Match the following. List – I List – II (A) The number of polynomials f(x) with non – negative integer coefficients of degree ≤ 2 , satisfying f(0) =0 and ∫ 0 1 f x d x = 1 ,   i s (P) 8 (B) The number of point in the interval -   13 ,   13 at which f x…

Question

Match the following. List &ndash; I List &ndash; II (A) The number of polynomials f(x) with non &ndash; negative integer coefficients of degree &#8804; 2 , satisfying f(0) =0 and &#8747; 0 1 f x d x = 1 , &#160; i s (P) 8 (B) The number of point in the interval - &#160; 13 , &#160; 13 at which f x = sin &#8289; x 2 + cos &#8289; x 2 attains its maximum value, is (Q) 2 (C) &#8747; - 2 2 3 x 2 1 + e x d x &#160; equals (R) 4 (D) &#8747; - 1 2 1 2 cos &#8289; 2 x &#160; . log &#8289; 1 + x 1 - x d x &#8747; 0 1 2 cos &#8289; 2 x &#160; . log &#8289; 1 + x 1 - x d x (S) 0

Aicharya · Accepted Answer

P f x = a x 2 + b x , ∫ 0 1 f x d x = 1 ⇒ 2 a + 3 b = 6 ⇒ a , b ≡ 0 , 2 a n d 3 , 0 Q f x = 2 cos ⁡ x 2 - π 4 For maximum value, x 2 - π 4 = 2 n π ⇒ x 2 = 2 n π + π 4 ⇒ x = ± π 4 , ± 9 π 4 as x ϵ - 3 , 13 R ∫ 0 2 3 x 2 1 + e x + 3 x 2 1 + e - x d x = ∫ 0 2 3 x 2 d x = 8 S ∫ - 1 2 1 2 cos ⁡ 2 x I n 1 + x 1 - x d x = 0 as it is an odd function.

Match the following. List – I List – II (A) The number of polynomials f(x) with non –…

Solution

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

Match the following. List &ndash; I List &ndash; II (A) The number of polynomials f(x) with non &ndash;…

Solution

Match the following. List – I List – II (A) The number of polynomials f(x) with non –…