Factorising determinantsAQA A-Level Further Maths: Subtopic test

1. 1
   Let $\Delta=\begin{vmatrix} x & 1 & 1 \\ 1 & x & 1 \\ 1 & 1 & x \end{vmatrix}$.

   (a)

   Find the value of $\Delta$ when $x=1$.

   [1 mark]

   • A$0$
   • B$1$
   • C$3$
   • D$-1$

   (b)

   Which of the following is a linear factor of $\Delta$?

   [1 mark]

   • A$x+1$
   • B$x-2$
   • C$x+2$
   • D$x+3$

   (c)

   Use a column operation to show that $(x+2)$ is a factor of $\Delta$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $\Delta=\begin{vmatrix} 1 & 1 & 1 \\ x & 2 & 3 \\ x^2 & 4 & 9 \end{vmatrix}$.

   (a)

   For which of these values of $x$ are the first two columns of the determinant identical?

   [1 mark]

   • A$1$
   • B$2$
   • C$4$
   • D$6$

   (b)

   The operations $C_2\to C_2-C_1$ and $C_3\to C_3-C_1$ are applied to $\Delta$. What is the first row of the new determinant?

   [1 mark]

   • A$1,\ 1,\ 1$
   • B$1,\ 2,\ 2$
   • C$0,\ 0,\ 0$
   • D$1,\ 0,\ 0$

   (c)

   Explain why $(x-2)$ is a factor of $\Delta$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Let $\Delta=\begin{vmatrix} x+1 & 2 & 3 \\ 1 & x+2 & 3 \\ 1 & 2 & x+3 \end{vmatrix}$.

   (a)

   Show that $(x+6)$ is a factor of $\Delta$.

   [3 marks]

   (b)

   Hence factorise $\Delta$ completely.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $a$, $b$ and $c$ be constants, and let $\Delta_1=\begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^2 & b^2 & c^2 \end{vmatrix}$ and $\Delta_2=\begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ a^3 & b^3 & c^3 \end{vmatrix}$.

   (a)

   Show that $\Delta_1=(a-b)(b-c)(c-a)$.

   [6 marks]

   (b)

   Show that $\Delta_2=(a+b+c)(a-b)(b-c)(c-a)$.

   [6 marks]

   Total for question 4: 12 marks

End of questions