This is part of a collection of videos showing step-by-step solutions for CIE A-Level Mathematics past papers.

This page covers Questions and Worked Solutions for CIE Pure Maths Paper 3 February/March 2021, 9709/32.

**Related Pages**

More A Levels Past Papers

CIE March 2021 9709 Pure Maths Paper 3 (pdf)

- Solve the equation ln(x
^{3}− 3) = 3 ln x − ln 3. Give your answer correct to 3 significant figure. - The polynomial ax
^{3}+ 5x^{2}− 4x + b, where a and b are constants, is denoted by p(x). It is given that (x + 2) is a factor of p(x) and that when p(x) is divided by (x + 1) the remainder is 2.

Find the values of a and b. - By first expressing the equation (tanx + 45°) = 2 cot x + 1 as a quadratic equation in tan x, solve the equation for 0° < x < 180°
- The variables x and y satisfy the differential equation

It is given that y = 4 when x = π.

(a) Solve the differential equation, obtaining an expression for y in terms of x

(b) Sketch the graph of y against x for 0 < x < 2π - (a) Express √7 sin x + 2 cos x in the form R sin(x + 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 √7 sin 2θ + 2 cos 2θ = 1, for 0° < θ < 180°.

- Let f(x) = , where a is a positive constant.

(a) Express f(x) in partial fractions.

(b) Hence show that - Two lines have equations r =

(a) Show that the lines are skew.

(b) Find the acute angle between the directions of the two lines - The complex numbers u and v are defined by u = −4 + 2i and v = 3 + i.

(a) Find u/v in the form x + iy, where x and y are real.

(b) Hence express u/v in the form re^{iθ}, where r and θ are exact

In an Argand diagram, with origin O, the points A, B and C represent the complex numbers u, v and 2u + v respectively.

(c) State fully the geometrical relationship between OA and BC

(d) Prove that angle AOB = 3/4 π - Let f(x) = , for x > 0.

(a) The equation x = f(x) has one root, denoted by α.

Verify by calculation that α lies between 1 and 1.5

(b) Use an iterative formula based on the equation in part (a) to determine a correct to 2 decimal places. Give the result of each iteration to 4 decimal place

(c) Find f′(x). Hence find the exact value of x for which f′(x) = −8. - The diagram shows the curve y = sin 2x cos
^{2}x for 0 ≤ x ≤ 1/2 π, and its maximum point M.

(a) Using the substitution u = sin x, find the exact area of the region bounded by the curve and the x-axis.

(b) Find the exact x-coordinate of M

Try the free Mathway calculator and
problem solver below to practice various math topics. Try the given examples, or type in your own
problem and check your answer with the step-by-step explanations.

We welcome your feedback, comments and questions about this site or page. Please submit your feedback or enquiries via our Feedback page.