Question

Let the domain of the function $ f(x) = \log_2 \log_4 \log_6 (3 + 4x - x^2) $ be (a, b). If $ \int_0^{a+b} [x^2] dx = p - q\sqrt{r} $, $ p, q, r \in \mathbb{N} $, gcd(p, q, r) = 1, where [.] is the greatest integer function, then p + q + r is equal to

• A. 10
• B. 8
• C. 11
• D. 9

Hint:

Find the domain of the function by solving the inequalities and then evaluate the integral by splitting it into intervals where \( [x^2] \) is constant.

Answer 1

The Correct Option is A

\( \log_6(3 + 4x - x^2)>1 \) 

\( 3 + 4x - x^2>6 \) \( x^2 - 4x + 3<0 \) 

\( (x-1)(x-3)<0 \) \( x \in (1, 3) \) So \( a = 1 \) and \( b = 3 \) 

\( \Rightarrow \int_0^{2} [x^2] dx = ? \) \( I = \int_0^1 [x^2] dx + \int_1^{\sqrt{2}} [x^2] dx + \int_{\sqrt{2}}^{\sqrt{3}} [x^2] dx + \int_{\sqrt{3}}^{2} [x^2] dx \) 

\( = 0 + |x|_1^{\sqrt{2}} + 2|x|_{\sqrt{2}}^{\sqrt{3}} + 3|x|_{\sqrt{3}}^{2} \) \( = (\sqrt{2}-1) + 2(\sqrt{3}-\sqrt{2}) + 3(2-\sqrt{3}) \) \( = 5 - \sqrt{2} - \sqrt{3} \) \( \Rightarrow p + q + r = 10 \)