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Mathematics · 2023 · 4 marks
CBSE 2023 · Region 3 · Set 2 · Q37
An octagonal prism is a three-dimensional polyhedron bounded by two octagonal bases and eight rectangular side faces. It has 24 edges and 16…
An octagonal prism is a three-dimensional polyhedron bounded by two octagonal bases and eight rectangular side faces. It has $\displaystyle 24$ edges and $\displaystyle 16$ vertices.
The prism is rolled along the rectangular faces and number on the bottom face (touching the ground) is noted. Let X denote the number obtained on the bottom face and the following table give the probability distribution of X.
Based on the above information, answer the following questions :
| $\displaystyle \mathrm{X}:$ | $\displaystyle 1$ | $\displaystyle 2$ | $\displaystyle 3$ | $\displaystyle 4$ | $\displaystyle 5$ | $\displaystyle 6$ | $\displaystyle 7$ | $\displaystyle 8$ |
| $\displaystyle \mathrm{P}(\mathrm{X}):$ | p | $\displaystyle 2$ p | $\displaystyle 2$ p | p | $\displaystyle 2$ p | $\displaystyle \mathrm{p}^{2}$ | $\displaystyle 2 \mathrm{p}^{2}$ | $\displaystyle 7 \mathrm{p}^{2}+\mathrm{p}$ |
(i)
Find the value of p.
(ii)
Find $\displaystyle \mathrm{P}(\mathrm{X}>6)$.
(iii)
(a)
Find $\displaystyle \mathrm{P}(\mathrm{X}=3 \mathrm{~m})$, where m is a natural number.
OR
(b)
Find the mean E(X). Case Study - $\displaystyle 3$
Marking-scheme solution
$$\begin{aligned}
& \text { (i) } 10 \mathrm{p}^{2}+9 \mathrm{p}=1 \\
& \Rightarrow \mathrm{p}=\frac{1}{10}
\end{aligned}
$$
$$\begin{aligned}
& \text { (ii) } \mathrm{P}(\mathrm{X}>6)=9 \mathrm{p}^{2}+\mathrm{p} \\
& =\frac{9}{100}+\frac{1}{10} \\
& =\frac{19}{100}
\end{aligned}
$$
$$\begin{aligned}
& \text { (iii)(a) } \mathrm{P}(\mathrm{X}=3 \mathrm{~m})=\mathrm{P}(3)+\mathrm{P}(6) \\
& \Rightarrow 2 \mathrm{p}+\mathrm{p}^{2}=\frac{21}{100}
\end{aligned}
$$
(iii)(b)
$$\begin{aligned}
& E(\mathrm{X})=\sum \mathrm{X} \mathrm{P}(\mathrm{X})=\mathrm{p}+4 \mathrm{p}+6 \mathrm{p}+4 \mathrm{p}+10 \mathrm{p}+6 \mathrm{p}^{2}+14 \mathrm{p}^{2}+56 \mathrm{p}^{2}+8 \mathrm{p} \\
& =33 \mathrm{p}+76 \mathrm{p}^{2} \\
& =\frac{406}{100} \text { or } \frac{203}{50}
\end{aligned}
$$
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CBSE Class 12 Mathematics past-paper question from the 2023 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.