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Mathematics · 2023 · 5 marks
CBSE 2023 · Region 6 · Set 3 · Q32
The sum of first seven terms of an A.P. is 182. If its $\displaystyle 4^{\text {th }}$ term and the $\displaystyle 17{ }^{\text {th }}$ term are in the ratio $\displaystyle 1$ : $\displaystyle 5$, find the A.P.The sum of first q terms of an A.P. is $\displaystyle 63 \mathrm{q}-3 \mathrm{q}^{2}$. If its $\displaystyle \mathrm{p}^{\text {th }}$ term is -$\displaystyle 60$, find the value of p . Also, find the $\displaystyle 11^{\text {th }}$ term of this A.P.
The sum of first seven terms of an A.P. is 182. If its $\displaystyle 4^{\text {th }}$ term and the $\displaystyle 17{ }^{\text {th }}$ term are in the ratio $\displaystyle 1$ : $\displaystyle 5$, find the A.P.
The sum of first q terms of an A.P. is $\displaystyle 63 \mathrm{q}-3 \mathrm{q}^{2}$. If its $\displaystyle \mathrm{p}^{\text {th }}$ term is -$\displaystyle 60$, find the value of p . Also, find the $\displaystyle 11^{\text {th }}$ term of this A.P.
Marking-scheme solution
Let a be the first term and d be the common difference.
\[\begin{aligned}
& \quad \mathrm{S}_{7}=182 \Rightarrow \frac{7}{2}(2 \mathrm{a}+6 \mathrm{~d})=182 \\
& \Rightarrow 2 \mathrm{a}+6 \mathrm{~d}=\frac{182 \times 2}{7}=52 \\
& \mathrm{a}+3 \mathrm{~d}=26 \\
& \frac{a_{4}}{a_{17}}=\frac{1}{5} \Rightarrow \frac{a+3 \mathrm{~d}}{\mathrm{a}+16 \mathrm{~d}}=\frac{1}{5} \Rightarrow 5 \mathrm{a}+15 \mathrm{~d}=\mathrm{a}+16 \mathrm{~d} \\
& 4 \mathrm{a}=\mathrm{d}
\end{aligned}
\]
Solving (i) and (ii)
\[\begin{aligned}
& \mathrm{a}=2 \text { and } \mathrm{d}=8 \\
& \therefore \mathrm{AP} \text { is } 2,10,18,26, \ldots .
\end{aligned}
\]
\(\displaystyle \mathrm{Sq}=63 \mathrm{q}-3 \mathrm{q}^{2}\)
\[\begin{aligned}
& \therefore S_{1}=60 \Rightarrow a_{1}=60\left(1^{\text {st }} \text { term }\right) \\
& S_{2}=63(2)-3(2)^{2}=126-12=114 \\
& a_{1}+a_{2}=114 \Rightarrow a_{2}=114-60=54 \\
& d=a_{2}-a_{1}=54-60=-6 \\
& a_{p}=-60 \\
& 60+(p-1) d=-60 \\
& (p-1)(-6)=-120 \Rightarrow p=21 \\
& a_{11}=a+10 d=60+10(-6)=0
\end{aligned}
\]
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CBSE Class 10 Mathematics past-paper question from the 2023board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.