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

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

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

Q33(B) · 5 marks · Long answerArithmetic Progressions

In an A.P. of 40 terms, the sum of first 9 terms is 153 and the sum of last 6 terms is 687. Determine the first term and common difference of A.P. Also, find the sum of all the terms of the A.P.

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

The last 6 terms of a 40-term A.P. are the 35th to the 40th terms.

Hint 2

Write each of $a_{35}, \dots, a_{40}$ as $a + (n - 1) d$ and add them. Also turn $S_{9} = 153$ into a linear equation in $a$ and $d$ .

Hint 3

Solve the two linear equations together, then use $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ with $n = 40$ .

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 $d$ . The A.P. has $n = 40$ terms.
Sum of the first 9 terms: $S_{9} = \frac{9}{2} [2 a + 8 d] = 153$
$\Rightarrow 2 a + 8 d = 34 \Rightarrow a + 4 d = 17$ ... (i)
1 mark
The last 6 terms are $a_{35}, a_{36}, \dots, a_{40}$ , so their sum is $S_{40} - S_{34} = 687$ .
Using $a_{n} = a + (n - 1) d$ :
$(a + 34 d) + (a + 35 d) + (a + 36 d) + (a + 37 d) + (a + 38 d) + (a + 39 d) = 687$
1 mark
$\Rightarrow 6 a + 219 d = 687 \Rightarrow 2 a + 73 d = 229$ ... (ii)
1 mark
From (i), $2 a + 8 d = 34$ . Subtract this from (ii):
$65 d = 195 \Rightarrow d = 3$
Then $a = 17 - 4 (3) = 5$ .
1 mark
Sum of all 40 terms: $S_{40} = \frac{40}{2} [2 (5) + 39 (3)] = 20 (10 + 117) = 20 \times 127 = 2540$
So first term $= 5$ , common difference $= 3$ , and the sum of all terms $= 2540$ .
1 mark

Answer: a = 5, d = 3, S₄₀ = 2540

Where the marks are

Uses $S_{9} = 153$ to get $a + 4 d = 17$ (1 mark)
Writes the sum of the last 6 terms as $a_{35}$ to $a_{40}$ and simplifies to $6 a + 219 d = 687$ , i.e. $2 a + 73 d = 229$ (2 marks)
Solves the two equations to get $a = 5$ , $d = 3$ (1 mark)
Finds $S_{40} = 2540$ (1 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(B)

Hints

Step-by-step solution

Revise the chapter

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