MathsCBSE Class 10 · Standard · Board exam 2025, set 30/3/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Standard 2025: question 33(a)

Set in the CBSE Class 10 board exam of 2025, paper 30/3/1, 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 the third term and the seventh term of an AP is 6 and their product is 8. Find the sum of the first sixteen terms of the AP.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 30/3/1.

Hints

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

Hint 1

Write $a_{3} = a + 2 d$ and $a_{7} = a + 6 d$ . The sum condition gives $a + 4 d = 3$ .

Hint 2

Substitute into the product condition to find d. There are two values, so find $S_{16}$ 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 $d$ .
$a_{3} + a_{7} = (a + 2 d) + (a + 6 d) = 6 \Rightarrow a + 4 d = 3 \Rightarrow a = 3 - 4 d$
½ mark
$a_{3} \cdot a_{7} = (a + 2 d) (a + 6 d) = 8$
½ mark
Substitute $a = 3 - 4 d$ : $(3 - 2 d) (3 + 2 d) = 8 \Rightarrow 9 - 4 d^{2} = 8 \Rightarrow d^{2} = \frac{1}{4} \Rightarrow d = \pm \frac{1}{2}$
1 mark
Case 1: $d = \frac{1}{2}$ gives $a = 3 - 2 = 1$ .
½ mark
$S_{16} = \frac{16}{2} [2 \times 1 + 15 \times \frac{1}{2}]$
½ mark
$= 8 \times 9.5 = 76$
½ mark
Case 2: $d = - \frac{1}{2}$ gives $a = 3 + 2 = 5$ .
½ mark
$S_{16} = \frac{16}{2} [2 \times 5 + 15 \times (- \frac{1}{2})]$
½ mark
$= 8 \times 2.5 = 20$
½ mark

Answer: $S_{16} = 76$ (when $d = \frac{1}{2}$ ) or $S_{16} = 20$ (when $d = - \frac{1}{2}$ )

Where the marks are

Forms $a + 4 d = 3$ from the sum condition (½ mark)
Forms the product equation $(a + 2 d) (a + 6 d) = 8$ (½ mark)
Solves to get $d = \pm \frac{1}{2}$ (1 mark)
For $d = \frac{1}{2}$ : $a = 1$ , correct substitution, $S_{16} = 76$ (½ + ½ + ½ marks)
For $d = - \frac{1}{2}$ : $a = 5$ , correct substitution, $S_{16} = 20$ (½ + ½ + ½ marks)

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 paper2025 board paper, set 30/3/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2025: question 33(a)

Hints

Step-by-step solution

Revise the chapter

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