CIE May 2020 9709 Pure Maths Paper 21

CIE May/June 2020 9709 Pure Maths Paper 21 (pdf)

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1. Solve the equation
   ln(x + 1) − ln x = 2 ln 2.
2. The polynomial p(x) is defined by
   p(x) = 6x³ + ax² + 9x + b,
   where a and b are constants. It is given that (x − 2) and (2x + 1) are factors of p(x).
   Find the values of a and b.
3. A curve has parametric equations
   x = e^t − 2e^-t, y = 3e^2t + 1.
   Find the equation of the tangent to the curve at the point for which t = 0.
4. (a) Sketch, on the same diagram, the graphs of y = |3x + 2a| and y = |3x − 4a|, where a is a positive constant.
   Give the coordinates of the points where each graph meets the axes.
   (b) Find the coordinates of the point of intersection of the two graphs.
   (c) Deduce the solution of the inequality |3x + 2a| < |3x − 4a|.
5. The diagram shows part of the curve with equation y = x³cos 2x. The curve has a maximum at the point M.
   (a) Show that the x-coordinate of M satisfies the equation x
   (b) Use the equation in part (a) to show by calculation that the x-coordinate of M lies between 0.59 and 0.60.
   (c) Use an iterative formula, based on the equation in part (a), to find the x-coordinate of M correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

6. (a) Prove that
   sin 2θ(cosec θ − sec θ)= √8 co(θ + 1/4 π).
   (b) Solve the equation
   sin 2θ(cosec θ − sec θ) = 1
   for 0 < θ < 1/2 π. Give the answer correct to 3 significant figures.
   (c) Find
   
7. (a) Find the quotient when 9x³ − 6x² − 20x + 1 is divided by (3x + 2), and show that the remainder is 9.
   (b) Hence find
   (c) Find the exact root of the equation 9e^9y − 6e^6y − 20e^3y − 8 = 0.

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