JEE Maths PYQ Practice, chapter-wise

JEE Advanced 2024 Paper 1Question 321

Let $\gamma\in\mathbb{R}$ be such that the lines $L_1:\frac{x+11}{1}=\frac{y+21}{2}=\frac{z+29}{3}$ and $L_2:\frac{x+16}{3}=\frac{y+11}{2}=\frac{z+4}{\gamma}$ 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. $$\begin{array}{ll}\textbf{List-I} & \textbf{List-II}\\ (P)\ \gamma \text{ equals} & (1)\ -\hat{i}-\hat{j}+\hat{k}\\ (Q)\ \text{A possible choice for } \hat{n} \text{ is} & (2)\ \sqrt{\tfrac{3}{2}}\\ (R)\ \overrightarrow{OR_1} \text{ equals} & (3)\ 1\\ (S)\ \text{A possible value of } \overrightarrow{OR_1}\cdot\hat{n} \text{ is} & (4)\ \tfrac{1}{\sqrt{6}}\hat{i}-\tfrac{2}{\sqrt{6}}\hat{j}+\tfrac{1}{\sqrt{6}}\hat{k}\\ & (5)\ \sqrt{\tfrac{2}{3}}\end{array}$$

3D Geometry

(P)→(3), (Q)→(4), (R)→(1), (S)→(2)(P)→(5), (Q)→(4), (R)→(1), (S)→(2)(P)→(3), (Q)→(4), (R)→(1), (S)→(5)(P)→(3), (Q)→(1), (R)→(4), (S)→(5)

JEE Advanced 2024 Paper 1Question 322

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 $$S=\left\{X\in\mathbb{R}^3: (dist(X,P))^2-(dist(X,Q))^2=50\right\}$$ and $$T=\left\{Y\in\mathbb{R}^3: (dist(Y,Q))^2-(dist(Y,P))^2=50\right\}.$$ Then which of the following statements is (are) TRUE?

3D Geometry

More than one option may be correct.

There is a triangle whose area is 1 and all of whose vertices are from $S$.There are two distinct points $L$ and $M$ in $T$ such that each point on the line segment $LM$ is also in $T$.There are infinitely many rectangles of perimeter 48, two of whose vertices are from $S$ and the other two vertices are from $T$.There is a square of perimeter 48, two of whose vertices are from $S$ and the other two vertices are from $T$.

JEE Main 2023 (Online) 30th January Evening ShiftQuestion 323

If a plane passes through the points $(-1,k,0)$, $(2,k,-1)$, $(1,1,2)$ and is parallel to the line $\frac{x-1}1=\frac{2y+1}2=\frac{z+1}{-1}$, then the value of $\frac{k^{2}+1}{(k-1)(k-2)}$ is

3D Geometry

$\frac{17}5$$\frac5{17}$$\frac6{13}$$\frac{13}6$

JEE Main 2023 (Online) 31st January Morning ShiftQuestion 324

Let the shortest distance between the lines $L:\frac{x-5}{-2}=\frac{y-\lambda}{0}=\frac{z+\lambda}{1}$, $\lambda\ge0$ and $L_1: x+1=y-1=4-z$ be $2\sqrt6$. If $(\alpha,\beta,\gamma)$ lies on $L$, then which of the following is NOT possible?

3D Geometry

$\alpha+2\gamma=24$$2\alpha+\gamma=7$$2\alpha-\gamma=9$$\alpha-2\gamma=19$

JEE Main 2025 (Online) 29th January Evening ShiftQuestion 325

Let a straight line $L$ pass through the point $P(2,-1,3)$ and be perpendicular to the lines $\frac{x-1}2=\frac{y+1}1=\frac{z-3}{-2}$ and $\frac{x-3}1=\frac{y-2}3=\frac{z+2}4$. If the line $L$ intersects the $yz$-plane at the point $Q$, then the distance between the points $P$ and $Q$ is :

3D Geometry

$\sqrt{10}$$2\sqrt3$$2$$3$