1. 1
   The functions $f$, $g$ and $h$ are defined for $x\in\mathbb{R}$ by $f(x)=x^{3}-4x$, $g(x)=x^{4}+2\cos x$ and $h(x)=x^{2}+x$.

   (a)

   Which of the functions $f$, $g$ and $h$ are even?

   [1 mark]

   • A$g$ only
   • B$f$ only
   • C$f$ and $g$
   • D$g$ and $h$

   (b)

   The function $p$ is defined by $p(x)=f(x)\,g(x)$. Which statement about $p$ is true?

   [1 mark]

   • A$p$ is even
   • B$p$ is both odd and even
   • C$p$ is odd
   • D$p$ is neither odd nor even

   (c)

   The function $q$ is defined by $q(x)=f(x)+bx^{2}+cx+d$, where $b,c,d\in\mathbb{R}$. Given that $q$ is an odd function, find the value of $b$ and the value of $d$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $k$ is defined for $x\in\mathbb{R}$ by $k(x)=\sin 2x+x\cos x$.

   (a)

   Which statement about $k$ is true?

   [1 mark]

   • A$k$ is even
   • B$k$ is neither odd nor even
   • C$k$ is both odd and even
   • D$k$ is odd

   (b)

   Find $k\left(-\frac{\pi}{4}\right)$.

   [1 mark]

   • A$1+\frac{\pi\sqrt{2}}{8}$
   • B$-1-\frac{\pi\sqrt{2}}{8}$
   • C$-1+\frac{\pi\sqrt{2}}{8}$
   • D$1-\frac{\pi\sqrt{2}}{8}$

   (c)

   Given that $k(a)=5$ for some real number $a$, find the value of $k(a)+k(-a)+k(0)$. Justify your answer.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The height, in metres, of a stone above the ground $t$ seconds after it is thrown upwards from the top of a cliff is modelled by $h(t)=11+6t-t^{2}$, for $0\le t\le T$, where $T$ is the time at which the stone hits the ground.

   (a)

   Express $h(t)$ in the form $p-(t-q)^{2}$, and hence explain why $h$ does not have an inverse function on the domain $0\le t\le T$.

   [3 marks]

   (b)

   The function $g$ is the restriction of $h$ to the domain $3\le t\le T$. Find $g^{-1}(x)$, stating its domain, and hence find the time at which the falling stone is $11$ m above the ground.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The function $g$ is defined by $g(x)=\dfrac{3x+5}{x-3}$, for $x\in\mathbb{R}$, $x\neq3$. The function $r$ is defined by $r(x)=\dfrac{px+7}{2x+q}$, for $x\neq-\frac{q}{2}$, where $p,q\in\mathbb{R}$.

   (a)

   (i) Show that $(g\circ g)(x)=x$.\n(ii) Hence write down $g^{-1}(x)$, and state the domain and the range of $g^{-1}$.

   [6 marks]

   (b)

   Find an expression for $r^{-1}(x)$ in terms of $p$ and $q$. Hence find $q$ in terms of $p$ for which $r$ is self-inverse, and write down $r(x)$ in this case when $p=4$.

   [6 marks]

   Total for question 4: 12 marks

End of questions