CIE March 2021 9709 Pure Maths Paper 3 (pdf)
- Solve the equation ln(x3 − 3) = 3 ln x − ln 3. Give your answer correct to 3 significant figure.
- The polynomial ax3 + 5x2 − 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 reiθ, 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 cos2x 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
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