JEE Advanced2014MathematicsDefinite IntegrationActual
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 = sin ⁡ x 2 + cos ⁡ x 2 attains its maximum value, is (Q) 2 (C) ∫ - 2 2 3 x 2 1 + e x d x equals (R) 4 (D) ∫ - 1 2 1 2
Options
- Aa-r;b-p;c-s;d-q;
- Ba-q;b-r;c-p;d-s;
- Ca-p;b-r;c-q;d-s;
- Da-q;b-p;c-s;d-r;
Correct answer
B. a-q;b-r;c-p;d-s;
Step-by-step solution
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.