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Q1
Answer
A circle has centre $C$ and equation $x^2 + y^2 -2x + 10y - 19 = 0$.
  1. Find the coordinates of $C$ and the radius of the circle..
  2. Verify that the point $(7, -2)$ lies on the circle.
  3. Find the equation of the tangent to the circle at the point $(7,-2)$ giving your answer in the form $ax + by + c = 0$ where $a$, $b$ and $c$ are integers.
  1. $(1, -5)$ radius $\sqrt{45}$
  2. Substitute coordinates in
  3. $2x + y - 12 = 0$
Q2
Answer
A curve has equation $y = x^2 + 5$. On the curve, $A$ has $x$ coordinate $1$, $B$ has $x$ coordinate $b$, where $b > 1$, and $C$ lies between $A$ and $B$.
  1. Find the gradient of the curve at $A$.
  2. The line segment $AB$ has gradient $2.3$. Find the value of $b$.
  3. Suggest a possible value of the gradient of $AC$.
  1. $2$
  2. $1.3$
  3. Between $2$ and $2.3$
Q3
Answer
Solve $7^{x-3} - 4 = 180$ correct to 3 significant figures.
$5.68$
Q4
Answer
Solve $x - 8\sqrt{x} + 13 = 0$
$19\pm8\sqrt{3}$
Q5
Answer
Prove that $2^n - 1$ is prime for all prime numbers $n$ less than $11$.
Try $n = 2, 3, 5, 7$