CIE Oct 2020 9709 Pure Maths Paper 12

CIE Oct 2020 9709 Pure Maths Paper 12 (pdf)

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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).

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