1. 1
   Consider the system of equations: (1) $x + y + z = 6$, (2) $2x - y + z = 3$, (3) $x + 2y - z = 2$.

   (a)

   Which equation in $y$ and $z$ only is obtained by eliminating $x$ from equations (1) and (2)?

   [1 mark]

   • A$3y + z = 15$
   • B$3y - z = 9$
   • C$3y + z = 9$
   • D$-3y - z = 9$

   (b)

   Find the solution of the system.

   [1 mark]

   • A$x = 3,\ y = 2,\ z = 1$
   • B$x = 1,\ y = 2,\ z = 3$
   • C$x = 2,\ y = 1,\ z = 3$
   • D$x = 1,\ y = 3,\ z = 2$

   (c)

   Equation (3) is replaced by $x + 2y + kz = 2$, where $k \in \mathbb{R}$, and equations (1) and (2) are unchanged. Find the value of $k$ for which the new system has no solution.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   At a café, a coffee costs $c$ dirhams, a tea costs $t$ dirhams and a juice costs $j$ dirhams. Order P is 2 coffees, 1 tea and 3 juices and costs 62 dirhams. Order Q is 1 coffee, 3 teas and 1 juice and costs 44 dirhams. Order R is 3 coffees, 2 teas and 2 juices and costs 74 dirhams.

   (a)

   Which equation represents order R?

   [1 mark]

   • A$7(c + t + j) = 74$
   • B$2c + 3t + 2j = 74$
   • C$3c + 2t + 2j = 62$
   • D$3c + 2t + 2j = 74$

   (b)

   Use your graphic display calculator to find the price of one juice, in dirhams.

   [1 mark]

   • A$9$
   • B$7$
   • C$14$
   • D$10$

   (c)

   A group orders 4 coffees, 2 teas and 5 juices and pays with a 150-dirham note. Find the change they receive.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider the system of equations: (1) $x + 2y - z = 3$, (2) $2x + 5y + z = 10$, (3) $x + 3y + 2z = 7$.

   (a)

   Use row reduction to show that the system has infinitely many solutions.

   [3 marks]

   (b)

   Find the general solution of the system, and interpret it geometrically.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Consider the system of equations $x + y + z = 2$, $x + 2y + 3z = 5$ and $2x + 3y + az = b$, where $a, b \in \mathbb{R}$.

   (a)

   Find the conditions on $a$ and $b$ for which the system has (i) a unique solution; (ii) infinitely many solutions; (iii) no solution.

   [6 marks]

   (b)

   Let $a = 4$ and $b = 7$. Find the general solution of the system. Hence find the solution for which $x^{2} + y^{2} + z^{2}$ takes its least value.

   [6 marks]

   Total for question 4: 12 marks

End of questions