CIE May 2020 9709 Pure Maths Paper 12

CIE May 2020 9709 Pure Maths Paper 12 (pdf)

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1. (a) Find the coefficient of x² in the expansion of
   (b) Find the coefficient of x² in the expansion of 2 (a) Express the equation 3 cos θ = 8 tan θ as a quadratic equation in sin θ
   (b) Hence find the acute angle, in degrees, for which 3 cos θ = 8 tan θ.
2. A weather balloon in the shape of a sphere is being inflated by a pump. The volume of the balloon is increasing at a constant rate of 600 cm³ per second. The balloon was empty at the start of pumping.
   (a) Find the radius of the balloon after 30 seconds.
   (b) Find the rate of increase of the radius after 30 seconds
3. The nth term of an arithmetic progression is 1/2(3n − 15).
   Find the value of n for which the sum of the first n terms is 84.
4. The function f is defined for by
   where a is a constant.
   (a) Express ff(x) and f⁻¹(x) in terms of a and x
   (b) Given that ff(x) = f⁻¹(x), find x in terms a

6. The equation of a curve is y = 2x² + kx + k − 1, where k is a constant.
   (a) Given that the line y = 2x + 3 is a tangent to the curve, find the value of k
   It is now given that k = 2.
   (b) Express the equation of the curve in the form y = 2(x + a)² + b, where a and b are constants, and hence state the coordinates of the vertex of the curve
7. In the diagram, OAB is a sector of a circle with centre O and radius 2r, and angle AOB = 1/6 π radians.
   The point C is the midpoint of OA.
   (a) Show that the exact length of BC is
   (b) Find the exact perimeter of the shaded region
   (c) Find the exact area of the shaded region
8. The diagram shows part of the curve y = 6/x. The points (1, 6) and (3, 2) lie on the curve. The shaded region is bounded by the curve and the lines y = 2 and x = 1.
   (a) Find the volume generated when the shaded region is rotated through 360° about the y-axis.
   (b) The tangent to the curve at a point X is parallel to the line y + 2x = 0. Show that X lies on the line y = 2x.
9. Functions f and g are such that
   (a) State the ranges of f and g.
   The diagram below shows the graph of y = f(x).
   (b) Sketch, on this diagram, the graph of y = g(x).
   The function h is such that
   (c) Describe fully a sequence of transformations that maps the curve y = f(x) on to y = h(x)
10. The equation of a curve is y = 54x − (2x − 7)²
    (a) Find
    (b) Find the coordinates of each of the stationary points on the curve
    (c) Determine the nature of each of the stationary point
11. The equation of a circle with centre C is x² + y² − 8x + 4y − 5 = 0.
    (a) Find the radius of the circle and the coordinates of C.
    The point P(1, 2) lies on the circle.
    (b) Show that the equation of the tangent to the circle at P is 4y = 3x + 5
    The point Q also lies on the circle and PQ is parallel to the x-axis.
    (c) Write down the coordinates of Q
    The tangents to the circle at P and Q meet at T.
    (d) Find the coordinates of T.

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