Aahana being a plant lover decides to convert her balcony into beautiful garden full of plants. She bought few plants with pots for her balcony. She placed the pots in such a way that number of pots in the first row is 2, second row is 5, third row is 8 and so on. Based on the above information, answer the following questions : (i) Find the number of pots placed in the 10^th row. (ii) Find the difference in the number of pots placed in 5^th row and 2^nd row. (iii) If Aahana wants to place 100 pots in total, then find the total number of rows formed in the arrangement. (iii) If Aahana has sufficient space for 12 rows, then how many total number of pots are placed by her with the same arrangement ?

MathsCBSE Class 10 · Basic · Board exam 2023, set 430/1/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Basic 2023: question 36

Set in the CBSE Class 10 board exam of 2023, paper 430/1/1, section E, question 36, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q36 · 4 marks · Case-basedCompetency-basedArithmetic Progressions

Aahana being a plant lover decides to convert her balcony into beautiful garden full of plants. She bought few plants with pots for her balcony. She placed the pots in such a way that number of pots in the first row is 2, second row is 5, third row is 8 and so on.

Based on the above information, answer the following questions :

(i)
Find the number of pots placed in the $1 0^{th}$ row.
1 mark
(ii)
Find the difference in the number of pots placed in $5^{th}$ row and $2^{nd}$ row.
1 mark
(iii)
If Aahana wants to place 100 pots in total, then find the total number of rows formed in the arrangement.
2 marks
OR
(iii)
If Aahana has sufficient space for 12 rows, then how many total number of pots are placed by her with the same arrangement ?
2 marks

Figure in the paper: Pot arrangement in rows: the top row has 2 flower pots; the middle row has 5 pots; the bottom row has 8 pots. Pots are drawn as flower pots, and a dashed line marks the ground below the bottom row. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 430/1/1.

Hints

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

Hint 1

The rows go 2, 5, 8, …, which is an A.P. Find its first term and common difference.

Hint 2

Use $a_{n} = a + (n - 1) d$ for a single row, and $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ for a total.

Hint 3

For 100 pots, set $S_{n} = 100$ and solve the quadratic in $n$ . Keep only the natural-number root.

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 pots in the rows are 2, 5, 8, … This is an A.P. with $a = 2$ and $d = 5 - 2 = 3$ .
(i) $a_{10} = a + 9 d = 2 + 9 (3) = 29$ . So 29 pots are placed in the 10th row.
1 mark
(ii) $a_{5} - a_{2} = (a + 4 d) - (a + d) = 3 d = 3 \times 3 = 9$ . So the 5th row has 9 more pots than the 2nd row.
1 mark
(iii) Set $S_{n} = 100$ : $\frac{n}{2} [2 (2) + (n - 1) 3] = 100$ . This gives $n (3 n + 1) = 200$ , i.e. $3 n^{2} + n - 200 = 0$ .
1 mark
(iii) Factorise: $(3 n + 25) (n - 8) = 0$ , so $n = 8$ or $n = - \frac{25}{3}$ . Reject $n = - \frac{25}{3}$ , since the number of rows must be a natural number. Hence 8 rows are formed.
1 mark
(iii) OR: $S_{12} = \frac{12}{2} [2 (2) + 11 (3)] = 6 (4 + 33)$
1 mark
(iii) OR: $S_{12} = 6 \times 37 = 222$ . So she places 222 pots in total.
1 mark

Answer: (i) 29 pots; (ii) 9; (iii) 8 rows; OR (iii) 222 pots

Where the marks are

(i) $a_{10} = a + 9 d = 29$ (1 mark)
(ii) $a_{5} - a_{2} = 3 d = 9$ (1 mark)
(iii) Forms $\frac{n}{2} [4 + 3 (n - 1)] = 100$ (1 mark)
(iii) Solves $3 n^{2} + n - 200 = 0$ to get $n = 8$ , rejecting the negative root (1 mark)
(iii) OR: Correct substitution in $S_{12}$ (1 mark)
(iii) OR: $S_{12} = 222$ (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 430/1/1 (Basic): every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 10 Mathematics Basic 2023: question 36

Hints

Step-by-step solution

Revise the chapter

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