CIE October 2020 9709 Pure Maths Paper 3 (pdf)
- Solve the inequality 2 − 5x > 2|x − 3|.
- On a sketch of an Argand diagram, shade the region whose points represent complex numbers z
satisfying the inequalities |z| ≥ 2 and |z − 1 + i| ≤ 1.
- The parametric equations of a curve are
x = 3 − cos 2θ, y = 2θ + sin 2θ,
for 0 < 1 < 1/2 π
Show that dy/dx = cot θ.
- Solve the equation
log10(2x + 1) = 2 log10(x + 1) − 1.
Give your answers correct to 3 decimal place
- (a) By sketching a suitable pair of graphs, show that the equation cosec x = 1 + e-1/2x has exactly two
roots in the interval 0 < x < π.
(b) The sequence of values given by the iterative formula
converges to one of these roots.
Use the formula to determine this root correct to 2 decimal places. Give the result of each
iteration to 4 decimal places.
- (a) Express √6 cos θ + 3 sin θ in the form R cos(θ − α), where R > 0 and 0° < α < 90°. State the exact
value of R and give α correct to 2 decimal places
(b) Hence solve the equation √6 cos 1/3x + 3 sin 1/3x = 2.5, for 0° < x < 360°.
- (a) Verify that −1 + √5i is a root of the equation 2x3 + x2 + 6x − 18 = 0.
(b) Find the other roots of this equation.
- The coordinates (x, y) of a general point of a curve satisfy the differential equation
for x > 0. It is given that y = 1 when x = 1.
Solve the differential equation, obtaining an expression for y in terms of x.
- 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.
- The diagram shows the curve y = (2 − x)e-1/2x, and its minimum point M.
(a) Find the exact coordinates of M
(b) Find the area of the shaded region bounded by the curve and the axes. Give your answer in terms
- Two lines have equations r = i + 2j + k + λ(ai + 2j − k) and r = 2i + j − k + μ(2i − j + k), where α is a
(a) Given that the two lines intersect, find the value of a and the position vector of the point of
(b) Given instead that the acute angle between the directions of the two lines is cos−1(1/6), find the
two possible values of a.
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