CIE May 2021 9709 Pure Maths Paper 12

CIE May/June 2021 9709 Pure Maths Paper 12 (pdf)

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1. (a) Express 16x² − 24x + 10 in the form (4x + a)² + b
   (b) It is given that the equation 16x² − 24x + 10 = k, where k is a constant, has exactly one root. Find the value of this root
2. (a) The graph of y = f(x) is transformed to the graph of y = 2f(x − 1).
   Describe fully the two single transformations which have been combined to give the resulting transformation.
   (b) The curve y = sin 2x − 5x is reflected in the y-axis and then stretched by scale factor 1/3 in the x-direction.
   Write down the equation of the transformed curve.
3. The equation of a curve is y = (x − 3)√(x + 1) + 3. The following points lie on the curve. Non-exact values are rounded to 4 decimal places
4. The coefficient of x in the expansion of (4x + 10/x)³ is p
5. The function f is defined by f(x) = 2x² + 3 for x ge; 0.
   (a) Find and simplify an expression for ff(x)
6. Points A and B have coordinates (8, 3) and (p, q) respectively. The equation of the perpendicular bisector of AB is y = −2x + 4.
   Find the values of p and q

7. The point A has coordinates (1, 5) and the line l has gradient −2/3 and passes through A. A circle has centre (5, 11) and radius √52.
   (a) Show that l is the tangent to the circle at A.
8. The first, second and third terms of an arithmetic progression are a, 3/2a and b respectively, where a and b are positive constants. The first, second and third terms of a geometric progression are a, 18 and b + 3 respectively.
   (a) Find the values of a and b
9. The diagram shows part of the curve with equation y² = x − 2 and the lines x = 5 and y = 1. The shaded region enclosed by the curve and the lines is rotated through 360° about the x-axis.
   Find the volume obtained
10. (a) Prove the identity
11. The gradient of a curve is given by
12. The diagram shows a cross-section of seven cylindrical pipes, each of radius 20 cm, held together by a thin rope which is wrapped tightly around the pipes. The centres of the six outer pipes are A, B, C, D, E and F. Points P and Q are situated where straight sections of the rope meet the pipe with centre A.

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