1. 1
   A factory's production planning region R is defined by the inequalities $x \geq 0$, $y \geq 0$ and $x + y \leq 10$.

   (a)

   Does the point $(3, 4)$ lie within region R?

   [1 mark]

   • AYes, all three inequalities are satisfied
   • BNo, because $x + y > 10$
   • CNo, because $x < 0$
   • DNo, because $y < 0$

   (b)

   Does the point $(6, 6)$ lie within region R?

   [1 mark]

   • ANo, because $x + y > 10$
   • BYes
   • CNo, because $x < 0$
   • DNo, because $y < 0$

   (c)

   What is the gradient of the boundary line $x + y = 10$?

   [1 mark]

   • A$1$
   • B$-1$
   • C$10$
   • D$-10$

   Total for question 1: 3 marks

2. 2
   Region S is defined by the inequalities $y \leq 2x + 3$, $y \geq -1$ and $x \leq 4$.

   (a)

   Determine whether the point $(2, 5)$ lies within region S, showing your check against each of the three inequalities.

   [2 marks]

   (b)

   Determine whether the point $(5, 2)$ lies within region S, showing your check against each of the three inequalities.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Region T is bounded by the lines $y = x$, $y = 6 - x$ and $y = 1$.

   (a)

   Find the coordinates of the point where the lines $y = x$ and $y = 6 - x$ intersect.

   [3 marks]

   (b)

   Region T is the set of points satisfying $y \geq 1$, $y \leq x$ and $y \leq 6 - x$. Determine whether the point $(4, 2)$ lies within region T, showing your working for each inequality.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A gardener has 40 metres of fencing to enclose a rectangular vegetable patch of length $x$ metres and width $y$ metres, where $x > 0$ and $y > 0$. The perimeter must not exceed 40 metres, and the area must be at least 75 square metres.

   (a)

   Write down the two inequalities representing the fencing (perimeter) constraint and the area constraint, simplifying the perimeter inequality fully.

   [4 marks]

   (b)

   The gardener chooses $x = 15$. Using the inequality $x + y \leq 20$ from part (a), find the maximum possible integer value of $y$.

   [4 marks]

   (c)

   Check whether $x = 15$, $y = 5$ satisfies both inequalities from part (a), and hence determine whether the area constraint ($xy \geq 75$) is met, stating the resulting area.

   [5 marks]

   Total for question 4: 13 marks

End of questions