1. 1
   Let $f(x) = 2x^{2} - 5x - 3$, $x \in \mathbb{R}$.

   (a)

   Find the solutions of the equation $f(x) = 0$.

   [1 mark]

   • A$x = -\frac{1}{2}$ or $x = 3$
   • B$x = \frac{1}{2}$ or $x = -3$
   • C$x = -1$ or $x = \frac{3}{2}$
   • D$x = -1$ or $x = 6$

   (b)

   Find the value of the discriminant of $f(x)$.

   [1 mark]

   • A$1$
   • B$49$
   • C$-49$
   • D$7$

   (c)

   Solve the inequality $f(x) < 0$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Consider the equation $x^{2} + (k - 2)x + 4 = 0$, where $k \in \mathbb{R}$.

   (a)

   Find an expression, in terms of $k$, for the discriminant of this equation.

   [1 mark]

   • A$k^{2} - 4k + 20$
   • B$k^{2} - 12$
   • C$k^{2} + 4k - 12$
   • D$k^{2} - 4k - 12$

   (b)

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

   [1 mark]

   • A$k = 6$ only
   • B$k = 2$ or $k = -6$
   • C$k = -2$ or $k = 6$
   • D$k = 4$ or $k = -4$

   (c)

   Find the set of values of $k$ for which the equation has no real roots.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A farmer has 40 m of fencing to make a rectangular pen against a long straight wall. The wall forms one side of the pen and the fencing forms the other three sides. Each of the two sides perpendicular to the wall has length $x$ metres.

   (a)

   Find the values of $x$ for which the area of the pen is $150\ \text{m}^{2}$.

   [3 marks]

   (b)

   By considering the discriminant of a suitable quadratic equation, find the maximum possible area of the pen. Justify your answer.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C$ has equation $y = x^{2} - 6x + 14$. For each $k \in \mathbb{R}$, the line $L_{k}$ has equation $y = kx + 5$.

   (a)

   (i) Write $x^{2} - 6x + 14$ in the form $(x - h)^{2} + q$, where $h, q \in \mathbb{Z}$.
   (ii) Hence write down the coordinates of the vertex of $C$ and explain why $C$ does not meet the $x$-axis.
   (iii) Show that the $x$-coordinates of any points where $C$ and $L_{k}$ meet satisfy $x^{2} - (6 + k)x + 9 = 0$.

   [6 marks]

   (b)

   (i) Find the values of $k$ for which $L_{k}$ is a tangent to $C$.
   (ii) Find the set of values of $k$ for which $L_{k}$ does not meet $C$.

   [6 marks]

   Total for question 4: 12 marks

End of questions