CIE Oct 2020 9709 Pure Maths Paper 12 (pdf)
- The coefficient of x3 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 = 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
- The sum, Sn, of the first n terms of an arithmetic progression is given by
Sn = n2 + 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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