List I List II (P) The number of polynomials f(x) with non‑negative integer coefficients of degree \leq 2, satisfying f(0) = 0 and \int_{0}^{1} f \left(\right. x \left.\right) \textrm{ } dx = 1, is (1) 8 (Q) The number of points in the interval \left[\right. - \sqrt{13} , \textrm{ } \sqrt{13} \left]\right. at which f(x) = sin (x2) + cos (x2) attains its maximum value, is (2) 2 (R) \int_{- 2}^{2} \frac{3 x^{2}}{\left(\right. 1 + e^{x} \left.\right)} \textrm{ } dx equals (3) 4 (S) \frac{\left(\right. \int_{\frac{- 1}{2}}^{\frac{1}{2}} cos 2 x \textrm{ } log \textrm{ } \left(\right. \frac{1 + x}{1 - x} \left.\right) \textrm{ } dx \left.\right)}{\left(\right. \int_{0}^{\frac{1}{2}} cos 2 x \textrm{ } log \textrm{ } \left(\right. \frac{1 + x}{1 - x} \left.\right) \textrm{ } dx \left.\right)} equals (4) 0

Engineering

Mathematics

Properties of Definite Integral

Question

List I	List II
(P) The number of polynomials f(x) with non‑negative integer coefficients of degree $\leq$ 2, satisfying f(0) = 0 and $\int_{0}^{1} f (x) dx$ = 1, is	(1) 8
(Q) The number of points in the interval $[- \sqrt{13}, \sqrt{13}]$ at which f(x) = sin (x²) + cos (x²) attains its maximum value, is	(2) 2
(R) $\int_{- 2}^{2} \frac{3 x^{2}}{(1 + e^{x})} dx$ equals	(3) 4
(S) $\frac{(\int_{\frac{- 1}{2}}^{\frac{1}{2}} \cos 2 x \log (\frac{1 + x}{1 - x}) dx)}{(\int_{0}^{\frac{1}{2}} \cos 2 x \log (\frac{1 + x}{1 - x}) dx)}$ equals	(4) 0

P → 3

Q → 2

R → 4

S → 1

P → 2

Q → 3

R → 1

S → 4

P → 2

Q → 3

R → 4

S → 1

P → 3

Q → 2

R → 1

S → 4

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P. f (x) = ax² + bx

Now $\int_{0}^{1} f (x) dx = 1 \Rightarrow$ 2a + 3b = b

$∴$ a = 0 ⇒ b = 2

or b = 0 ⇒ a = 3

So, f (x) = 2x ; f (x) = 3x²

^{$R . I . = \int_{- 2}^{2} \frac{3 x^{2}}{(1 + e^{x})} dx = \int_{0}^{2} (\frac{3 x^{2}}{1 + e^{x}} + \frac{3 x^{2}}{1 + e^{- x}}) dx = \int_{0}^{2} 3 x^{2} dx = 8$}