Complex numbers — AS & A Level Mathematics (9709) questions by topic

Find the complex numbers z for which (z + 5i)/(z − 5) is real and z = √17. Give your answers in the form z = x + iy, where x and y are real.

It is given that z₁ = 3 e^((1/4)πi), z₂ = (3/2) e^((1/6)πi) and ω = 2 e^((1/2)πi). State the values of ωz₁ and ωz₂. Give your answers in the form r e^(iθ), where r 0 and −π < θ ⩽…

On an Argand diagram shade the region whose points represent complex numbers z which satisfy both the inequalities z - 3i <= 2 and (1/4)pi <= arg(z - 1 - 2i) <= (3/4)pi. [5]

The square roots of -1 - 4 sqrt(5) i can be expressed in the Cartesian form x + iy, where x and y are real and exact. By first forming a quartic equation in x or y, find the…

It is given that z1 = r1 e^(iθ1) and z2 = r2 e^(iθ2). Show that (z1 z2) = z1 z2.

Find the complex numbers z for which (z + 4)/(z + 4i) is real and z = √10. Give your answers in the form z = x + iy, where x and y are real.

The complex numbers s and t are given by s = 5(cos 0.25 + i sin 0.25) and t = 6e^(3i). Express s/t in the form r e^(iθ), where −π < θ <= π and r 0. [2]

The diagram shows the locus of points representing the complex numbers, z, satisfying z + 5 − 4i = 3. The locus is a circle of radius 3 centred at the point representing −5 + 4i…

The complex number u is given by u = −1 − i√3. Express u in the form r(cos θ + i sin θ), where r 0 and −π < θ ≤ π. Give the exact values of r and θ.

On a single Argand diagram sketch the loci given by the equations z − 3 + 2i = 2 and w − 3 + 2i = w + 3 − 4i , where z and w are complex numbers.

The complex numbers z and w are defined by z = 1 - i and w = -3 + 3sqrt(3) i. Express zw in the form a + bi, where a and b are real and in exact surd form. [1]

The square roots of 24 − 7i can be expressed in the Cartesian form x + iy, where x and y are real and exact. By first forming a quartic equation in x or y, find the square roots…

On an Argand diagram shade the region whose points represent complex numbers z which satisfy both the inequalities z − 4 − 3i ⩽ 2 and arg(z − 2 − i) ⩾ ⅓π.

Given that z = 1 + yi and that y is a real number, express 1/z in the form a + bi, where a and b are functions of y.

The square roots of 6 − 8i can be expressed in the Cartesian form x + iy, where x and y are real and exact. By first forming a quartic equation in x or y, find the square roots of…

The complex number u is given by u = (cos ⅐π + i sin ⅐π)⁴ / (cos ⅐π − i sin ⅐π). Find the exact value of arg u.

The complex number z satisfies z = 2 and 0 ≤ arg z ≤ (1/4)π. On the Argand diagram below, sketch the locus of the points representing z.

Find the complex number z satisfying the equation (z − 3i) / (z + 3i) = (2 − 9i) / 5. Give your answer in the form x + iy, where x and y are real.

On an Argand diagram, sketch the locus of points representing complex numbers z satisfying z + 3 − 2i = 2.

The complex number 2 + yi is denoted by a, where y is a real number and y < 0. It is given that f(a) = a³ − a² − 2a. Find a simplified expression for f(a) in terms of y.

On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities z − 3 − i ⩽ 3 and z ⩾ z − 4i .

The complex number z is defined by z = (5a − 2i) / (3 + ai), where a is an integer. It is given that arg z = −¼π. Find the value of a and hence express z in the form x + iy, where…

On an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities z − 2i ≤ z + 2 − i and 0 ≤ arg(z + 1) ≤ (1/4)π.

The complex number u is defined by u = (3 + 2i) / (a − 5i), where a is real. Express u in the Cartesian form x + iy, where x and y are in terms of a.

On a sketch of an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities z − 4 − 3i ≤ 2 and Re z ≤ 3.

It is given that (2 + 3ai) / (a + 2i) = λ(2 − i), where a and λ are real constants. Show that 3a² + 4a − 4 = 0.

On an Argand diagram, shade the region whose points represent complex numbers z satisfying the inequalities z − 1 + 2i ⩽ z and z − 2 ⩽ 1.

Solve the quadratic equation (3 + i)w² − 2w + 3 − i = 0, giving your answers in the form x + iy, where x and y are real.