CAPF AC 2025 Q57: General Mental Ability – Quantitative Aptitude
- (a) 1
- (b) 3
- (c) 76
- (d) 785
Answer: (c) 76
Explanation
Least integer value of n + √n + ∛n
Given: n is an integer larger than 1, and the expression is n + n^(1/2) + n^(1/3).
(The exponents were lost in the copy of the question; the options 3 and 76 only fit this reading.)
How to think: the sum is an integer only if √n and ∛n are both whole numbers, so n must be a perfect square and a perfect cube, that is a sixth power.
| n | √n | ∛n | Sum | Allowed? |
|---|---|---|---|---|
| 1 = 1⁶ | 1 | 1 | 3 | No, n must be larger than 1 |
| 64 = 2⁶ | 8 | 4 | 64 + 8 + 4 = 76 | Yes, the smallest allowed |
| 729 = 3⁶ | 27 | 9 | 765 | Larger |
The option 3 is the trap: it belongs to n = 1, which the question rules out.
Answer: (c) 76.
Why each option is right or wrong
- (a) Incorrect. 1 is not a possible value of the sum.
- (b) Incorrect. 3 is the sum for n = 1, which the question excludes.
- (c) CORRECT. The smallest allowed n is 64, giving 76.
- (d) Incorrect. 785 is not the sum for n = 64 or any smaller sixth power.
Key point
n + √n + ∛n is an integer only when n is a sixth power. n = 1 gives 3 (excluded). n = 64 gives 76. n = 729 gives 765.
Remember
Square root and cube root both whole: n is a sixth power, so n = 64 gives 76.
Common mistakes
- Choosing 3 by using n = 1 although n must be larger than 1.
- Looking only for a perfect square or only a perfect cube.
- Taking the smallest perfect square above 1, which is 4, without checking the cube root.
Exam tip
Educated guess: 3 and 76 are the only options that look like sums of three terms, and 3 is for n = 1. The question says larger than 1, so 76.
Sources
- NCERT Class IX Mathematics, Chapter 1 (Number Systems: laws of exponents for real numbers)
- NCERT Class VIII Mathematics, Chapter 6 (Squares and Square Roots) and Chapter 7 (Cubes and Cube Roots)