1
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
2
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
$$ (P) \rightarrow(5) \quad(Q) \rightarrow(1) \quad(R) \rightarrow(4) \quad(S) \rightarrow(2) $$
3
JEE Advanced 2017 Paper 2 Offline
MCQ (Single Correct Answer)
+3
-1
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
4
JEE Advanced 2017 Paper 2 Offline
MCQ (Single Correct Answer)
+3
-1
The equation of the plane passing through the point (1, 1, 1) and perpendicular to the planes 2x + y $$-$$ 2z = 5 and 3x $$-$$ 6y $$-$$ 2z = 7 is
A
14x + 2y $$-$$ 15z = 1
B
$$-$$14x + 2y + 15z = 3
C
14x $$-$$ 2y + 15z = 27
D
14x + 2y + 15z = 31
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