CIE Mar 2020 9709 Pure Maths Paper 32

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

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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

6. 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.
7. 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.
8. 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.
   
9. 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
10. (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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