Click here to do these on a digital whiteboard. Note that these are a repeat of the holiday questions.
Q1
Answer
The equation $kx^2 + (3k-1)x - 4 = 0$ has no real roots. Find the set of possible values of $k$.
$-1 < x < -\dfrac{1}{9}$
Q2
Answer
$$\mathrm{f}(x) = 2x^3 + 3x^2 - 17x + 6$$
  1. Find the quotient and remainder when $\mathrm{f}(x)$ is divided by $(x-3)$.
  2. Find the quotient and remainder when $\mathrm{f}(x)$ is divided by $(x-2)$.
  3. Hence determine the number of real roots of the equation $\mathrm{f}(x) = 0$, giving a reason.
  1. quotient $2x^2 + 9x + 10$ remainder $36$
  2. quotient $2x^2 + 7x - 3$ remainder $0$
  3. $b^2 - 4ac > 0$ three real roots
Q3
Answer
A curve $C$ has equation $y = 7 + 6x - x^2$.
  1. Find the equation of the tangent to $C$ at the point $P$ where $x = 5$ giving your answer in the form $ax + by + c = 0$ where $a$, $b$ and $c$ are integers.
  2. The tangent meets the $x$ axis at the point $Q$. Find the midpoint of $PQ$.
  3. State the set of values for $x$ for which the curve is increasing.
  1. $4x + y - 32 = 0$
  2. $\left(\dfrac{13}{2}, 6\right)$
  3. $x < 3$
Q4
Answer
Express $x^2 - 12x + 1$ in the form $(x-p)^2 + q$.
$(x-6)^2 - 35$