CIE October 2020 9709 Pure Maths Paper 33

CIE October 2020 9709 Pure Maths Paper 3 (pdf)

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1. Solve the inequality 2 − 5x > 2|x − 3|.
2. 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.
3. 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.
4. Solve the equation
   log₁₀(2x + 1) = 2 log₁₀(x + 1) − 1.
   Give your answers correct to 3 decimal place
5. (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.

6. (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°.
7. (a) Verify that −1 + √5i is a root of the equation 2x³ + x² + 6x − 18 = 0.
   (b) Find the other roots of this equation.
8. 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.
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. 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.
11. 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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