Three-Dimensional Geometry

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

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

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

Let $\overrightarrow{OP} = \dfrac{\alpha-1}{\alpha}\hat{i} + \hat{j} + \hat{k}$, $\overrightarrow{OQ} = \hat{i} + \dfrac{\beta-1}{\beta}\hat{j} + \hat{k}$ and $\overrightarrow{OR} = \hat{i} + \hat{j} + \dfrac{1}{2}\hat{k}$ be three vectors, where $\alpha, \beta \in \mathbb{R} - \{0\}$ and $O$ denotes the origin. If $\left(\overrightarrow{OP} \times \overrightarrow{OQ}\right) \cdot \overrightarrow{OR} = 0$ and the point $(\alpha, \beta, 2)$ lies on the plane $3x + 3y - z + l = 0$, then the value of $l$ is _____.

Let $\gamma \in \mathbb{R}$ be such that the lines $L_1 : \dfrac{x+11}{1} = \dfrac{y+21}{2} = \dfrac{z+29}{3}$ and $L_2 : \dfrac{x+16}{3} = \dfrac{y+11}{2} = \dfrac{z+4}{\gamma}$ 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.

The correct option is

A straight line drawn from the point $P(1,3,2)$, parallel to the line $\dfrac{x-2}{1} = \dfrac{y-4}{2} = \dfrac{z-6}{1}$, intersects the plane $L_1 : x - y + 3z = 6$ at the point $Q$. Another straight line which passes through $Q$ and is perpendicular to the plane $L_1$ intersects the plane $L_2 : 2x - y + z = -4$ at the point $R$. Then which of the following statements is (are) TRUE?

Let $Q$ be the cube with the set of vertices $\{(x_1, x_2, x_3) \in \mathbb{R}^3 : x_1, x_2, x_3 \in \{0, 1\}\}$. Let $F$ be the set of all twelve lines containing the diagonals of the six faces of the cube $Q$. Let $S$ be the set of all four lines containing the main diagonals of the cube $Q$; for instance, the line passing through the vertices $(0,0,0)$ and $(1,1,1)$ is in $S$. For lines $\ell_1$ and $\ell_2$, let $d(\ell_1, \ell_2)$ denote the shortest distance between them. Then the maximum value of $d(\ell_1, \ell_2)$, as $\ell_1$ varies over $F$ and $\ell_2$ varies over $S$, is

Let $P$ be the plane $\sqrt3x + 2y + 3z = 16$ and let

$$S = \left\{\alpha\hat i + \beta\hat j + \gamma\hat k : \alpha^2 + \beta^2 + \gamma^2 = 1 \text{ and the distance of } (\alpha, \beta, \gamma) \text{ from the plane } P \text{ is } \frac72\right\}.$$

Let $\vec u$, $\vec v$ and $\vec w$ be three distinct vectors in $S$ such that $|\vec u - \vec v| = |\vec v - \vec w| = |\vec w - \vec u|$. Let $V$ be the volume of the parallelepiped determined by vectors $\vec u$, $\vec v$ and $\vec w$. Then the value of $\dfrac{80}{\sqrt3}V$ is _____.

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 $d(H_0)$ 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.

The correct option is:

Let $P_1$ and $P_2$ be two planes given by
$$P_1:\ 10x + 15y + 12z - 60 = 0, \qquad P_2:\ -2x + 5y + 4z - 20 = 0.$$
Which of the following straight lines can be an edge of some tetrahedron whose two faces lie on $P_1$ and $P_2$?

Let $S$ be the reflection of a point $Q$ with respect to the plane given by
$$\vec r = -(t+p)\hat i + t\hat j + (1+p)\hat k$$
where $t, p$ are real parameters and $\hat i, \hat j, \hat k$ are the unit vectors along the three positive coordinate axes. If the position vectors of $Q$ and $S$ are $10\hat i + 15\hat j + 20\hat k$ and $\alpha\hat i + \beta\hat j + \gamma\hat k$ respectively, then which of the following is/are TRUE?

Let a line having direction ratios $1,-4,2$ intersect the lines $\dfrac{x-7}3=\dfrac{y-1}{-1}=\dfrac{z+2}1$ and $\dfrac x2=\dfrac{y-7}3=\dfrac z1$ at the points $A$ and $B$. Then $(AB)^2$ is equal to ______.

If the shortest distance between the lines $\vec r=\left(-\hat i+3\hat k\right)+\lambda\left(\hat i-a\hat j\right)$ and $\vec r=\left(-\hat j+2\hat k\right)+\mu\left(\hat i-\hat j+\hat k\right)$ is $\sqrt{\dfrac23}$, then the integral value of $a$ is equal to ______.

If the shortest distance between the lines $\dfrac{x-1}2=\dfrac{y-2}3=\dfrac{z-3}\lambda$ and $\dfrac{x-2}1=\dfrac{y-4}4=\dfrac{z-5}5$ is $\dfrac1{\sqrt3}$, then the sum of all possible values of $\lambda$ is:

Let the points on the plane P be equidistant from the points $(-4,2,1)$ and $(2,-2,3)$. Then the acute angle between the plane P and the plane $2x+y+3z=1$ is

Let Q be the mirror image of the point P(1, 0, 1) with respect to the plane $S:x+y+z=5$. If a line L passing through $(1,-1,-1)$, parallel to the line PQ meets the plane S at R, then $QR^2$ is equal to :

Let the lines
$$L_1:\vec r=\lambda\left(\hat i+2\hat j+3\hat k\right),\ \lambda\in\mathbb R$$
$$L_2:\vec r=\left(\hat i+3\hat j+\hat k\right)+\mu\left(\hat i+\hat j+5\hat k\right);\ \mu\in\mathbb R,$$
intersect at the point S. If a plane $ax+by-z+d=0$ passes through S and is parallel to both the lines $L_1$ and $L_2$, then the value of $a+b+d$ is equal to ______.

Let $P$ be the plane passing through the intersection of the planes $\vec r\cdot\left(\hat i+3\hat j-\hat k\right)=5$ and $\vec r\cdot\left(2\hat i-\hat j+\hat k\right)=3$, and the point $(2,1,-2)$. Let the position vectors of the points $X$ and $Y$ be $\hat i-2\hat j+4\hat k$ and $5\hat i-\hat j+2\hat k$ respectively. Then the points

Let $l_1$ be the line in $xy$-plane with $x$ and $y$ intercepts $\dfrac18$ and $\dfrac1{4\sqrt2}$ respectively, and $l_2$ be the line in $zx$-plane with $x$ and $z$ intercepts $-\dfrac18$ and $-\dfrac1{6\sqrt3}$ respectively. If $d$ is the shortest distance between the line $l_1$ and $l_2$, then $d^{-2}$ is equal to ______.

If the two lines $l_1:\dfrac{x-2}3=\dfrac{y+1}{-2}$, $z=2$ and $l_2:\dfrac{x-1}1=\dfrac{2y+3}\alpha=\dfrac{z+5}2$ are perpendicular, then an angle between the lines $l_2$ and $l_3:\dfrac{1-x}3=\dfrac{2y-1}{-4}=\dfrac z4$ is :

Let the plane $2x+3y+z+20=0$ be rotated through a right angle about its line of intersection with the plane $x-3y+5z=8$. If the mirror image of the point $\left(2,-\dfrac12,2\right)$ in the rotated plane is B$(a,b,c)$, then :

If the plane $2x+y-5z=0$ is rotated about its line of intersection with the plane $3x-y+4z-7=0$ by an angle of $\dfrac\pi2$, then the plane after the rotation passes through the point :