| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2010 |
| Session | June |
| Marks | 12 |
| Topic | Linear combinations of normal random variables |
| Difficulty | Standard +0.8 This question tests linear combinations of normal random variables with three parts of increasing sophistication. Part (i) is standard (H-P distribution), part (ii) requires forming 2P+4H, both routine A-level Further Maths. Part (iii) requires H>2P, equivalent to H-2P>0, which is slightly less standard but still a direct application. The multi-part structure and need to correctly identify variance formulas (Var(aX)=a²Var(X)) elevates this above average difficulty, but it remains a textbook-style question without novel insight. |
| Spec | 5.04a Linear combinations: E(aX+bY), Var(aX+bY) |
**(i)** $X - Y \sim \text{N}(3, 3)$ — B1B1
$z = \frac{1-3}{\sqrt{3}} = -1.155$ — M1
$P(X - Y > 1) = 0.876$ — A1 [4]
**(ii)** $P_1 + P_2 + H_1 + H_2 + H_3 + H_4 \sim \text{N}(24, 10.5)$ — B1B1
$z = \frac{20 - 24}{\sqrt{10.5}} = -1.234$ — M1
$P(< 20) = 0.109$ — A1 [4]
**(iii)** $H - 2P \sim \text{N}(1, 5.25)$ — B1B1
$z = \frac{0-1}{\sqrt{5.25}} = -0.436...$ — M1
$P(H > 2P) = 0.669$ — A1 [4]
**[Total: 12]**11 The thickness of a randomly chosen paperback book is $P \mathrm {~cm}$ and the thickness of a randomly chosen hardback is $H \mathrm {~cm}$, where $P$ and $H$ have distributions $\mathrm { N } ( 2.0,0.75 )$ and $\mathrm { N } ( 5.0,2.25 )$ respectively. When more than one book is selected, any book is selected independently of all other books.\\
(i) Calculate the probability that a randomly chosen hardback is more than 1 cm thicker than a randomly chosen paperback.\\
(ii) Calculate the probability that 2 paperbacks and 4 hardbacks, randomly chosen, have a combined thickness of less than 20 cm .\\
(iii) Find the probability that a randomly chosen hardback is more than twice the thickness of a randomly chosen paperback.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2010 Q11 [12]}}