While preparing for a competitive examination, Akbar came across a match-stick pattern based question. The pattern is given below: Fig. (1), Fig. (2), Fig. (3), ... Based on the above information, answer the following questions: (i) Write first term and common difference of the A.P. formed by number of squares in each figure. (ii) Write first term and common difference of the A.P. formed by number of sticks used in each figure. (iii) (a) How many squares are there in Fig. (10)? Also, write the number of sticks used in Fig. (10). (iii) (b) If 88 sticks are used to make m^th figure (Fig. (m)), find the value of m. How many squares are formed in this figure?

MathsCBSE Class 10 · Basic · Board exam 2024, set 430/3/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Basic 2024: question 36

Set in the CBSE Class 10 board exam of 2024, paper 430/3/1, 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

While preparing for a competitive examination, Akbar came across a match-stick pattern based question. The pattern is given below: Fig. (1), Fig. (2), Fig. (3), ...

Based on the above information, answer the following questions:

(i)
Write first term and common difference of the A.P. formed by number of squares in each figure.
1 mark
(ii)
Write first term and common difference of the A.P. formed by number of sticks used in each figure.
1 mark
(iii)
(a) How many squares are there in Fig. (10)? Also, write the number of sticks used in Fig. (10).
2 marks
OR
(iii)
(b) If $88$ sticks are used to make $m^{th}$ figure (Fig. (m)), find the value of $m$ . How many squares are formed in this figure?
2 marks

Figure in the paper: Match-stick figures of squares. Fig. (1) is a single square. Fig. (2) is a row of three squares in a cross-like arrangement with one square above and one below the middle square. Fig. (3) is a larger cross-shaped arrangement of squares, with a row of squares horizontally and a column vertically, extended with more squares. A dotted ellipsis follows Fig. (3). See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 430/3/1.

Hints

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

Hint 1

Count the squares and the sticks in Figs. (1), (2) and (3), and see how much each figure grows by.

Hint 2

Each count forms an A.P.; use $a_{n} = a + (n - 1) d$ .

Hint 3

For (iii)(b), put $a_{m} = 88$ in the sticks A.P. to find $m$ , then use that $m$ in the squares A.P.

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) Count the squares in each figure: Fig. (1) has 1, Fig. (2) has 5, Fig. (3) has 9. Each new figure adds 4 squares.
So the A.P. is $1, 5, 9, \dots$ with first term $a = 1$ and common difference $d = 4$ .
1 mark
(ii) Count the sticks: Fig. (1) uses 4, Fig. (2) uses 16, Fig. (3) uses 28. Each new figure adds 12 sticks.
So the A.P. is $4, 16, 28, \dots$ with first term $a = 4$ and common difference $d = 12$ .
1 mark
(iii)(a) Use $a_{n} = a + (n - 1) d$ .
Squares in Fig. (10): $a_{10} = 1 + 9 \times 4 = 37$
1 mark
(iii)(a) Sticks in Fig. (10): $a_{10} = 4 + 9 \times 12 = 112$
1 mark
(iii)(b) For the sticks: $88 = 4 + (m - 1) \times 12$
1 mark
(iii)(b) $(m - 1) \times 12 = 84 \Rightarrow m - 1 = 7 \Rightarrow m = 8$
½ mark
(iii)(b) Squares in Fig. (8): $1 + 7 \times 4 = 29$
½ mark

Answer: (i) a = 1, d = 4; (ii) a = 4, d = 12; (iii)(a) 37 squares, 112 sticks; (iii)(b) m = 8, 29 squares

Where the marks are

(i) First term 1 and common difference 4 (1 mark)
(ii) First term 4 and common difference 12 (1 mark)
(iii)(a) 37 squares in Fig. (10) (1 mark)
(iii)(a) 112 sticks in Fig. (10) (1 mark)
(iii)(b) Sets up $88 = 4 + (m - 1) 12$ (1 mark)
(iii)(b) Gets $m = 8$ (½ mark)
(iii)(b) Gets 29 squares (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 paper2024 board paper, set 430/3/1 (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 2024: question 36

Hints

Step-by-step solution

Revise the chapter

More questions from Arithmetic Progressions