1. 1
   The points $A(2, 5)$ and $B(8, -7)$ lie on a straight line $L_1$.

   (a)

   Find the gradient of $L_1$.

   [1 mark]

   • A$-2$
   • B$2$
   • C$-\frac12$
   • D$\frac12$

   (b)

   The line $L_2$ passes through the point $(0, 3)$ and is perpendicular to $L_1$. Find the equation of $L_2$ in the form $y = mx + c$.

   [1 mark]

   • A$y = -2x + 3$
   • B$y = \frac12x + 3$
   • C$y = -\frac12x + 3$
   • D$y = \frac12x$

   (c)

   Find the coordinates of the point of intersection of $L_1$ and $L_2$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The quadratic function $f$ is defined by $f(x) = 2x^2 - 12x + 16$, for $x \in \mathbb{R}$.

   (a)

   Write $f(x)$ in the form $a(x-h)^2+k$, and state the coordinates of the vertex of the graph of $f$.

   [1 mark]

   • A$(3, 7)$
   • B$(-3, -2)$
   • C$(3, -2)$
   • D$(6, -20)$

   (b)

   State the equation of the axis of symmetry of the graph of $f$.

   [1 mark]

   • A$x = -3$
   • B$x = 12$
   • C$x = 1.5$
   • D$x = 3$

   (c)

   Write $f(x)$ in the form $a(x-p)(x-q)$, stating the values of $p$ and $q$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider the equation $3kx^2 + 2x + k = 0$, where $k$ is a non-zero real constant.

   (a)

   Find the set of values of $k$ for which the equation has two distinct real roots.

   [3 marks]

   (b)

   Given that $k = \frac14$, find the roots of the equation in exact (surd) form.

   [4 marks]

   Total for question 3: 7 marks

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

   (a)

   Find the value of $x$ for which $f(x) = g(x)$, giving your answer in exact form.

   [6 marks]

   (b)

   Hence, or otherwise, find the range of values of $x$ for which $f(x) > g(x)$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The function $f$ is defined by $f(x) = \sqrt{2x+7}$, for $x \ge -\frac72$.

   (a)

   State the largest possible domain of $f$.

   [1 mark]

   • A$x \ge -\frac72$
   • B$x \ge \frac72$
   • C$x \ge -7$
   • D$x > -\frac72$

   (b)

   Find $f^{-1}(x)$, stating its domain.

   [1 mark]

   • A$f^{-1}(x) = x^2 - 7$, $x\ge0$
   • B$f^{-1}(x) = \frac{x^2-7}{2}$, $x\ge0$
   • C$f^{-1}(x) = \frac{x^2+7}{2}$, $x\ge0$
   • D$f^{-1}(x) = \frac{x^2-7}{2}$, $x\in\mathbb{R}$

   (c)

   State the range of $f$ and the range of $f^{-1}$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The function $h$ is defined by $h(x) = \dfrac{3x+4}{x-2}$, for $x \ne 2$.

   (a)

   Find the equation of the horizontal asymptote of the graph of $h$.

   [1 mark]

   • A$y = -2$
   • B$y = -4$
   • C$y = 3$
   • D$x = 2$

   (b)

   Find the $x$-intercept of the graph of $h$.

   [1 mark]

   • A$x = \frac43$
   • B$x = -2$
   • C$x = -\frac32$
   • D$x = -\frac43$

   (c)

   State the equations of the vertical and horizontal asymptotes of the graph of $h$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The functions $f$ and $g$ are defined by $f(x) = 2x - 5$ and $g(x) = x^2 + 1$, for $x \in \mathbb{R}$.

   (a)

   Find $(f\circ g)(x)$, and hence find the value of $(f\circ g)(3)$.

   [3 marks]

   (b)

   The graph of $y=g(x)$ is transformed to the graph of $y=k(x)$ by a vertical stretch with scale factor 3, followed by a translation of 2 units in the positive $x$-direction. Write down an expression for $k(x)$, and state the coordinates of the minimum point of the graph of $y=k(x)$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A ball is thrown vertically. Its height above the ground, $h$ metres, $t$ seconds after it is thrown, is modelled by $h(t) = -5t^2 + 20t + 1$, for $0 \le t \le T$, where $T$ is the time at which the ball lands.

   (a)

   Find the maximum height reached by the ball and the time at which it occurs. Find also the value of $T$, correct to 3 significant figures.

   [6 marks]

   (b)

   A second ball is thrown from the same position with the same initial vertical velocity, but starting 3 m higher. Write down an expression for its height function $h_2(t)$ in terms of $h(t)$, describing the transformation involved, and hence find the maximum height reached by the second ball and the time $T_2$ at which it lands, correct to 3 significant figures.

   [6 marks]

   Total for question 8: 12 marks

End of questions