cubic-numbers MCQs for UPSC Prelims

30 practice questions on cubic-numbers from the A Square and A Cube section of the UPSC Prelims syllabus. 30 come with a written explanation. Try the sample set below - the answer stays hidden until you ask for it.

9 Easy 15 Medium 6 Hard

Sample questions

Q1
medium

Assertion (A): The cube of $4$ is $64$. Reason (R): $64$ is also 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: B - Both A and R are true, but R is not the correct explanation of A

A is true since $4 \times 4 \times 4 = 64$. R is also true because $8 \times 8 = 64$. But being a perfect square is a separate fact and does nothing to explain why $4^3$ equals $64$, so R is not the explanation.

Q2
medium

Consider: (i) The cube of an odd number is always odd. (ii) The cube of an even number is always even. Which is true?

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

Correct answer: B - Both (i) and (ii)

An odd number times an odd number stays odd, so $3^3 = 27$ is odd and (i) holds. Any product containing an even factor is even, so $4^3 = 64$ is even and (ii) holds too. Since both are true, the other three choices fail.

Q3
medium

Assertion (A): $5^3$ is greater than $5^2$. Reason (R): The cube of a number equals its square multiplied by the number again, and $5$ is greater than $1$.

  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: A - Both A and R are true, and R is the correct explanation of A

A is true because $125 > 25$. R is true as well, since $5^3 = 5^2 \times 5$, and multiplying by a number bigger than $1$ makes a value larger. That is precisely why the cube beats the square, so R explains A.

Q4
easy

What is $4^3 - 4^2$?

  1. A $12$
  2. B $16$
  3. C $48$
  4. D $80$
Show answer and explanation

Correct answer: C - $48$

Here $4^3 = 64$ and $4^2 = 16$, so the difference is $48$. The value $16$ is just the square, $80$ comes from adding instead of subtracting, and $12$ comes from working out $4 \times 3 - 4 \times 2$.

Q5
medium

How many times as large as $6^2$ is $6^3$?

  1. A $36$ times
  2. B $216$ times
  3. C $3$ times
  4. D $6$ times
Show answer and explanation

Correct answer: D - $6$ times

Here $6^3 = 216$ and $6^2 = 36$, and $216 \div 36 = 6$, so the cube is $6$ times the square. The value $36$ is the square itself and $216$ is the cube itself, neither of which is the ratio. The answer $3$ comes from comparing the small numbers $3$ and $2$ instead of dividing.

Q6
easy

What is the value of $\sqrt[3]{125}$?

  1. A $15$
  2. B $25$
  3. C $75$
  4. D $5$
Show answer and explanation

Correct answer: D - $5$

The cube root asks which number used three times gives $125$, and $5 \times 5 \times 5 = 125$, so the answer is $5$. The value $25$ is the square of $5$, $15$ is $5 \times 3$, and $75$ is $125 \times \frac{3}{5}$; none of them cubes to $125$.

Q7
hard

The edge of a cubical tank is doubled. How many times as much water will the new tank hold?

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

Correct answer: A - $8$ times

Doubling the edge multiplies the volume by $2 \times 2 \times 2 = 8$; a cube of edge $2\ \text{m}$ holds $8\ \text{m}^3$ while one of edge $4\ \text{m}$ holds $64\ \text{m}^3$. Only the edge doubles, so $2$ is wrong. The factor $4$ applies to areas, and $6$ comes from counting the six faces.

Q8
medium

A cubical block of side $6\ \text{cm}$ is cut into small cubes of side $2\ \text{cm}$. How many small cubes are formed?

  1. A $216$
  2. B $3$
  3. C $9$
  4. D $27$
Show answer and explanation

Correct answer: D - $27$

Three small cubes fit along each edge, so the count is $3 \times 3 \times 3 = 27$. Checking with volumes, $216 \div 8 = 27$ as well. The value $3$ counts only along one edge, $9$ counts only one layer, and $216$ is the volume of the big block in cubic centimetres.

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