# CIE 0606/2020/W/11Q

1. The diagram shows the graph of $y=|p(x)|$, where $p(x)$ is a cubic function. Find the two possible expressions for $p(x)$. 

2. (a) Write down the amplitude of  $1+4\cos \left(\dfrac{x}{3}\right).$

(b) Write down the period of  $1+4\cos \left(\dfrac{x}{3}\right).$

(c) On the axes below, sketch the graph of   $1+4\cos \left(\dfrac{x}{3}\right)$ for   -180°$\le x\le$180°.

3. (a) Write $\dfrac{\sqrt p(qr^2)^{\frac 13}}{(q^3p)^{-1}r^3}$ in the form $p^aq^br^c$, where $a, b$ and $c$ are constants. 

(b) Solve $6x^{\frac 23}-5x^{\frac 13}+1=0$.

4. It is given that $y=\dfrac{\tan 3x}{\sin x}$.

(a) Find the exact value of $\dfrac{dy}{dx}$ when $x=\dfrac{\pi}{3}.$ 

(b) Hence find the approximate change in $y$ as $x$ increases from $\dfrac{\pi}{3}$ to $\dfrac{\pi}{3}+h,$ where $h$ is small. 

(c) Given that $x$ is increasing at the rate of 3 units per second, find the corresponding rate of change in $y$ when $x =\dfrac{\pi}{3}$, giving your answer in its simplest surd form. 

5. (a) (i) Find how many different 4-digit numbers can be formed using the digits 1, 3, 4, 6, 7 and 9. Each digit may be used once only in any 4-digit number. 

(ii) How many of these 4-digit numbers are even and greater than 6000? 

(b) A committee of 5 people is to be formed from 6 doctors, 4 dentists and 3 nurses. Find the number of different committees that could be formed if

(i) there are no restrictions, 

(ii) the committee contains at least one doctor, 

(iii) the committee contains all the nurses. 

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6. A particle P is initially at the point with position vector $\cvec{30}{10}$  and moves with a constant speed of 10ms$^{-1}$ in the same direction as $\cvec{-4}{3}.$

(a) Find the position vector of $P$ after $t$ s. 

As $P$ starts moving, a particle $Q$ starts to move such that its position vector after $t$ s is given by $\cvec{-80}{90}+t\cvec{5}{12}.$

(b) Write down the speed of $Q.$ 

(c) Find the exact distance between $P$ and $Q$ when $t= 10$ , giving your answer in its simplest surd form. 

7. It is given that $f(x)=5\ln(2x+3)$ for $x>-\dfrac 32$

(a) Write down the range of $f.$ 

(b) Find $f^{-1}$ and state its domain. 

(c) On the axes below, sketch the graph of $y=f(x)$ and the graph of  $y=f^{-1}(x)$. Label each curve and state the intercepts on the coordinate axes.

8. (a) (i) Show that $\dfrac{1}{(1+\mbox{cosec }\theta)(\sin\theta-\sin^2\theta)}= \mbox{sec }^2\theta.$ 

(ii) Hence solve $(1+\mbox{cosec }\theta)(\sin\theta-\sin^2\theta)=\dfrac 34$ for $-180^{\circ}\le\theta\le 180^{\circ}.$ 

(b) Solve $\sin\left(3\phi+\dfrac{2\pi}{3}\right)= \cos\left(3\phi+\dfrac{2\pi}{3}\right)$ for $0\le \phi\le \dfrac{2\pi}{3}$ radians, giving your answers in terms of p; $\pi.$ 

9. (a) Given that $\displaystyle \int_1^a\left(\dfrac 1x-\dfrac{1}{2x+3}\right) dx= \ln 3,$ where $a>0$, find the exact value of $a$, giving your answer in simplest surd form. 

(b) Find the exact value of  $\displaystyle\int_0^{\dfrac{\pi}{3}}\left( \sin\left(2x+\dfrac{\pi}{3}\right)-1+\cos 2x\right) dx.$ 

10. (a) An arithmetic progression has a second term of 8 and a fourth term of 18. Find the least number of terms for which the sum of this progression is greater than 1560

(b) A geometric progression has a sum to infinity of 72. The sum of the first 3 terms of this progression is $\dfrac{333}{8}.$

(i) Find the value of the common ratio. 

(ii) Hence find the value of the first term.