JEE Mathematics 2025 Question with Solution

Given: $f(x) = \log_2 \log_4 \log_6 (3 + 4x - x^2)$ and the domain is $(a,b)$.

Find: $p+q+r$ from the value of the integral.

For the nested logarithm to be defined, we need

which gives

and hence

So,

which is equivalent to

Now factorizing,

Therefore,

Hence the domain is $(1,3)$, so $a=1$ and $b=3$.

Using the working shown in the solution, the required integral is taken as

This matches the extracted solution steps, although the question text shows $a+b$. The solution concludes using $b-a$ and gives the correct option accordingly.

Now split the interval where $[x^2]$ remains constant:

On these intervals,

• for $0 \le x < 1$, $[x^2]=0$
• for $1 \le x < \sqrt{2}$, $[x^2]=1$
• for $\sqrt{2} \le x < \sqrt{3}$, $[x^2]=2$
• for $\sqrt{3} \le x < 2$, $[x^2]=3$

Therefore,

So,

Simplifying,

Comparing with the form $p - \sqrt{q} - \sqrt{r}$, we get

Thus,

Therefore, the correct option is A.

Given: Domain of $f(x)=\log_2\log_4\log_6(3+4x-x^2)$.

Find: $p+q+r$.

From the nested logarithm condition,

so

Hence $a=1$ and $b=3$.

Using the extracted solution's interpretation, evaluate

Then directly use the values of $[x^2]$ on the standard intervals:

So,

which gives

Therefore $p=5, q=2, r=3$ and hence

So the correct option is A.