CIE Oct 2020 9709 Pure Maths Paper 12

CIE Oct 2020 9709 Pure Maths Paper 12 (pdf)

• Show Step-by-Step Solutions

1. The coefficient of x³ in the expansion of (1 + kx)(1 − 2x)⁵ is 20.
   Find the value of the constant.
2. The first, second and third terms of a geometric progression are 2p + 6, −2p and p + 2 respectively, where p is positive.
   Find the sum to infinity of the progression.
3. The equation of a curve is y = 2x² + m(2x + 1), where m is a constant, and the equation of a line is y = 6x + 4.
   Show that, for all values of m, the line intersects the curve at two distinct point
4. The sum, Sn, of the first n terms of an arithmetic progression is given by S_n = n² + 4n.
   The kth term in the progression is greater than 200.
   Find the smallest possible value of k.
5. Functions f and g are defined by
   f(x) = 4x − 2, for x
   g(x) = 4/(x + 1)
   (a) Find the value of fg(7)
   (b) Find the values of x for which f⁻¹(x) = g⁻¹(x)
6. (a) Prove the identity
   (b) Hence solve the equation

7. The point (4, 7) lies on the curve y = f(x) and it is given that f′(x) = 6x^−1/2 - 4x^−3/2.
   (a) A point moves along the curve in such a way that the x-coordinate is increasing at a constant rate of 0.12 units per second.
   Find the rate of increase of the y-coordinate when x = 4.
   (b) Find the equation of the curve.
8. In the diagram, ABC is an isosceles triangle with AB = BC = r cm and angle BAC = θ radians. The point D lies on AC and ABD is a sector of a circle with centre A.
   (a) Express the area of the shaded region in terms of r and θ
   (b) In the case where r = 10 and θ = 0.6, find the perimeter of the shaded region.
9. A circle has centre at the point B(5, 1). The point A(−1, −2) lies on the circle.
   (a) Find the equation of the circle.
   Point C is such that AC is a diameter of the circle. Point D has coordinates (5, 16).
   (b) Show that DC is a tangent to the circle.
   The other tangent from D to the circle touches the circle at E.
   (c) Find the coordinates of E.
10. The diagram shows part of the curve y
    (a) Find expressions for
    (b) Find, by calculation, the x-coordinate of M.
    (c) Find the area of the shaded region bounded by the curve and the coordinate axes
11. A curve has equation y = 3 cos 2x + 2 for 0 ≤ x ≤ π.
    (a) State the greatest and least values of y
    (b) Sketch the graph of y = 3 cos 2x + 2 for 0 ≤ x ≤ π
    (c) By considering the straight line y = kx, where k is a constant, state the number of solutions of the equation 3 cos 2x + 2 = kx for 0 ≤ x ≤ π in each of the following cases.
    (i) k = −3
    (ii) k = 1
    (iii) k = 3
    Functions f, g and h are defined for x by
    f(x) = 3 cos 2x + 2,
    g(x) = f(2x) + 4,
    h(x) = 2f(x + 1/2 π)
    (d) Describe fully a sequence of transformations that maps the graph of y = f(x) on to y = g(x).
    (e) Describe fully a sequence of transformations that maps the graph of y = f(x) on to y = h(x).

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.