$\text { Let } f(x)=a \sin b x \text {, for } x \in \mathbb{R} \text {. The following diagram shows part of the graph of } f \text {. }$


1. 1. Write down the amplitude of f .   
2. Find the value of a .   
2. 1. Write down the period of f .   
2. Find the value of b .   

  Let f(x)=a $\cos b x$ , for $x \in \mathbb{R}$ , where b>0 .
Part of the graph of f is shown on the diagram below.

1. Find the value of a .   
2. Find:
1. the period of f ;   
2. the value of b .   

  Let f(x)=$a \cos b x+d$ , for $x \in \mathbb{R}$ , where b>0 . Part of the graph of f is shown on the diagram below.



1. Find the value of a .   
2. 1. Write down the period of f .   
2. Find the value of b .   
3. Find the value of d .   

  The following diagram shows the curve g/z a-pdqf e5p5o af)f5 2dac8;e+wtpy=a cos (k(x-d))+c where a, k, d and c are all positive constants. The curve has apwg z ffe/a8; ) -d+paa525copdq5etf minimum point at (1.5,2) and a maximum point at (3.5,7) .



1. Write down the value of a and the value of c . a =    c =   
2. Find the value of k .   
3. Find the smallest possible value of d , given d>0 .   

  Given that $\sin x=\frac{3}{5}$ , where x is an acute angle, find:
1. cos x   
2. sin 2 x ;   
3. cos 2 x .   

  Let sin x=$\frac{2}{3}$ , where x is obtuse.
1. Find cos x .   
2. Find cos 2 x .   

Let f(x)=3 $\sin (\pi x)$+1 .
1. Write down the amplitude of f .
2. Find the period of f .
3. On the following grid, sketch the graph of f , for 0 $\leq x \leq 4 $.

  Let f(x)=p $\sin (q x)+r$ , for $x \in \mathbb{R} $.
Part of the graph of f is shown on the diagram below.

The graph of f has a minimum at $\mathrm{A}(3,-1)$ and a maximum at $\mathrm{B}(9,5)$ .
1. 1. Find the period of f .   
2. Hence find the value of q .   
2. Find the values of:
1. p ;   
2. r .   
3. Solve f(x)=4 , for 0 $\leq x \leq 15$ .      

  The following diagram shows the curve y=a sin (k(x-d))+ dll 9tr2vxp3srl z/33 j+njm+c where a, k, d and c are all positive constants. The curve has a minimumplmnx/ z3r2 vl39jrsj ++ltd3 point at (1.5,1) and a maximum point at (3,7) .



1. Write down the value of a and the value of c . a =    c =   
2. Find the value of k .   
3. Find the smallest possible value of d , given d>0 .   

  Let $f(x)=\sin \left(x+\frac{\pi}{6}\right)+q$ . The graph of f passes through the point $\left(\frac{\pi}{3}, 5\right)$ .
1. Find the value of q .   
2. Find the maximum value of f .

Let $g(x)=\sin x$ . The graph of g is translated to the graph of f by the vector $\binom{a}{b}$ .   
3. Write down the values of a and b . a =    b =   

  $\text { Solve } \sqrt{3} \tan x+1=0 \text {, for } 0 \leq x \leq \pi \text {. }$   

  The height, h metres, of a seat on a Ferris wheecefjg)2 s7l,7am wvq + ldih4)l after t minutes is given by)mg7+) jsdfeaw4vilhc2q7l ,

$h(t)=-23.5 \cos (0.4 t)+25, \text { for } t \geq 0 \text {. }$

1. Find the initial height of the seat.

Once a passenger's seat is more than 30 m above the ground, there are no trees in view and they can take unobstructed photographs of a nearby city.   
2. Given that passengers only complete one rotation on the Ferris wheel, calculate how long they can take unobstructed photographs of the nearby city.   

  $\text { Solve } \frac{\sin 2 x}{\cos x}-1=0 \text {, for } \frac{\pi}{2} \leq x \leq \pi \text {. }$   

  Given that sin x=$\frac{4}{5}$ , where x is an obtuse angle, find the value of:
1. cos x   
2. cos 2 x ;   
3. tan 2 x .   

  The Singapore Flyer is a gifdg 3 5n.7 ggf,6bicitant observation wheel in Singapore with diameter of 150 metres. The wheel rotates at a constant speed and completes one rotation in 32 minutes. The bottom of the wheel is d metres aboti3c6n5fg ,f7bi g. gdve the ground.


A seat starts at the bottom of the wheel.
1. After 16 minutes, the seat is 165 metres above the ground. Find d .   

After t minutes, the height of the seat above the ground is given by

$h(t)=90+a \cos \left(\frac{\pi}{16} t\right), \text { for } 0 \leq t \leq 64 $.

2. Find the value of a .   
3. Find when the seat is 60 metres above the ground for the third time.    min

  Given that sin x=$\frac{2}{3} $, for $ \frac{\pi}{2} \leq x \leq \pi$, find the value of:
1. cos x ≈   
2. tan 2 x .≈   

  A function is define-cr6/cpzz5 +4heqlff d by f(x)=a cos (k x)+c , for $-\pi \leq x \leq \pi $, where k>0 . Part of the graph of y=f(x) is shown on the diagram below.

1. Find the value of:
1. a ;   
2. k ;   
3. c .   
2. Solve f(x)=2 , for 0$ \leq x \leq \pi$ .         

  Let f(x)=a sin (k(x-d)t8 *a/s w. v,qhj*qrnr)+c , for $1 \leq x \leq 7$.
The graph of f is shown on the diagram below.

The graph of f has a maximum at $ \mathrm{A}(2,5) $ and a minimum at $\mathrm{B}(4,1)$ .
1. Find the value of:
1. a ;   
2. c.   
2. 1. Find the value of k .   
2. Find the smallest possible value of d , given d>0 .

The graph of f is translated by a vector $\binom{0}{b}$ to give the graph of a function g such that g(x)=3 has only one solution in the given domain.   
3. Find the value of b .   

Let $f(x)=\sin \left(\frac{\pi}{3} x\right)+\cos \left(\frac{\pi}{3} x\right)$ , for $0 \leq x \leq 8 $.
1. Find the values of x where f has a positive rate of change.

The function f can be written in the form $f(x)=a \cos \left(\frac{\pi}{3}(x-d)\right)$ where $6 \leq d \leq 9$ .
2. Find the value of:
1. a ;
2. d .
3. Solve f(x)=1 , for $0 \leq x \leq 8$ .

  $\text { Solve the equation } \cos 2 x-\sin ^{2} x=\cos ^{2} x+3 \cos x \text {, for } 0 \leq x \leq 2 \pi \text {. }$      

  The expression 8 sin x cos x b12/ o5 0mahoehhew-j c+ (qaf can be written in the form p sin q x .
1. Find the value of p and the value of q . p =    q =   
2. Hence, or otherwise, solve the equation 8 sin x cos x=2 $\sqrt{3}$ , for $\frac{\pi}{4} \leq x \leq \frac{\pi}{2}$ .   

  The following diagram shows trianmosu+fgy w*g0txs+g 9z;fi + .gle A B C . Point D lies on [A C] so that [DB] bisects CEAA. The area of the trm0 g+i9s*xuf +tf osg +;zwy.giangle A B C is 3$ \mathrm{~cm}^{2}$ .


$\mathrm{AB}=2 \sqrt{7} \mathrm{~cm}, \mathrm{BC}=x \mathrm{~cm}$ , and $\mathrm{CBD}=\theta$ , where $ \sin \theta=\frac{3}{4} $ Find the value of x in the form of $ \frac{a}{b}$ where a and b are positive integers. x =   

$\text { Find all solutions to the equation } \tan x-\sin 2 x=0 \text { where } 0^{\circ} \leq x \leq 360^{\circ} \text {. }$

Let f(x)=3 cos (2 x)+5 ymtf* gm(p7z+q6shad1 0k.yr , for $x \in \mathbb{R}$ .
The range of f is $p \leq f(x) \leq q$ .
1. Find the value of p and the value of q .

Let g(x)=4 f(3 x) , for x $\in \mathbb{R}$ .
2. Find the range of g .

The function g can be written in the form g(x)=12 cos (k x)+c .
3. 1. Find the value of k and the value of c .
2. Find the period of g .

The equation g(x)=10 has two solutions for $\frac{2 \pi}{3} \leq x \leq \pi $.
4. Find both solutions.

  $\text { Given that } \cos x=\frac{3}{4} \text {, where } 0 \lt x \lt \frac{\pi}{2} \text {, find the value of } \sin 4 x \text {. }$   

  $\text { Solve } \log _{\sqrt{3}}(\sin x)-\log _{\sqrt{3}}(\cos x)=1 \text {, for } 0 \lt x \lt \frac{\pi}{2} \text {. }$   

Consider the function f(x)=sin x , for 0 rm7z:ut o2xqbe;(i4cqti 1j$x \in \mathbb{R}$ , where x is in radians.
1. Write down:
1. the maximum value of f ;
2. the smallest positive value of x for which the maximum of f occurs.

Let $g(x)=2 \sin \left(x+\frac{\pi}{4}\right)$ , for $ x \in \mathbb{R}$ , where x is in radians.
2. 1. Determine the two transformations the graph of f undergoes to form the graph of g .
2. Hence find the maximum value of g and the smallest positive value of x for which this maximum occurs.

Let $h(x)=\frac{4}{2 \sin \left(x+\frac{\pi}{4}\right)-3}$, for $x \in \mathbb{R}$, where x is in radians.
3. Determine if the graph of h has a vertical asymptote. Justify your answer.
4. Find the range of h .

$\text { The following diagram shows the graph of } f(x)=a \sin k x+c \text {, for } 0 \leq x \leq 16 \text {. }$



The graph of f has a minimum at $\mathrm{P}(4,8)$ and a maximum at $\mathrm{Q}(12,16)$ .
1. 1. Find the value of c .
2. Show that $k=\frac{\pi}{8}$ .
3. Find the value of a .

The graph of g is obtained from the graph of f by a translation of $\binom{d}{0} $.
The minimum point on the graph of g has coordinates (6.5,8) .
2. 1. Write down the value of d .
2. Find g(x) .

The graph of g changes from concave-up to concave-down when x=\nu .
3. 1. Find $\nu$ .
2. Hence, or otherwise, find the maximum positive rate of change of g .

  1. Given that $\theta=18^{\circ}$ satisfies the equation $4 \sin ^{2} \theta+2 \sin \theta-1=0$ , find the value of $\sin 18^{\circ}$ .   
2. Hence find the value of $\cos 36^{\circ}$ .   

The following diagram shows the triangle ABC where $ \mathrm{AB}=4 \mathrm{~cm}, \mathrm{BC}=5 \mathrm{~cm}$ and $\mathrm{CBA}=36^{\circ}$ .

$\text { 3. Find } \mathrm{AC} \text { in the form } \sqrt{a+b \sqrt{5}} \mathrm{~cm} \text { where } a, b \in \mathbb{Z} \text {. }$   

Consider the function s,;o;-s yplih$f(x)=x^{2} \arcsin (x) , for -1 \leq x \leq 1 $.
1. Sketch the graph of y=f(x) .
2. Write down the range of f .
3. Solve the inequality $ \left|x^{2} \arcsin (x)\right|>0.5$ .

  $\text { Find all solutions to the equation } \frac{1}{\tan x}+\frac{1}{\tan 2 x}=0 \text { where } 0 \lt x\lt 2 \pi \text { and } x \neq \pi \text {. }$      

  $\text { It is given that } \sec \theta=-\frac{6}{5} \text {, where } \pi<\theta<2 \pi \text {. Find the exact value of } \tan \theta \text {. }$   

Consider the functions f(x)=3 cos (xs ;wn; w :*cxv8kp1lac)+$\frac{9}{2} $ and $g(x)=3 \cos \left(x+\frac{\pi}{3}\right)+A$ , where $x \in \mathbb{R}$ and $A<\frac{9}{2}$ .
1. Describe a sequence of two transformations that transforms the graph of f to the graph of g .

The y -intercept of the graph g is at the point $\left(0, \frac{9}{2}\right)$
2. Find the range of g .

$\text { Solve the equation } \cos 2 x+\cos x=1+\sin 2 x-\sin x \text {, for } x \in[-\pi, \pi] \text {. }$

Let $g(x)=-4 x-\frac{\pi}{3}$ and h(x)=$5 \cos x-1$ , for $x \in \mathbb{R}$ .
1. Show that $(h \circ g)(x)=5 \cos \left(-4 x-\frac{\pi}{3}\right)-1$ .
2. Find the range of $h \circ g$ .
3. Given that $(h \circ g)\left(\frac{5 \pi}{12}\right)=4$ , find the next value of x , greater than $\frac{5 \pi}{12} $, for which $(h \circ g)(x)=4$.
4. The graph of $y=(h \circ g)(x) $ can be obtained by applying five transformations to the graph of $ y=\cos x$. State what the five transformations represent geometrically and give the order in which they are applied.

  $\text { The diagram below shows the graph of } f(x)=a \sin (k(x-d))+c \text {, for } 2 \leq x \leq 14 \text {. }$



The graph of f has a maximum at $\mathrm{P}(5,15)$ and a minimum at $\mathrm{Q}(11,-5)$ .
1. Write down the value of:
1. a ;   
2. c .   
2. 1. Show that $ k=\frac{\pi}{6}$ .   
2. Find the smallest possible value of d , given d>0 .   
3. Find $f^{\prime}(x)$ .
4. At a point R , the gradient is $-\frac{5 \pi}{3}$ . Find the x -coordinate of R .   

Let $f(x)=\sin x-\sqrt{3} \cos x, 0 \leq x \leq 2 \pi$ .
The following diagram shows the graph of f .

The curve crosses the x -axis at A and C and has a maximum at point B .
1. Find the exact coordinates of A and of C .
2. Find $f^{\prime}(x) $.
3. Find the coordinates of B .

$\text { Find all solutions to the equation } \tan 2 x-3 \tan x=0 \text { where } 0^{\circ} \leq x<360^{\circ} \text {. }$

In a triangle $\mathrm{ABC}, $\mathrm{AB}=2 \mathrm{~cm}$, $\mathrm{CBA}=\frac{\pi}{4}$ and $\mathrm{B} \hat{\mathrm{A}} \mathrm{C}=\theta $.
1. Show that $\mathrm{AC}=\frac{2}{\cos \theta+\sin \theta}$ .
2. Given that A C has a minimum value, find the value of $\theta$ for which this occurs.

The London Eye is an observation wheel vwu/ vs sj8:y*in England with diameter of 120 metres. The wheel rotates at a constant speed and completes 2.5 rotations every hour. The bottom of the wheel is 15 metres above the groundv sj:*u8/svw y.


A seat starts at the bottom of the wheel.
1. Find the maximum height above the ground of the seat.

After t minutes, the height h metres above the ground of the seat is given by

h(t)=75+a cos (b t), b>0 .

2. 1. Show that the period of h is 24 minutes.
2. Write down the exact value of b .
3. Find the value of a .
4. Sketch the graph of h , for $0 \leq t \leq 48$ .
5. In one rotation of the wheel, find the probability that a randomly selected seat is at least 110 metres above the ground.

  $\text { It is given that } \operatorname{cosec} \theta=-\frac{5}{3} \text {, where } \frac{\pi}{2}<\theta<\frac{3 \pi}{2} \text {. Find the exact value of } \cot \theta \text {. }$   

  1. Show that $2 x-9+\frac{5}{x+1}=\frac{2 x^{2}-7 x-4}{x+1}, x \in \mathbb{R}, x \neq-1$ .  (代数式) 
2. Hence, solve the equation $2 \sin 2 \theta-9+\frac{5}{\sin 2 \theta+1}=0$ for $0 \leq \theta \leq \pi, \theta \neq \frac{3 \pi}{4}$ .      

1. Show that $\log _{16}(\cos 2 x+7)=\log _{4} \sqrt{\cos 2 x+7}$ .
2. Hence, or otherwise, solve $\log _{4}(\sqrt{10} \cos x)=\log _{16}(\cos 2 x+7) $, for $0\lt x \lt\frac{\pi}{2}$ .

  The following table shows the probability distribution of a discrete random c udkkf8/a9+j+ ti2z xvariable Z, in terms of an angle θ. k+k /982cjzf+xit ua d



1. Show that $\cos \theta=\frac{3}{4}$ .   
2. Find $\tan \theta $, given that $\tan \theta>0$ .

Let $f(x)=\frac{1}{\cos x}$ , for $0 \lt x \lt\frac{\pi}{2}$ .
The graph of y=f(x) between $x=\theta$ and $x=\frac{\pi}{4}$ is rotated $360^{\circ}$ about the x -axis.   
3. Find the volume of the solid formed.   

  Given that $\sin x-\cos x=\frac{1}{4}$, find $ \cos 8 x$ , rounding your answer to 3 significant figures.   

A function, f , is defin jslpgh-a mn66 /n1j4zed by $f(x)=6.2 \sin \left(\frac{\pi}{9}(x-7.5)\right)+c $, for $0 \leq x \leq 15$ , where $c \in \mathbb{R}$ .
1. Find the period of f .

The function f has a minimum at $\mathrm{A}(3,11.8)$ and a maximum at $\mathrm{B}(12,24.2)$ .
2. 1. Find the value of c .
2. Hence find the value of f(9) .

A second function, g , is defined by $g(x)=a \sin \left(\frac{2 \pi}{15}(x+2.25)\right)+b $, for $0 \leq x \leq 15$ , where a, b$ \in \mathbb{R}$ .
The function g passes through the points $ \mathrm{P}(1.5,14.5)$ and $ \mathrm{Q}(14,10.2)$ .
3. Find the value of a and the value of b .
4. Find the value of x for which the functions have the greatest difference.

1. Show that $\log _{2} \sqrt{3-\cos 2 x}=\log _{4}(3-\cos 2 x)$ .
2. Hence, or otherwise, solve $\log _{4}(3 \sin x)+\frac{1}{4}=\log _{2} \sqrt{3-\cos 2 x}$ , for $0

  The following diagram shows a ball attached to the enqtwc91 5 u)/ p:n*y dq/lvtxsxd of a spring, which is suspended from a cs/cwlq1n qx)p*d9 5xvutt :/yeiling.



The height, h metres of the ball above the ground at time t seconds after being released can be modelled by the function h(t)=0.5 $\cos (\pi t)+2.2$, where $t \geq 0 $.
1. Find the height of the ball above the ground when it is released.   
2. Find the minimum height of the ball above the ground.   
3. Show that the ball takes 2 seconds to return to its initial height above the ground for the first time.   
4. For the first 2 seconds of its motion, determine the amount of the time that the ball is less than $2.2+0.25 \sqrt{2}$ meters above the ground.   
5. Find the rate of change of the ball's height above the ground when $t=\frac{1}{3}$ . Give your answer in the form p $\pi \sqrt{q} \mathrm{~ms}^{-1} $, where $p \in \mathbb{Q}$ and $q \in \mathbb{Z}^{+}$    $\mathrm{ms}^{-1}$

The first three terms of a geometric sequence are hm hy5ir9yz.5

$\sqrt{2} \cos x, \sin 2 x$, $2 \sqrt{2} \sin ^{2} x \cos x$,$-\frac{\pi}{2} \lt x\lt\frac{\pi}{2}$ .

1. Find the common ratio.
2. Find the set of values of x for which the geometric series below converges.

$\sqrt{2} \cos x+\sin 2 x+2 \sqrt{2} \sin ^{2} x \cos x+\cdots$

Consider $x=\arcsin \left(\frac{1}{2 \sqrt{2}}\right), 0 \lt x \lt \frac{\pi}{2}$ .
3. Show that the sum to infinity of this series is $\sqrt{7}$ .

A physicist is studying the a9xq9e3 c*pmy6b, .3 sxxuiqfmotion of two separate particles moving in a straight line. She measures the displacement of each particle from a fixed origin over the course of 10 seconds. The physicist found that the displacement of particle A, s_{A} \mathrm{~cm} , at time t 3p, 9yex9mqi xcq .a63xufsb * seconds can be modelled by the function $ s_{A}(t)=7 t+9$ , where $ 0 \leq t \leq 10$ .
The physicist found that the displacement of particle B,$ s_{B} \mathrm{~cm}$ , at time t seconds can be modelled by the function $s_{B}(t)=\cos (3 t+5)+8 t+4$ .
1. Use the physicist's models to find the initial displacement of
1. Particle A ;
2. Particle B correct to three significant figures.
2. Find the values of t when s_{A}(t)=s_{B}(t) .
3. For t>6 , prove that particle B was always further away from the fixed origin than particle A .
4. For $0 \leq t \leq 10 $, find the total amount of time that the velocity of particle A was greater than the velocity of particle B .

Consider the function shv9i , 3k0, ssv-u2.pzc pimcu+)y av$f(x)=\sin \left(\frac{\pi}{12}-\frac{x}{4}\right)$ for $ x \in \mathbb{R}$ .
1. Show that the y -intercept of f(x) is $\frac{\sqrt{6}-\sqrt{2}}{4} $
2. Find the least positive value of x for which $ f(x)=\frac{\sqrt{3}}{2}$ .

A curve is given by theue1 f9kuw:0g0my -wur zy:t5l equation $y=\cos (2 \pi \sin x)$ .
Find the coordinates of all the points on the curve for which $\frac{\mathrm{d} y}{\mathrm{~d} x}=0,0 \leq x \leq \pi$ .

A curve is given by the ga5 t,/ h */mvjio:zfsl-4ef eequation $ y=\cos (2 \pi \cos x)$.
Find the coordinates of all the points on the curve for which $\frac{\mathrm{d} y}{\mathrm{~d} x}=0,0 \leq x \leq \pi $.

The first two terms of an infinite g,, gb++ hm/np vvu;eckeometric sequence are $ u_{1}=20$ and $u_{2}=16 \sin ^{2} \theta$, where $0 \lt \theta \lt 2 \pi$ , and $\theta \neq \pi$ .
1. 1. Find an expression for r in terms of $\theta$ .
2. Find the possible values of r .
2. Show that the sum of the infinite sequence is $\frac{100}{3+2 \cos 2 \theta}$
3. Find the values of $\theta$ which give the greatest value of the sum.

Consider the functionu.(o h guw)4wy defined by $f(x)=\frac{7}{x^{2}+6 x-7}$ for $x \in \mathbb{R}$,$ x \neq-7$,$x \neq 1$ .
1. Sketch the graph of y=f(x) , showing the values of any axes intercepts, the coordinates of any local maxima and minima, and the graphs of any asymptotes.

Next, consider the function g defined by $g(x)=\frac{7}{x^{2}+6 x-7}$ for $x \in \mathbb{R}$, x>1 .
2. Show that $ g^{-1}(x)=-3+\sqrt{\frac{16 x+7}{x}}$ .
3. State the domain of $g^{-1}$ .

Now, consider the function h defined by $h(x)=\arccos \left(\frac{x}{7}\right)$ .
4. Given that $(h \circ g)(a)=\frac{\pi}{3}$ , find the value of a . Give your answer in the form $ p+q \sqrt{2}$ where p, q$\in \mathbb{Z}$ .

Let $z=\cos \theta+\mathrm{i} \sin \theta $, for $-\frac{\pi}{4}<\theta<\frac{\pi}{4} $.
1. 1. Find $z^{3}$ using the binomial theorem.
2. Use de Moivre's theorem to show that $\cos 3 \theta=4 \cos ^{3} \theta-3 \cos \theta $ and $\sin 3 \theta=3 \sin \theta-4 \sin ^{3} \theta$ .
2. Hence show that $\frac{\sin 3 \theta-\sin \theta}{\cos 3 \theta+\cos \theta}=\tan \theta $.
3. Given that $\sin \theta=\frac{1}{3}$ , find the exact value of $\tan 3 \theta $.

1. Solve $2 \sin \left(x+120^{\circ}\right)=\sqrt{3} \cos \left(x+60^{\circ}\right) $, for x $\in\left[0,180^{\circ}\right]$ .
2. Show that $\sin 75^{\circ}+\cos 75^{\circ}=\frac{\sqrt{6}}{2}$ .
3. Let $z=\sin 4 \theta+\mathrm{i}(1-\cos 4 \theta)$ , for $z \in \mathbb{C}, \theta \in\left[0,90^{\circ}\right] $.
1. Find the modulus and argument of z in terms of $\theta$ .
2. Hence find the fourth roots of z in modulus-argument form.

A water truck tank which is 3 metres bc 6;k3 dbkecro)+sf)r6 ct-along has a uniform cross-section in the shape of a major segment. The tank is divided into two equal parts and is partially filled with water as shown in the following diagram of the cross-section. The centre of the circle is O, the angle AOB is α radians, and the angle AOF is β k -6)osrbf )kc;3eacr b6cdt+radians.



1. Given that $\alpha=\frac{\pi}{4}$ , calculate the amount of water, in litres, in the right part
of the water tank. Give your answer correct to the nearest integer.
2. Find an expression for the volume of water V , in $\mathrm{m}^{3}$, in the left part of the water tank in terms of $\beta$ .

The left part of the tank is now being filled with water at a constant rate of 0.001 $\mathrm{~m}^{3}$ per second.
3. Calculate $\frac{\mathrm{d} \beta}{\mathrm{d} t}$ when \$beta=\frac{3 \pi}{5}$ . Round your answer to 3 significant figures.
4. Calculate the amount of time it will take for the left part of the tank to be fully filled with water. Give your answer in minutes and correct to the nearest integer.

In a triangle ABC , ,.uje 4i ;ngtd

$\begin{array}{l}
5 \sin (\mathrm{ABC})-6 \cos (\mathrm{B} \hat{\mathrm{C}} \mathrm{A})=7, \\
6 \sin (\mathrm{B} \hat{\mathrm{C}} \mathrm{A})-5 \cos (\mathrm{A} \hat{\mathrm{B}})=\sqrt{2} .
\end{array}$

1. Show that $\sin (\mathrm{ABC}+\mathrm{BCA})=\frac{1}{6}$.

James claims that CÂB can have two possible values.
2. Show that James is wrong by proving that CÂB has only one possible value.

Consider the equation vv/c6ey zw4+yz9qo9f

$\frac{\sqrt{\sqrt{2}-1}}{\sin x}+\frac{\sqrt{\sqrt{2}+1}}{\cos x}=2 \sqrt[4]{8}, \quad 0 \lt x \lt\frac{\pi}{2}$ .

1. Given that $\sin \left(\frac{\pi}{8}\right)=\frac{1}{2} \sqrt{2-\sqrt{2}} $ and $\cos \left(\frac{\pi}{8}\right)=\frac{1}{2} \sqrt{2+\sqrt{2}}$ , verify that $x=\frac{\pi}{8}$ is a solution to the equation.
2. Hence find the other solution to the equation.

Let $ f(x)=x^{4}-0.4 x^{3}-2.85 x^{2}+0.9 x+1.35$ , for $x \in \mathbb{R}$ .
1. Find the solutions for f(x)>0 .
2. For the graph of y=f(x) ,
1. find the coordinates of local minimum and maximum points.
2. find the x -coordinates of the points of inflexion.

The domain of f is now restricted to [a, b] where a, b $\in \mathbb{R}^{+}$ .
3. 1. Write down the smallest value of a>0 and the largest value of b>0 for which f has an inverse. Give your answers correct to three significant figures.
2. For these values of a and b , sketch the graphs of y=f(x) and $y=f^{-1}(x)$ on the same set of axes, showing clearly the coordinates of the end points of each curve.
3. Solve f^{-1}(x)=0.5 .

Let $g(x)=\frac{2}{3} \sin (2 x-1)+\frac{1}{2}$, $\frac{1}{2}-\frac{\pi}{4} \leq x \leq \frac{1}{2}+\frac{\pi}{4}$ .
4. 1. Find an expression for g^{-1} and state its domain.
2. Solve $\left(f^{-1} \circ g\right)(x)<0.5$ .

1. Solve the equation l 4o +1vhbg m/utl+t qhr*5c0r $\sin \left(x+90^{\circ}\right)=2 \cos \left(x-60^{\circ}\right)$, $0^{\circ} \lt x\lt360^{\circ} $.
2. Show that $\sin 15^{\circ}+\cos 15^{\circ}=\frac{\sqrt{6}}{2}$ .
3. Let $z=1-\cos 4 \theta-\mathrm{i} \sin 4 \theta$ , for $ z \in \mathbb{C}, 0<\theta<\frac{\pi}{2}$ .
1. Find the modulus and argument of z . Express each answer in its simplest form.
2. Hence find the fourth roots of z in modulus-argument form.

Consider the function v(9*bjwttku ,$f(x)=-2 \sin ^{2} x+3 \sin 2 x+\tan x-3,0 \leq x<\frac{\pi}{2}$ .
1. 1. Determine an expression for $f^{\prime}(x)$ in terms of x .
2. Sketch the graph of $y=f^{\prime}(x)$ for $ 0 \leq x<\frac{\pi}{2}$ .
3. Find the x -coordinate(s) of the point(s) of inflexion of the graph of y=f(x) , labelling these clearly on the graph of $y=f^{\prime}(x)$ .
2. Let $u=\tan x$ .
1. Express sin x in terms of u .
2. Express sin 2 x in terms of u .
3. Show that f(x)=0 can be expressed as $u^{3}-5 u^{2}+7 u-3=0 $.
3. Solve the equation f(x)=0 , giving your answers in the form $\arctan p$ , where $ p \in \mathbb{Z}$ .

A function f(x) is d obs9apg*ypypl )+t;6 efined by $f(x)=\arccos \left(\frac{x^{2}-1}{x^{2}+1}\right)$, $x \in \mathbb{R}$
1. Show that f is an even function.
2. Find the equation of the horizontal asymptote to the graph of y=f(x) .
3. 1. Show that $f^{\prime}(x)=-\frac{2 x}{\sqrt{x^{2}}\left(x^{2}+1\right)}$ for $x \in \mathbb{R}$,$ x \neq 0 $.
2. Using the expression for $f^{\prime}(x)$ and the result $ \sqrt{x^{2}}=|x|$ , show that f is increasing for x<0 .

A function g is defined by $g(x)=\arccos \left(\frac{x^{2}-1}{x^{2}+1}\right)$, $x \in \mathbb{R}$, $x \geq 0$ .
4. Find the range of g .
5. Find an expression for $g^{-1}(x)$ .
6. State the domain of $g^{-1}(x)$ .
7. Sketch the graph of $y=g^{-1}(x)$ . Clearly indicating any asymptotes with their equations and stating the values of any axes intercepts.

The following diagramasi/m)vi: ei1 shows the graph of $y=\arctan (2 x-3)+\frac{3 \pi}{4}$ for $ x \in \mathbb{R}$ , with asymptotes at $ y=\frac{\pi}{4} $ and $y=\frac{5 \pi}{4}$ .



1. Describe a sequence of transformations that transforms the graph of $ y=\arctan x$ to the graph of $y=\arctan (2 x-3)+\frac{3 \pi}{4}$ for $ x \in \mathbb{R}$ .
2. Show that $\arctan p-\arctan q \equiv \arctan \left(\frac{p-q}{1+p q}\right)$ .
3. Verify that $\arctan (x+2)-\arctan (x+1)=\arctan \left(\frac{1}{(x+1)^{2}+(x+1)+1}\right) $.
4. Using mathematical induction and the results from part (b) and (c), prove that

$\sum_{r=1}^{n} \arctan \left(\frac{1}{r^{2}+r+1}\right)=\arctan (n+1)-\frac{\pi}{4} \quad \text { for } n \in \mathbb{Z}^{+}$