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This page covers Questions and Worked Solutions for CIE Pure Maths Paper 3 October/November 2020, 9709/33.

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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

log_{10}(2x + 1) = 2 log_{10}(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 2x
^{3}+ x^{2}+ 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 x^{2}. - 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 of e. - 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
constant.

(a) Given that the two lines intersect, find the value of a and the position vector of the point of intersection.

(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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