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

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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 ω ≠ 1 be a complex cube roots of unity. If 3 - 3 ω + 2 ω 2 4 n + 3 + 2 + 3 ω - 3 ω 2 4 n + 3 + - 3 + 2 ω + 3 ω 2 4 n + 3 = 0 then possible value (s) of n is (are) R 3 ( D ) Let the harmonic mean of two positive real numbers a and b 4. If q is a positive real number such that a, 5, q, b is an arithmetic progression, then the value (s) of |q – a| is (are) S 4 (T) 5

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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 + b = 2 ⇒ 5 - α 2 5 + α 5 - α 2 + 5 + α = 2 ⇒ α = 5 , - 2 α = 5 , 2

Column I	Column II
$(A)$	In $R^{2}$ , if the magnitude of the projection vector of the vector $α \hat{i} + β \hat{j}$ on $\sqrt{3} \hat{i} + \hat{j}$ is $\sqrt{3}$ and if $α = 2 + \sqrt{3} β,$ then possible value (s) of $\| α \|$ is (are)	$(P)$	1
$(B)$	Let a and b be real numbers such that the function $f (x) = \{\begin{matrix} - 3 a x^{2} - 2, & x < 1 \\ b x + a^{2}, & x \geq 1 \end{matrix}$ is differentiable for all $x \in R .$ Then possible value (s) of a is (are)	$(Q)$	2
$(C)$	Let $ω \neq 1$ be a complex cube roots of unity. If ${(3 - 3 ω + 2 ω^{2})}^{4 n + 3} + {(2 + 3 ω - 3 ω^{2})}^{4 n + 3} + {(- 3 + 2 ω + 3 ω^{2})}^{4 n + 3} = 0$ then possible value (s) of n is (are)	$(R)$	3
$(D)$	Let the harmonic mean of two positive real numbers a and b 4. If q is a positive real number such that a, 5, q, b is an arithmetic progression, then the value (s) of \|q – a\| is (are)	$(S)$	4
		$(T)$	5

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