JEE Advanced · Mathematics · 5. Sequences & Series

Column I Column II
(A) In \(R ^2\), if the magnitude of the projection vector of the vector \(\alpha \hat{ i }+\beta \hat{ j }\) on \(\sqrt{3} \hat{ i }+\hat{ j }\) is \(\sqrt{3}\) and if \(\alpha=2+\sqrt{3} \beta\), then possible value (s) of \(|\alpha|\) is (are) (P) 1
(B) Let a and b be real numbers such that the function \(f ( x )=\left\{\begin{array}{cc}-3 ax ^2-2, & x <1 \\ bx + a ^2, & x \geq 1\end{array}\right.\) is differentiable for all \(x \in R\). Then possible value ( s ) of a is (are) (Q) 2
(C)Let \(\omega \neq 1\) be a complex cube roots of unity. If \(\left(3-3 \omega+2 \omega^2\right)^{4 n+3}\) \(+~\left(2+3 \omega-3 \omega^2\right)^{4 n+3}\) \(~+\left(-3+2 \omega+3 \omega^2\right)^{4 n+3}=0\) 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

A a-t;b-t;c-q;d-s;
B a-s,t;b-r,s,t;c-q,r,s;d-t;
C a-p,q;b-p,q;c-p,q,s,t;d-q,t;
D a-r,s;b-s,t;c-s,t;d-q,r,s,t;

Verified Solution

Answer & Solution

Correct Answer

(C) a-p,q;b-p,q;c-p,q,s,t;d-q,t;

Step-by-step Solution

Detailed explanation

(A)

|\frac{\sqrt{3} α + β}{2}| = \sqrt{3} \Rightarrow |\sqrt{3} α + β| = 2 \sqrt{3}

...(i)
And, also

α = 2 + \sqrt{3} β

...(ii)
Solving (i) & (ii), we get

\Rightarrow 2 \sqrt{3} + 4 β = \pm 2 \sqrt{3}

\Rightarrow β = 0, - \sqrt{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}

\Rightarrow a^{2} - 3 a + 2 = 0

\Rightarrow a = 1, 2

(C)

Here,

{(3 - 3 ω + 2 ω^{2})}^{4 n + 3} \{1 + ω^{4 n + 3} + {(ω^{2})}^{4 n + 3}\}

\Rightarrow {(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 = \frac{α}{2}

a = 5 - \frac{α}{2}, b = a + \frac{3 α}{2} = 5 + α

Now,

\frac{a b}{a + b} = 2

\Rightarrow \frac{(5 - \frac{α}{2}) (5 + α)}{5 - \frac{α}{2} + 5 + α} = 2

\Rightarrow α = 5, - 2

|α| = 5, 2

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Column I	Column II
(A) In \(R ^2\), if the magnitude of the projection vector of the vector \(\alpha \hat{ i }+\beta \hat{ j }\) on \(\sqrt{3} \hat{ i }+\hat{ j }\) is \(\sqrt{3}\) and if \(\alpha=2+\sqrt{3} \beta\), then possible value (s) of \(\|\alpha\|\) is (are)	(P) 1
(B) Let a and b be real numbers such that the function \(f ( x )=\left\{\begin{array}{cc}-3 ax ^2-2, & x <1 \\ bx + a ^2, & x \geq 1\end{array}\right.\) is differentiable for all \(x \in R\). Then possible value ( s ) of a is (are)	(Q) 2
(C)Let \(\omega \neq 1\) be a complex cube roots of unity. If \(\left(3-3 \omega+2 \omega^2\right)^{4 n+3}\) \(+~\left(2+3 \omega-3 \omega^2\right)^{4 n+3}\) \(~+\left(-3+2 \omega+3 \omega^2\right)^{4 n+3}=0\) 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