MathsCBSE Class 10 · Basic · Board exam 2025, set 430/1/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Basic 2025: question 37

Set in the CBSE Class 10 board exam of 2025, paper 430/1/1, section E, question 37, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q37 · 4 marks · Case-basedCompetency-basedArithmetic Progressions

Case Study – 2. In a garden, saplings of rose flowers were planted at equal intervals to form a spiral pattern. The spiral is made up of successive semicircles, with centres alternatively at A and B, starting with centre at A, of radii 50 cm, 100 cm, 150 cm, ....... as shown in the figure given below. Spiral 1 has 10 flowers, Spiral 2 has 20 flowers, Spiral 3 has 30 flowers and so on. Based on the above information, answer the following questions :

(i)
What is the radius of the 13th spiral ?
1 mark
(ii)
If the radius of the nth spiral is 500 cm, find the value of n.
1 mark
(iii)(a)
Find the total number of saplings till the 11th spiral.
2 marks
OR
(iii)(b)
Till which spiral, will there be a total of 450 saplings ?
2 marks

Figure in the paper: A horizontal line with points A and B close together near the centre. Semicircular arcs labelled l1 (above, small, centred at A), l2 (below, centred at B), l3 (above, larger), l4 (below, larger) forming an expanding spiral; a dotted continuation arc is shown at the upper right. Photograph of rose saplings in a spiral pattern is also shown. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 430/1/1.

Hints

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

Hint 1

Both the radii and the numbers of flowers increase by a fixed amount each time, so think of them as APs.

Hint 2

For the radius use $a_{n} = a + (n - 1) d$ . For the total number of saplings use $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ .

Hint 3

In (iii)(b), set $S_{n} = 450$ and solve the resulting quadratic in $n$ , rejecting the negative 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.

(i) The radii 50, 100, 150, … form an AP with $a = 50$ and $d = 50$ .
$a_{13} = a + 12 d = 50 + 12 \times 50 = 650$
So the radius of the 13th spiral is 650 cm.
1 mark
(ii) Set $a_{n} = 500$ :
$50 + (n - 1) \times 50 = 500 \Rightarrow 50 n = 500 \Rightarrow n = 10$
1 mark
(iii)(a) The numbers of saplings 10, 20, 30, … form an AP with $a = 10$ and $d = 10$ . Then
$S_{11} = \frac{11}{2} [2 \times 10 + (11 - 1) \times 10] = \frac{11}{2} [20 + 100]$
1½ marks
(iii)(a) $S_{11} = \frac{11}{2} \times 120 = 660$
So there are 660 saplings till the 11th spiral.
½ mark
(iii)(b) With $a = 10$ and $d = 10$ , set $S_{n} = 450$ :
$\frac{n}{2} [20 + (n - 1) 10] = 450$
1 mark
(iii)(b) Simplify: $5 n (n + 1) = 450$ , so $n^{2} + n - 90 = 0$ .
½ mark
(iii)(b) Factorise: $(n + 10) (n - 9) = 0$ , so $n = 9$ ( $n = - 10$ is rejected because $n$ must be a natural number).
So there are 450 saplings in total till the 9th spiral.
½ mark

Answer: (i) 650 cm; (ii) n = 10; (iii)(a) 660 saplings; (iii)(b) 9th spiral

Where the marks are

(i) Finds $a_{13} = 650$ cm using $a = 50$ , $d = 50$ (1 mark)
(ii) Solves $50 + (n - 1) 50 = 500$ to get $n = 10$ (1 mark)
(iii)(a) Writes $S_{11} = \frac{11}{2} [20 + 10 \times 10]$ (1½ marks)
(iii)(a) Gets 660 saplings (½ mark)
(iii)(b) Forms $\frac{n}{2} [20 + (n - 1) 10] = 450$ (1 mark)
(iii)(b) Reduces it to $n^{2} + n - 90 = 0$ (½ mark)
(iii)(b) Gets $n = 9$ (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 paper2025 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 2025: question 37

Hints

Step-by-step solution

Revise the chapter

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