3D Geometry · JEE Advanced previous year questions

Let $L_1$ be the line of intersection of the planes given by the equations $$2x+3y+z=4\text{and}x+2y+z=5.$$ Let $L_2$ be the line passing through the point $P(2,-1,3)$ and parallel to $L_1$. Let $M$ denote the plane given by the equation $$2x+y-2z=6.$$ Suppose that the line $L_2$ meets the plane $M$ at the point $Q$. Let $R$ be the foot of the perpendicular drawn from $P$ to the plane $M$. Then which of the following statements is (are) TRUE?

1. A.The length of the line segment $PQ$ is $9\sqrt{3}$
2. B.The length of the line segment $QR$ is 15
3. C.The area of $\Delta PQR$ is $\frac{3}{2}\sqrt{234}$
4. D.The acute angle between the line segments $PQ$ and $PR$ is ${\cos }^{-1}\left(\frac{1}{2\sqrt{3}}\right)$

For any two points $M$ and $N$ in the $XY$-plane, let $\overrightarrow{MN}$ denote the vector from $M$ to $N$, and $\vec{0}$ denote the zero vector. Let $P,Q$ and $R$ be three distinct points in the $XY$-plane. Let $S$ be a point inside the triangle $\Delta PQR$ such that $$\overrightarrow{SP}+5\overrightarrow{SQ}+6\overrightarrow{SR}=\vec{0}.$$ Let $E$ and $F$ be the mid-points of the sides $PR$ and $QR$, respectively. Then the value of $$\frac{\text{length of the line segment }EF}{\text{length of the line segment }ES}$$ is ____.

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

Let $A_1,B_1,C_1$ be three points in the $xy$-plane. Suppose that the lines $A_1{C}_1$ and $B_1{C}_1$ are tangents to the curve $y^2=8x$ at $A_1$ and $B_1$, respectively. If $O=(0,0)$ and $C_1=(-4,0)$, then which of the following statements is (are) TRUE?

1. A.The length of the line segment $O{A}_1$ is $4\sqrt{3}$
2. B.The length of the line segment $A_1{B}_1$ is 16
3. C.The orthocenter of the triangle $A_1{B}_1{C}_1$ is $(0,0)$
4. D.The orthocenter of the triangle $A_1{B}_1{C}_1$ is $(1,0)$

Let $Q$ be the cube with the set of vertices $\left\{(x_1,x_2,x_3)\in {\mathbb{R}}^3:x_1,x_2,x_3\in \{0,1\}\right\}$. 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

1. A.$\frac{1}{\sqrt{6}}$
2. B.$\frac{1}{\sqrt{8}}$
3. C.$\frac{1}{\sqrt{3}}$
4. D.$\frac{1}{\sqrt{12}}$

Let $P$ be the plane $\sqrt{3}x+2y+3z=16$ and let $$S=\left\{\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}:{\alpha }^2+{\beta }^2+{\gamma }^2=1\text{ and the distance of }(\alpha ,\beta ,\gamma )\text{ from the plane }P\text{ is }\frac{7}{2}\right\}.$$ 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 $\frac{80}{\sqrt{3}}V$ 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. $$\begin{array}{ll}\text{List-I}& \text{List-II}\\ \text{(P) The value of }d(H_0)\text{ is}& \text{(1) }\sqrt{3}\\ \text{(Q) The distance of the point }(0,1,2)\text{ from }{H}_0\text{ is}& \text{(2) }\frac{1}{\sqrt{3}}\\ \text{(R) The distance of origin from }{H}_0\text{ is}& \text{(3) }0\\ \text{(S) The distance of origin from the point of intersection of planes }y=z,x=1\text{ and }{H}_0\text{ is}& \text{(4) }\sqrt{2}\\ & \text{(5) }\frac{1}{\sqrt{2}}\end{array}$$ The correct option is:

1. A.$(P)\to (2)(Q)\to (4)(R)\to (5)(S)\to (1)$
2. B.$(P)\to (5)(Q)\to (4)(R)\to (3)(S)\to (1)$
3. C.$(P)\to (2)(Q)\to (1)(R)\to (3)(S)\to (2)$
4. D.$(P)\to (5)(Q)\to (1)(R)\to (4)(S)\to (2)$