CIE Feb/Mar 2020 9709 Pure Maths Paper 32 (pdf)
- (a) Sketch the graph of y = |x − 2|
(b) Solve the inequality |x − 2| < 3x − 4.
- Solve the equation ln 3 + ln(2x + 5) = 2 ln(x + 2). Give your answer in a simplified exact form.
- (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
- (a) Show that
(b) Hence solve the equation
- 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.
- 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.
- 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.
- 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
- (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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