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Mathematics · 2023 · 5 marks
CBSE 2023 · Region 2 · Set 3 · Q32
The mode of the following frequency distribution is 55. Find the missing frequencies 'a' and 'b'. Class Interval $\displaystyle 0$-$\displaystyle 15$ $\displaystyle 15$-$\displaystyle 30$ $\displaystyle 30$-$\displaystyle 45$ $\displaystyle 45$-$\displaystyle 60$ $\displaystyle 60$-$\displaystyle 75$ $\displaystyle 75$-$\displaystyle 90$ Total Frequency $\displaystyle 6$ $\displaystyle 7$ a $\displaystyle 15$ $\displaystyle 10$ b $\displaystyle 51$
| Class Interval | $\displaystyle 0$-$\displaystyle 15$ | $\displaystyle 15$-$\displaystyle 30$ | $\displaystyle 30$-$\displaystyle 45$ | $\displaystyle 45$-$\displaystyle 60$ | $\displaystyle 60$-$\displaystyle 75$ | $\displaystyle 75$-$\displaystyle 90$ | Total |
| Frequency | $\displaystyle 6$ | $\displaystyle 7$ | a | $\displaystyle 15$ | $\displaystyle 10$ | b | $\displaystyle 51$ |
Marking-scheme solution
Modal Class: $\displaystyle 45$-$\displaystyle 60$
Mode = $\displaystyle 55$
\(\displaystyle 55=45+\frac{15-\mathrm{a}}{30-(\mathrm{a}+10)} \times 15\)
\(\displaystyle \Rightarrow \mathrm{a}=5\)
\(\displaystyle 6+7+\mathrm{a}+15+10+\mathrm{b}=51\)
\(\displaystyle \Rightarrow \mathrm{a}+\mathrm{b}=13\)
\(\displaystyle \Rightarrow \mathrm{b}=13-5=8\)
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