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The diagram shows the graph of $y=|p(x)|$, where $p(x)$ is a cubic function. Find the two possible expressions for $p(x)$. [3]
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(a) Write down the amplitude of $1+4\cos \left(\dfrac{x}{3}\right).$
(b) Write down the period of $1+4\cos \left(\dfrac{x}{3}\right).$[1]
(c) On the axes below, sketch the graph of $1+4\cos \left(\dfrac{x}{3}\right)$ for -180°$\le x\le $180°.
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(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. [3]
(b) Solve $6x^{\frac 23}-5x^{\frac 13}+1=0$.
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It is given that $y=\dfrac{\tan 3x}{\sin x}$.
(a) Find the exact value of $\dfrac{dy}{dx}$ when $x=\dfrac{\pi}{3}.$ [4]
(b) Hence find the approximate change in $y$ as $x$ increases from $\dfrac{\pi}{3}$ to $\dfrac{\pi}{3}+h,$ where $h$ is small. [1]
(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. [2]
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(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. [1]
(ii) How many of these 4-digit numbers are even and greater than 6000? [3]
(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, [1]
(ii) the committee contains at least one doctor, [2]
(iii) the committee contains all the nurses. [1]
$\def\cvec#1#2{\left(\begin{array}{c}#1\\#2\end{array}\right)}$
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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. [3]
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 $Q.$ [1]
(c) Find the exact distance between $P$ and $Q$ when $t= 10$ , giving your answer in its simplest surd form. [3]
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It is given that $f(x)=5\ln(2x+3)$ for $x>-\dfrac 32$
(a) Write down the range of $f.$ [1]
(b) Find $f^{-1}$ and state its domain. [3]
(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.
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(a) (i) Show that $\dfrac{1}{(1+\mbox{cosec }\theta)(\sin\theta-\sin^2\theta)}= \mbox{sec }^2\theta.$ [4]
(ii) Hence solve $(1+\mbox{cosec }\theta)(\sin\theta-\sin^2\theta)=\dfrac 34$ for $-180^{\circ}\le\theta\le 180^{\circ}.$ [4]
(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; $\pi.$ [4]
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(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. [6]
(b) Find the exact value of $\displaystyle\int_0^{\dfrac{\pi}{3}}\left( \sin\left(2x+\dfrac{\pi}{3}\right)-1+\cos 2x\right) dx.$ [5]
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(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. [5]
(ii) Hence find the value of the first term. [1]
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