CIE March 2021 9709 Pure Maths Paper 3

CIE March 2021 9709 Pure Maths Paper 3 (pdf)

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1. Solve the equation ln(x³ − 3) = 3 ln x − ln 3. Give your answer correct to 3 significant figure.
2. 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.
3. 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°
4. 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
5. (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°.

6. Let f(x) = , where a is a positive constant.
   (a) Express f(x) in partial fractions.
   (b) Hence show that
7. Two lines have equations r =
   (a) Show that the lines are skew.
   (b) Find the acute angle between the directions of the two lines
8. 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
9. 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.
   
10. 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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