The random variable X takes values in the range 1 ≤ x ≤ p, where p is a constant. The graph of the probability density function of X is shown in the diagram. [Figure X.X]

Mark scheme

Marking points from the official mark scheme (May/June 2021). The mark or mark type is shown in brackets.

1. [M1] (a) 1/2 p(p – 1) = 1; For area =1; For verification methods accept 1/2 × 2 × 1 = 1 or 1/2 × 2 × (p-1) = 1 or = 1/2 × 1 × p = 1 as indication that area=1
2. [A1] (a) p = 2; AG - Convincing method and answer. Must see quadratic rearranged to =0 and no errors seen. N.B. Accept convincing verification methods (e.g. statement such as 'assume p = 2' or ‘if p = 2' or 'using p = 2' or showing by clear substitution that p = 2 fits 1/2 p(p – 1) = 1 with clear conclusion)
3. [M1] (b) Gradient = 2; equation of line is y = 2x + c; line passes through (1, 0), hence c = −2; y = 2x - 2; May be seen in (a); M1 can be implied by correct answer
4. [M1] (b) ∫[1,2] x f(x)dx; For attempting ∫x f(x)dx. Ignore limits, FT their equation.
5. [A1 FT] (b) 2 ∫[1,2] (x² - x)dx; 2 [x^3/3 - x^2/2]^2_1; Correct integration FT their f(x)and correct limits
6. [A1] (b) 5/3 or 1.67 (3 sf)