FP1 Complex Numbers Past Paper Questions

FP1 January 2009

1. f(x) = 2x3 – 8x2 + 7x – 3.

Given that x = 3 is a solution of the equation f(x) = 0, solve f(x) = 0 completely.(5)

9. Given that = 3 + 2i and = ,

(a) find in the form a + ib, where a and b are real.(2)

(b) Show, on an Argand diagram, the point P representing and the point Q representing .

(2)

(c) Given that O is the origin, show that POQ = .(2)

The circle passing through the points O, P and Q has centre C. Find

(d) the complex number represented by C,(2)

(e) the exact value of the radius of the circle.(2)

FP1 June 2009

1. The complex numbers and are given by

= 2 – i and = –8 + 9i

(a) Show and on a single Argand diagram.(1)

Find, showing your working,

(b) the value of ,(2)

(c) the value of arg , giving your answer in radians to 2 decimal places,

(2)

(d) in the form a + bi, where a and b are real.

(3)

3. f(x) = (x2 + 4)(x2 + 8x + 25)

(a) Find the four roots of f(x) = 0.(5)

(b) Find the sum of these four roots.(2)

FP1 January 2010

1. The complex numbers and are given by

= 2 + 8i and = 1 – i

Find, showing your working,

(a) in the form a + bi, where a and b are real,

(3)

(b) the value of ,

(2)

(c) the value of arg , giving your answer in radians to 2 decimal places.

(2)

6. Given that 2 and 5 + 2i are roots of the equation

x3 − 12x2 + cx + d = 0, c, d ∈ℝ,

(a) write down the other complex root of the equation.(1)

(b) Find the value of c and the value of d.(5)

(c) Show the three roots of this equation on a single Argand diagram.(2)

FP1 June 2010

1. z = 2 – 3i

(a) Show that z2 = −5 −12i.(2)

Find, showing your working,

(b) the value of |z2|,(2)

(c) the value of arg (z2), giving your answer in radians to 2 decimal places.(2)

(d) Show z and z2 on a single Argand diagram. (1)

4. f(x) = x3 + x2 + 44x + 150.

Given that f (x) = (x + 3)(x2 + ax + b), where a and b are real constants,

(a) find the value of a and the value of b.(2)

(b) Find the three roots of f(x) = 0.(4)

(c) Find the sum of the three roots of f (x) = 0.(1)

Mark Scheme

FP1 January 2009

FP1 June 2009

FP1 January 2010

FP1 June 2010