JEE Advanced · Mathematics · 30. Vector Algebra

Consider the vectors
$\vec{x}=\hat{i}+2 \hat{j}+3 \hat{k}, \quad \vec{y}=2 \hat{i}+3 \hat{j}+\hat{k}, \quad \text { and } \quad \vec{z}=3 \hat{i}+\hat{j}+2 \hat{k}$
For two distinct positive real numbers $\alpha$ and $\beta$, define
$\vec{X}=\alpha \vec{x}+\beta \vec{y}-\vec{z}, \quad \vec{Y}=\alpha \vec{y}+\beta \vec{z}-\vec{x}, \quad \text { and } \quad \vec{Z}=\alpha \vec{z}+\beta \vec{x}-\vec{y}$
If the vectors $\vec{X}, \vec{Y}$, and $\vec{Z}$ lie in a plane, the value of $\alpha+\beta-3$ is ______ .

A -5
B -2
C -1
D 5

Verified Solution

Answer & Solution

Correct Answer

(B) -2

Step-by-step Solution

Detailed explanation

$\Rightarrow\left|\begin{array}{ccc}\alpha & \beta & -1 \\ -1 & \alpha & \beta \\ \beta & -1 & \alpha\end{array}\right| \underbrace{\left|\begin{array}{ccc}1 & 2 & 3 \\ 2 & 3 & 1 \\ 3 & 1 & 2\end{array}\right|}_{-0}=0$
$\begin{aligned} & \Rightarrow\left(\alpha^3+\beta^3-1\right)-(-\alpha \beta-\alpha \beta-\alpha \beta)=0 \\ & \Rightarrow \alpha^3+\beta^3+3 \alpha \beta=1 \\ & \Rightarrow \alpha^3+\beta^3+(-1)^3=3(\alpha)(\beta)(-1) \\ & \Rightarrow \alpha+\beta-1=0\end{aligned}$
So, $\alpha+\beta-3=-2$

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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From JEE Advanced

Let

ℓ_{1}

and

ℓ_{2}

be the lines

{\vec{r}}_{1} = λ (\hat{i} + \hat{j} + \hat{k})

and

{\vec{r}}_{2} = (\hat{j} - \hat{k}) + μ (\hat{i} + \hat{k})

, respectively, Let

X

be the set of all the planes

H

that contain the line

ℓ_{1}

. For a plane

H

, let

d (H)

denote the smallest possible distance between the points of

ℓ_{2}

and

H

. Let

H_{0}

be a plane in

X

for which

d (H_{0})

is the maximum value of

d (H)

H

varies over all planes in

X

. Match each entry in List-I to the correct entries in List-II.

	List-I		List-II
$(P)$	The value of $d (H_{0})$ is	$(1)$	$\sqrt{3}$
$(Q)$	The distance of the point $(0, 1, 2)$ from $H_{0}$ is	$(2)$	$\frac{1}{\sqrt{3}}$
$(R)$	The distance of origin from $H_{0}$ is	$(3)$	$0$
$(S)$	The distance of origin from the point of intersection of planes $y = z, x = 1$ and $H_{0}$ is	$(4)$	$\sqrt{2}$
		$(5)$	$\frac{1}{\sqrt{2}}$

The correct option is

JEE Advanced 2023 Medium

One mole of a monatomic ideal gas undergoes the cyclic process $\mathrm{J} \rightarrow \mathrm{K} \rightarrow \mathrm{L} \rightarrow \mathrm{M} \rightarrow \mathrm{J}$, as shown in the P-T diagram.

Match the quantities mentioned in List-I with their values in List-II and choose the correct option.
[$\mathcal{R}$ is the gas constant.]

List-I	List-II
(P) Work done in the complete cyclic process	(1) $R T_0-4 R T_0 \ln 2$
(Q) Change in the internal energy of the gas in the process JK	(2) $0$
(R) Heat given to the gas in the process KL	(3) $3 K T_0$
(S) Change in the internal energy of the gas in the process MJ	(4) $-2 R T_0 \ln 2$
	$(5)-3 R T_0 \ln 2$

JEE Advanced 2024 Medium

Paragraph:
The hydrogen-like species $\mathrm{Li}^{2+}$ is in a spherically symmetric state $S_1$ with one radial node. Upon absorbing light the ion undergoes transition to a state $S_2$. The state $S_2$ has one radial node and its energy is equal to the ground state energy of the hydrogen atom.
Question:
Energy of the state $S_1$ in units of the hydrogen atom ground state energy is

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