CAPF AC 2025 Q69: General Mental Ability – Quantitative Aptitude
If a^{2x} = b^{3y} = c^{5z} and b = a²c, then which one of the following is true?
- (a) 2/x + 3/y = 5/z
- (b) 1/x − 3/5z = 1/3y
- (c) 1/x + 1/5z = 1/3y
- (d) 2/x − 5/z = 3/y
Answer: (c) 1/x + 1/5z = 1/3y
Explanation
Equal powers and a relation among a, b, c
Given: a^(2x) = b^(3y) = c^(5z) and b = a² c.
How to think: call the common value k.
Then a = k^(1/2x), b = k^(1/3y) and c = k^(1/5z).
Put these into b = a² c and compare the powers of k.
| Step | Working | Result |
|---|---|---|
| Write a, b, c with powers of k | a = k^(1/2x), b = k^(1/3y), c = k^(1/5z) | Roots of the common value |
| a² | (k^(1/2x))² = k^(1/x) | Square doubles the power |
| b = a² c | k^(1/3y) = k^(1/x) · k^(1/5z) | k^(1/3y) = k^(1/x + 1/5z) |
| Equal bases, equal powers | 1/(3y) = 1/x + 1/(5z) | 1/x + 1/5z = 1/3y |
That is option (c).
Answer: (c) 1/x + 1/5z = 1/3y.
Why each option is right or wrong
- (a) Incorrect. 2/x + 3/y = 5/z does not follow from b = a²c.
- (b) Incorrect. It has a minus sign; the powers of a and c are added, not subtracted.
- (c) CORRECT. The powers of k give 1/x + 1/5z = 1/3y.
- (d) Incorrect. 2/x − 5/z = 3/y is not what the powers of k give.
Key point
Common value k: a = k^(1/2x), b = k^(1/3y), c = k^(1/5z). b = a²c gives 1/(3y) = 2/(2x) + 1/(5z) = 1/x + 1/(5z).
Remember
Set the common value k and add the exponents: 1/3y = 1/x + 1/5z.
Common mistakes
- Taking the square of a as k^(1/2x) instead of k^(1/x).
- Subtracting the powers instead of adding them when multiplying a² and c.
- Misreading b = a²c as b = a raised to the power 2c.
Exam tip
Educated guess: multiplying a² by c adds the exponents of k. Only one option has a plus sign joining 1/x and 1/5z on one side.
Sources
- NCERT Class IX Mathematics, Chapter 1 (Number Systems: laws of exponents)
- NCERT Class VIII Mathematics, Chapter 12 (Exponents and Powers)