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

CBSE Class 10 Mathematics Standard 2023: question 35(b)

Set in the CBSE Class 10 board exam of 2023, paper 30/1/1, section D, question 35(b), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q35(b) · 5 marks · Long answerCompetency-basedArithmetic Progressions

250 logs are stacked in the following manner: 22 logs in the bottom row, 21 in the next row, 20 in the row next to it and so on (as shown by an example). In how many rows, are the 250 logs placed and how many logs are there in the top row?

Figure in the paper: Example stack of cylindrical logs: a bottom row of circles, with each higher row having one fewer log, forming a triangular-like pile (labelled 'Example'). See it in the official paper.

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

Hints

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

Hint 1

The row counts 22, 21, 20, … form an A.P. What are $a$ and $d$ ?

Hint 2

Set $S_{n} = 250$ and simplify to a quadratic in $n$ .

Hint 3

You will get two values of $n$ . Check which one gives a negative number of logs in the top row.

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.

The numbers of logs in the rows (from the bottom) form an A.P. $22, 21, 20, \dots$ with $a = 22$ , $d = - 1$ . Let the number of rows be $n$ , so $S_{n} = 250$ .
1 mark
$S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ , which gives $250 = \frac{n}{2} [44 + (n - 1) (- 1)] = \frac{n}{2} (45 - n)$
1 mark
$500 = 45 n - n^{2}$ , so $n^{2} - 45 n + 500 = 0$
1 mark
$(n - 20) (n - 25) = 0$ , so $n = 20$ or $n = 25$ .
For $n = 25$ : $a_{25} = 22 + 24 (- 1) = - 2$ . A row cannot have a negative number of logs, so $n = 25$ is rejected. Hence $n = 20$ .
1 mark
Logs in the top row: $a_{20} = a + 19 d = 22 + 19 (- 1) = 3$ .
So the 250 logs are placed in 20 rows and the top row has 3 logs.
1 mark

Answer: 20 rows; 3 logs in the top row

Where the marks are

Identifies the A.P. with $a = 22$ , $d = - 1$ , $S_{n} = 250$ (1 mark)
Substitutes correctly in the sum formula (1 mark)
Forms $n^{2} - 45 n + 500 = 0$ (1 mark)
Solves and rejects $n = 25$ with a reason, giving $n = 20$ (1 mark)
Finds the top row $a_{20} = 3$ (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 paper2023 board paper, set 30/1/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2023: question 35(b)

Hints

Step-by-step solution

Revise the chapter

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