The diagram shows the graph of the probability density function, f, of a random variable X. The graph is a straight line from (0, a) to (2, b), where a and b are positive constants. Elsewhere, f(x) = 0.

Mark scheme

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

1. [M1] (a+b)/2 x 2 = 1 or ½(b - a) x 2 + 2a = 1 or 2b - 1/2 x 2(b - a) = 1 or ∫[0 to 2] (a+(b-a)/2 x)dx = 1 and attempt to integrate
2. [A1] eg a + b = 1 or 2a + b – a = 1 or 2b – b + a = 1 ⇒ b = 1 - a Must see correct intermediate step and answer correctly obtained no errors seen.
3. [B1] y = a + (1-2a)/2 x or y = a + (b-a)/2 x Could be seen in 4(a)(i).
4. [M1] ∫[0 to 2] (ax+(1-2a)/2 x²)dx or ∫[0 to 2] (ax+(b-a)/2 x²)dx Attempt to integrate xf(x) ignore limits, ft their line equation (of form y = mx + c and in terms of a or a and b).
5. [A1] [ax²/2 + (1-2a)/6 x³][0 to 2] or [ax²/2 + (b-a)/6 x³][0 to 2] [=1.2] Correct integration, ignore limits.
6. [M1] [2a+(1-2a)*8/6]=1.2 or [2a+(1-2a)*4/3]=1.2 Substitute correct limits into their integral in terms of a, not a and b (their integral must come from xf(x)) and equated to 1.2.
7. [A1] [4-2a / 3 = 1.2] a = 0.2
8. [M1] ∫[-c to c] ½cos t dt = ½ or ∫[0 to c] ½cos t dt = ¼ Attempt integrate cos t with correct limits and RHS.
9. [A1] [½sin t][-c to c] = ½ or [½sin t][0 to c] = ¼ OE. Correct integral with correct limits and RHS.
10. [A1] sin c = ½
11. [A1] c = π/6 Alone. Accept c ≈ 0.5236.