Column I Column II ( A ) In R 2 , if the magnitude of the projection vector of the vector α i ^ + β j ^ on 3 i ^ + j ^ is 3 and if α = 2 + 3 β , then possible value (s) of | α | is (are) P 1 ( B ) Let a and b be real numbers such that the function f x = - 3 a x 2 - 2 , x < 1 b x + a 2 , x ≥ 1 is differentiable for all x ∈ R . Then possible value (s) of a is (are) Q 2 ( C ) Let

Step-by-step solution

A As 3 α + β 2 = 3 ⇒ 3 α + β = 2 3 ...(i) And, also α = 2 + 3 β ...(ii) Solving (i) & (ii), we get ⇒ 2 3 + 4 β = ± 2 3 ⇒ β = 0 , - 3 So, α = 2 and - 1 i.e, α = 2 , 1 ( B ) Here - 3 a - 2 = b + a 2 ...(i) And - 6 a = b ...(ii) Solving (i) and (ii), - 3 a - 2 = - 6 a + a 2 ⇒ a 2 - 3 a + 2 = 0 ⇒ a = 1 , 2 C Here, 3 - 3 ω + 2 ω 2 4 n + 3 1 + ω 4 n + 3 + ω 2 4 n + 3 ⇒ 3 - 3 ω + 2 ω 2 4 n + 3 1 + ω 4 n + ω 8 n = 0 So, n = 1 , 2 , 4 , 5 D q - a = 2 d Let q - q = α , d = α 2 a = 5 - α 2 , b = a + 3 α 2 = 5 + α Now, a b a +