Q1
Answer
Given that $2x^2 - 12x + p \equiv q(x-r)^2 + 10$, find the constants $p$, $q$ and $r$.
$p = 28$, $q = 2$, $r = 3$
Q2
Answer
  1. Without finding the roots, find the number of real roots of the equation $-2x^2 + 7x + 3 = 0$.
  2. The equation $2x^2 + px + x + 8 = 0$ has equal roots. Find the possible values of $p$.
  1. Discriminant is $73$, two real roots
  2. $p = -9, 7$
Q3
Answer
  1. Express $x^2 + 3x$ in the form $(x+a)^2 + b$
  2. Express $y^2 - 4y - \dfrac{11}{4}$ in the form $(y + p)^2 + q$
  1. $\left(x+\frac{3}{2}\right)^2 - \frac{9}{4}$
  2. $(y - 2)^2 - \frac{27}{4}$
Q4
Answer
  1. Solve $x^2 - 8x + 11 = 0$ giving your answers in exact form
  2. Sketch the curve $y = x^2 - 8x + 11$ labelling the coordinates of the points where it crosses the axes
  3. Solve $y - 8\sqrt{y} + 11 = 0$, giving your answers in the form $a \pm b\sqrt{5}$
  1. $4\pm\sqrt{5}$
  2. Correct sketch
  3. $y = 21 \pm 8\sqrt{5}$
Q5
Answer
Solve $x^{\frac{2}{3}} + 3x^{\frac{1}{3}} - 10 = 0$
$x = 8$, $x = -125$
Q6
Answer
  1. Write $2x^2 - 24x + 80$ in the form $a(x-b)^2 + c$
  2. State the equation of the line of symmetry of the curve $y = 2x^2 - 24x + 80$
  3. State the equation of the tangent to the curve $y = 2x^2 - 24x + 80$ at its minimum point
  1. $2(x-6)^2 + 8$
  2. $x = 6$
  3. $y = 8$
Q7
Answer
Given that $3x^2 + bx + 10 \equiv a(x + 3)^2 + c$, find the constants $a$, $b$ and $c$.
$a = 3$, $b = 18$, $c = -17$
Q8
Answer
Solve the simultaneous equations $$x^2 - 3y + 11 = 0$$ $$2x - y + 1 = 0$$
$x = 2, y = 5 \quad x = 4, y = 9$
Q9
Answer
  1. Solve the simultaneous equations $$y = x^2 - 5x + 15$$ $$5x - y = 10$$
  2. What can you deduce about the line $5x - y = 10$ and the curve $y = x^2 - 5x + 15$
  3. Hence find the equation of the normal to the curve $y = x^2 - 5x + 15$ at the point $(5,15)$ giving your answer in the form $ax + by + c = 0$ where $a$, $b$ and $c$ are integers
  1. $x = 5$ and $y = 15$
  2. The line is tangent to the curve
  3. $x + 5y = 80$
Q10
Answer
The length of a rectangle is $10$ m more than its width. The perimeter is greater than $64$ m. The area is less than $299$ m$^2$. Determine the set of possible values for the width of the rectangle.
$11 < x < 13$ where $x$ is the width
Q11
Answer
Solve the inequalities
  1. $3(x-5) \leq 24$
  2. $5x^2 - 2 > 78$
  1. $x \leq 13$
  2. $x > 4$, $x < -4$
Q12
Answer
Solve the inequality $x^2 + 8x + 10 \geq 0$
$x \leq -4 - \sqrt{6}$, $x\geq -4 + \sqrt{6}$