The polynomial p(x) is defined by p(x) = ax⁴ + bx³ +13x²-35x+15, where a and b are constants. It is given that (2x−1) and (x-3) are factors of p(x).

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

1. [M1] Substitute x = 1/2 and x = 3, and equate each to zero to produce two equations SC B1 for a correct equation, if M0 otherwise.
2. [A1] Obtain 1/16 a + 1/8 b = -15/4 or a+2b=-60 OE a+2b+12=0
3. [A1] Obtain 81a +27b=-27 OE 3a+b+1=0
4. [A1] Solve simultaneous equations to obtain a = 2 and b=-7
5. [M1] Divide by 2x² −7x+3 or successively by 2x−1 and x-3 OE method, such as inspection.
6. [A1] Obtain quotient x² +5 Condone 2(x² +5) from synthetic division.
7. [A1] State fully factorised form (2x-1)(x-3)(x² +5)
8. [M1] Attempt solution of at least cot 2θ=3
9. [A1] Obtain tan 2θ = 1/3 and hence θ=0.161 Or greater accuracy 0.16087...