CIE Oct/Nov 2021 9709 Pure Maths Paper 12

CIE Oct/Nov 2021 9709 Pure Maths Paper 12 Questions (pdf)

CIE Oct/Nov 2021 9709 Pure Maths Paper 12 Mark Scheme (pdf)

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1. Solve the equation 2 cos 1 = 7 − 3/cos θ for −90° < 1 < 90°
2. The graph of y = f(x) is transformed to the graph of y = 2f(x) - 3.
   (a) Describe fully the two single transformations which have been combined to give the resulting transformation.
   The point P(5, 6) lies on the transformed curve y = f(2x) − 3.
   (b) State the coordinates of the corresponding point on the original curve y = f(x)
3. The function f is defined as follows:
   f(x) = (x + 3)/(x − 1) for x > 1.
   (a) Find the value of ff(5).
   (b) Find an expression for f⁻¹(x)
4. A curve is such that
5. The first, third and fifth terms of an arithmetic progression are 2 cos x, −6√3 sin x and 10 cos x respectively, where 1/2 π < x < π.
   (a) Find the exact value of x
   (b) Hence find the exact sum of the first 25 terms of the progression
6. The second term of a geometric progression is 54 and the sum to infinity of the progression is 243.
   The common ratio is greater than 1/2.
   Find the tenth term, giving your answer in exact form.

7. In the diagram the lengths of AB and AC are both 15 cm. The point P is the foot of the perpendicular from C to AB. The length CP = 9 cm. An arc of a circle with centre B passes through C and meets AB at Q.
   (a) Show that angle ABC = 1.25 radians, correct to 3 significant figures.
   (b) Calculate the area of the shaded region which is bounded by the arc CQ and the lines CP and PQ.
8. (a) It is given that in the expansion of (4 + 2x)(2 − ax)⁵, the coefficient of x² is −15.
   Find the possible values of
   (b) It is given instead that in the expansion of (4 + 2x)(2 − ax)⁵, the coefficient of x² is k. It is also given that there is only one value of a which leads to this value of k.
   Find the values of k and a.
9. The volume V m³ of a large circular mound of iron ore of radius r m is modelled by the equation
   (a) Find the rate at which the radius of the mound is increasing at the instant when the radius is 5.5 m.
   (b) Find the volume of the mound at the instant when the radius is increasing at 0.1 m per second.
10. The function f is defined by f(x)
    (a) Given that the curve with equation y = f(x) has a stationary point when x = 2, find
    (b) Determine the nature of the stationary point
    (c) Given that this is the only stationary point of the curve, find the range of f.
11. The diagram shows the line x
12. The diagram shows the circle with equation x² + y² − 6x + 4y − 27 = 0 and the tangent to the circle at the point P(5, 4).
    (a) The tangent to the circle at P meets the x-axis at A and the y-axis at B.
    Find the area of triangle OAB, where O is the origin.
    (b) Points Q and R also lie on the circle, such that PQR is an equilateral triangle.
    Find the exact area of triangle PQR

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