MathsCBSE Class 10 · Standard · Board exam 2024, set 30/1/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Standard 2024: question 33(A)

Set in the CBSE Class 10 board exam of 2024, paper 30/1/1, section D, question 33(A), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q33(A) · 5 marks · Long answerArithmetic Progressions

The sum of first and eighth terms of an A.P. is 32 and their product is 60. Find the first term and common difference of the A.P. Hence, also find the sum of its first 20 terms.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 30/1/1.

Hints

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

Hint 1

Write the 8th term as $a + 7 d$ and form two equations, one for the sum and one for the product.

Hint 2

Replace $a + 7 d$ by $32 - a$ in the product equation to get a quadratic in $a$ .

Hint 3

Both roots give valid A.P.s, so find $S_{20}$ for each.

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.

Let the first term be $a$ and the common difference be $d$ , so $a_{8} = a + 7 d$ .
Sum: $a + (a + 7 d) = 32 \Rightarrow 2 a + 7 d = 32$ ...(1)
1 mark
Product: $a (a + 7 d) = 60$ ...(2)
1 mark
From (1), $a + 7 d = 32 - a$ . Substituting in (2): $a (32 - a) = 60 \Rightarrow a^{2} - 32 a + 60 = 0 \Rightarrow (a - 2) (a - 30) = 0$ .
So $a = 2$ or $a = 30$ .
If $a = 2$ : $7 d = 32 - 4 = 28 \Rightarrow d = 4$ .
If $a = 30$ : $7 d = 32 - 60 = - 28 \Rightarrow d = - 4$ .
So the first term and common difference are 2 and 4, or 30 and −4.
2 marks
$S_{20} = \frac{20}{2} [2 a + 19 d]$ .
For $a = 2$ , $d = 4$ : $S_{20} = 10 (4 + 76) = 800$ .
½ mark
For $a = 30$ , $d = - 4$ : $S_{20} = 10 (60 - 76) = - 160$ .
½ mark

Answer: $a = 2, d = 4$ giving $S_{20} = 800$ ; or $a = 30, d = - 4$ giving $S_{20} = - 160$

Where the marks are

Forms $2 a + 7 d = 32$ (1 mark)
Forms $a (a + 7 d) = 60$ (1 mark)
Solves to get $a = 2, d = 4$ or $a = 30, d = - 4$ (2 marks)
$S_{20} = 800$ for the first case (½ mark)
$S_{20} = - 160$ for the second case (½ mark)

Revise the chapter

Chapter: Arithmetic Progressions (notes, formulas, where marks are lost and every past question)
Topic: Finding nth Term from the Sum of n Terms

Practise this chapterBoard-style questions from Arithmetic Progressions, one at a time, marked against the answer key and marking points.Back to the full paper2024 board paper, set 30/1/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2024: question 33(A)

Hints

Step-by-step solution

Revise the chapter

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