CIE Oct 2021 9709 Pure Maths Paper 3 (pdf)
- Find the value of x for which 3(21−x = 7x. Give your answer in the form ln a/ln b, where a and b are integer
- Solve the inequality |3x − a| > 2|x + 2a|, where a is a positive constant
- (a) Given the complex numbers u = a + ib and w = c + id, where a, b, c and d are real, prove that
(u + w)* = u* + w*.
(b) Solve the equation (z + 2 + i)* + (2 + i)z = 0, giving your answer in the form x + iy where x and
y are real.
- Express in partial fractions.
- (a) On a sketch of an Argand diagram, shade the region whose points represent complex numbers z
satisfying the inequalities |z − 3 − 2i| ≤ 1 and Imz ≥ 2.
(b) Find the greatest value of arg z for points in the shaded region, giving your answer in degrees.
- (a) Using the expansions of sin(3x + 2x) and sin(3x − 2x), show that
1/2(sin 5x + sin x) ≡ sin 3x cos
(b) Hence show that (b) Hence show that
- The variables x and y satisfy the differential equation
- (a) By first expanding (cos2θ + sin2θ)2, show that
(b) Hence solve the equation
- The equation of a curve is ye2x − y2ex = 2.
(a) Show that
(b) Find the exact coordinates of the point on the curve where the tangent is parallel to the y-axis.
- With respect to the origin O, the position vectors of the points A and B are given by
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