A-LevelMathematicsComplex numbers9709/33 · May/June 2018 · Q99 Marks
9
- (a)Find the complex number z satisfying the equation 3z - iz* = 1 + 5i, where z* denotes the complex conjugate of z.[4]
- (b)On a sketch of an Argand diagram, shade the region whose points represent complex numbers z which satisfy both the inequalities |z| ≤ 3 and Im z ≥ 2, where Im z denotes the imaginary part of z. Calculate the greatest value of arg z for points in this region. Give your answer in radians correct to 2 decimal places.[5]
Mark scheme
Marking points from the official mark scheme (May/June 2018). The mark or mark type is shown in brackets.
- B1 Substitute and obtain a correct equation in x and y
- M1 Use i² =-1 and equate real and imaginary parts
- A1 Obtain two correct equations in x and y, e.g. 3x - y = 1 and 3y – x = 5
- A1 Solve and obtain answer z = 1 + 2(i)
- B1 Show a circle with radius 3
- B1 Show the line y = 2 extending in both quadrants
- B1 Shade the correct region
- M1 Carry out a complete method for finding the greatest value of arg z
- A1 Obtain answer 2.41
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