CIE October 2020 9709 Pure Maths Paper 32

CIE October 2020 9709 Pure Maths Paper 3 (pdf)

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1. Solve the equation
   ln(1 + e^−3x) = 2.
   Give the answer correct to 3 decimal places
2. (a) Expand in ascending powers of x, up to and including the term in x³ , simplifying the coefficients.
   (b) State the set of values of x for which the expansion is valid.
3. The variables x and y satisfy the relation 2^y = 3^1−2x.
   (a) By taking logarithms, show that the graph of y against x is a straight line. State the exact value of the gradient of this line.
   (b) Find the exact x-coordinate of the point of intersection of this line with the line y = 3x.
   Give your answer in the form ln a/ln b, where a and b are integers.
4. (a) Show that the equation tan(θ + 60°) = 2 cot 1 can be written in the form
   (b) Hence solve the equation tan(θ + 60°) = 2 cot 1, for 0° < θ < 180°
5. The diagram shows the curve with parametric equations
   x = tan θ, y = cos²θ,
   for −1/2π < θ < 1/2π.
   (a) Show that the gradient of the curve at the point with parameter θ is −2 sin θ cos³ θ
   The gradient of the curve has its maximum value at the point P.
   (b) Find the exact value of the x-coordinate of P.

6. The complex number u is defined by
   (a) Express u in the form x + iy, where x and y are real.
   (b) Show on a sketch of an Argand diagram the points A, B and C representing u, 7 + i and 1 − i respectively.
   (c) By considering the arguments of 7 + i and 1 − i, show that
7. The variables x and t satisfy the differential equation
   (a) Solve the differential equation and obtain an expression for x in terms of t.
   (b) State what happens to the value of x when t tends to infinity.
8. With respect to the origin O, the position vectors of the points A, B, C and D are given by
   (a) Show that AB = 2CD
   (b) Find the angle between the directions of AB and CD
   (c) Show that the line through A and B does not intersect the line through C and D
9. Let f(x) =
   (a) Express f(x) in partial fractions.
   (b) Hence find the exact value of
10. The diagram shows the curve y = √x cos x, for 0 ≤ x ≤ 3/2 π, and its minimum point M, where x = α.
    The shaded region between the curve and the x-axis is denoted by R.
    (a) Show that a satisfies the equation tan α = 1/2α.
    (b) The sequence of values given by the iterative formula
    Use this formula to determine α correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
    (c) Find the volume of the solid obtained when the region R is rotated completely about the x-axis. Give your answer in terms of π.

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