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Mathematics · 2023 · 5 marks
CBSE 2023 · Region 6 · Set 1 · Q34
(A) The ratio of the 11^th term to the 18^th term of an A.P. is 2: 3. Find the ratio of the 5^th term to the 21 ^st term. Also, find the ratio of the…
(A)
The ratio of the $\displaystyle 11^{\text {th }}$ term to the $\displaystyle 18^{\text {th }}$ term of an A.P. is $\displaystyle 2$ : 3. Find the ratio of the $\displaystyle 5^{\text {th }}$ term to the $\displaystyle 21$ $\displaystyle { }^{\text {st }}$ term. Also, find the ratio of the sum of first $\displaystyle 5$ terms to the sum of first $\displaystyle 21$ terms.
OR
(B)
If the sum of first $\displaystyle 6$ terms of an A.P. is $\displaystyle 36$ and that of the first $\displaystyle 16$ terms is $\displaystyle 256$, find the sum of first $\displaystyle 10$ terms.
Marking-scheme solution
\[\begin{aligned}
& \frac{a+10 \mathrm{~d}}{\mathrm{a}+17 \mathrm{~d}}=\frac{2}{3} \\
& 3 \mathrm{a}+30 \mathrm{~d}=2 \mathrm{a}+34 \mathrm{~d} \Rightarrow \mathrm{a}=4 \mathrm{~d} \\
& \text { Therefore, } \frac{a+4 \mathrm{~d}}{\mathrm{a}+20 \mathrm{~d}}=\frac{4 d+4 \mathrm{~d}}{4 \mathrm{~d}+20 \mathrm{~d}}=\frac{8 d}{24 d}=\frac{1}{3} \\
& \frac{S_{5}}{S_{21}}=\frac{\dfrac{5}{2}[2 a+4 \mathrm{~d}]}{\dfrac{21}{2}[2 a+20 \mathrm{~d}]}=\frac{5[8 d+4 \mathrm{~d}]}{21[8 d+20 \mathrm{~d}]} \\
& =\frac{5 \times 12 \mathrm{~d}}{21 \times 28 \mathrm{~d}}=\frac{5}{49} \text { or } S_{5}: S_{21}=5: 49
\end{aligned}
\]
\(\displaystyle \mathrm{S}_{6}=36 \Rightarrow \frac{6}{2}[2 \mathrm{a}+5 \mathrm{~d}]=36\)
\[\Rightarrow 2 \mathrm{a}+5 \mathrm{~d}=12 \quad \text {----------- (1) }
\]
\(\displaystyle \mathrm{S}_{16}=256 \Rightarrow \frac{16}{2}[2 \mathrm{a}+15 \mathrm{~d}]=256\)
\[\Rightarrow 2 \mathrm{a}+15 \mathrm{~d}=32 \quad \text {------------ (2) }
\]
Solving ($\displaystyle 1$) and ($\displaystyle 2$)
\[\begin{aligned}
d & =2 \\
a & =1 \\
S_{10} & =\frac{10}{2}[2(1)+9(2)] \\
& =100
\end{aligned}
\]
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CBSE Class 10 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.