CIE March 2021 9709 Pure Maths Paper 1

Question Paper Pure Maths Paper 1 March 2021, 9709/12

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Mark Scheme Pure Maths Paper 1 March 2021, 9709/12

Worked Solutions Pure Maths Paper 1 March 2021, 9709/12

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1. (a) Find the first three terms in the expansion, in ascending powers of x, of (1 + x)⁵
   (b) Find the first three terms in the expansion, in ascending powers of x, of (1 − 2x)⁶
   (c) Hence find the coefficient of x² in the expansion of (1 + x)⁵5(1 − 2x)⁶
2. By using a suitable substitution, solve the equation
3. Solve the equation
4. A line has equation y = 3x + k and a curve has equation y = x² + kx + 6, where k is a constant.
   Find the set of values of k for which the line and curve have two distinct points of intersection.
5. In the diagram, the graph of y = f(x) is shown with solid lines. The graph shown with broken lines is a transformation of y = f(x).
   (a) Describe fully the two single transformations of y = f(x) that have been combined to give the resulting transformation.
   (b) State in terms of y, f and x, the equation of the graph shown with broken lines.

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6. A curve is such that and A(1,-3) lies on the curve. A point is moving along the curve and at A the y-coordinate of the point is increasing at 3 units per second.
   (a) Find the rate of increase at A of the x-coordinate of the point.
   (b) Find the equation of the curve.
7. Functions f and g are defined as follows:
   (a) Express f(x) in the form (x + a)² + b and state the range of f.
   (b) Find an expression for f⁻¹(x)
   (c) Solve the equation gf(x) = 13.
8. The points A(7, 1), B(7, 9) and C(1, 9) are on the circumference of a circle.
   (a) Find an equation of the circle.
   (b) Find an equation of the tangent to the circle at B.
9. The first term of a progression is cos θ, where 0 < θ < 1/2 π
   (a) For the case where the progression is geometric, the sum to infinity is 1/cosθ
   (i) Show that the second term is cos θ sin²θ
   (ii) Find the sum of the first 12 terms when θ = 1/3 π, giving your answer correct to 4 significant figures.
   (b) For the case where the progression is arithmetic, the first two terms are again cos θ and cos θ sin²θ respectively.
   Find the 85th term when θ = 1/3 π
10. The diagram shows a sector ABC which is part of a circle of radius α. The points D and E lie on AB and AC respectively and are such that AD = AE = kα, where k < 1. The line DE divides the sector into two regions which are equal in area.
    (a) For the case where angle BAC = 1/6 π radians, find k correct to 4 significant figures.
    (b) For the general case in which angle BAC = θ radians, where 0 < θ < 1/2 π, it is given that θ/sin θ > 1.
    Find the set of possible values of k.
11. The diagram shows the curve with equation y = 9(x^-1/2 - 4x^-3/2). The curve crosses the x-axis at the point A.
    (a) Find the x-coordinate of A.
    (b) Find the equation of the tangent to the curve at A
    (c) Find the x-coordinate of the maximum point of the curve.
    (d) Find the area of the region bounded by the curve, the x-axis and the line x = 9.

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