1
JEE Advanced 2024 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2

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 $T$.
2
JEE Advanced 2022 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2
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
$$\frac{x}{1}=\frac{y-4}{-2}=\frac{z}{3}$$
3
JEE Advanced 2022 Paper 1 Online
MCQ (More than One Correct Answer)
+4
-2
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
$$3(\alpha+\beta+\gamma)=-121$$
4
JEE Advanced 2020 Paper 2 Offline
MCQ (More than One Correct Answer)
+4
-2
Let $$\alpha$$2 + $$\beta$$2 + $$\gamma$$2 $$\ne$$ 0 and $$\alpha$$ + $$\gamma$$ = 1. Suppose the point (3, 2, $$-$$1) is the mirror image of the point (1, 0, $$-$$1) with respect to the plane $$\alpha$$x + $$\beta$$y + $$\gamma$$z = $$\delta$$. Then which of the following statements is/are TRUE?
A
$$\alpha$$ + $$\beta$$ = 2
B
$$\delta$$ $$-$$ $$\gamma$$ = 3
C
$$\delta$$ + $$\beta$$ = 4
D
$$\alpha$$ + $$\beta$$ + $$\gamma$$ = $$\delta$$
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