If a ⊕ b is defined as aᵇ + bᵃ, then consider :
- I 2 ⊕ x = 100
- II 4 ⊕ x = 145
- III 3 ⊕ x = 145
- IV 6 ⊕ x = 100
For which of the above, is x smallest ?
I
II
III
IV
Answer is Right!
Answer is Wrong!
This question was previously asked in
UPSC CAPF – 2009
I: 2 ⊕ x = 100 => 2ˣ + x² = 100.
Testing integer values for x:
If x=1, 2¹ + 1² = 2 + 1 = 3
If x=2, 2² + 2² = 4 + 4 = 8
If x=3, 2³ + 3² = 8 + 9 = 17
If x=4, 2⁴ + 4² = 16 + 16 = 32
If x=5, 2⁵ + 5² = 32 + 25 = 57
If x=6, 2⁶ + 6² = 64 + 36 = 100. So, x = 6 for I.
II: 4 ⊕ x = 145 => 4ˣ + x⁴ = 145.
Testing integer values for x:
If x=1, 4¹ + 1⁴ = 4 + 1 = 5
If x=2, 4² + 2⁴ = 16 + 16 = 32
If x=3, 4³ + 3⁴ = 64 + 81 = 145. So, x = 3 for II.
III: 3 ⊕ x = 145 => 3ˣ + x³ = 145.
Testing integer values for x:
If x=1, 3¹ + 1³ = 3 + 1 = 4
If x=2, 3² + 2³ = 9 + 8 = 17
If x=3, 3³ + 3³ = 27 + 27 = 54
If x=4, 3⁴ + 4³ = 81 + 64 = 145. So, x = 4 for III.
IV: 6 ⊕ x = 100 => 6ˣ + x⁶ = 100.
Testing integer values for x:
If x=1, 6¹ + 1⁶ = 6 + 1 = 7
If x=2, 6² + 2⁶ = 36 + 64 = 100. So, x = 2 for IV.
The values of x are 6 (for I), 3 (for II), 4 (for III), and 2 (for IV). The smallest value of x is 2, which occurs in case IV.