CIE May 2021 9709 Pure Maths Paper 1

CIE May/June 2021 9709 Pure Maths Paper 13 (pdf)

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1. A curve with equation y = f(x) is such that f′(x) = 6x² − 8/x². It is given that the curve passes through the point (2, 7). Find f
2. The function f is defined by f(x) = 1/3(2x − 1)^3/2 − 2x for 1/2 < x < a. It is given that f is a decreasing function. Find the maximum possible value of the constant a.
3. A line with equation y = mx − 6 is a tangent to the curve with equation y = x² − 4x + 3. Find the possible values of the constant m, and the corresponding coordinates of the points at which the line touches the curve.
4. (a) Show that the equation
   (b) Hence express cos x in terms of k
5. The diagram shows a triangle ABC, in which angle ABC = 90° and AB = 4 cm. The sector ABD is part of a circle with centre A. The area of the sector is 10 cm². (a) Find angle BAD in radians.
   (b) Find the perimeter of the shaded region

6. Functions f and g are both defined for x ∈ R and are given by
   f(x) = x² − 2x + 5,
   g(x) = x² + 4x + 13.
   (a) By first expressing each of f(x) and g(x) in completed square form, express g(x) in the form f(x + p) + q, where p and q are constants.
   (b) Describe fully the transformation which transforms the graph of y = f(x) to the graph of y = g(x).
7. (a) Write down the first four terms of the expansion, in ascending powers of x, of (a − x)⁶
   (b) Given that the coefficient of x² in the expansion of
8. Functions f and g are defined as follows:
   (a) Solve the equation fg(x) = 3.
   
   (b) Find an expression for (fg)^-1(x)
   
9. (a) A geometric progression is such that the second term is equal to 24% of the sum to infinity.
   Find the possible values of the common ratio.
   (b) An arithmetic progression P has first term a and common difference d. An arithmetic progression
   Q has first term 2(a + 1) and common difference (d + 1). It is given that
10. Points A (−2, 3), B (3, 0) and C (6, 5) lie on the circumference of a circle with centre D.
    (a) Show that angle ABC = 90°
    (b) Hence state the coordinates of D.
    (c) Find an equation of the circle.
    The point E lies on the circumference of the circle such that BE is a diameter.
    (d) Find an equation of the tangent to the circle at E.
11. The diagram shows part of the curve with equation

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