OCR GCE AS Level Further Maths B (MEI) Core Pure November 2020 question paper Y410

In this question you must show detailed reasoning.

Fig. 2 shows two complex numbers z₁ and z₂ represented on an Argand diagram.

On the copy of Fig. 2 in the Printed Answer Booklet, mark points representing each of the following complex numbers.

• z₁*

• z₂ - z₁

In this question you must show detailed reasoning.

The roots of the equation x² - 2x + 4 = 0 are α and β.

Find α and β in modulus-argument form.

Calculate det M.

State two geometrical consequences of this value for the transformation associated

with M.

Describe fully the transformation associated with M.

You are given that u₁ = 5 and u_n+1 = u_n + 2n + 4.

Prove by induction that u_n = n² + 3n + 1 for all positive integers n.

Investigate whether M and N are commutative under multiplication.

You are now given that MN = I.

Write down a relationship between det M and det N.

Given that λ > 0, find the exact values of λ and μ.

Hence verify your answer to part (i).

In the quartic equation 2x⁴ - 20x³ + ax² + bx + 250 = 0, the coefficients a and b are real. One root of the equation is 2 + i.

Find the other roots.

Find M².

Write down the transformation represented by M.

Hence state the geometrical significance of the result of part (i).

Using determinants, investigate whether N can represent a reflection.

k = - 1