MathsCBSE Class 10 · Standard · Board exam 2023, set 30/6/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Standard 2023: question 34(A)

Set in the CBSE Class 10 board exam of 2023, paper 30/6/1, section D, question 34(A), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q34(A) · 5 marks · Long answerArithmetic Progressions

The ratio of the 11th term to the 18th term of an A.P. is 2 : 3. Find the ratio of the 5th term to the 21st term. Also, find the ratio of the sum of first 5 terms to the sum of first 21 terms.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 30/6/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Write the 11th and 18th terms using $a_{n} = a + (n - 1) d$ and find a relation between a and d.

Hint 2

Substitute a = 4d into the required terms and into $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ . The d cancels.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

$\frac{a _{11}}{a _{18}} = \frac{a + 10 d}{a + 17 d} = \frac{2}{3}$ .
1 mark
$3 a + 30 d = 2 a + 34 d \Rightarrow a = 4 d$ .
1 mark
$\frac{a _{5}}{a _{21}} = \frac{a + 4 d}{a + 20 d} = \frac{4 d + 4 d}{4 d + 20 d} = \frac{8 d}{24 d} = \frac{1}{3}$ , so $a_{5} : a_{21} = 1 : 3$ .
1 mark
$\frac{S _{5}}{S _{21}} = \frac{\frac{5}{2} [ 2 a + 4 d ]}{\frac{21}{2} [ 2 a + 20 d ]} = \frac{5 [ 8 d + 4 d ]}{21 [ 8 d + 20 d ]}$ .
1 mark
$= \frac{5 \times 12 d}{21 \times 28 d} = \frac{60}{588} = \frac{5}{49}$ , so $S_{5} : S_{21} = 5 : 49$ .
1 mark

Answer: $a_{5} : a_{21} = 1 : 3$ ; $S_{5} : S_{21} = 5 : 49$

Where the marks are

Forms $\frac{a + 10 d}{a + 17 d} = \frac{2}{3}$ (1 mark)
Gets a = 4d (1 mark)
Ratio of 5th to 21st term = 1 : 3 (1 mark)
Substitutes into $S_{n}$ for $S_{5}$ and $S_{21}$ (1 mark)
Gets 5 : 49 (1 mark)

Revise the chapter

Chapter: Arithmetic Progressions (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Arithmetic Progressions, one at a time, marked against the answer key and marking points.Back to the full paper2023 board paper, set 30/6/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2023: question 34(A)

Hints

Step-by-step solution

Revise the chapter

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