1
JEE Advanced 2024 Paper 1 Online
MCQ (Single Correct Answer)
+3
-1
Change Language

Let $\gamma \in \mathbb{R}$ be such that the lines $L_1: \frac{x+11}{1}=\frac{y+21}{2}=\frac{z+29}{3}$ and $L_2: \frac{x+16}{3}=\frac{y+11}{2}=\frac{z+4}{\gamma}$ intersect. Let $R_1$ be the point of intersection of $L_1$ and $L_2$. Let $O=(0,0,0)$, and $\hat{n}$ denote a unit normal vector to the plane containing both the lines $L_1$ and $L_2$.

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

List-I List-II
(P) $\gamma$ equals (1) $-\hat{i} - \hat{j} + \hat{k}$
(Q) A possible choice for $\hat{n}$ is (2) $\sqrt{\frac{3}{2}}$
(R) $\overrightarrow{OR_1}$ equals (3) $1$
(S) A possible value of $\overrightarrow{OR_1} \cdot \hat{n}$ is (4) $\frac{1}{\sqrt{6}} \hat{i} - \frac{2}{\sqrt{6}} \hat{j} + \frac{1}{\sqrt{6}} \hat{k}$
(5) $\sqrt{\frac{2}{3}}$

The correct option is :
A
$(\mathrm{P}) \rightarrow(3) \quad(\mathrm{Q}) \rightarrow(4) \quad(\mathrm{R}) \rightarrow(1) \quad(\mathrm{S}) \rightarrow(2)$
B
$(\mathrm{P}) \rightarrow(5) \quad(\mathrm{Q}) \rightarrow(4) \quad(\mathrm{R}) \rightarrow(1) \quad(\mathrm{S}) \rightarrow(2)$
C
$(\mathrm{P}) \rightarrow(3) \quad$ (Q) $\rightarrow(4) \quad(\mathrm{R}) \rightarrow(1) \quad$ (S) $\rightarrow(5)$
D
$(\mathrm{P}) \rightarrow(3) \quad(\mathrm{Q}) \rightarrow(1) \quad(\mathrm{R}) \rightarrow(4) \quad$ (S) $\rightarrow(5)$
2
JEE Advanced 2023 Paper 2 Online
MCQ (Single Correct Answer)
+3
-1
Change Language
Let the position vectors of the points $P, Q, R$ and $S$ be $\vec{a}=\hat{i}+2 \hat{j}-5 \hat{k}, \vec{b}=3 \hat{i}+6 \hat{j}+3 \hat{k}$, $\vec{c}=\frac{17}{5} \hat{i}+\frac{16}{5} \hat{j}+7 \hat{k}$ and $\vec{d}=2 \hat{i}+\hat{j}+\hat{k}$, respectively. Then which of the following statements is true?
A
The points $P, Q, R$ and $S$ are NOT coplanar
B
$\frac{\vec{b}+2 \vec{d}}{3}$ is the position vector of a point which divides $P R$ internally in the ratio $5: 4$
C
$\frac{\vec{b}+2 \vec{d}}{3}$ is the position vector of a point which divides $P R$ externally in the ratio $5: 4$
D
The square of the magnitude of the vector $\vec{b} \times \vec{d}$ is 95
3
JEE Advanced 2023 Paper 1 Online
MCQ (Single Correct Answer)
+3
-1
Change Language
Let $\ell_1$ and $\ell_2$ be the lines $\vec{r}_1=\lambda(\hat{i}+\hat{j}+\hat{k})$ and $\vec{r}_2=(\hat{j}-\hat{k})+\mu(\hat{i}+\hat{k})$, respectively. Let $X$ be the set of all the planes $H$ that contain the line $\ell_1$. For a plane $H$, let $d(H)$ denote the smallest possible distance between the points of $\ell_2$ and $H$. Let $H_0$ be a plane in $X$ for which $d\left(H_0\right)$ is the maximum value of $d(H)$ as $H$ 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 $d\left(H_0\right)$ 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:
A
$$ (P) \rightarrow(2) \quad(Q) \rightarrow(4) \quad(R) \rightarrow(5) \quad(S) \rightarrow(1) $$
B
$$ (P) \rightarrow(5) \quad(Q) \rightarrow(4) \quad(R) \rightarrow(3) \quad(S) \rightarrow(1) $$
C
$$ (P) \rightarrow(2) \quad(Q) \rightarrow(1) \quad(R) \rightarrow(3) \quad(S) \rightarrow(2) $$
D
$$ (P) \rightarrow(5) \quad(Q) \rightarrow(1) \quad(R) \rightarrow(4) \quad(S) \rightarrow(2) $$
4
JEE Advanced 2017 Paper 2 Offline
MCQ (Single Correct Answer)
+3
-1
Change Language
Let O be the origin and let PQR be an arbitrary triangle. The point S is such that

$$\overrightarrow{OP}$$ . $$\overrightarrow{OQ}$$ + $$\overrightarrow{OR}$$ . $$\overrightarrow{OS}$$ = $$\overrightarrow{OR}$$ . $$\overrightarrow{OP}$$ + $$\overrightarrow{OQ}$$ . $$\overrightarrow{OS}$$ = $$\overrightarrow{OQ}$$ . $$\overrightarrow{OR}$$ + $$\overrightarrow{OP}$$ . $$\overrightarrow{OS}$$

Then the triangle PQR has S as its
A
centroid
B
orthocentre
C
incentre
D
circumcentre
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