CIE Mar 2020 9709 Pure Maths Paper 32

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This page covers Questions and Worked Solutions for CIE Pure Maths Paper 3 February/March 2020, 9709/32.

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CIE Feb/Mar 2020 9709 Pure Maths Paper 32 (pdf)

  1. (a) Sketch the graph of y = |x − 2|
    (b) Solve the inequality |x − 2| < 3x − 4.
  2. Solve the equation ln 3 + ln(2x + 5) = 2 ln(x + 2). Give your answer in a simplified exact form.
  3. (a) By sketching a suitable pair of graphs, show that the equation sec x = 2 − 1/2 x has exactly one root in the interval 0 ≤ x < 1/2 π
    (b) Verify by calculation that this root lies between 0.8 and 1.
    (c) Use the iterative formula to determine the root correct to 2 decimal places.
    Give the result of each iteration to 4 decimal places
  4. Find
  5. (a) Show that
    (b) Hence solve the equation

  1. The variables x and y satisfy the differential equation
    It is given that y = 0 when x = 1.
    (a) Solve the differential equation, obtaining an expression for y in terms of x
    (b) State what happens to the value of y as x tends to infinity.
  2. The equation of a curve is x3 + 3xy2 − y3 = 5.
    (a) Show that
    (b) Find the coordinates of the points on the curve where the tangent is parallel to the y-axis.
  3. In the diagram, OABCDEFG is a cuboid in which OA = 2 units, OC = 3 units and OD = 2 units. Unit vectors i, j and k are parallel to OA, OC and OD respectively. The point M on AB is such that MB = 2AM. The midpoint of FG is N.
    (a) Express the vectors OM and MN in terms of i, j and k.
    (b) Find a vector equation for the line through M and N.
    (c) Find the position vector of P, the foot of the perpendicular from D to the line through M and N.
  4. Let f(x)
    (a) Express f(x) in partial fractions
    (b) Hence obtain the expansion of f(x) in ascending powers of x, up to and including the term in x2
  5. (a) The complex numbers v and w satisfy the equations
    v + iw = 5 and (1 + 2i)v − w = 3i.
    Solve the equations for v and w, giving your answers in the form x + iy, where x and y are real
    (b) (i) On an Argand diagram, sketch the locus of points representing complex numbers z satisfying |z − 2 − 3i| = 1
    (ii) Calculate the least value of arg z for points on this locus.

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