## Tuesday, December 25, 2018

### Graph (CIE)

$\def\D{\displaystyle}$
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. 
(ii) Solve $\D |2x - 5| = 3 .$ 

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. 
(ii) Solve $|x^2 - x - 6| = 6.$ 

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 $\D c.$ 
(b) Given that $\D f(x) = 5\cos3x + 1,$ for all $\D x,$ state
(i) the period of $\D f,$ 
(ii) the amplitude of $\D f.$ 

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. 
(ii) Solve the equation $\D |3 + 5x| = 2.$ 

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 $\D y = 1 + \cos 2x.$ 
(ii) Write down the solutions of the equation $\D \sin 2x - \cos 2x = 1,$ for $\D 0 \le x \le \pi.$ 
(b) (i) Write down the amplitude and period of $\D 5 \cos 4x - 3.$ 
(ii) Write down the period of $\D 4 \tan 3x.$ 

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 $\D y = 1 + 3\sin 2x.$ 
Write down
(ii) the coordinates of the stationary points on the curve $\D y = 1 + 3\sin 2x$ for $\D 0 \le x \le \pi,$ 
(iii) the number of solutions of the equation $\D \tan x = 1 + 3\sin 2x$ for $\D 0 \le x \le \pi.$ 

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 $\D y = 2 + \cos x.$ 
(ii) State the number of values of $\D x$ for which $\D 3 \sin 2x = 2 + \cos x$ in the interval $\D 0^{\circ} \le x \le 360^{\circ}.$ 

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) $\D y = \cos x - 1,$ 
(ii) $\D y = \sin 2x.$ 

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

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. 
(ii) Find the length of time for which the particle is travelling with constant velocity. 
(iii) Find the total distance travelled by the particle. 

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. 
(ii) Solve the equation $\D |4x - 2| = x.$ 

11 (CIE 2013, s, paper 22, question 3)
(i) Write down the letter of each graph which does not represent a function. 
(ii) Write down the letter of each graph which represents a function that does not have an inverse. 
(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. 
(ii) State the amplitude of $\D f.$ 
(iii) State the period of $\D f.$ 
(b) Given that $\D \cos x = p ,$ where $\D 270^{\circ} < x < 360^{\circ},$ find  cosec $\D x$ in terms of $\D p.$ 

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 $\D c.$ 

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. 
(ii) Find the coordinates of the stationary point on the curve $\D y = |(x - 2) (x + 3)| .$ 
(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. 

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 $\D B.$ 
(ii) Given that the point $\D \left(\frac{\pi}{8},7\right)$  also lies on the graph, find the value of $\D C.$ 
(b) Given that $\D f (x) = 8 - 5 \cos 3x,$ state the period and the amplitude of $\D f.$ 

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 $\D P.$ 
(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 $\D Q.$ 

(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. 
(ii) Find the value of $\D t$ for which the acceleration of the particle $\D R$ is 0.25ms$\D ^{-1}.$ 

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. 
(ii) Find the non-zero values of $\D k$ for which the equation $\D |(2x+1)(x-2)| = k$ has two solutions only.


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. 
(ii) Find the set of values of k for which ^x - 4h^x + 2h = k has four solutions. 

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

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 $\D c.$ 
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 $\D AB.$ 

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. $\D 2\sqrt{17}$

1. 2. 3. 