3D Geometry · JEE Advanced previous year questions

2023 3 questions

1. Q52023 · 4 Jun 2023 · Paper 1

   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.$\frac{1}{\sqrt{12}}$
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   Answer: A

2. Q122023 · 4 Jun 2023 · Paper 1· Numerical answer

   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 ____.

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   Answer: 45

3. Q162023 · 4 Jun 2023 · Paper 1

   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.$(P)\to (5)(Q)\to (1)(R)\to (4)(S)\to (2)$
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   Answer: B