CIE Oct 2021 9709 Pure Maths Paper 32

CIE Oct 2021 9709 Pure Maths Paper 3 (pdf)

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1. Find the value of x for which 3(2^1−x = 7^x. Give your answer in the form ln a/ln b, where a and b are integer
2. Solve the inequality |3x − a| > 2|x + 2a|, where a is a positive constant
3. (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.
4. Express in partial fractions.
5. (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.
6. (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
7. The variables x and y satisfy the differential equation
8. (a) By first expanding (cos²θ + sin²θ)², show that
   (b) Hence solve the equation
9. The equation of a curve is ye^2x − y²e^x = 2. (a) Show that
   (b) Find the exact coordinates of the point on the curve where the tangent is parallel to the y-axis.
10. With respect to the origin O, the position vectors of the points A and B are given by

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