square-numbers MCQs for UPSC Prelims

35 practice questions on square-numbers from the A Square and A Cube section of the UPSC Prelims syllabus. 35 come with a written explanation. Try the sample set below - the answer stays hidden until you ask for it.

10 Easy 18 Medium 7 Hard

Sample questions

Q1
medium

Assertion (A): $3600$ is a perfect square. Reason (R): A number ending in two zeros is always a perfect square.

  1. A Both A and R are true, and R is the correct explanation of A
  2. B Both A and R are true, but R is not the correct explanation of A
  3. C A is true, but R is false
  4. D A is false, but R is true
Show answer and explanation

Correct answer: C - A is true, but R is false

A is true because $60 \times 60 = 3600$. R is false: $200$ ends in two zeros but is not a square, since $14^2 = 196$ and $15^2 = 225$. So a true A sits with a false R, which rules out the first two choices and the last one.

Q2
easy

What is $45^2$?

  1. A $1625$
  2. B $2020$
  3. C $2025$
  4. D $1620$
Show answer and explanation

Correct answer: C - $2025$

Multiply the tens digit by the next whole number, $4 \times 5 = 20$, then write $25$ after it to get $2025$. Using $4 \times 4 = 16$ gives the wrong $1625$. Every square of a number ending in $5$ must end in $25$, so $2020$ and $1620$ cannot be right.

Q3
easy

The sum of the first $n$ odd numbers is always equal to which of these?

  1. A $2n$
  2. B $n^2$
  3. C $n^3$
  4. D $n + 1$
Show answer and explanation

Correct answer: B - $n^2$

Adding odd numbers from $1$ builds a square: $1 + 3 = 4$, $1 + 3 + 5 = 9$, so the sum of the first $n$ of them is $n^2$. Doubling gives $2n$, which fails already at $n = 3$ since $2 \times 3 = 6$, not $9$. Neither $n^3$ nor $n + 1$ matches these values.

Q4
medium

Consider: (i) Any number ending in $6$ is a perfect square. (ii) A perfect square can end in $6$. Which is true?

  1. A Neither (i) nor (ii)
  2. B Only (i)
  3. C Both (i) and (ii)
  4. D Only (ii)
Show answer and explanation

Correct answer: D - Only (ii)

Squares such as $16$ and $36$ end in $6$, so (ii) is true. But $26$ ends in $6$ and is not a square, so (i) fails. Ending in $6$ is allowed for a square, not a guarantee of one.

Q5
hard

Priya writes numbers that end in the digit $3$, such as $53$ and $483$. She claims none of them can ever be a perfect square. Is she right?

  1. A Yes, because no perfect square ends in $3$
  2. B No, because $9$ is a perfect square and $9 = 3^2$
  3. C Yes, because all such numbers are odd
  4. D No, because $1369$ is a perfect square
Show answer and explanation

Correct answer: A - Yes, because no perfect square ends in $3$

The units digit of a square can only be $0, 1, 4, 5, 6$ or $9$, so a number ending in $3$ is never a square and Priya is right. The number $9$ is a square but it does not end in $3$. Many odd numbers such as $49$ are squares, so oddness is not the reason. And $1369 = 37^2$ ends in $9$, not $3$.

Q6
easy

Which sentence describes a square number correctly?

  1. A A number formed by doubling a whole number
  2. B A number that has exactly four factors
  3. C A number formed by multiplying a whole number by itself
  4. D A number formed by multiplying a whole number by $4$
Show answer and explanation

Correct answer: C - A number formed by multiplying a whole number by itself

A square number is $n \times n$, such as $6 \times 6 = 36$. Doubling gives $2n$, so $12$ would count, but $12$ is not a square. Multiplying by $4$ gives numbers like $20$, which is not a square either. The count of factors is unrelated: $36$ has nine factors.

Q7
medium

The square of a whole number ending in $4$ must end in which digit?

  1. A $4$
  2. B $6$
  3. C $8$
  4. D $2$
Show answer and explanation

Correct answer: B - $6$

The units digit of the square comes from $4 \times 4 = 16$, so it is $6$, as in $14^2 = 196$ and $24^2 = 576$. Keeping the digit $4$ or doubling it to $8$ ignores the carry, and no square ends in $2$.

Q8
medium

Read the two statements. (i) $15^2$ means $15 \times 15$. (ii) $15^2$ means $15 + 15$. Which is correct?

  1. A Only (ii)
  2. B Both (i) and (ii)
  3. C Neither (i) nor (ii)
  4. D Only (i)
Show answer and explanation

Correct answer: D - Only (i)

A small $2$ written above means the number is used twice as a factor, so $15^2 = 15 \times 15 = 225$. Statement (ii) describes doubling, which gives $30$, so it is wrong. Hence only (i) holds and the last two choices fail.

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