Q1
Answer
The straight lines $l_1$ and $l_2$ have equations $l_1: 2x+y=10$ and $l_2: 3x-4y=10$.
  1. Sketch $l_1$ and $l_2$ on a single set of axes. The sketch must include the coordinates of all the points where each of these straight lines meet the coordinate axes.
  2. Use algebra to determine the exact coordinates of the point of intersection.
  1. Intersections: $(0,10)$ and $(5,0)$; $\left(0, -\frac{5}{2}\right)$ and $\left(\frac{10}{3}, 0\right)$
  2. $\left(\dfrac{50}{11}, \dfrac{10}{11}\right)$
Q2
Answer
Shown below is the curve $y = x^3 - 12x^2 + 45x - 34$ and the line $y = 6x-6$.
  1. Show that the curve and the line intersect at $(1,0)$ and $(4, 18)$.
  2. Find the exact area enclosed by the curve and the line.
  1. Solve simultaneously
  2. Integrate the curve and subtract the triangle: $\dfrac{81}{4}$
Q3
Answer
At time $t$ seconds, the surface area of a cube is $A$ cm$^2$ and the volume is $V$ cm$^3$. The surface area of the cube is expanding at a constant rate of 2 cm$^2$ s$^{-1}$.
  1. Write an expression for $V$ in terms of $A$.
  2. Hence show that $\dfrac{\mathrm{d}V}{\mathrm{d}t} = \dfrac{1}{2}V^{\frac{1}{3}}$.
  1. $V = \left(\dfrac{A}{6}\right)^{\frac{3}{2}}$
  2. $\dfrac{\mathrm{d}V}{\mathrm{d}t} = \dfrac{\mathrm{d}V}{\mathrm{d}A} \times \dfrac{\mathrm{d}A}{\mathrm{d}t}$