 2015, \mathrm{~s}$, paper 11, question 1$)$
(i) State the period of $\sin 2 x$.[1]
(ii) State the amplitude of $1+2 \cos 3 x$.$[1]$
(iii) On the axes below, sketch the graph of
(a) $y=\sin 2 x \quad$ for $0^{\circ} \leqslant x \leqslant 180^{\circ}$,[1]
(b) $y=1+2 \cos 3 x \quad$ for $0^{\circ} \leqslant x \leqslant 180^{\circ}$.$[2]$ 

(iv) State the number of solutions of $\sin 2 x-2 \cos 3 x=1$ for $0^{\circ} \leqslant x \leqslant 180^{\circ}$.$[1]$

$24($ CIE $2015, \mathrm{~s}$, paper 21, question 2$)$

(a)The diagram shows the graph of $y=|\mathrm{f}(x)|$ passing through $(0,4)$ and touching the $x$ -axis at $(2,0)$. Given that the graph of $y=\mathrm{f}(x)$ is a straight line, write down the two possible expressions for $\mathrm{f}(x)$.
(b) On the axes below, sketch the graph of $y=\mathrm{e}^{-x}+3$, stating the coordinates of any point of intersection with the coordinate axes.

25 (CIE 2015, s, paper 22, question 10)
(a) The function $\mathrm{f}$ is defined by $\mathrm{f}: x \mapsto|\sin x|$ for $0^{\circ} \leqslant x \leqslant 360^{\circ}$. On the axes below, sketch the graph of $y=f(x)$ 
(b) The functions $g$ and hg are defined, for $x \geqslant 1$, by
(i) Show that $\mathrm{h}(x)=\frac{\mathrm{e}^{x}+3}{4}$.$[2]$
(ii)The diagram shows the graph of $y=\mathrm{g}(x)$. Given that $\mathrm{g}$ and $\mathrm{h}$ are inverse functions, sketch, on the same diagram, the graph of $y=\mathrm{h}(x)$. Give the coordinates of any point where your graph meets the coordinate axes. $\quad[2]$
(iii) State the domain of $\mathrm{h}$.$[1]$
(iv) State the range of h.[1]

26 (CIE 2016, march, paper 12 , question 4)
(i) On the axes below, sketch the graphs of $y=2-x$ and $y=|3+2 x|$.[4]










(ii) Solve $|3+2 x|=2-x$.$[3]$

27 (CIE 2016 , march, paper 22 , question 4)
(a) $\mathrm{f}(x)=a \cos b x+c$ has a period of $60^{\circ}$, an amplitude of 10 and is such that $\mathrm{f}(0)=14$. State the values of $a, b$ and $c .$
(b) Sketch the graph of $y=3 \sin 4 x-2$ for $0^{\circ} \leqslant x \leqslant 180^{\circ}$ on the axes below.[3] 

28 (CIE $2016, \mathrm{~s}$, paper 12, question $11 \mathrm{con})$
(b) The diagram shows a velocity-time graph of a particle $Q$ moving in a straight line with velocity $y \mathrm{~ms}^{-1}$, at time $t$ s after leaving a fixed point.

The displacement of $Q$ at time $t \mathrm{~s}$ is $s \mathrm{~m}$. On the axes below, draw the corresponding displacement-time graph for $Q$$[2]$

(c) The velocity, $v \mathrm{~ms}^{-1}$, of a particle $R$ moving in a straight line, $t$ s after passing through a fixed point $O$, is given by $v=4 \mathrm{e}^{2 t}+6$
(i) Explain why the particle is never at rest.$[1]$
(ii) Find the exact value of $t$ for which the acceleration of $R$ is $12 \mathrm{~ms}^{-2}$.$[2]$
(iii) Showing all your working, find the distance travelled by $R$ in the interval between $t=0.4$ and $t=0.5 . \quad[4]$

29 (CIE 2016, s, paper 12 , question 11 plus $)$
(a)
The diagram shows the velocity-time graph of a particle $P$ moving in a straight line with velocity $v \mathrm{~ms}^{-1}$ at time $t$ s after leaving a fixed point.
(i) Find the distance travelled by the particle $P$.[2]
(ii) Write down the deceleration of the particle when $t=30$.$[1]$

30 (CIE $2016, \mathrm{~s}$, paper 21 , question 9)
(a) Given that $y=a \tan b x+c$ has period $\frac{\pi}{4}$ radians and passes through the points $(0,-2)$ and $\left(\frac{\pi}{16}, 0\right)$, find the value of each of the constants $a, b$ and $c$.
(b) (i) On the axes below, draw the graph of $y=2 \cos 3 x+1$ for $-\frac{2 \pi}{3} \leqslant x \leqslant \frac{2 \pi}{3}$ radians.$[3]$

(ii) Using your graph, or otherwise, find the exact solutions of $(2 \cos 3 x+1)^{2}=1$ for $-\frac{2 \pi}{3} \leqslant x \leqslant \frac{2 \pi}{3}$ radians.
31 (CIE 2016, w, paper 13 , question 1)
On the axes below, sketch the graph of $y=|2 \cos 3 x|$ for $0 \leqslant x \leqslant 180^{\circ}$.

32 (CIE 2017, march, paper 12, question 2)
(i) On the axes below sketch, for $0^{\circ} \leqslant x \leqslant 360^{\circ}$, the graph of $y=1+3 \cos 2 x$

(ii) Write down the coordinates of the point where this graph first has a minimum value.$[1]$

33 (CIE 2017, s, paper 11 , question 8)
(i) On the axes below sketch the graphs of $y=|2 x-5|$ and $9 y=80 x-16 x^{2}$. $[5]$

(ii) Solve $|2 x-5|=4$.$[3]$
(iii) Hence show that the graphs of $y=|2 x-5|$ and $9 y=80 x-16 x^{2}$ intersect at the points where $y=4$
(iv) Hence find the values of $x$ for which $9|2 x-5| \leqslant 80 x-16 x^{2}$.$[2]$

34 (CIE 2017, s, paper 13 , question 3)
(i) On the axes below, sketch the graph of $y=3 \sin x-2$ for $0^{\circ} \leqslant \theta \leqslant 360^{\circ}$. [3]

(ii) Given that $0 \leqslant|3 \sin x-2| \leqslant k \quad$ for $0^{\circ} \leqslant x \leqslant 360^{\circ}$, write down the value of $k$. $[1]$

35 (CIE 2017, s, paper 21 , question 4 )
(a) Given that $y=7 \cos 10 x-3$, where the angle $x$ is measured in degrees, state
(i) the period of $y$, $[1]$
(ii) the amplitude of $y$. $[1]$
(b) 
Find the equation of the curve shown, in the form $y=a g(b x)+c$, where $g(x)$ is a trigonometric function and $a, b$ and $c$ are integers to be found.$[4]$

36 (CIE 2017, s, paper 23, question 2)
The four graphs above are labelled $\mathbf{A}, \mathbf{B}, \mathbf{C}$ and $\mathbf{D}$.
(i) Write down the letter of each graph that represents a function, giving a reason for your choice. [2]
(ii) Write down the letter of each graph that represents a function which has an inverse, giving a reason for your choice.[2]

37 (CIE 2017, w, paper 12, question 2)
The graph of $y=a \sin (b x)+c$ has an amplitude of 4 , a period of $\frac{\pi}{3}$ and passes through the point $\left(\frac{\pi}{12}, 2\right)$. Find the value of each of the constants $a, b$ and $c$.

38 (CIE 2017, w, paper 12, question 9) 
(a)The diagram shows the displacement-time graph of a particle $P$ which moves in a straight line such that, $t$ s after leaving a fixed point $O$, its displacement from $O$ is $x \mathrm{~m}$. On the axes below, draw the velocity-time graph of $P$.
velocity$\mathrm{ms}$$[3]$
(b) A particle $Q$ moves in a straight line such that its velocity, $v \mathrm{~ms}^{-1}, t$ s after passing through a fixed
point $O$, is given by $v=3 \mathrm{e}^{-5 \mathrm{t}}+\frac{3 t}{2}$, for $t \geqslant 0$.
(i) Find the velocity of $Q$ when $t=0$.$[1]$
(ii) Find the value of $t$ when the acceleration of $Q$ is zero.$[3]$
(iii) Find the distance of $Q$ from $O$ when $t=0.5$.$[4]$

39 (CIE 2018, march, paper 22, question 4)
(a) (i) State the amplitude of $15 \sin 2 x-5$. $[1]$
(ii) State the period of $15 \sin 2 x-5$. $[1]$
(b)

The diagram shows the graph of $y=|\mathrm{f}(x)|$ for $-180^{\circ} \leqslant x^{\circ} \leqslant 180^{\circ}$, where $\mathrm{f}(x)$ is a trigonometric function.
(i) Write down two possible expressions for the trigonometric function $\mathrm{f}(x)$.[2]
 (ii) State the number of solutions of the equation $|f(x)|=1$ for $-180^{\circ} \leqslant x^{6} \leqslant 180^{*}$.$[1]$

40 CIE 2018, s, paper 11, question 3)
Diagrams A to D show four different graphs. In each case the whole graph is shown and the scales on the two axes are the same.


Place ticks in the boxes in the table to indicate which descriptions, if any, apply to each graph. There may be more than one tick in any row or column of the table. $[4]$
$\begin{array}{|l|l|l|l|l|}\hline &A & B & C & D \\\hline \text{Not a function} & & & & \\\hline \text{One-one function} & && & \\\hline \text{A function that is its own inverse} & && & \\\hline \text{A function with no inverse} & & && \\\hline\end{array}$

41 (CIE 2018, s, paper 11, question 4)
(i) The curve $y=a+b \sin c x$ has an amplitude of 4 and a period of $\frac{\pi}{3}$. Given that the curve passes through the point $\left(\frac{\pi}{12}, 2\right)$, find the value of each of the constants $a, b$ and $c$. $[4]$
(ii) Using your values of $a, b$ and $c$, sketch the graph of $y=a+b \sin c x$ for $0 \leqslant x \leqslant \pi$ radians. [3]


42 (CIE 2018, s, paper 12 , question 1 ) It is given that $y=1+\tan 3 x$.
(i) State the period of $y$. $[1]$
(ii) On the axes below, sketch the graph of $y=1+\tan 3 x$ for $0^{\circ} \leqslant x^{\circ} \leqslant 180^{\circ}$. $[3]$ 


43 (CIE 2018, s, paper 21 , question 9)
(i) Express $5 x^{2}-14 x-3$ in the form $p(x+q)^{2}+r$, where $p, q$ and $r$ are constants. $[3]$
(ii) Sketch the graph of $y=\left|5 x^{2}-14 x-3\right|$ on the axes below. Show clearly any points where your graph meets the coordinate axes. $[4]$

(iii) State the set of values of $k$ for which $\left|5 x^{2}-14 x-3\right|=k \quad$ has exactly four solutions. [2]




Answers


1. (i)
(ii) $\D x = 1, 4$
2. (i)
(ii) $\D x = 0, 1$
3. (a) $\D a = 3, b = 8, c = 7$
(b) $\D 2\pi/3, 5$
4. (i)
(ii) $\D x = -1=5,-1$
5. (ai)
(ii) $\D x = \pi/4, \pi/2$
(b)(i)Amp=5,Period= $\D \pi/2$
(ii)Period= $\D \pi/3$
6. (i)
(ii) $\D (\pi/4, 4), (3\pi/4,-2)$
(iii) $\D 3$
7. (i)
(ii) 4
8. (ii)
(iii) 3
9. 3.2,15,312
10. (ii) 2/5
11. (ai) A,E
(ii) C,D
(b)
12. (a)(i)
(ii)3
(iii)180
(b) $\D \frac{-1}{\sqrt{1-p^2}}$
13. $\D a = 3, b = 2, c = 1$
14. (i)
(ii) $\D (-0.5, 25/4)$
(iii) $\D k > 25/4$
15. (a)(i) $\D A = 3,B = 2$
(ii) $\D C = 4$
(b) 120,5
16. (a) 480
17. (b)
(c)  3,7
18. (i)
(ii) $\D k > 25/8$
19. (i)
(ii) $\D 0 < k < 9$
20. (a)
(b) $\D 4,\pi/3$
21. $\D a = 3, b = 2, c = 4$
22. $\D 2\sqrt{17}$

23. (i) $180, \pi$ (ii) 2
(iii)

(iv) 3
24. (a) $2 x-4$ or $-2 x+4$
(b) 

25 . 



26. (i)

(ii) $x=-1/3,-5$
27 (a) $a=10,b=6,c=4$
(b) 

28 .(b) 

(ci) always positive
(ii) $t=.5\ln 1.5,$
(iii) $1.59$
29 (a)(i) 1275 (ii) 1.5
30 (a) $a_2,b=4,c=-2$
(b)(i) 

(ii) $\pm \pi / 2, \pm \pi / 6, \pm \pi / 3$
31 


$32 .$ (i) 

(ii) $(90,-2)$
33 (i) 

(ii) $9 / 2,1 / 2$
(iv) $1 / 2 \leq x \leq 9 / 2$
$34 .$ (i) 

(ii)5
35. (a)36.7 (b) $5 \sin 4 x+7$
36. (i)B,C (ii) B
37. $a=4, b=6, c=-2$
38. (a) 

(bi) $3,(\mathrm{ii}) t=0.461$
(iii)s $=0.738$
39. $15,180, \tan x,-\tan x, 4$
$40 .$ 




42. (i) $\frac{\pi}{3}$ or $60^{\circ}$

43 $5(x-7/5)^2-64/5$



$0<k<64 / 5$


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