1. 1
   The polynomial $p(x) = 2x^{3} + ax^{2} + bx - 6$, where $a, b \in \mathbb{R}$, has a factor $(x - 2)$. When $p(x)$ is divided by $(x + 1)$ the remainder is $-6$.

   (a)

   Which equation follows from the fact that $(x - 2)$ is a factor of $p(x)$?

   [1 mark]

   • A$4a - 2b = 22$
   • B$4a + 2b = 10$
   • C$4a + 2b = -10$
   • D$4a + 2b = -16$

   (b)

   Which equation follows from the remainder theorem?

   [1 mark]

   • A$a - b = 2$
   • B$a - b = 14$
   • C$a + b = -2$
   • D$a - b = -4$

   (c)

   Find the value of $a$ and the value of $b$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The equation $3x^{4} - 6x^{3} + kx^{2} + 5x - 12 = 0$, where $k \in \mathbb{R}$, has roots $\alpha$, $\beta$, $\gamma$ and $\delta$.

   (a)

   Find $\alpha + \beta + \gamma + \delta$.

   [1 mark]

   • A$-2$
   • B$2$
   • C$6$
   • D$\frac{1}{2}$

   (b)

   Find $\alpha\beta\gamma\delta$.

   [1 mark]

   • A$-\frac{1}{4}$
   • B$4$
   • C$-12$
   • D$-4$

   (c)

   Find the sum and the product of the numbers $2\alpha$, $2\beta$, $2\gamma$ and $2\delta$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A cubic function $f$ has zeros at $x = -2$, $x = 1$ and $x = 3$, and the graph of $y = f(x)$ meets the $y$-axis at $(0, 12)$. The function $g$ is defined by $g(x) = f(x) + px$, where $p \in \mathbb{R}$.

   (a)

   Find $f(x)$, giving your answer in the form $ax^{3} + bx^{2} + cx + d$.

   [3 marks]

   (b)

   Given that $(x - 2)$ is a factor of $g(x)$, find the value of $p$ and find the other two zeros of $g$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A company makes closed boxes in the shape of cuboids. Box P has edges of length $x$ cm, $(x + 2)$ cm and $(x + 5)$ cm, and a volume of $120\ \text{cm}^{3}$. For box Q, the lengths in cm of its three different edges are the roots of the equation $2x^{3} - 27x^{2} + 118x - 168 = 0$.

   (a)

   Show that $x^{3} + 7x^{2} + 10x - 120 = 0$. Hence use the factor theorem to find the dimensions of box P, justifying that there is only one possible box.

   [6 marks]

   (b)

   Without solving the equation, find the volume of box Q and the total length of its twelve edges. Given that one edge of box Q is 4 cm long, find the lengths of the other two edges.

   [6 marks]

   Total for question 4: 12 marks

End of questions