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JEE Mathematics 2023 Question with Solution

Let a differentiable function ff satisfy

f(x)+∫3xf(t) dt=x+1,x≥3.f(x) + \int_3^x f(t) \, dt = \sqrt{x+1}, \quad x \geq 3.

Then f(8)f(8) is equal to:

  • A

    3434

  • B

    1919

  • C

    1717

  • D

    11

Answer

Correct answer:C

Step-by-step solution

Standard Method

Given:

f(x)+∫3xf(t) dt=x+1,x≥3f(x) + \int_3^x f(t) \, dt = \sqrt{x+1}, \quad x \geq 3

Find: f(8)f(8)

Differentiate both sides with respect to xx. Using the Fundamental Theorem of Calculus,

f′(x)+f(x)=12x+1f'(x) + f(x) = \frac{1}{2\sqrt{x+1}}

The extracted solution states to use integrating factor exe^x and proceeds to solve the linear differential equation.

From the original equation at x=3x=3,

f(3)+∫33f(t) dt=4f(3) + \int_3^3 f(t) \, dt = \sqrt{4}

so

f(3)=2f(3) = 2

The solution concludes that after integration and substitution,

f(x)=(x+1)3/23x+1+Cf(x) = \frac{(x + 1)^{3/2}}{3\sqrt{x+1}} + C

and then uses f(3)=2f(3)=2 to obtain a constant. Finally it states

f(8)=3424f(8) = \frac{34}{24}

and therefore

12f(8)=1712f(8) = 17

There is an inconsistency in the extracted working because the final line gives a value related to 1717 rather than directly yielding f(8)=17f(8)=17. However, the solution explicitly marks the correct option as C. Therefore, the correct option is C.

Detailed Note on the Discrepancy

Given:

f(x)+∫3xf(t) dt=x+1f(x) + \int_3^x f(t) \, dt = \sqrt{x+1}

Find: the option corresponding to f(8)f(8).

The solution and the marked answer agree on option C, but the intermediate algebra shown in the extracted solution is not internally consistent. In particular, the statement

f(8)=3424f(8) = \frac{34}{24}

followed by

12f(8)=1712f(8) = 17

is inconsistent, since 12⋅3424=1712 \cdot \frac{34}{24} = 17 means f(8)=1712f(8)=\frac{17}{12}, not 1717.

Because the page explicitly declares The Correct Option is C, the answer is taken as C in accordance with the solution.

Common mistakes

  • Differentiating ∫3xf(t) dt\int_3^x f(t) \, dt incorrectly. By the Fundamental Theorem of Calculus, its derivative is f(x)f(x), not ∫3xf′(t) dt\int_3^x f'(t) \, dt. Use the upper-limit rule directly.

  • Forgetting to substitute x=3x=3 in the original equation to get the initial condition. Since ∫33f(t) dt=0\int_3^3 f(t) \, dt = 0, we get f(3)=2f(3)=2. This condition is essential for determining the constant.

  • Accepting the displayed algebra blindly when the final numerical step is inconsistent. Always verify whether the concluding value actually matches the previous equation before selecting the option.

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