Maths Q17 – JEE-MAIN 2026

Home Maths Differential EquationsQuestion #174

Let

f

be a real polynomial of degree

n

such that

f (x) = f^{'} (x) f^{″} (x)

, for all

x \in R

. If

f (0) = 0

, then

36 (f^{'} (2) + f^{″} (2) + \int_{0}^{2} f (x) d x)

is equal to:

Clue 1 of 3: Degree Analysis

The problem involves a differential equation where the function is a polynomial. The first step is to determine the degree of this polynomial by comparing the degrees of both sides of the equation.

🧠 Full Solution Path

Step 1: Determine the Polynomial

f (x)

✦ Active

Let $f (x)$ be a polynomial of degree $n$ . By comparing the degrees of both sides of the equation $f (x) = f^{'} (x) f^{″} (x)$ , we get $n = (n - 1) + (n - 2)$ , which simplifies to $n = 3$ .

Let $f (x) = a x^{3} + b x^{2} + c x + d$ . Comparing the leading coefficients in the equation gives $a = (3 a) (6 a) = 18 a^{2}$ , which implies $a = 1 / 18$ (since $a \neq 0$ ).

The condition $f (0) = 0$ implies $d = 0$ . By substituting the general form of $f (x)$ into the differential equation and comparing the remaining coefficients, we find that $b = 0$ and $c = 0$ .

f (x) = \frac{1}{18} x^{3}

Step 2: Calculate the Required Values○ Expand

Now we find the values of the derivatives at $x = 2$ and the definite integral.

f^{'} (x) = \frac{d}{d x} (\frac{1}{18} x^{3}) = \frac{1}{6} x^{2} ⟹ f^{'} (2) = \frac{1}{6} (2^{2}) = \frac{2}{3}

f^{″} (x) = \frac{d}{d x} (\frac{1}{6} x^{2}) = \frac{1}{3} x ⟹ f^{″} (2) = \frac{1}{3} (2) = \frac{2}{3}

\int_{0}^{2} f (x) d x = \int_{0}^{2} \frac{1}{18} x^{3} d x = \frac{1}{18} {[\frac{x^{4}}{4}]}_{0}^{2} = \frac{1}{18} (\frac{16}{4}) = \frac{4}{18} = \frac{2}{9}

Step 3: Compute the Final Expression○ Expand

Substitute the calculated values into the given expression.

36 (f^{'} (2) + f^{″} (2) + \int_{0}^{2} f (x) d x) = 36 (\frac{2}{3} + \frac{2}{3} + \frac{2}{9})

= 36 (\frac{4}{3} + \frac{2}{9}) = 36 (\frac{12 + 2}{9}) = 36 (\frac{14}{9}) = 4 \times 14 = 56

✦ STEM Console utilizes AI models to generate step-by-step explanations and math clues. AI can make mistakes.

Maths Question 17 – JEE-MAIN 2026

Clue 1 of 3: Degree Analysis

🧠 Full Solution Path