JEE Advanced Maths: Logarithms / Sequences questions with solutions

Q1. Let N1 = 5⁷ + 5⁷ + 5⁷ + 5⁷ + 5⁷ (five terms) and N2 = 10³ + 10³ +... + 10³ (ten terms). Find log base 250 of (N1 * N2).

1. 4
2. 8
3. 10
4. 21

Answer: 4

N1 = 5*5⁷ = 5⁸. N2 = 10*10³ = 10⁴ = 2⁴*5⁴. N1*N2 = 5¹² * 2⁴. 250 = 2*5³. log₂₅₀(5¹² * 2⁴). Let x = log₂₅₀(5¹² * 2⁴). 250^x = 5¹² * 2⁴. (2*5³)^x = 5¹² * 2⁴. 2^x * 5^(3x) = 2⁴ * 5¹². So x=4 and 3x=12 => x=4. Both conditions give x=4. Answer: 4. But option 4 exists. Checking: 250⁴ = (2*5³)⁴ = 2⁴ * 5¹². N1*N2 = 5⁸ * 2⁴ * 5⁴ = 2⁴ * 5¹². Yes! So log₂₅₀(N1*N2) = 4.

Q2. How many integral values of x satisfy the equation |log_(sqrt(3))(x - 2)| - |log₃(x - 2)| = 2?

1. 1
2. 2
3. 3
4. 4

Answer: 1

Setting t = log₃(x-2), the equation simplifies to |t| = 2, giving x = 11 (integer) and x = 2 + 1/9 (not integer), so there is exactly 1 integral solution.