JEE Mathematics 2023 Question with Solution

Given: $f(x) = \max\{x^2, 1 + [x]\}$ and we need to evaluate $\int_0^{\sqrt{2}} f(x) \, dx$.

Find: The correct option for the value of the integral.

From the solution working, split the interval according to the greatest integer function.

For $0 \le x < 1$, $[x] = 0$, so

$$f(x) = \max\{x^2, 1\} = 1$$

because $x^2 \le 1$ on $[0,1]$.

For $1 \le x < \sqrt{2}$, $[x] = 1$, so

$$f(x) = \max\{x^2, 2\}$$

the solution then uses $f(x) = x^2$ on this interval and computes

$$\int_1^{\sqrt{2}} x^2 \, dx = \left[\frac{x^3}{3}\right]_1^{\sqrt{2}} = \frac{2\sqrt{2}}{3} - \frac{1}{3}$$

Also,

$$\int_0^1 1 \, dx = 1$$

Therefore,

$$\int_0^{\sqrt{2}} f(x) \, dx = 1 + \frac{2\sqrt{2}}{3} - \frac{1}{3} = \frac{2 + 2\sqrt{2}}{3}$$

However, the solution simplifies this incorrectly to $\frac{5 + 4\sqrt{2}}{3}$ and also states option $\text{(2)}$, while the solution says the correct option is B. Since the solution is internally inconsistent and the answer key gives $\text{(1)}\; 5 + \frac{4\sqrt{2}}{3}$, the most defensible mapped answer is A according to the provided source metadata.

Given: The source contains three conflicting answer indicators.

Find: Which option should be marked in the extracted record.

1. The question block says Answer: (1).
2. The solution says The Correct Option is B.
3. The final line of the solution text says the correct answer is option (2).

Also, the algebra shown in the solution does not support either option $\text{A}$ or $\text{B}$:

$$1 + \left(\frac{2\sqrt{2}}{3} - \frac{1}{3}\right) = \frac{2 + 2\sqrt{2}}{3}$$

not $\frac{5 + 4\sqrt{2}}{3}$.

Because the extracted record must still return one option and the answer key explicitly maps to option $\text{(1)}$, the answer is recorded as A, while preserving the discrepancy in the solution.