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NCERT Solutions Ganita Prakash (Part 1) Chapter 1 In-text Questions — Square Numbers

Book page 3 Updated on2026-09-05

Q1.
Write the locker numbers that remain open.
Answer

The lockers with square numbers between 1 and 100:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100

That is 10 lockers, namely 12, 22, 32, …, 102. The next square, 112 = 121, is beyond 100.

Tip: The count of open lockers up to 100 is 10 because 102 = 100 — in general the number of squares from 1 to N is the whole-number part of √N.
Q2.
Which are these five lockers?
Answer

The passcode lockers are the ones touched exactly twice, so their numbers have exactly two factors — the prime numbers.

2 → 1, 2
3 → 1, 3
5 → 1, 5
7 → 1, 7
11 → 1, 11

The first five such lockers are 2, 3, 5, 7 and 11, so the code is 2-3-5-7-11.

Why it happens: A locker toggled exactly twice is opened by person 1 and closed by exactly one other person. That means the number has 1 and itself as factors and nothing else — the definition of a prime. Note that 1 itself is touched only once, so it is not on the list.
Q3.
Can we have a square of sidelength 3/5 or 2.5 units?
Answer

Yes. A side length need not be a whole number; squaring works for fractions and decimals in exactly the same way.

(3/5)2 = (3/5) × (3/5) = 9/25 sq units
(2.5)2 = 2.5 × 2.5 = 6.25 sq units
Why it happens: The area of a square is side × side whatever the side is. Squaring a fraction squares the numerator and the denominator separately, because (a/b) × (a/b) = (a×a)/(b×b).
Tip: 9/25 and 6.25 are squares of fractions, but they are not perfect squares — that name is kept for squares of natural numbers.
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