1
JEE Advanced 2024 Paper 2 Online
MCQ (More than One Correct Answer)
+4
-2
Change Language
A straight line drawn from the point $P(1,3,2)$, parallel to the line $\frac{x-2}{1}=\frac{y-4}{2}=\frac{z-6}{1}$, intersects the plane $L_1: x-y+3 z=6$ at the point $Q$. Another straight line which passes through $Q$ and is perpendicular to the plane $L_1$ intersects the plane $L_2: 2 x-y+z=-4$ at the point $R$. Then which of the following statements is (are) TRUE?
A
The length of the line segment $P Q$ is $\sqrt{6}$
B
The coordinates of $R$ are $(1,6,3)$
C
The centroid of the triangle $P Q R$ is $\left(\frac{4}{3}, \frac{14}{3}, \frac{5}{3}\right)$
D
The perimeter of the triangle $P Q R$ is $\sqrt{2}+\sqrt{6}+\sqrt{11}$
2
JEE Advanced 2024 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2
Change Language

Let $\mathbb{R}^3$ denote the three-dimensional space. Take two points $P=(1,2,3)$ and $Q=(4,2,7)$. Let $\operatorname{dist}(X, Y)$ denote the distance between two points $X$ and $Y$ in $\mathbb{R}^3$. Let

$$ \begin{gathered} S=\left\{X \in \mathbb{R}^3:(\operatorname{dist}(X, P))^2-(\operatorname{dist}(X, Q))^2=50\right\} \text { and } \\ T=\left\{Y \in \mathbb{R}^3:(\operatorname{dist}(Y, Q))^2-(\operatorname{dist}(Y, P))^2=50\right\} . \end{gathered} $$

Then which of the following statements is (are) TRUE?

A
There is a triangle whose area is 1 and all of whose vertices are from $S$.
B
There are two distinct points $L$ and $M$ in $T$ such that each point on the line segment $L M$ is also in $T$.
C
There are infinitely many rectangles of perimeter 48 , two of whose vertices are from $S$ and the other two vertices are from $T$.
D
There is a square of perimeter 48 , two of whose vertices are from $S$ and the other two vertices are from $T$.
3
JEE Advanced 2022 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2
Change Language
Let $$P_{1}$$ and $$P_{2}$$ be two planes given by

$$ \begin{aligned} &P_{1}: 10 x+15 y+12 z-60=0 \\\\ &P_{2}:-2 x+5 y+4 z-20=0 \end{aligned} $$

Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on $$P_{1}$$ and $$P_{2}$$ ?
A
$$\frac{x-1}{0}=\frac{y-1}{0}=\frac{z-1}{5}$$
B
$$\frac{x-6}{-5}=\frac{y}{2}=\frac{z}{3}$$
C
$$\frac{x}{-2}=\frac{y-4}{5}=\frac{z}{4}$$
D
$$\frac{x}{1}=\frac{y-4}{-2}=\frac{z}{3}$$
4
JEE Advanced 2022 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2
Change Language
Let $$S$$ be the reflection of a point $$Q$$ with respect to the plane given by

$$ \vec{r}=-(t+p) \hat{\imath}+t \hat{\jmath}+(1+p) \hat{k} $$

where $$t, p$$ are real parameters and $$\hat{\imath}, \hat{\jmath}, \hat{k}$$ are the unit vectors along the three positive coordinate axes. If the position vectors of $$Q$$ and $$S$$ are $$10 \hat{\imath}+15 \hat{\jmath}+20 \hat{k}$$ and $$\alpha \hat{\imath}+\beta \hat{\jmath}+\gamma \hat{k}$$ respectively, then which of the following is/are TRUE ?
A
$$3(\alpha+\beta)=-101$$
B
$$3(\beta+\gamma)=-71$$
C
$$3(\gamma+\alpha)=-86$$
D
$$3(\alpha+\beta+\gamma)=-121$$
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