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This page covers Questions and Worked Solutions for CIE Pure Maths Paper 12 October/November 2020, 9709/12.

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CIE Oct 2020 9709 Pure Maths Paper 12 (pdf)

- The coefficient of x
^{3}in the expansion of (1 + kx)(1 − 2x)^{5}is 20.

Find the value of the constant. - 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. - The equation of a curve is y = 2x
^{2}+ 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 - The sum, Sn, of the first n terms of an arithmetic progression is given by
S
_{n}= n^{2}+ 4n.

The kth term in the progression is greater than 200.

Find the smallest possible value of k. - 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) - (a) Prove the identity

(b) Hence solve the equation

- 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. - 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. - 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. - 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 - 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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