AQA A-Level Maths: Pure Mathematics 1

A circle $C$C has equation $x^2 + y^2 - 6x + 4y - 12 = 0.$x2+y2−6x+4y−12=0.

(a) By completing the square, find the coordinates of the centre of $C$C and the length of its radius. (3 marks)

(b) The line $l$l has equation $y = -x + 2$y=−x+2. Show that $l$l meets $C$C at two distinct points, and find the coordinates of these two points. (4 marks)

(c) One of the points of intersection found in part (b) is $P(7, -5)$P(7,−5). Find an equation of the tangent to $C$C at $P$P, giving your answer in the form $ax + by + c = 0$ax+by+c=0 where $a$a, $b$b and $c$c are integers. (5 marks)

(a) Find the first four terms, in ascending powers of $x$x, of the binomial expansion of $(2 + 3x)^5$(2+3x)5, giving each coefficient as an integer. (4 marks)

(b) A geometric series has first term $27$27 and second term $18$18.

  (i) Find the common ratio of the series, and hence find the fourth term. (3 marks)

  (ii) Explain why the series converges, and find its sum to infinity. (3 marks)

A curve has equation $y = 2x^3 - 9x^2 + 12x + 1.$y=2x3−9x2+12x+1.

(a) Find the coordinates of the two stationary points of the curve, and determine the nature of each. (5 marks)

(b) Find an equation of the tangent to the curve at the point where $x = 3$x=3. (3 marks)

Solve, for $0 \leq \theta < 360^{\circ}$0≤θ<360∘, the equation $2\sin^2\theta + 5\cos\theta - 4 = 0,$2sin2θ+5cosθ−4=0, giving all solutions to the nearest degree where appropriate. (6 marks)

Solve the equation $\log_2(x + 3) + \log_2(x - 3) = 4.$log2​(x+3)+log2​(x−3)=4. (5 marks)

Prove that $2x^2 - 4x + 5 > 0$2x2−4x+5>0 for all real values of $x$x. (4 marks)