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. [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]

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. $\D 2\sqrt{17}$

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