CIE Oct 2020 9709 Pure Maths Paper 13

CIE Oct 2020 9709 Pure Maths Paper 13 (pdf)

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1. (a) Express x² + 6x + 5 in the form (x + a)² + b, where a and b are constant
   (b) The curve with equation y = x² is transformed to the curve with equation y = x² + 6x + 5.
   Describe fully the transformation(s) involve
2. The function f is defined by
   (a) Find
   (b) The equation of a curve is such
3. Solve the equation
4. A curve has equation y = 3x² − 4x + 4 and a straight line has equation y = mx + m − 1, where m is a constant.
   Find the set of values of m for which the curve and the line have two distinct points of intersection.
5. In the expansion of (a + bx)⁷, where a and b are non-zero constants, the coefficients of x, x² and x⁴ are the first, second and third terms respectively of a geometric progression.
   Find the value of a/b
6. The function f is defined by
   (a) Find an expression for f⁻¹(x)
   (b) Show that
   (c) State the range of

7. The first and second terms of an arithmetic progression are
   (a) Show that the common difference is
   (b) Find the exact value of the 13th term when θ = 1/6 π
8. The equation of a curve is
   (a) Find
   (b) Find the coordinates of the stationary point and determine the nature of the stationary point
   
9. In the diagram, arc AB is part of a circle with centre O and radius 8 cm. Arc BC is part of a circle with centre A and radius 12 cm, where AOC is a straight line.
   (a) Find angle BAO in radians
   (b) Find the area of the shaded region
   (c) Find the perimeter of the shaded region.
10. A curve has equation
    (a) It is given that when x = 1/4, the gradient of the curve is 3.
    Find the value of k.
    (b) It is given instead that
    Find the value of k.
11. A circle with centre C has equation (x − 8)² + (y − 4)² = 100.
    (a) Show that the point T(−6, 6) is outside the circle.
    Two tangents from T to the circle are drawn.
    (b) Show that the angle between one of the tangents and CT is exactly 45°
    The two tangents touch the circle at A and B.
    (c) Find the equation of the line AB, giving your answer in the form y = mx + c.
    (d) Find the x-coordinates of A and B.

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