1. 1
   Point $A$ has coordinates $(-3, 2)$ and point $B$ has coordinates $(5, 8)$.

   (a)

   What is the midpoint of $AB$?

   [1 mark]

   • A$(1,5)$
   • B$(2,6)$
   • C$(1,3)$
   • D$(4,10)$

   (b)

   What is the length of $AB$?

   [1 mark]

   • A$8.0$
   • B$10.0$
   • C$14.0$
   • D$6.0$

   (c)

   What is the gradient of the line segment $AB$?

   [1 mark]

   • A$\frac{4}{3}$
   • B$\frac{3}{2}$
   • C$\frac{3}{4}$
   • D$-\frac{3}{4}$

   Total for question 1: 3 marks

2. 2
   A straight line has equation $y=3x-7$.

   (a)

   Write down the gradient and the $y$-intercept of the line.

   [2 marks]

   (b)

   Find the equation of the line parallel to $y=3x-7$ that passes through the point $(2,5)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A quadratic curve has equation $y=x^2-6x+5$.

   (a)

   Find the coordinates of the points where the curve crosses the $x$-axis.

   [3 marks]

   (b)

   Find the coordinates of the turning point of the curve, by completing the square.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A model rocket's height above the ground, $h$ metres, $t$ seconds after launch is given by $h=30t-5t^2$ for $0\leq t\leq6$.

   (a)

   Find $\frac{dh}{dt}$, and hence find the time at which the rocket reaches its maximum height.

   [4 marks]

   (b)

   Find the maximum height reached by the rocket, and confirm it is a maximum using the second derivative.

   [4 marks]

   (c)

   Estimate the gradient of the height-time graph at $t=1$ second by finding $\frac{dh}{dt}$ at this instant, and interpret what this value represents in the context of the rocket's motion, including its sign.

   [5 marks]

   Total for question 4: 13 marks

5. 5
   Three inequalities are being considered: $2x-5<9$, $-3\leq2x+1<9$, and the inequality shown by a closed circle at $-2$ and an open circle at $4$ on a number line, with the region between shaded.

   (a)

   What is the solution to $2x-5<9$?

   [1 mark]

   • A$x<2$
   • B$x<7$
   • C$x>7$
   • D$x<14$

   (b)

   What is the solution to $-3\leq2x+1<9$?

   [1 mark]

   • A$-2\leq x<4$
   • B$-2<x\leq4$
   • C$-4\leq x<8$
   • D$-2\leq x<8$

   (c)

   Which inequality is represented by a closed circle at $-2$ and an open circle at $4$, with the region between shaded?

   [1 mark]

   • A$-2<x\leq4$
   • B$-2\leq x\leq4$
   • C$-2<x<4$
   • D$-2\leq x<4$

   Total for question 5: 3 marks

6. 6
   A water tank is being drained at a constant rate. After 4 minutes it contains 180 litres, and after 10 minutes it contains 60 litres.

   (a)

   Find the rate at which the tank is being drained, in litres per minute.

   [2 marks]

   (b)

   Find the volume of water in the tank at the moment draining began ($t=0$).

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Line $L_1$ has equation $y=-\frac{1}{2}x+4$. Line $L_2$ passes through the points $(1,2)$ and $(5,10)$.

   (a)

   Find the equation of the line perpendicular to $L_1$ that passes through the point $(4,1)$.

   [3 marks]

   (b)

   Find the gradient of $L_2$ and determine whether $L_2$ is parallel to, perpendicular to, or neither parallel nor perpendicular to $L_1$.

   [3 marks]

   Total for question 7: 6 marks

8. 8
   A curve has equation $y=x^2-4x-5$. Points $C$ and $D$ are the points where the curve crosses the $x$-axis.

   (a)

   Find the coordinates of points $C$ and $D$.

   [4 marks]

   (b)

   Find the midpoint of $CD$ and the length of $CD$.

   [4 marks]

   (c)

   Find $\frac{dy}{dx}$, and hence find the coordinates of the turning point of the curve, stating whether it is a minimum or maximum.

   [5 marks]

   Total for question 8: 13 marks

End of questions