Q JEE Main 2024

The square of the distance of the image of the point $(6,1,5)$ in the line $\frac{x-1}{3}=\frac{y}{2}=\frac{z-2}{4}$, from the origin is

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Q JEE Main 2024

Between the following two statements : Statement-I : Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}}$ and $\vec{b}=2 \hat{\imath}+\hat{\jmath}-\hat{k}$. Then the vector $\overrightarrow{\mathrm{r}}$ satisfying...

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Q JEE Main 2024

Consider the line L passing through the points $(1,2,3)$ and $(2,3,5)$. The distance of the point $\left(\frac{11}{3}, \frac{11}{3}, \frac{19}{3}\right)$ from...

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Q JEE Main 2024

Consider a line L passing through the points $\mathrm{P}(1,2,1)$ and $\mathrm{Q}(2,1,-1)$. If the mirror image of the point $\mathrm{A}(2,2,2)$ in...

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Q JEE Main 2024

Let P the point of intersection of the lines $\frac{x-2}{1}=\frac{y-4}{5}=\frac{z-2}{1}$ and $\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-3}{2}$. Then, the shortest distance of $P$ from the...

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Q JEE MAIN 2024

Let $\vec{a}=\hat{\imath}+\hat{\jmath}+\hat{k}, \vec{b}=-\hat{\imath}-8 \hat{\jmath}+2 \hat{k}$ and $\overrightarrow{\mathrm{c}}=4 \hat{\imath}+\mathrm{c}_2 \hat{\jmath}+\mathrm{c}_3 \hat{\mathrm{k}}$ be three vectors such that $\vec{b} \times \vec{a}=\vec{c} \times \vec{a}$....

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Q JEE Main 2024

If $\lambda>0$, let $\theta$ be the angle between the vectors $\vec{a}=\hat{\imath}+\lambda \hat{\jmath}-3 \hat{k}$ and $\vec{b}=3 \hat{\imath}-\hat{\jmath}+2 \hat{k}$. If the vectors...

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Q Jee Main 2024

Let $O$ be the origin, and $M$ and $N$ be the points on the lines $\frac{x-5}{4}=\frac{y-4}{1}=\frac{z-5}{3}$ and $\frac{x+8}{12}=\frac{y+2}{5}=\frac{z+11}{9}$ respectively such...

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Q Jee Main 2024

Let a unit vector $\hat{\mathrm{u}}=\mathrm{x} \hat{\mathrm{i}}+\mathrm{y} \hat{\mathrm{j}}+\mathrm{z} \hat{\mathrm{k}}$ make angles $\frac{\pi}{2}, \frac{\pi}{3}$ and $\frac{2 \pi}{3}$ with the vectors $...

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Q JEE MAIN 2024

Let a line passing through the point $(-1,2,3)$ intersect the lines $\mathrm{L}_1: \frac{\mathrm{x}-1}{3}=\frac{\mathrm{y}-2}{2}=\frac{\mathrm{z}+1}{-2}$ at $M(\alpha, \beta, \gamma)$ and $L_2: \frac{x+2}{-3}=\frac{y-2}{-2}=\frac{z-1}{4}$...

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Q JEE-Main 2024

The lines $\frac{x-2}{2}=\frac{y}{-2}=\frac{z-7}{16}$ and $\frac{x+3}{4}=\frac{y+2}{3}=\frac{z+2}{1}$ intersect at the point $P$. If the distance of P from the line $\frac{x+1}{2}= \frac{\mathrm{y}-1}{3}=\frac{\mathrm{z}-1}{1}$...

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Q Jee Main 2024

A line passes through $\mathrm{A}(4,-6,-2)$ and $\mathrm{B}(16,-2,4)$. The point $\mathrm{P}(\mathrm{a}, \mathrm{b}, \mathrm{c})$ where $\mathrm{a}, \mathrm{b}, \mathrm{c}$ are non-negative integers, on...

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Q Jee Main 2024

Let $a=3 \hat{\imath}+2 \hat{\jmath}+k^{\wedge}, b^{+}=2 \hat{\imath}-\hat{\jmath}+3 k^{\wedge}$ and $c$ be a vector such that $...

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Q JEE MAIN 2024

If the mirror image of the point $\mathrm{P}(3,4,9)$ in the line $\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-2}{1}$ is $(\alpha, \beta, \gamma)$, then $14(\alpha+\beta+\gamma)$ is :

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Q JEE MAIN 2024

Let $L_1: \vec{r}=(\hat{\imath}-\hat{\jmath}+2 \hat{k})+\lambda(\hat{\imath}-\hat{\jmath}+2 \hat{k}), \lambda \in R \mathrm{~L}_2: \overrightarrow{\mathrm{r}}=(\hat{\jmath}-\hat{\mathrm{k}})+\mu(3 \hat{\imath}+\hat{\jmath}+\mathrm{p} \hat{\mathrm{k}}), \mu \in \mathrm{R}$ and $...

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Q JEE-Main 2024

Let the position vectors of the vertices $A, B$ and $C$ of a triangle be $...

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Q JEE-Main 2024

The position vectors of the vertices $A, B$ and $C$ of a triangle are $...

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Q Jee Main 2024

The shortest distance between lines $\mathrm{L}_1$ and $\mathrm{L}_2$, where $\mathrm{L}_1: \frac{\mathrm{x}-1}{2}=\frac{\mathrm{y}+1}{-3}=\frac{z+4}{2}$ and $\mathrm{L}_2$ is the line passing through the points...

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Q Jee Main 2024

Let $\overrightarrow{\mathrm{OA}}=\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{OB}}=12 \overrightarrow{\mathrm{a}}+4 \overrightarrow{\mathrm{~b}}$ and $\overrightarrow{\mathrm{OC}}=\overrightarrow{\mathrm{b}}$, where O is the origin. If $S$ is the parallelogram with adjacent sides...

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Q JEE MAIN 2024

Consider a $\triangle \mathrm{ABC}$ where $\mathrm{A}(1,2,3), \mathrm{B}(-2,8,0)$ and $\mathrm{C}(3,6,7)$. If the angle bisector of $\angle \mathrm{BAC}$ meets the line $...

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Q JEE MAIN 2024

Let $P$ and $Q$ be the points on the line $\frac{x+3}{8}=\frac{y-4}{2}=\frac{z+1}{2}$ which are at a distance of 6 units from...

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Q Jee Main 2024

Let $(\alpha, \beta, \gamma)$ be mirror image of the point $(2,3,5)$ in the line $-\underline{-2} \quad-\underline{-3} x_{-} y_{-} z$. Then...

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Q JEE MAIN 2024

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{b}|=1$ and $|\vec{b} \times \vec{a}|=2$. Then $|(\dot{b} \times \vec{a})-\vec{b}|^2$ is equal...

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Q Jee Main 2024

Let $P(3,2,3), Q(4,6,2)$ and $R(7,3,2)$ be the vertices of $\triangle \mathrm{PQR}$. Then, the angle $\angle \mathrm{QPR}$ is

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Q JEE-Main 2024

Let the image of the point $(1,0,7)$ in the line $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$ be the point $(\alpha, \beta, \gamma)$. Then which one...

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Q JEE Main 2024

Let $\vec{a}=\hat{\imath}+\alpha \hat{\jmath}+\beta \hat{k}, \alpha, \beta \in R$. Let a vector $\vec{b}$ be such that the angle between $\vec{a}$ and...

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Q JEE Main 2026

Let $\vec{a}=2 \hat{\mathrm{i}}-5 \hat{\mathrm{j}}+5 \hat{\mathrm{k}}$ and $\vec{b}=\hat{\mathrm{i}}-\hat{\mathrm{j}}+3 \hat{\mathrm{k}}$. If $\vec{c}$ is a vector such that $2(\vec{a} \times \vec{c})+3(\vec{b} \times \vec{c})=\overrightarrow{0}$...

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Q JEE MAIN 2026

If the distance of the point $\mathrm{P}(43, \alpha, \beta), \beta

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Q JEE MAIN 2026

Let $Q(a, b, c)$ be the image of the point $P(3,2,1)$ in the line $\frac{x-1}{1}=\frac{y}{2}=\frac{z-1}{1}$. Then the distance of $Q$...

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Q JEE MAIN 2026

Let $P Q R$ be a triangle such that $\overrightarrow{P Q}=-2 \hat{i}-\hat{j}+2 \hat{k}$ and $...

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Q JEE MAIN 2026

For three unit vectors $\vec{a}, \vec{b}, \vec{c}$ satisfying $|\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}|^2+|\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{c}}|^2+|\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}|^2=9$ and $|2 \overrightarrow{\mathrm{a}}+\mathrm{k} \overrightarrow{\mathrm{b}}+\mathrm{k} \overrightarrow{\mathrm{c}}|=3$, the positive value of k is...

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Q JEE MAIN 2026

If the distances of the point $(1,2, a)$ from the line $\frac{x-1}{1}=\frac{y}{2}=\frac{z-1}{1}$ along the lines $\mathrm{L}_1: \frac{x-1}{3}=\frac{y-2}{4}=\frac{z-a}{b}$ and $\mathrm{L}_2: \frac{x-1}{1}=\frac{y-2}{4}=\frac{z-a}{c}$...

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Q JEE MAIN 2026

Let $P$ be a point in the plane of the vectors $\overrightarrow{A B}=3 \hat{i}+\hat{j}-\hat{k}$ and $\overrightarrow{A C}=\hat{i}-\hat{j}+3 \hat{k}$ such that...

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Q JEE MAIN_2026_

The plot of $\log _{10} \mathrm{~K} v s \frac{1}{\mathrm{~T}}$ gives a straight line. The intercept and slope respectively are (where...

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Q JEE MAIN 2026

If the image of the point $\mathrm{P}(a, 2, a)$ in the line $\frac{x}{2}=\frac{y+a}{1}=\frac{z}{1}$ is Q and the image of Q...

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Q JEE MAIN 2026

Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors such that $\vec{a} \times \vec{b}=2(\vec{a} \times \vec{c})$. If $|\vec{a}|=1,|\vec{b}|=4,|\vec{c}|=2$, and the angle between...

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Q JEE MAIN 2026

Let $\vec{a}=\hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}, \vec{b}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}, \vec{c}=\lambda \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\vec{v}=\vec{a} \times \vec{b}$. If $\vec{v} \cdot \vec{c}=11$ and the length of...

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Q JEE MAIN 2026

Let a line L passing through the point $\mathrm{P}(1,1,1)$ be perpendicular to the lines $\frac{x-4}{4}=\frac{y-1}{1}=\frac{z-1}{1}$ and $\frac{x-17}{1}=\frac{y-71}{1}=\frac{z}{0}$. Let the line...

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Q JEE MAIN_2026_

Let the direction cosines of two lines satisfy the equations: $4 l+m-n=0$ and $2 m n+10 n l+3 l m=0$....

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Q JEE MAIN 2026

Let $\vec{a}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}, \vec{b}=\hat{\mathrm{i}}+\hat{\mathrm{j}}$ and $\vec{c}=\vec{a} \times \vec{b}$. Let $\vec{d}$ be a vector such that $|\vec{d}-\vec{a}|=\sqrt{11},|\vec{c} \times \vec{d}|=3$ and...

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Q JEE MAIN 2026

Let the lines $\mathrm{L}_1: \vec{r}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}+\lambda(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}), \lambda \in \mathbb{R} \quad$ and $...

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Q JEE MAIN_2026

The vertices $B$ and $C$ of a triangle $A B C$ lie on the line $\frac{x}{1}=\frac{1-y}{-2}=\frac{z-2}{3}$. The coordinates of $A$...

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Q JEE MAIN 2026

Let a vector $\overrightarrow{\mathrm{a}}=\sqrt{2} \hat{i}-\hat{j}+\lambda \hat{k}, \lambda>0$, make an obtuse angle with the vector $\overrightarrow{\mathrm{b}}=-\lambda^2 \hat{i}+4 \sqrt{2} \hat{j}+4 \sqrt{2} \hat{k}$...

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Q JEE MAIN 2026

Let L be the line $\frac{x+1}{2}=\frac{y+1}{3}=\frac{z+3}{6}$ and let S be the set of all points $(\mathrm{a}, \mathrm{b}, \mathrm{c})$ on L...

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Q JEE MAIN 2026

Let $\vec{a}=2 \hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=\lambda \hat{j}+2 \hat{k}, \lambda \in Z$ be two vectors. Let $\vec{c}=\vec{a} \times \vec{b}$ and $\vec{d}$ be...

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Q JEE MAIN_2026

Let $\overrightarrow{\mathrm{a}}=-\hat{i}+\hat{j}+2 \hat{k}, \overrightarrow{\mathrm{~b}}=\hat{i}-\hat{j}-3 \hat{k}, \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{d}}=\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}}$. Then $(\vec{a}-\vec{b}) \cdot \vec{d}$ is equal to:

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Q JEE MAIN_2026_

If the image of the point $\mathrm{P}(1,2, a)$ in the line $\frac{x-6}{3}=\frac{y-7}{2}=\frac{7-\mathrm{z}}{2}$ is $\mathrm{Q}(5, b, \mathrm{c})$, then $a^2+b^2+c^2$ is equal...

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Q JEE MAIN_2026_

Let $\mathrm{P}(\alpha, \beta, \gamma)$ be the point on the line $\frac{x-1}{2}=\frac{y+1}{-3}=z$ at a distance $4 \sqrt{14}$ from the point $(1,-1,0)$...

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Q JEE MAIN_2026_

Let $\overrightarrow{\mathrm{AB}}=2 \hat{i}+4 \hat{j}-5 \hat{k}$ and $\overrightarrow{\mathrm{AD}}=\hat{i}+2 \hat{j}+\lambda \hat{k}, \lambda \in \mathbb{R}$. Let the projection of the vector $\vec{v}=\hat{i}+\hat{j}+\hat{k}$ on...

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Q JEE MAIN 2026

For a triangle A B C, let $\vec{p}=\overrightarrow{B C}, \vec{q}=\overrightarrow{C A}$ and $\vec{r}=\overrightarrow{B A}$. If $|\vec{p}|=2 \sqrt{3},|\vec{q}|=2$ and $\cos \theta=\frac{1}{\sqrt{3}}$,...

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Q JEE MAIN 2026

Let the line $\mathrm{L}_1$ be parallel to the vector $-3 \hat{i}+2 \hat{j}+4 \hat{k}$ and pass through the point $(2,6,7)$, and...

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Q JEE MAIN 2026

Let the line L pass through the point $(-3,5,2)$ and make equal angles with the positive coordinate axes. If the...

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Q JEE MAIN_2026_

Use the following data : One mole each of $\mathrm{A}_2(\mathrm{~g})$ and $\mathrm{B}_2(\mathrm{~g})$ are taken in a 1 L closed flask...

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Q JEE MAIN_2026

Which of the following graphs between pressure 'p' versus volume 'V' represents the maximum work done?

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Q JEE MAIN 2026

Let $(\alpha, \beta, \gamma)$ be the co-ordinates of the foot of the perpendicular drawn from the point $(5,4,2)$ on the...

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Q JEE MAIN 2026

Let $\vec{c}$ and $\vec{d}$ be vectors such that $|\vec{c}+\vec{d}|=\sqrt{29}$ and $\vec{c} \times(2 \hat{i}+3 \hat{j}+4 \hat{k})=(2 \hat{i}+3 \hat{j}+4 \hat{k}) \times \vec{d}$....

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Q JEE MAIN 2026

Let $\overrightarrow{\mathrm{a}}=-\hat{i}+2 \hat{j}+2 \hat{k}, \overrightarrow{\mathrm{~b}}=8 \hat{i}+7 \hat{j}-3 \hat{k}$ and $\overrightarrow{\mathrm{c}}$ be a vector such that $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{b}}$. If $...

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Q JEE MAIN_2019_

If $Q(0,-1,-3)$ is the image of the point $P$ in the plane $3 x-y+4 z=2$ and $R$ is the point...

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Q JEE MAIN_2019_

Let $A(3,0-1), B(2,10,6)$ and $C(1,2,1)$ be the vertices of a triangle and $M$ be the mid point of $A C$....

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Q JEE MAIN_2019_

If the length of the perpendicular from the point ( $\beta$, $0, \beta)(\beta \neq 0)$ to the line, $\frac{x}{1}=\frac{y-1}{0}=\frac{z+1}{-1}$ is...

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Q JEE MAIN 2019

Let $\vec{\alpha}=3 \hat{i}+\hat{j}$ and $\vec{\beta}=2 \hat{i}-\hat{j}+3 \hat{k}$. If $\vec{\beta}=\vec{\beta}_1-\vec{\beta}_2$ , where $\vec{\beta}_1$ is parallel to $\vec{\alpha}$ and $\vec{\beta}_2$ is perpendicular...

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Q JEE MAIN 2019

If the line, $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-2}{4}$ meets the plane, x + 2y + 3z = 15 at a point P, then the...

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Q JEE MAIN 2020


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Q JEE MAIN 2019

The length of the perpendicular drawn from the point (2, 1, 4) to the plane containing the lines

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Q JEE MAIN 2019

Let α ∈ R and the three vectors

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Q JEE MAIN 2019

A plane which bisects the angle between the two given planes 2x – y + 2z – 4 = 0...

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Q JEE MAIN 2020

A plane passing through the point $(3,1,1)$ contains two lines whose direction ratios are $1,-2,2$ and $2,3,-1$ respectively. If this...

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Q JEE Main 2020

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three unit vectors such that $|\vec{a}-\vec{b}|^2+|\vec{a}-\vec{c}|^2=8$. Then $|\vec{a}+2 \vec{b}|^2+|\vec{a}+2 \vec{c}|^2$ is equal to

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Q JEE Main 2020

The plane passing through the points (1, 2, 1), (2, 1, 2) and parallel to the line, 2x = 3y,...

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Q JEE-MAIN 2019

The sum of the distinct real values of $\mu$, for which the vectors, $\mu \hat{j}+\hat{j}+\hat{k}, \hat{i}+\mu \hat{j}+\hat{k}, \hat{i}+\hat{j}+\mu \hat{k}$ are...

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Q JEE-MAIN 2019

A tetrahedron has vertices $P(1,2,1), Q(2,1,3), R(-1,1,2)$ and $O(0,0,0)$. The angle between the faces $O P Q$ and $...

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Q JEE MAIN 2019

The perpendicular distance from the origin to the plane containing the two lines, $\frac{x+2}{3}=\frac{y-2}{5}=\frac{z+5}{7}$ and $\frac{x-1}{1}=\frac{y-4}{4}=\frac{z+4}{7}$ is

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Q JEE MAIN 2019

The equation of the plane containing the straight line $\frac{x}{2}=\frac{y}{3}=\frac{z}{4}$ and perpendicular to the plane containing the straight lines $\frac{x}{3}=\frac{y}{4}=\frac{z}{2}$...

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Q JEE MAIN 2020

If the vectors, $\vec{p}=(a+1) \hat{i}+a \hat{j}+a \hat{k}$, $\overrightarrow{\mathrm{q}}=\mathrm{a} \hat{\mathrm{i}}+(\mathrm{a}+1) \hat{\mathrm{j}}+\mathrm{a} \hat{\mathrm{k}}$, and $\vec{r}=a \hat{i}+a \hat{j}+(a+1) \hat{k}(a \in R)$ are coplanar...

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Q JEE MAIN 2020

The projection of the line segment joining the point (1, –1, 3) and (2, –4, 11) on the line joining...

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Q JEE MAIN 2019

If the lines x = ay + b, z = cy + d and x = a′z + b′, y...

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Q JEE MAIN 2020

If the line, $2 x-y+3=0$ is at a distance $\frac{1}{\sqrt{5}}$ and $\frac{2}{\sqrt{5}}$ from the lines $4 x-2 y+\alpha=0$ and $...

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Q JEE MAIN 2020

If the volume of a parallelopiped, whose coterminus edges are given by the vectors $\vec{a}=\hat{i}+\hat{j}+n \hat{k}, \vec{b}=2 \hat{i}+4 \hat{j}-n \hat{k}$...

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Q JEE MAIN 2020

If $(a, b, c)$ is the image of the point $(1,2,-3)$ in the line, $\frac{x+1}{2}=\frac{y-2}{-2}=\frac{z}{-1}$, then $a+b+c$ is equal to...

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Q JEE MAIN 2020

If $\vec{a}$ and $\vec{b}$ are unit vectors, then the greatest value of $\sqrt{3}|\vec{a}+\vec{b}|+|\vec{a}-\vec{b}|$ is

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Q JEE MAIN 2020

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Q JEE MAIN 2019

If an angle between the line, $\frac{x+1}{2}=\frac{y-2}{1}=\frac{z-3}{-2}$ and the plane, $x-2 y-k z=3$ is $\cos ^{-1}\left(\frac{2 \sqrt{2}}{3}\right)$

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Q JEE MAIN 2019

Let $S$ be the set of all real values of $\lambda$ such that a plane passing through the points $...

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Q JEE MAIN 2020

A plane P meets the coordinate axes at A, B and C respectively. The centroid of ΔABC is given to...

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Q JEE MAIN 2019

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three unit vectors, out of which vectors $\vec{b}$ and $\vec{c}$ are non-parallel. If $\alpha$...

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Q JEE MAIN 2020

The shortest distance between the lines $\frac{x-1}{0}=\frac{y+1}{-1}=\frac{z}{1}$ and $x+y+z+1=0,2 x-y+z+3=0$ is :

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Q JEE MAIN 2019

The equation of the plane containing the straight line $\frac{x}{2}=\frac{y}{3}=\frac{z}{4}$ and perpendicular to the plane containing the straight lines $\frac{x}{3}=\frac{y}{4}=\frac{z}{2}$...

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Q JEE MAIN 2019

Let $\ddot{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\sqrt{2} \hat{\mathrm{k}}, \ddot{\mathrm{b}}=\mathrm{b}_1 \hat{\mathrm{i}}+\mathrm{b}_2 \hat{\mathrm{j}}+\sqrt{2} \hat{\mathrm{k}}$ and $\ddot{\mathrm{c}}=5 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\sqrt{2} \hat{\mathrm{k}}$ be three vectors such that the projection vector of...

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Q JEE MAIN 2020

Let the vectors $\vec{a}, \vec{b}, \vec{c}$, be such that $|\vec{a}|=2,|\vec{b}|=4$ and $|\vec{c}|=4$ and $|\vec{c}|=4$. If the projection of $\vec{b}$ on...

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Q JEE MAIN 2019

The direction ratios of normal to the plane through the points $(0,-1,0)$ and $(0,0,1)$ and making an angle $\frac{\pi}{4}$ with...

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Q JEE MAIN 2020

The shortest distance between the lines $\frac{x-3}{3}=\frac{y-8}{-1}=\frac{z-3}{1}$ and $\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-6}{2}$ is

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Q JEE MAIN 2020

Let the volume of a parallelopiped whose coterminous edges are given by $\overrightarrow{\mathrm{u}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\lambda \hat{\mathrm{k}}, \quad \overrightarrow{\mathrm{v}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{w}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$...

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Q JEE MAIN 2019

Let $\vec{a}=\hat{i}+2 \hat{j}+4 \hat{k}, \vec{b}=\hat{i}+\lambda \hat{j}+4 \hat{k}$ and $\overrightarrow{\mathrm{c}}=2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+\left(\lambda^2-1\right) \hat{\mathrm{k}}$ be coplanar vectors. Then the non-zero vector $...

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Q JEE MAIN 2019

The plane containing the line $\frac{x-3}{2}=\frac{y+2}{-1}=\frac{z-1}{3}$ and also containing its projection on the plane $2 x+3 y-z=$ 5 , contains...

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Q JEE MAIN 2019

The distance of the point having position vector $-\hat{i}+2 \hat{j}+6 \hat{k}$ from the straight line passing through the point (...

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Q JEE MAIN 2019

If the plane $2 x-y+2 z+3=0$ has the distances $\frac{1}{3}$ and $\frac{2}{3}$ units from the planes $4 x-2 y+4 z+\lambda=0$...

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Q JEE MAIN 2019

A perpendicular is drawn from a point on the line $\frac{x-1}{2}=\frac{y+1}{-1}=\frac{z}{1}$ to the plane $x+y+z=3$ such that the foot of...

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Q JEE MAIN 2020

If $\vec{a}=2 \hat{i}+\hat{j}+2 \hat{k}$, then the value of $|\hat{i} \times(\vec{a} \times \hat{i})|^2+|\hat{j} \times(\vec{a} \times \hat{j})|^2+|\hat{k} \times(\vec{a} \times \hat{k})|^2$ is equal...

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Q JEE Main 2019

Let $A$ be a point on the line $\vec{r}=(1-3 \mu) \hat{i}+(\mu-1) \hat{j}+(2+5 \mu) \hat{k}$ and $B(3,2,6)$ be a point in...

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Q JEE Main 2019

Let $\vec{a}=2 \hat{i}+\lambda_1 \hat{j}+3 \hat{k}, \vec{b}=4 \hat{i}+\left(3-\lambda_2\right) \hat{j}+6 \hat{k}$ and $\vec{c}=3 \hat{i}+6 \hat{j}+\left(\lambda_3-1\right) \hat{k}$ be three vectors such that $...

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Q JEE Main 2019

The plane passing through the point $(4,-1,2)$ and parallel to the lines $\frac{x+2}{3}=\frac{y-2}{-1}=\frac{z+1}{2}$ and $\frac{x-2}{1}=\frac{y-3}{2}=\frac{z-4}{3}$ also passes through the point:

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Q JEE MAIN 2020

If the distance between the plane, $23 x-10 y-2 z+ 48=0$ and the plane containing the lines $$ \frac{x+1}{2}=\frac{y-3}{4}=\frac{z+1}{3} $$...

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Q JEE MAIN 2020

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Q JEE MAIN_2021_

The equation of the plane passing through the line of intersection of the planes $\vec{r} \cdot(\hat{i}+\hat{j}+\hat{k})=1$ and $...

JEE Main Mathematics Medium

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Q JEE MAIN 2020

If the equation of a plane P , passing through the intersection of the planes $x+4 y-z+7=0$ and $...

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Q JEE MAIN 2020

Let $x_0$ be the point of local maxima of $f(x)=\vec{a} \cdot(\vec{b} \times \vec{c})$, where $...

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Q JEE MAIN 2020

The distance of the point $(1,-2,3)$ from the plane $x-y+z=5$ measured parallel to the line $\frac{x}{2}=\frac{y}{3}=\frac{z}{-6}$ is :

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Q JEE MAIN 2019

Let $\sqrt{3} \hat{i}+\hat{j}, \hat{i}+\sqrt{3} \hat{j}$ and $\beta \hat{i}+(1-\beta) \hat{j}$ respectively be the position vectors of the points $A, B$ and...

JEE Main Mathematics Medium

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Q JEE MAIN 2019

If the point $(2, \alpha, \beta)$ lies on the plane which passes through the points $(3,4,2)$ and $(7,0,6)$ and is...

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Q JEE-MAIN 2020

Let a plane P contain two lines $$ \overrightarrow{\mathrm{r}}=\hat{\mathrm{i}}+\lambda(\hat{\mathrm{i}}+\hat{\mathrm{j}}), \lambda \in \mathrm{R} \text { and } \overrightarrow{\mathrm{r}}=-\hat{\mathrm{i}}+\mu(\hat{\mathrm{j}}-\hat{\mathrm{k}}), \mu \in \mathrm{R}...

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Q JEE MAIN 2020

The plane which bisects the line joining, the points (4, –2, 3) and (2, 4, –1) at right angles also...

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Q JEE MAIN 2019

Two lines $\frac{x-3}{1}=\frac{y+1}{3}=\frac{z-3}{4}$ and $\frac{x+5}{7}=\frac{y-2}{-6}=\frac{z-3}{4}$ intersect at the point $R$. The reflection of $R$ in the $x y$-plane has coordinates...

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Q JEE MAIN 2020

The mirror image of the point $(1,2,3)$ in a plane is $\left(-\frac{7}{3},-\frac{4}{3},-\frac{1}{3}\right)$. Which of the following points lies on this...

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Q JEE MAIN 2019

$f$ the lines $x=a y+b, z=c y+d$ and $x=a^{\prime} z+b^{\prime}, y=c^{\prime} z+d^{\prime}$ are perpendicular, then

JEE Main Mathematics Easy

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Q JEE MAIN 2020

Let $\vec{a}=\hat{i}-2 \hat{j}+\hat{k}$ and $\vec{b}=\hat{i}-\hat{j}+\hat{k}$ be two vectors. If $\vec{c}$ is a vector such that $\vec{b} \times \vec{c}=\vec{b} \times \vec{a}$...

JEE Main Mathematics Medium

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Q JEE-MAIN 2020

Let $a, b c \in R$ be such that $a^2+b^2+c^2=1$. If $a \cos \theta=b \cos \left(\theta+\frac{2 \pi}{3}\right)=c \cos \left(\theta+\frac{4 \pi}{3}\right)$,...

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Q JEE MAIN 2019

Let P be the plane, which contains the line of intersection of the planes, x + y + z –...

JEE Main Mathematics Medium

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Q JEE MAIN 2019

If a unit vector $\vec{a}$ makes angles $\frac{\pi}{3}$ with $\hat{i}, \frac{\pi}{4}$ with $\hat{j}$ and $\theta \in(0, \pi)$ with $\hat{k}$, then...

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Q JEE MAIN 2019

The vertices $B$ and $C$ of a $\triangle A B C$ lie on the line, $\frac{x+2}{3}=\frac{y-1}{0}=\frac{z}{4}$ such that $B C=5$...

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Q JEE Main 2021

Suppose the line $\frac{x-2}{\alpha}=\frac{y-2}{-5}=\frac{z+2}{2}$ lies on the plane $x+3 y-2 z+\beta=0$. Then $(\alpha+\beta)$ is equal to $\_\_\_\_$。

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Q JEE MAIN 2021

The square of the distance of the point of intersection of the line $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z+1}{6}$ and the plane $2 x-y+z=6$ from...

JEE Main Mathematics Easy

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Q JEE MAIN 2020

A vector $\vec{a}=\alpha \hat{i}+2 \hat{j}+\beta \hat{k}(\alpha, \beta \in R)$ lies in the plane of the vectors, $\vec{b}=\hat{i}+\hat{j}$ and $\vec{c}=\hat{i}-\hat{j}+4 \hat{k}$....

JEE Main Mathematics Hard

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Q JEE MAIN 2020

Let P be a plane through the points (2, 1, 0), (4, 1, 1) and (5, 0, 1) and R...

JEE Main Mathematics Easy

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Q JEE MAIN 2019

On which of the following lines lies the point of intersection of the line, $\frac{x-4}{2}=\frac{y-5}{2}=\frac{z-3}{1}$ and the plane, $...

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Q JEE MAIN 2019

The plane which bisects the line segment joining the points (–3, –3, 4) and (3, 7, 6) at right angles,...

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Q JEE MAIN 2019

Let $\alpha=(\lambda-2) \mathrm{a}+\mathrm{b}$ and $\beta=(4 \lambda-2) \mathrm{a}+3 \mathrm{~b}$ be two given vectors where vectors a and b are non-collinear. The...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|2 \vec{a}+3 \vec{b}|=|3 \vec{a}+\vec{b}|$ and the angle between $\vec{a}$ and $\vec{b}$...

JEE Main Mathematics Easy

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Q JEE Main 2021

The distance of the point $(-1,2,-2)$ from the line of intersection of the planes $2 x+3 y+2 z=0$ and $...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Let the equation of the plane, that passes through the point $(1,4,-3)$ and contains the line of intersection of the...

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Q JEE Main 2021

Let $\vec{a}, \vec{b}, \vec{c}$ be three vectors mutually perpendicular to each other and have same magnitude. If a vector $\vec{r}$...

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Q JEE MAIN 2021

Equation of a plane at a distance $\sqrt{\frac{2}{21}}$ from the origin, which contains the line of intersection of the planes...

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Q JEE MAIN 2021

The distance of the point (1, –2, 3) from the plane x – y + z = 5 measured parallel...

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Q JEE Main 2021

Let the line $L$ be the projection of the line $\frac{x-1}{2}=\frac{y-3}{1}=\frac{z-4}{2}$ in the plane $x-2 y-z=3$. If $d$ is the...

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Q JEE Main 2021

A plane P contains the line x + 2y + 3z + 1 = 0 = x – y –...

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Q JEE Main 2021

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=\hat{j}-\hat{k}$. If $\vec{c}$ is a vector-such that $\vec{a} \times \vec{c}=\vec{b}$ and $\vec{a} \cdot \vec{c}=3$, then $...

JEE Main Mathematics Easy

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Q JEE MAIN_2021_

The angle between the straight lines, whose direction cosines are given by the equations $2 \ell+2 \mathrm{~m}-\mathrm{n}=0$ and $...

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Q JEE MAIN_2021_

Let $S$ be the mirror image of the point $Q(1,3,4)$ with respect to the plane $2 x-y+z+3=0$ and let $...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Let $\vec{a}=\hat{i}+5 \hat{j}+\alpha \hat{k}, \vec{b}=\hat{i}+3 \hat{j}+\beta \hat{k}$ and $\vec{c}=-\hat{i}+2 \hat{j}+3 \hat{k}$ be three vectors such that, $|\vec{b} \times \vec{c}|=5 \sqrt{3}$...

JEE Main Mathematics Medium

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Q JEE Main 2019

The equation of the line passing through $(-4,3,1)$, parallel to the plane $x+2 y-z-5=0$ and intersecting the line $\frac{x+1}{-3}=\frac{y-3}{2}=\frac{z-2}{-1}$ is...

JEE Main Mathematics Medium

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Q JEE MAIN 2019

The length of the perpendicular from the point ( $2,-$ 1,4) on the straight line, $\frac{x+3}{10}=\frac{y-2}{-7}=\frac{z}{1}$ is

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Let $Q$ be the foot of the perpendicular from the point $P(7,-2,13)$ on the plane containing the lines $\frac{x+1}{6}=\frac{y-1}{7}=\frac{z-3}{8}$ and...

JEE Main Mathematics Medium

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Q JEE MAIN 2019

The magnitude of the projection of the vector $2 \hat{i}+3 \hat{j}+\hat{k}$ on the vector perpendicular to the plane containing the...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

If the projection of the vector $\hat{i}+2 \hat{j}+\hat{k}$ on the sum of the two vectors $2 \hat{i}+4 \hat{j}-5 \hat{k}$ and...

JEE Main Mathematics Easy

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Q JEE MAIN 2019

The equation of a plane containing the line of intersection of the planes $2 x-y-4=0$ and $y+2 z -4=0$ and...

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Q JEE Main 2019

The plane through the intersection of the planes x + y + z = 1 and 2x + 3y –...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Let $P$ be the plane passing through the point $(1,2,3)$ and the line of intersection of the planes $...

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Q JEE MAIN 2021

A hall has a square floor of dimension $10 \mathrm{~m} \times 10 \mathrm{~m}$ (see the figure) and vertical walls. If...

JEE Main Mathematics Medium

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Q JEE Main 2019

Let $\vec{a}=\vec{i}-\vec{j}, \vec{b}=\vec{i}+\vec{j}+\vec{k}$ and $\vec{c}$ be a vector such that $\vec{a} \times \vec{c}+\vec{b}=\overrightarrow{0}$ and $\vec{a} \cdot \vec{c}=4$, then $|\vec{c}|^2$ is...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Let $\vec{a}-=2 \hat{i}-\hat{j}+2 \hat{k}$ and $\vec{b}=\hat{i}--2 \hat{j}-\hat{k}$. Let a vector $\vec{v}$ be in the plane containing $\vec{a}$ and $\vec{b}$. If...

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Q JEE MAIN 2021

The distance of line 3y – 2z – 1 = 0 = 3x – z + 4 from the point...

JEE Main Mathematics Medium

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Q _JEE MAIN_2021

Let $\vec{p}=2 \hat{i}+3 \hat{j}+\hat{k}$ and $\vec{q}=\hat{i}+2 \hat{j}+\hat{k}$ be two vectors. If a vector $\vec{r}=(\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k})$ is perpendicular to...

JEE Main Mathematics Medium

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Q _JEE MAIN_2021_

Let the foot of perpendicular from a point $P(1,2,-1)$ to the straight line $L: \frac{x}{1}=\frac{y}{0}=\frac{z}{-1}$ be $N$. Let a line...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Let the acute angle bisector of the two planes x – 2y – 2z + 1 = 0 and 2x...

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Q JEE MAIN 2021 S2

The distance of the point $\mathrm{P}(3,4,4)$ from the point of intersection of the line joining the points. $\mathrm{Q}(3,-4,-5)$ and $R(2,-3,1)$...

JEE Main Mathematics Easy

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Q JEE MAIN 2021 S2

Let $\vec{a}=\hat{i}-\alpha \hat{j}+\beta \hat{k}, \vec{b}=3 \hat{i}+\beta \hat{j}+\alpha \hat{k}$ and $\vec{c}=-\alpha \hat{i}+2 \hat{j}+\hat{k}$, where $\alpha$ and $\beta$ are integers. If $...

JEE Main Mathematics Medium

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Q JEE MAIN 2021 S2

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three vectors such that $\vec{a}=\vec{b} \times(\vec{b} \times \vec{c})$. If magnitudes of the vectors $...

JEE Main Mathematics Medium

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Q JEE MAIN 2021 S2

For real numbers $\alpha$ and $\beta \neq 0$, if the point of intersection of the straight lines $\frac{x-\alpha}{1}=\frac{y-1}{2}=\frac{z-1}{3}$ and $\frac{x-4}{\beta}=\frac{y-6}{3}=\frac{z-7}{3}$,...

JEE Main Mathematics Medium

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Q JEE MAIN 2019

If a point R(4, y, z) lies on the line segment joining the points P(2, –3, 4) and Q(8, 0,...

JEE Main Mathematics Easy

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Q _JEE MAIN_2021

Let the vectors $(2+a+b) \hat{i}+(a+2 b+c) \hat{j}-(b+c) \hat{k},(1+b) \hat{i}+2 b \hat{j}-b \hat{k}$ and $...

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Q JEE MAIN 2021 S2

If the lines $\frac{x-k}{1}=\frac{y-2}{2}=\frac{z-3}{3}$ and $\frac{x+1}{3}=\frac{y+2}{2}=\frac{z+3}{1}$ are co-planar, then the value of $k$ is

JEE Main Mathematics Easy

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Q JEE MAIN 2021 S2

If $(\vec{a}+3 \vec{b})$ is perpendicular to $(7 \vec{a}-5 \vec{b})$ and $(\vec{a}-4 \vec{b})$ is perpendicular to $(7 \vec{a}-2 \vec{b})$, then the...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

The force is given in terms of time t and displacement x by the equation F = A cos Bx...

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Q JEE MAIN 2019

The vector equation of plane through the line of intersection of the planes $x+y+z=1$ and $2 x+3 y+4 z=5$ which...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Let a plane $P$ pass through the point $(3,7,-7)$ and contain the line, $\frac{x-2}{-3}=\frac{y-3}{2}=\frac{z+2}{1}$. If distance of the plane P...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

If the mirror image of the point $(1,3,5)$ with respect to the plane $4 x-5 y+2 z=8$ is $...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}$ and $\vec{c}=\hat{j}-\hat{k}$ be three vectors such that $\vec{a} \times \vec{b}=\vec{c}$ and $\vec{a} \cdot \vec{b}=1$. If the length...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Let the plane passing through the point (–1, 0, –2) and perpendicular to each of the planes 2x + y...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Let $L$ be a line obtained from the intersection of two planes $x+2 y+z=6$ and $y+2 z=4$. If point $...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

If $|\vec{a}|=2, \mid \vec{b}=5$ and $|\vec{a} \times \vec{b}|=8$, then $|\vec{a} \cdot \vec{b}|$ is equal to:

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Q JEE MAIN 2021

Let $a, b$ and $c$ be distinct positive numbers. If the vectors $a-a \hat{i}+a \hat{j}+c \hat{k}, \hat{i}+\hat{k}$ and $...

JEE Main Mathematics Easy

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Q JEE MAIN_2021_

If the shortest distance between the straight lines $3(x-1)=6(y-2)=2(z-1)$ and $4(x-2)=2(y-\lambda)=(z-3)$, $\lambda \in \mathrm{R}$ is $\frac{1}{\sqrt{38}}$, then the integral value...

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Q JEE MAIN 2021

Let $\vec{a}=\hat{i}+\hat{j}+2 \hat{k}$ and $\vec{b}=-\hat{i}+2 \hat{j}+3 \hat{k}$. Then the vector product $(\vec{a}+\vec{b}) \times((\vec{a} \times((\vec{a}-\vec{b}) \times \vec{b})) \times \vec{b})$ is equal...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

If the shortest distance between the lines $\vec{r}_1=\alpha \hat{i}+2 \hat{j}+2 \hat{k}+\lambda(\hat{i}-2 \hat{j}+2 \hat{k}), \lambda \in R, \alpha>0$ and $...

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Q JEE MAIN 2021

Let $P$ be a plane passing through the points $(1,0,1),(1,-2,1)$ and $(0,1,-2)$. Let a vector $\overrightarrow{\mathbf{a}}=\alpha \hat{\mathbf{i}}+\beta \hat{\mathbf{j}}+\gamma \hat{\mathbf{k}}$ be...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Let $\vec{a}, \vec{b}, \vec{c}$-be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle $\theta$, with...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Let $\vec{a}-=2 \hat{i}+\hat{j}-2 \hat{k}$ and $\vec{b}-=\hat{i}+\hat{j}$. If $\vec{c}-$ is a vector such that $\vec{a} \cdot \vec{c}=|\vec{c}|,|\vec{c}-\vec{a}|=2 \sqrt{2}$ and the angle...

JEE Main Mathematics Medium

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Q JEE-MAIN 2021

Let P be an arbitrary point having sum of the squares of the distance from the planes $\mathrm{x}+\mathrm{y}+\mathrm{z}=0, l \mathrm{x}-\mathrm{nz}=0$...

JEE Main Mathematics Hard

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Q JEE-MAIN 2021

Let $\vec{x}$ be a vector in the plane containing vectors $\vec{a}=2 \hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}+2 \hat{j}-\hat{k}$. If the vector $\vec{x}$ is...

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Q JEE MAIN 2021

A line l passing through origin is perpendicular to the lines $$ \begin{aligned} & I_1: r=(3+t) \hat{i}+(-1+2 t) \hat{j}+(4-2 t)...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Let ${ }^{\mathrm{a}}=\hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}+\alpha \hat{\mathrm{j}}+\hat{\mathrm{k}}$. If the area of the parallelogram whose adjacent sides are represented...

JEE Main Mathematics Medium

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Q JEE MAIN_2021_

Let three vectors $\overline{\mathrm{a}}, \overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ be such that $\overline{\mathrm{a}} \times \overline{\mathrm{b}}=\overline{\mathrm{c}}, \overline{\mathrm{b}} \times-\overline{\mathrm{c}}=\overline{\mathrm{a}}$ and $|\overline{\mathrm{a}}|=2$. Then which one...

JEE Main Mathematics Easy

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Q JEE MAIN_2021

Let a vector $\bar{a}$ be coplanar with vectors $\bar{b}=2 \hat{i}+\hat{j}+\hat{k}$ and $\bar{c}=\hat{i}-\hat{j}+\hat{k}$. If $\bar{a}$ is perpendicular to $...

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Q JEE MAIN_2021

Let $L$ be the line of intersection of planes $\vec{r} \cdot(\hat{i}-\hat{j}+2 \hat{k})=2$ and $\vec{r} \cdot(2 \hat{i}+\hat{j}-\hat{k})=2$. If $P(\alpha, \beta, \gamma)$...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Let $P$ be $a$ plane containing the line $\frac{x-1}{3}=\frac{y+6}{4}=\frac{z+5}{2}$ and parallel to the line $\frac{x-3}{4}=\frac{y-2}{-3}=\frac{z+5}{7}$. If the point $(1,-1, \alpha)$...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Let the mirror image of the point $(1,3, a)$ with respect to the plane $\overrightarrow{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}})-\mathrm{b}=0$ be $(-3,5$, 2)....

JEE Main Mathematics Medium

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Q JEE-MAIN 2021

If the equation of plane passing through the mirror image of a point $(2,3,1)$ with respect to line $\frac{x+1}{2}=\frac{y-3}{1}=\frac{z+2}{-1}$ and...

JEE Main Mathematics Easy

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Q JEE-MAIN 2021

Let O be the origin. Let $\overline{\mathrm{OP}}=x \hat{\mathrm{i}}+y \hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\overline{\mathrm{PQ}}=\sqrt{20} \overline{\mathrm{OQ}}=-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 x \hat{\mathrm{k}}, x, y \in \mathrm{R}, x>0$,...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Let $\overline{\mathrm{a}}$ and $\overline{\mathrm{b}}$ be two non-zero vectors perpendicular to each other and $|\overline{\mathrm{a}}|=|\overline{\mathrm{b}}|$. If $|\overline{\mathrm{a}} \times \overline{\mathrm{b}}|=|\overline{\mathrm{a}}|$, then the...

JEE Main Mathematics Medium

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Q JEE MAIN_2021_

If the equation of the plane passing through the line of intersection of the planes $...

JEE Main Mathematics Easy

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Q JEE MAIN_2021_

If $\bar{a}=\alpha \hat{i}+b \hat{j}+3 \hat{k}, \bar{b}=\beta \hat{i}-\alpha \hat{j}-\hat{k}$ and $\bar{c}=\hat{i}-2 \hat{j}-\hat{k}$ such that $\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}=1$ and $\overline{\mathrm{b}} \cdot \overline{\mathrm{c}}=-3$...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

In a triangle ABC, if $|\overrightarrow{\mathrm{BC}}|=8,|\overrightarrow{\mathrm{CA}}|=7,|\overrightarrow{\mathrm{AB}}|=10$, then the projection of the vector $\overrightarrow{\mathrm{AB}}$ on $\overrightarrow{\mathrm{AC}}$ is equal to:

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Q JEE MAIN 2021

For $\mathrm{p}>0$, a vector $\overrightarrow{\mathrm{v}}_2=2 \hat{\mathrm{i}}+(\mathrm{p}+1) \hat{\mathrm{j}}$ is obtained by rotating the vector $\overrightarrow{\mathrm{v}}_1=\sqrt{3} p \hat{\mathrm{i}}+\hat{\mathrm{j}}$ by an angle $\theta$...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Let three vectors $\vec{a}, \vec{b}$ and $\vec{c}$ be such that $\vec{c}$ is coplanar with $\vec{a}$ and, $\vec{b}, \vec{a} \cdot \vec{c}=7$...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

In a triangle ABC , if $|\overrightarrow{\mathrm{BC}}|=3,|\overrightarrow{\mathrm{CA}}|=5$ and $|\overrightarrow{\mathrm{BA}}|=7$, then the projection of the vector $\overrightarrow{\mathrm{BA}}$ on $\overrightarrow{\mathrm{BC}}$ is equal...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

If the distance of the point $(1,-2,3)$ from the plane $x+2 y-3 z+10=0$ measured parallel to the line, $\frac{x-1}{3}=\frac{2-y}{m}=\frac{z+3}{1}$ is...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Let $\vec{a}=\hat{i}+2 \hat{j}-3 \hat{k}$ and $\vec{b}=2 \hat{i}-3 \hat{j}+5 \hat{k}$ If $\vec{r} \times \vec{a}=\vec{b} \times \vec{r} \cdot(a \hat{i}+2 \hat{j}+\hat{k})=3$ and $...

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Q JEE MAIN 2021

If the foot of the perpendicular from point $(4,3,8)$ on the line $\mathrm{L}_1: \frac{\mathrm{x}-\mathrm{a}}{\ell}=\frac{\mathrm{y}-2}{3}=\frac{\mathrm{z}-\mathrm{b}}{4}, \ell \neq 0$ is $(3,5$, 7),...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Consider the line $L$ given by the equation $\frac{x-3}{2}=\frac{y-1}{1}=\frac{z-2}{1}$. Let $Q$ be the mirror image of the point $(2,3$, -1)...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

If $(x, y, z)$ be an arbitrary point lying on a plane $P$ which passes through the point $(42,0,0),(0,42,0)$ and...

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Q JEE MAIN 2021

The lines $x=a y-1=z-2$ and $x=3 y-2=b z-2,(a b \neq 0)$ are coplanar, if .

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Q JEE MAIN_2021

The equation of the plane which contains the y-axis and passes through the point (1, 2, 3) is :

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Q _JEE MAIN_2021

Let $\vec{a}=2 \hat{i}-3 \hat{j}+4 \hat{k}$ and $\vec{b}=7 \hat{i}+\hat{j}-6 \hat{k}$ If $\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{a}}=\overrightarrow{\mathrm{r}} \times \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}})=-3$ then $...

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Q JEE MAIN 2021

Let $\lambda$ be an interger. If the shortest distance between the lines $x-\lambda=2 y-1=-2 z$ and $x=y+ 2 \lambda=z-\lambda$ is...

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Q JEE MAIN 2021

The vector equation of the plane passing through the intersection of the planes $\vec{r} \cdot(\hat{i}+\hat{j}+\hat{k})=1$ and $\vec{r} \cdot(\hat{i}-2 \hat{j})=-2$, and...

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Q JEE MAIN 2021

Let $\mathrm{a}, \mathrm{b} \in \mathrm{R}$. If the mirror image of the point $\mathrm{P}(\mathrm{a}, 6$, 9) with respect to the line...

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Q JEE MAIN 2021

The equation of the planes parallel to the plane x – 2y + 2z – 3 = 0 which are...

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Q JEE MAIN 2021

Let the plane $a x+b y+c z+d=0$ bisect the line joining the points $(4,-3,1)$ and $(2,3,-5)$ at the right angles....

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Q JEE MAIN 2021

A vector $\vec{a}$ has components $3 p$ and 1 with respect to a rectangular cartesian system. This system is rotated...

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Q JEE MAIN 2021

Let $P$ be a plane $l x+m y+n z=0$ containing the line, $\frac{1-x}{1}=\frac{y+4}{2}=\frac{z+2}{3}$. IF plane $P$ divides the line segment...

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Q JEE MAIN 2021

Let the position vectors of two points $P$ and $Q$ be $3 \hat{i}+\hat{j}+2 \hat{k}$ and $\hat{i}+2 \hat{j}-4 \hat{k}$, respectively. Let...

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Q JEE MAIN 2021

Let $(\lambda, 2,1)$ be a point on the plane which passes through the point $(4,-2,2)$. If the plane is perpendicular...

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Q JEE MAIN 2021

If for $\mathrm{a}>0$, the feet of perpendiculars from the points $\mathrm{A}(\mathrm{a},-2 \mathrm{a}, 3)$ and $\mathrm{B}(0,4,5)$ on the plane $l \mathrm{x}+\mathrm{my}+\mathrm{nz}=0$...

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Q JEE MAIN 2021

Let a vector $\alpha \hat{i}+\beta \hat{j}$ be obtained by rotating the vector $\sqrt{3} \hat{i}+\hat{j}$ by an angle $45^{\circ}$ about the...

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Q JEE MAIN 2021

Consider the three planes $$ \begin{aligned} & P_1: 3 x+15 y+21 z=9 \\ & P_2: x \cdot 3 y \cdot...

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Q JEE MAIN 2021

If $(1,5,35),(7,5,5),(1, \lambda, 7)$ and $(2 \lambda, 1,2)$ are coplanar, then the sum of all possible values of $\lambda$ is

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Q JEE MAIN 2021

The distance of the point $(1,1,9)$ from the point of intersection of the line $\frac{x-3}{1}=\frac{y-4}{2}=\frac{z-5}{2}$ and the plane $x+y+z=17$ is

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Q JEE MAIN 2021

Let $\vec{a}=\hat{i}+2 \hat{j}-\hat{k}, \vec{b}=\hat{i}-\hat{j}$ and $\vec{c}=\hat{i}-\hat{j}-\hat{k}$ be three given vectors. If $\vec{r}$ is a vector such that $...

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Q JEE MAIN 2021

The equation of the plane passing through the point (1, 2, –3) and perpendicular to the planes 3x + y...

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Q JEE MAIN 2021

If $\vec{a} \quad$ and $\vec{b}$ are perpendicular, then $\overrightarrow{\mathbf{a}} \times(\overrightarrow{\mathbf{a}} \times(\overrightarrow{\mathbf{a}} \times(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})))$ is equal to.

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Q JEE MAIN 2021

Let $\alpha$ be the angle between the lines whose direction cosines satisfy the equations $l+\mathrm{m}-\mathrm{n}=0$ and $l^2+\mathrm{m}^2-\mathrm{n}^2 =0$. Then then...

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Q JEE MAIN 2021

The equation of the line through the point $(0,1,2)$ and perpendicular to the line $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{-2}$ is:

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Q JEE MAIN_2022

Let Q and R be two points on the line $\frac{x+1}{2}=\frac{y+2}{3}=\frac{z-1}{2}$ at a distance $\sqrt{26}$ from the point $P(4,2,7)$. Then...

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Q JEE MAIN_2022

Let $\vec{a}=\alpha \hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=2 \hat{i}+\hat{j}-\alpha \hat{k}, \alpha>0$. If the projection of $\vec{a} \times \vec{b}$ on the vector $...

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Q JEE MAIN_2022

The length of the perpendicular from the point (1, –2, 5) on the line passing through (1, 2, 4) and...

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Q JEE MAIN 2022

The line of shortest distance between the lines $\frac{x-2}{0}=\frac{y-1}{1}=\frac{z}{1}$ and $\frac{x-3}{2}=\frac{y-5}{2}=\frac{z-1}{1}$ makes an angle of $\cos ^{-1}\left(\sqrt{\frac{2}{27}}\right)$ with the plane...

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Q JEE MAIN 2022

Let ABC be a triangle such that $\overrightarrow{\mathrm{BC}}=\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{CA}}=\overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{c}},|\overrightarrow{\mathrm{a}}|=6 \sqrt{2},|\overrightarrow{\mathrm{~b}}|=2 \sqrt{3}$ and $\overrightarrow{\mathrm{b}} \cdot \overrightarrow{\mathrm{c}}=12$ Consider the statements: $...

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Q JEE MAIN 2022

Let P be the plane containing the straight line $\frac{x-3}{9}=\frac{y+4}{-1}=\frac{z-7}{-5}$ and perpendicular to the plane containing the straight lines $\frac{x}{2}=\frac{y}{3}=\frac{z}{5}$...

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Q JEE MAIN 2022

Let $\vec{a}$ and $\vec{b}$ be the vectors along the diagonal of a parallelogram having area $2 \sqrt{2}$. Let the angle...

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Q JEE MAIN 2022

The shortest distance between the lines $\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-1}{-1}$ and $\frac{x+3}{2}=\frac{y-6}{1}=\frac{z-5}{3}$ is :

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Q JEE MAIN 2022

Let $\vec{a}=\hat{i}-2 \hat{j}+3 \hat{k}, \vec{b}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{c}$ be a vector such that $\vec{a}+(\vec{b} \times \vec{c})=\overrightarrow{0}$ and $\vec{b} \cdot \vec{c}=5$. Then,...

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Q JEE MAIN 2022

Let the image of the point $P(1,2,3)$ in the line $L: \frac{x-6}{3}=\frac{y-1}{2}=\frac{z-2}{3}$ be $Q$. let $R(\alpha, \beta, \gamma)$ be a...

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Q JEE MAIN 2022

Let the foot of the perpendicular from the point $(1,2,4)$ on the line $\frac{x+2}{4}=\frac{y-1}{2}=\frac{z+1}{3}$ be $P$. Then the distance of...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

Let $\vec{a}$ be a vector which is perpendicular to the vector $3 \hat{i}+\frac{1}{2} \hat{j}+2 \hat{k}$. If $...

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Q JEE MAIN 2022

Let the plane ax + by + cz = d pass through (2, 3, –5) and is perpendicular to the...

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Q JEE MAIN 2022

Let $Q$ be the mirror image of the point $P(1,2,1)$ with respect to the plane $x+2 y+2 z=16$. Let $T$...

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Q JEE MAIN 2022

Let $\frac{x-2}{3}=\frac{y+1}{-2}=\frac{z+3}{-1}$ lie on the plane $p x-q y+z=5$, for some $p, q \in \mathbb{R}$. The shortest distance of the...

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Q JEE MAIN 2022

Let $\vec{a}=\alpha \hat{i}+2 \hat{j}-\hat{k}$ and $\vec{b}=-2 \hat{i}+\alpha \hat{j}+\hat{k}$, where $a \in \mathbf{R}$. If the area of the parallelogram whose adjacent...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

Let $l_1$ be the line in xy - plane with x and y intercepts $\frac{1}{8}$ and $\frac{1}{4 \sqrt{2}}$ respectively, and...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

Let $\vec{a}=\hat{i}+\hat{j}+2 \hat{k}, \vec{b}=2 \hat{i}-3 \hat{j}+\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}+\hat{k}$ be three given vectors. Let $\vec{v}$ be a vector in the plane...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

Let $\vec{b}=\hat{i}+\hat{j}+\lambda \hat{k}, \lambda \in R$. If $\vec{a}$ is a vector such that $\vec{a} \times \vec{b}=13 \hat{i}-\hat{j}-4 \hat{k}$ and $...

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Q JEE MAIN 2022

If the lines and $\vec{r}=(\hat{i}-\hat{j}+\hat{k})+\lambda(3 \hat{j}-\hat{k})$ and $\vec{r}=(\alpha \hat{i}-\hat{j})+\mu(2 \hat{i}-3 \hat{k})$ are co-planar, then distance of the plane containing these...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

If the plane 2x + y – 5z = 0 is rotated about its line of intersection with the plane...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

Let $\hat{a}$ and $\hat{b}$ be two unit vectors such that $|(\hat{a}+\hat{b})+2(\hat{a} \times \hat{b})|=2$. If $\theta \in(0, \pi)$ is the angle...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

Let the points on the plane P be equidistant from the points (.4, 2, 1) and (2, .2, 3). Then...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

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

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Q JEE MAIN 2022

Let $P_1: \bar{r} .(2 \hat{i}+\hat{j}-3 \hat{k})=4$ be a plane. Let $P_2$ be another plane which passes through the points $(2,-3,2)(2$,...

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Q JEE MAIN 2022

Let $d$ be the distance between the foot of perpendiculars of the points $P(1,2-1)$ and $Q(2,-1,3)$ on the plane $-x+y+z=1$....

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Q JEE MAIN 2022

Let $P$ be the plane passing through the intersection of the planes $\vec{r} \cdot(\hat{i}+3 \hat{j}-\hat{k})=5$ and $\vec{r} \cdot(2 \hat{i}-\hat{j}+\hat{k})=3$, and...

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Q JEE MAIN 2022

Let the mirror image of the point $(a, b, c)$ with respect to the plane $3 x-4 y+12 z+19=0$ be...

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Q JEE MAIN 2022

If $\vec{a}=2 \hat{i}+\hat{j}+3 \hat{k}, \vec{b}=3 \hat{i}+3 \hat{j}+\hat{k}$ and $\vec{c}=c_1 \hat{i}+c_2 \hat{j}+c_3 \hat{k}$ are coplanar vectors and $...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

Let the plane $P: \vec{r} \cdot \vec{a}=d$ contain the line of intersection of two planes $\vec{r} \cdot(\hat{i}+3 \hat{j}-\hat{k})=6$ and $...

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Q JEE MAIN 2022

The acute angle between the planes $\mathrm{P}_1$ and $\mathrm{P}_2$, when $\mathrm{P}_1$ and $\mathrm{P}_2$ are the planes passing through the intersection...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

If two distinct point $Q, R$ lie on the line of intersection of the planes $-x+2 y-z=0$ and $...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}}=2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$. Then the number of vectors $\overrightarrow{\mathrm{b}}$ such that $\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{a}}$ and $...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

If two straight lines whose direction cosines are given by the relations $I+m-n=0,3 I^2+m^2+c n l=0$ are parallel, then the...

JEE Main Mathematics Hard

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Q JEE MAIN 2022

Let $\vec{a}=\alpha \hat{i}+3 \hat{j}-\hat{k}, \vec{b}=3 \hat{i}-\beta \hat{j}+4 \hat{k}$ and $\vec{c}=\hat{i}+2 \hat{j}-2 \hat{k}$ where $\alpha, \beta \in R$, be three vectors....

JEE Main Mathematics Medium

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Q JEE MAIN 2022

If the mirror image of the point (2, 4, 7) in the plane 3x – y + 4z = 2...

JEE Main Mathematics Easy

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Q JEE MAIN 2022

If $\vec{a} \cdot \vec{b}=1, \vec{b} \cdot \vec{c}=2$ and $\vec{c} \cdot \vec{a}=3$, then the value of $...

JEE Main Mathematics Medium

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Q

Let the lines intersect at the point $S$. If a plane $a x+$ by $-z+d=0$ passes through $S$ and is...

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Q JEE MAIN_2022

Let the plane $2 x+3 y+z+20=0$ be rotated through a right angle about its line of intersection with the plane...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

If the two lines $\ell_1: \frac{x-2}{3}=\frac{y+1}{-2}, z=2$ and $\ell_2: \frac{x-1}{1}=\frac{2 y+3}{\alpha}=\frac{z+5}{2}$ perpendicular, then an angle between the lines $l_2$ and...

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Q JEE MAIN 2022

Let $\theta$ be the angle between the vectors $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ where $|\overrightarrow{\mathrm{a}}|=4,|\overrightarrow{\mathrm{~b}}|=3 \theta \in\left(\frac{\pi}{4}, \frac{\pi}{3}\right)$. Then

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Q JEE MAIN 2022

If the shortest distance between the line $\vec{r}=(-\hat{i}+3 \hat{k})+\lambda(\hat{i}-a \hat{j})$ and $\vec{r}=(-\hat{j}+2 \hat{k})+\mu(\hat{i}-\hat{j}+\hat{k})$ is $\sqrt{\frac{2}{3}}$ then the integral value of...

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Q JEE MAIN 2022

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...

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Q JEE MAIN 2022

Let a line having direction ratios $1,-4,2$ intersect the lines $\frac{x-7}{3}=\frac{y-1}{-1}=\frac{z+2}{1}$ and $\frac{x}{2}=\frac{y-7}{3}=\frac{z}{1}$ at the point $A$ and $B$. Then...

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Q JEE MAIN 2022

Let $\vec{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k} \quad a_i>0, i=1,2,3$ be a vector which makes equal angles with the coordinates axes OX,...

JEE Main Mathematics Hard

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Q JEE MAIN 2022

Let $\hat{a}, \hat{b}$ be unit vectors. If $\vec{c}$ be a vector such that the angle between $\hat{a}$ and $\hat{c}$ is...

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Q JEE MAIN

Let $P_1$ be the plane $3 x-y-7 z=11$ and $P_2$ be the plane passing through the points $(2,-1,0),(2,0,-1)$, and $(5,1,1)$....

JEE Main Mathematics Hard

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Q JEE-Main 2023

Let the vectors $\overrightarrow{\mathrm{u}}_1=\hat{\mathrm{i}}+\hat{\mathrm{j}}+a \hat{\mathrm{k}}, \overrightarrow{\mathrm{u}}=\hat{\mathrm{i}}+b \hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{u}}_3=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}$ be coplanar. If the vectors $...

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Q JEE MAIN 2025

Let the area of the triangle formed by the lines $x+2=y-1=z, \frac{x-3}{5}=\frac{y}{-1}=\frac{z-1}{1}$ and $\frac{x}{-3}=\frac{y-3}{3}=\frac{z-2}{1}$ be $A$. Then $A^2$ is equal...

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Q JEE-Main 2023

For $\mathrm{a}, \mathrm{b} \in \mathrm{Z}$ and $|\mathrm{a}-\mathrm{b}| \leq 10$, let the angle between the plane $\mathrm{P}: \mathrm{ax}+\mathrm{y}-\mathrm{z}=\mathrm{b}$ and the line...

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Q JEE MAIN 2023

Let the foot of perpendicular from the point $\mathrm{A}(4,3,1)$ on the plane $\mathrm{P}: \mathrm{x}-\mathrm{y}+2 \mathrm{z}+3=0$ be N . If $...

JEE Main Mathematics Medium

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Q JEE-Main 2023

The area of the quadrilateral ABCD with vertices A(2, 1, 1), B(1, 2, 5), C(–2, –3, 5) and D(1, –6,...

JEE Main Mathematics Easy

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Q JEE MAIN 2023

Let the line $=\frac{x}{1}=\frac{6-y}{2}=\frac{z+8}{5}$ intersect the lines $\frac{x-5}{4}=\frac{y-7}{3}=\frac{z+2}{1}$ and $\frac{x+3}{6}=\frac{3-y}{3}=\frac{z-6}{1}$ at the points $A$ and $B$ respectively. Then the distance...

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Q JEE MAIN 2023

If the lines and intersect, then the magnitude of the minimum value of 8is _________.

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Q JEE MAIN 2023

Let for a triangle ABC , $$ \begin{aligned} & \overrightarrow{\mathrm{AB}}=-2 \hat{\mathrm{i}}+\hat{\mathrm{j}}+3 \hat{\mathrm{k}} ; \\ & \overrightarrow{\mathrm{CB}}=\alpha \hat{\mathrm{i}}+\beta \hat{\mathrm{j}}+\gamma \hat{\mathrm{k}} \\...

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Q JEE MAIN 2023

Let the line L pass through the point $(0,1,2)$, intersect the line $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and be parallel to the plane $...

JEE Main Mathematics Hard

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Q JEE MAIN 2023

Let N be the foot of perpendicular from the point P(1, –2, 3) on the line passing through the points...

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Q JEE MAIN 2023

If the points $P$ and $Q$ are respectively the circumcentre and the orthocentre of a $\triangle A B C$, then...

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Q JEE MAIN 2023

Let $\vec{a}=2 \hat{i}+7 \hat{j}-\hat{k}, \vec{b}=3 \hat{i}+5 \hat{k}$ and $\vec{c}=\hat{i}-\hat{j}+2 \hat{k}$. Let $\vec{d}$ be a vector which is perpendicular to both...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Let $|\vec{a}|=2,|\vec{b}|=3$ and the angle between the vectors $\vec{a}$ and $\vec{b}$ be $\frac{\pi}{4}$. Then $|(\vec{a}+2 \vec{b}) \times(2 \vec{a}-3 \vec{b})|^2$ is...

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Q JEE-Main 2023

Let P be the plane passing through the line $\frac{x-1}{1}=\frac{y-2}{-3}=\frac{z+5}{7}$ and the point (2, 4, –3). If the image of...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let the values of $\lambda$ for which the shortest distance between the lines $\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $\frac{x-\lambda}{3}=\frac{y-4}{4}=\frac{z-5}{5}$ is $\frac{1}{\sqrt{6}}$ be $\lambda_1$...

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Q JEE MAIN 2023

The sum of all values of a, for which the points whose position vectors $...

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Q JEE MAIN 2023

The plane, passing through the points (0, –1, 2) and (–1, 2, 1) and parallel to the line passing through...

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Q JEE MAIN 2025

Let $\vec{a}=\hat{i}+2 \hat{j}+\hat{k}$ and $\vec{b}=2 \hat{i}+\hat{j}-\hat{k}$. Let $\hat{c}$ be a unit vector in the plane of the vectors $\vec{a}$ and...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Let the image of the point $\mathrm{P}(1,2,6)$ in the plane passing through the points $\mathrm{A}(1,2,0), \mathrm{B}(1,4,1)$ and $\mathrm{C}(0,5,1)$ be $...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let $\vec{a}=2 \hat{i}-3 \hat{j}+\hat{k}, \vec{b}=3 \hat{i}+2 \hat{j}+5 \hat{k}$ and a vector $\vec{c}$ be such that $...

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Q JEE MAIN 2025

Let $\vec{a}=\hat{i}+2 \hat{j}+\hat{k}, \vec{b}=3 \hat{i}-3 \hat{j}+3 \hat{k}, \vec{c}=2 \hat{i}-\hat{j}+2 \hat{k}$ and $\vec{d}$ be a vector such that $...

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Q JEE MAIN 2025

Let the three sides of a triangle ABC be given by the vectors $2 \hat{i}-\hat{j}+\hat{k}, \hat{i}-3 \hat{j}-5 \hat{k}$ and $...

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Q JEE MAIN 2025

Each of the angles $\beta$ and $\gamma$ that a given line makes with the positive $y$ - and $z$-axes, respectively,...

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Q JEE MAIN_2025

If the equation of the line passing through the point $\left(0,-\frac{1}{2}, 0\right)$ and perpendicular to the lines $...

JEE Main Physics Hard

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Q JEE MAIN_2025

Let $\vec{a}$ and $\vec{b}$ be the vectors of the same magnitude such that $\frac{|\vec{a}+\vec{b}|+|\vec{a}-\vec{b}|}{|\vec{a}+\vec{b}|-|\vec{a}-\vec{b}|}=\sqrt{2}+1$. Then $\frac{|\vec{a}+\vec{b}|^2}{|\vec{a}|^2}$ is :

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Q JEE MAIN 2025

Let $A$ be the point of intersection of the lines $L_1: \frac{x-7}{1}=\frac{y-5}{0}=\frac{z-3}{-1}$ and $L_2: \frac{x-1}{3}=\frac{y+3}{4}=\frac{z+7}{5}$. Let $B$ and $C$ be...

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Q JEE MAIN_2025

Consider the lines $\mathrm{L}_1: x-1=y-2=\mathrm{z}$ and $\mathrm{L}_2: x-2=y=\mathrm{z}-1$. Let the feet of the perpendiculars from the point $P(5,1,-3)$ on the...

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Q JEE MAIN 2025

The distance of the point $(7,10,11)$ from the line $\frac{x-4}{1}=\frac{y-4}{0}=\frac{z-2}{3}$ along the line $\frac{x-9}{2}=\frac{y-13}{3}=\frac{z-17}{6}$ is

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Q JEE MAIN 2025

The line $L_1$ is parallel to vector $\vec{a}=-3 \hat{i}+2 \hat{j}+4 \hat{k}$ and passes through the point $(7,6,2)$ and the line...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let the values of p , for which the shortest distance between the lines $\frac{x+1}{3}=\frac{y}{4}=\frac{z}{5}$ and $...

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Q JEE MAIN 2025

If the image of the point $\mathrm{P}(1,0,3)$ in the line joining the points $\mathrm{A}(4,7,1)$ and $\mathrm{B}(3,5,3)$ is $Q(\alpha, \beta, \gamma)$,...

JEE Main Mathematics Easy

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Q JEE MAIN 2023

Let $\vec{a}=3 \hat{i}+\hat{j}-\hat{k}$ and $\vec{c}=2 \hat{i}-3 \hat{j}+3 \hat{k}$. If $\vec{b}$ is a vector such that $\vec{a}=\vec{b} \times \vec{c}$ and $|\vec{b}|^2=50$,...

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Q JEE MAIN 2023

Let the image of the point $\left(\frac{5}{3}, \frac{5}{3}, \frac{8}{3}\right)$ in the plane $x-2 y+z-2=0$ be $P$. If the distance of...

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Q JEE MAIN 2023

Let the equation of plane passing through the line of intersection of the planes $x+2 y+a z=2$ and $x-y +z=3$...

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Q JEE MAIN 2023

Let the plane $P$ contain the line $2 x+y-z-3=0=5 x-3 y+4 z+9$ and be parallel to the line $\frac{x+2}{2}=\frac{3-y}{-4}=\frac{z-7}{5}$. Then...

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Q JEE MAIN 2023

Let S be the set of all $(\lambda, \mu)$ for which the vectors $\lambda \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\mu \hat{\mathrm{k}}$ and $...

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Q JEE MAIN 2023

Let the foot of perpendicular of the point $\mathrm{P}(3,-2-9)$ on the plane passing through the points $(-1,-2,-3),(9,3,4),(9,-2,1)$ be $...

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Q JEE MAIN 2023

The shortest distance between the lines $\frac{x+2}{1}=\frac{y}{-2}=\frac{z-5}{2}$ and $\frac{x-4}{1}=\frac{y-1}{-2}=\frac{z+3}{0}$ is

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Q JEE MAIN 2023

Let $\vec{a}=\hat{i}+4 \hat{j}+2 \hat{k}, \vec{b}=3 \hat{i}-2 \hat{j}+7 \hat{k}$ and $\vec{c}=2 \hat{i}-\hat{j}+4 \hat{k}$. If a vector $\vec{d}$ satisfies $...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Let $A B C D$ be a quadrilateral. If $E$ and $F$ are the mid points of the diagonals $...

JEE Main Mathematics Easy

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Q JEE-Main 2023

If the equation of the plane passing through the line of intersection of the planes 2x - y + z...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Let S be the set of all values of $\lambda$, for which the shortest distance between the lines $\frac{x-\lambda}{0}=\frac{y-3}{4}=\frac{x+6}{1}$ and...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

The distance of the point $(-1,2,3)$ from the plane $\vec{r} \cdot(\hat{i}-2 \hat{j}+3 \hat{k})=10$ parallel to the line of the shortest...

JEE Main Mathematics Medium

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Q JEE-Main 2023

Let $\vec{a}=2 \hat{i}+3 \hat{j}+4 \hat{k}, \vec{b}=2 \hat{i}-2 \hat{j}-2 \hat{k}$ and $\vec{c}=-\hat{i}+4 \hat{j}+3 \hat{k}$. If $\vec{d}$ is a vector perpendicular to...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Let a line $I$ pass through the origin and be perpendicular to the lines $...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Let the plane $x+3 y-2 z+6=0$ meet the co-ordinate axes at the points A,B,C. It the orthocentre of the triangle...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Let $\lambda_1, \lambda_2$ be the values of $\lambda$ for which the points $\left(\frac{5}{2}, 1, \lambda\right)$ and $(-2,0,1)$ are at equal...

JEE Main Mathematics Easy

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Q JEE MAIN 2023

Let $\vec{a}$ be a non-zero vector parallel to the line of intersection of the two planes described by $\hat{i}+\hat{j}, \hat{i}+\hat{k}$...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Let $(\alpha, \beta, \gamma)$ be the image of the point $\mathrm{P}(2,3,5)$ in the plane $2 \mathrm{x}+\mathrm{y}-3 \mathrm{z}=6$. Then $\alpha+\beta+\gamma$ is...

JEE Main Mathematics Easy

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Q JEE MAIN 2023

Let $\vec{a}=6 \hat{i}+9 \hat{j}+12 \hat{k}, \vec{b}=\alpha \hat{i}+11 \hat{j}-2 \hat{k}$, and $\vec{c}$ be vectors such that $...

JEE Main Mathematics Easy

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Q JEE MAIN 2023

Let the plane $P: 4 x-y+z=10$ be rotated by an angle $\frac{\pi}{2}$ about its line of intersection with the plane...

JEE Main Mathematics Easy

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Q JEE MAIN 2023

Let $\lambda \in Z, \vec{a}=\lambda \hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=3 \hat{i}-\hat{j}+2 \hat{k}$. Let $\vec{c}$ be a vector such that $...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

The shortest distance between the lines $\frac{x-4}{4}=\frac{y+2}{5}=\frac{z+3}{3}$ and $\frac{x-1}{3}=\frac{y-3}{4}=\frac{z-4}{2}$ is -

JEE Main Mathematics Easy

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Q JEE MAIN 2023

For any vector $\vec{a}=a_1 \hat{i}+a_2 \hat{j}+a_3 \hat{k}$, with $10\left|a_i\right|

JEE Main Mathematics Easy

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Q JEE MAIN 2023

Let $a, b, c$ be three distinct real numbers, none equal to one. If the vectors $a \hat{i}+\hat{j}+\hat{k}, \hat{i}+b \hat{j}+\hat{k}$...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

If the points with vectors $\alpha \hat{\mathrm{i}}+10 \hat{\mathrm{j}}+13 \hat{\mathrm{k}}, 6 \hat{\mathrm{i}}+11 \hat{\mathrm{j}}+11 \hat{\mathrm{k}}, \frac{9}{2} \hat{\mathrm{i}}+\beta \hat{\mathrm{j}}-8 \hat{\mathrm{k}}$ are collinear, then...

JEE Main Mathematics Easy

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Q JEE MAIN 2023

Let the lines $\ell_1: \frac{x+5}{3}=\frac{y+4}{1}=\frac{z-\alpha}{-2}$ and $\ell_2: 3 x+2 y+z-2=0=x-3 y+2 z-13$ be coplanar. If the point $\mathrm{P}(\mathrm{a}, \mathrm{b}, \mathrm{c})$...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

An are PQ of a circle subtends a right angle at its centre O . The mid point of the...

JEE Main Mathematics Easy

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Q JEE-Main 2023

Let the image of the point P(1, 2, 3) in the plane 2x – y + z = 9 be...

JEE Main Mathematics Easy

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Q JEE MAIN 2023

If equation of the plane that contains the point $(-2,3,5)$ and is perpendicular to each of the planes $...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Let two vertices of triangle ABC be $(2,4,6)$ and $(0,-2,-5)$, and its centroid be $(2,1,-1)$. If the image of third...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Let $P$ be the point of intersection of the line $\frac{x+3}{3}=\frac{y+2}{1}=\frac{1-z}{2}$ and the plane $x+y+z=2$. If the distance of the...

JEE Main Mathematics Easy

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Q JEE MAIN 2023

If the equation of the plane containing the line $x+2 y+3 z-4=0=2 x+y-z+5$ and perpendicular to the plane $...

JEE Main Mathematics Medium

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Q JEE-Main 2023

Let the position vectors of the points A, B, C and D be $...

JEE Main Mathematics Medium

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Q JEE-Main 2023

One vertex of a rectangular parallelopiped is at the origin O and the lengths of its edges along x, y...

JEE Main Mathematics Easy

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Q JEE MAIN 2023

Let $O$ be the origin and the position vector of the point $P$ be $-\hat{i}-2 \hat{j}+3 \hat{k}$. If the position...

JEE Main Mathematics Easy

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Q JEE-Main 2023

If the equation of the plane passing through the line of intersection of the planes 2x - y + z...

JEE Main Mathematics Medium

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Q JEE-Main 2023

Let $\vec{a}=2 \hat{i}+3 \hat{j}+4 \hat{k}, \vec{b}=2 \hat{i}-2 \hat{j}-2 \hat{k}$ and $\vec{c}=-\hat{i}+4 \hat{j}+3 \hat{k}$. If $\vec{d}$ is a vector perpendicular to...

JEE Main Mathematics Medium

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Q JEE MAIN

Let $...

JEE Main Mathematics Hard

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Q JEE-Main 2024

Let $\overrightarrow{\mathrm{a}}=2 \hat{\imath}-3 \hat{\jmath}+4 \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=3 \hat{\imath}+4 \hat{\jmath}-5 \hat{\mathrm{k}}$, and a vector $\vec{c}$ be such that $...

JEE Main Mathematics Easy

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Q JEE MAIN

Let three vectors $\vec{a}=\alpha \hat{\imath}+4 \hat{\jmath}+2 \hat{k}, \vec{b}=5 \hat{\imath}+3 \hat{\jmath}+4 \hat{k}, \vec{c}=x \hat{\imath}+y \hat{\jmath}+z \hat{k}$ from a triangle such that...

JEE Main Mathematics Easy

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Q JEE MAIN

The shortest distance between the line $\frac{x-3}{4}=\frac{y+7}{-11}=\frac{z-1}{5}$ and $\frac{x-5}{3}=\frac{y-9}{-6}=\frac{z+2}{1}$ is :

JEE Main Mathematics Easy

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Q JEE MAIN 2024

Let $\vec{a}=9 \hat{\imath}-13 \hat{\jmath}+25 k, b=3 \hat{\imath}+7 \hat{\jmath}-13 k$ and $\overrightarrow{\mathrm{c}}=17 \hat{\imath}-2 \hat{\jmath}+\mathrm{k}$ be three given vectros. If $\overrightarrow{\mathrm{r}}$ is...

JEE Main Mathematics Easy

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Q JEE MAIN 2024

Let $\vec{a}=\hat{i}-3 \hat{j}+7 k \hat{k}=2 \hat{i}-\hat{j}+k$ and $\vec{c}$ be a vector such that $(\vec{a}+2 \vec{b}) \times \vec{c}=3(\vec{c} \times \vec{a})$. If...

JEE Main Mathematics Easy

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Q JEE MAIN 2024

Let ABC be a triangle of area $15 \sqrt{2}$ and the vectors $\overrightarrow{\mathrm{AB}}=\hat{\imath}+2 \hat{\jmath}-7 \hat{k}, \overrightarrow{\mathrm{BC}}=a \hat{\imath}+b \hat{\jmath}+c \hat{k}$ and...

JEE Main Mathematics Easy

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Q JEE MAIN

Let $\overrightarrow{\mathrm{OA}}=2 \overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{OB}}=6 \overrightarrow{\mathrm{a}}+5 \overrightarrow{\mathrm{~b}}$ and $\overrightarrow{\mathrm{OC}}=3 \overrightarrow{\mathrm{~b}}$, where O is the origin. If the area of the parallelogram...

JEE Main Mathematics Easy

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Q JEE MAIN 2024

If the shortest distance between the lines $\frac{x+2}{2}=\frac{y+3}{3}=\frac{z-5}{4}$ and $\frac{x-3}{1}=\frac{y-2}{-3}=\frac{z+4}{2}$ is $\frac{38}{3 \sqrt{5}} \mathrm{k}$ and $\int_0^{\mathrm{k}}\left[\mathrm{x}^2\right] \mathrm{dx}=\alpha- \sqrt{\alpha}$, where $[\mathrm{x}]$...

JEE Main Mathematics Medium

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Q JEE MAIN 2024

If the line (2-x)/3=(3y-2)/(4λ+1)=4-z makes a right angle with the line (x+3)/3μ=(1-2y)/6=(5-z)/7, then 4λ+9μ is equal to :

JEE Main Mathematics Easy

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Q JEE MAIN 2024

The set of all $\alpha$, for which the vector $\vec{a}=\alpha t \hat{\imath}+6 \hat{\jmath}-3 \hat{k}$ and $...

JEE Main Mathematics Easy

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Q JEE MAIN 2024

If A(1,-1,2),B(5,7,-6),C(3,4,-10) and D(-1,-4,-2) are the vertices of a quadrilateral ABCD, then its area is

JEE Main Mathematics Easy

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Q JEE MAIN 2024

Let the point, on the line passing through the points $\mathrm{P}(1,-2,3)$ and $\mathrm{Q}(5,-4,7)$, farther from the origin and at a...

JEE Main Mathematics Easy

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Q JEE MAIN

Let the line L intersect the lines x-2=-y=z-1,2(x+1)=2(y-1)=z+1 and be parallel to the line $\frac{x-2}{3}=\frac{y-1}{1}=\frac{z-2}{2}$. Then which of the following...

JEE Main Mathematics Medium

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Q JEE-Main 2024

The shortest distance between the lines $\frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5}$ and $\frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3}$ is $\_\_\_\_$

JEE Main Mathematics Easy

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Q JEE MAIN 2024

Let a unit vector which makes an angle of $60^{\circ}$ with $2 \hat{\imath}+2 \hat{\jmath}-\hat{k}$ and an angle of $45^{\circ}$ with...

JEE Main Mathematics Easy

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Q JEE MAIN 2024

If the shortest distance between the lines L_1:r ⃗=(2+λ)i ˆ+(1-3λ)j ˆ+(3+4λ)k ˆ,λ∈R L_2:r ⃗=2(1+μ)i ˆ+3(1+μ)j ˆ+(5+μ)k ˆ,μ∈R is m/√n, where...

JEE Main Mathematics Easy

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Q JEE MAIN 2024

Let $\mathrm{P}(\mathrm{x}, \mathrm{y}, \mathrm{z})$ be a point in the first octant, whose projection in the xy -plane is the point...

JEE Main Mathematics Medium

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Q JEE-Main 2024

If $\mathrm{A}(3,1,-1), \mathrm{B}\left(\frac{5}{3}, \frac{7}{3}, \frac{1}{3}\right), \mathrm{C}(2,2,1)$ and $\mathrm{D}\left(\frac{10}{3}, \frac{2}{3}, \frac{-1}{3}\right)$ are the vertices of a quadrilateral ABCD , then its...

JEE Main Mathematics Easy

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Q JEE MAIN

Let $\vec{a}, \vec{b}$ and $\overrightarrow{\mathrm{c}}$ be three non-zero vectors such that $\vec{b}$ and $\vec{c}$ are non-collinear if $\vec{a}+5 \vec{b}$is collinear...

JEE Main Mathematics Medium

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Q JEE MAIN 2024

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}|=1,|\vec{b}|=4$ and $\vec{a} \cdot \vec{b}=2$. If $\vec{c}=(2 \vec{a} \times \vec{b})-3 \vec{b}$...

JEE Main Mathematics Easy

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Q JEE-Main 2024

Let the line of the shortest distance between the lines $\mathrm{L}_1: \vec{r}=(\hat{\imath}+2 \hat{\jmath}+3 \hat{k})+\lambda(\hat{\imath}-\hat{\jmath}+\hat{k})$ and $...

JEE Main Mathematics Easy

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Q JEE MAIN 2024

Let $Q$ and $R$ be the feet of perpendiculars from the point $P(a, a, a)$ on the lines $x=y, z=1$...

JEE Main Mathematics Easy

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Q JEE MAIN 2024

If d_1 is the shortest distance between the lines x+1=2y=-12z,x=y+2=6z-6 and d_2 is the shortest distance between the lines (x-1)/2=(y+8)/(-7)=(z-4)/5,(x-1)/2=(y-2)/1=(z-6)/(-3),...

JEE Main Mathematics Medium

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Q JEE MAIN 2024

Let A(2,3,5) and C(-3,4,-2) be opposite vertices of a parallelogram ABCD if the diagonal (BD) ⃗=i ˆ+2j ˆ+3k ˆ then...

JEE Main Mathematics Easy

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Q JEE MAIN

A line with direction ratios 2,1,2 meets the lines x=y+2=z and x+2=2y=2z respectively at the point P and Q. if...

JEE Main Mathematics Medium

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Q JEE-Main 2024

If the shortest distance between the lines $\frac{x-\lambda}{-2}=\frac{y-2}{1}=\frac{z-1}{1}$ and $\frac{x-\sqrt{3}}{1}=\frac{y-1}{-2}=\frac{z-2}{1}$ is 1 , then the sum of all possible values...

JEE Main Mathematics Easy

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Q JEE MAIN 2024

The distance of the point $Q(0,2,-2)$ form the line passing through the point $P(5,-4,3)$ and perpendicular to the lines $...

JEE Main Mathematics Medium

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Q JEE MAIN 2024

Let $\vec{a}=3 \hat{\imath}+\hat{\jmath}-2 \hat{k}, \vec{b}=4 \hat{\imath}+\hat{\jmath}+7 \hat{k}$ and $\overrightarrow{\mathrm{c}}=\hat{\imath}-3 \hat{\jmath}+4 \hat{k}$ be three vectors. If a vectors $\overrightarrow{\mathrm{p}}$ satisfies $...

JEE Main Mathematics Easy

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Q JEE MAIN

Let PQR be a triangle with R(-1,4,2). Suppose M(2,1,2) is the mid point of PQ. The distance of the centroid...

JEE Main Mathematics Easy

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Q JEE MAIN 2024

Let (α,β,γ) be the foot of perpendicular from the point (1,2,3) on the line (x+3)/5=(y-1)/2=(z+4)/3. then 19(α+β+γ) is equal to...

JEE Main Mathematics Easy

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Q JEE-Main 2024

Let $\vec{a}=-5 \hat{\imath}+\hat{\jmath}-3 \hat{k}, \vec{b}=\hat{\imath}+2 \hat{\jmath}-4 \hat{k}$ and $\vec{c}=(((\vec{a} \times \vec{b}) \times \hat{\imath}) \times \hat{\imath}) \times \hat{\imath}$. Then $\vec{c} \cdot(-\hat{\imath}+\hat{\jmath}+\hat{k})$...

JEE Main Mathematics Medium

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Q JEE MAIN

Let O be the origin and the position vector of A and B be $2 \hat{\imath}+2 \hat{\jmath}+\hat{k}$ and $...

JEE Main Mathematics Easy

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Q JEE MAIN 2024

Let $\vec{a}=a_i \hat{\imath}+a_2 \hat{\jmath}+a_3 \hat{k}$ and $\vec{b}=b_1 \hat{\imath}+b_2 \hat{\jmath}+b_3 \hat{k}$ be two vectors such that $|\vec{a}|=1 ; \vec{a} \cdot \vec{b}=2$...

JEE Main Mathematics Hard

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Q JEE MAIN 2024

The least positive integral value of $\alpha$, for which the angle between the vectors $\alpha \hat{\imath}-2 \hat{\jmath}+2 k$ and $...

JEE Main Mathematics Easy

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Q JEE MAIN 2024

Let $\overrightarrow{\mathrm{a}}=\hat{\imath}+2 \hat{\jmath}+\mathrm{k}, \overrightarrow{\mathrm{b}}=3(\hat{\imath}-\hat{\jmath}+\mathrm{k})$. Let $\vec{c}$ be the vector such that $\vec{a} \times \vec{c}=\vec{b}$ and $\vec{a} \cdot \vec{c}=3$. Then $...

JEE Main Mathematics Medium

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Q JEE MAIN 2024

If the shortest distance between the lines $\frac{x-4}{1}=\frac{y+1}{2}=\frac{z}{-3}$ and $\frac{x-\lambda}{2}=\frac{y+1}{4}=\frac{z-2}{-5}$ is $\frac{6}{\sqrt{5}}$, then the sum of all possible values of...

JEE Main Mathematics Easy

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Q JEE MAIN 2024

The distance, of the point (7,-2,11) from the line $\frac{x-6}{1}=\frac{y-4}{0}=\frac{z-8}{3}$ along the line $\frac{x-5}{2}=\frac{y-1}{-3}=\frac{z-5}{6}$, is :

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let $\hat{\mathrm{a}}$ be a unit vector perpendicular to the vectors $\overrightarrow{\mathrm{b}}=\hat{i}-2 \hat{j}+3 \hat{k}$ and $\overrightarrow{\mathrm{c}}=2 \hat{i}+3 \hat{j}-\hat{k}$, and makes an...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let P be the foot of the perpendicular from the point $(1,2,2)$ on the line $\mathrm{L}: \frac{x-1}{1}=\frac{y+1}{-1}=\frac{z-2}{2}$. Let the line...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

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...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

If the components of $\overrightarrow{\mathbf{a}}=\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$ along and perpendicular to $\overrightarrow{\mathbf{b}}=3 \hat{i}+\hat{j}-\hat{k}$ respectively, are $\frac{16}{11}(3 \hat{i}+\hat{j}-\hat{k})$ and $...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

If the shortest distance between the lines $\frac{{x - 1}}{2} = \frac{{y - 2}}{3} = \frac{{z - 3}}{4}$ and $...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let the angle $\theta ,0 < \theta < \frac{\pi }{2}$ between two-unit vectors $\hat a$ and $\hat b$ be $...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let $\mathrm{A}, \mathrm{B}, \mathrm{C}$ be three points in $\sqrt{3} \hat{i}+\hat{j}, \hat{i}+\sqrt{3} \hat{j}$ and $\mathrm{a} \hat{i}+(1-\mathrm{a}) \hat{j}$ respectively with respect to...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

The square of the distance of the point $\left(\frac{15}{7}, \frac{32}{7}, 7\right)$ from the line $\frac{x+1}{3}=\frac{y+3}{5}=\frac{z+5}{7}$ in the direction of the...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let the line L pass through (1,1,1) and intersect the lines $...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let the shortest distance between the lines $...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let A and B be two distinct points on the line $...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Consider two vectors $\vec u = 3\hat i - \hat j$ and $...

JEE Main Physics Easy

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Q JEE MAIN 2025

Let $...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Line ${L_1}$ passes through the point (1,2,3) and is parallel to z-axis. Line ${L_2}$ passes through the point $(\lambda ,5,6)$...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let a line passing through the point (4, 1, 0) intersect the line $...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let $P$ be the image of the point $Q(7,-2,5)$ in the line $L: \frac{x-1}{2}=\frac{y+1}{3}=\frac{z}{4}$ and $R(5, p, q)$ be a...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let the position vectors of three vertices of a triangle be $4 \vec{p}+\vec{q}-3 \vec{r},-5 \vec{p}+\vec{q}+2 \vec{r}$ and $2 \vec{p}-\vec{q}+2 \vec{r}$....

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let $\overrightarrow{\mathrm{a}}=3 \hat{i}-\hat{j}+2 \hat{k}, \overrightarrow{\mathrm{~b}}=\overrightarrow{\mathrm{a}} \times(\hat{i}-2 \hat{k})$ and $\overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{b}} \times \hat{k}$. Then the projection of $\overrightarrow{\mathrm{c}}-2 \hat{j}$ on $\vec{a}$ is:

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let ABCD be a tetrahedron such that the edges AB, AC and AD are mutually perpendicular. Let the areas of...

JEE Main Mathematics Hard

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Q JEE MAIN 2025

If $\vec a$ is a nonzero vector such that its projections on the vectors $...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let the vertices Q and R of the triangle PQR lie on the line $...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

The distance of the line $\frac{x-2}{2}=\frac{y-6}{3}=\frac{z-3}{4}$ from the point $(1,4,0)$ along the line $\frac{x}{1}=\frac{y-2}{2}=\frac{z+3}{3}$ is :

JEE Main Mathematics Easy

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Q JEE MAIN 2025

If the square of the shortest distance between the lines $\frac{x-2}{1}=\frac{y-1}{2}=\frac{z+3}{-3}$ and $\frac{x+1}{2}=\frac{y+3}{4}=\frac{z+5}{-5}$ is $\frac{m}{n}$, where $m, n$ are coprime...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let the point A divide the line segment joining the points $P(-1,-1,2)$ and $Q(5,5,10)$ internally in the ratio $r: 1(r>0)$....

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let ${{\rm{L}}_1}:\frac{{x - 1}}{1} = \frac{{y - 2}}{{ - 1}} = \frac{{z - 1}}{2}$ and $...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let $\overrightarrow {\rm{a}} = \hat i + 2\hat j + \hat k$ and $...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let $...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let $...

JEE Main Mathematics Hard

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Q JEE MAIN 2025

Let ${\rm{A}}(x,y,z)$ be a point in xy-plane, which is equidistant from three points (0,3,2),(2,0,3) and (0,0,1). Let $...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

If the image of the point (4,4,3) in the line $...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let in a $\Delta ABC$, the length of the side AC be 6, the vertex B be (1, 2, 3)...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let $...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let the line passing through the points (-1, 2, 1) and parallel to the line $...

JEE Main Mathematics Easy

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Q JEE MAIN 2023

Let P be the foot of the perpendicular from the point $Q(10, - 3, - 1)$ on the line $...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let the position vectors of the vertices A, B and C of a tetrahedron A, B, C, D be $...

JEE Main Mathematics Hard

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Q JEE MAIN 2025

Let the arc AC of a circle subtend a right angle at the centre O . If the point B...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

The perpendicular distance, of the line $\frac{x-1}{2}=\frac{y+2}{-1}=\frac{z+3}{2}$ from the point $\mathrm{P}(2,-10,1)$, is :

JEE Main Mathematics Easy

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Q JEE MAIN

Let $\mathrm{L}_1: \frac{x-1}{3}=\frac{y-1}{-1}=\frac{z+1}{0}$ and $\mathrm{L}_2: \frac{x-2}{2}=\frac{y}{0}=\frac{z+4}{\alpha}, \alpha \in \mathbf{R}$, be two lines, which intersect at the point $B$. If $P$...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let a line pass through two distinct points $\mathrm{P}(-2,-1,3)$ and Q , and be parallel to the vector $...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let $\vec{c}$ be the projection vector of $\vec{b}=\lambda \hat{i}+4 \hat{k}, \lambda>0$, on the vector $\vec{a}=\hat{i}+2 \hat{j}+2 \hat{k}$. If $|\vec{a}+\vec{c}|=7$, then...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that the angle between them is $\frac{\pi}{3}$. If $\lambda \vec{a}+2 \vec{b}$...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let ${{\text{L}}_{1}}:\frac{{x-1}}{2}=\frac{{y-2}}{3}=\frac{{z-3}}{4}$ and ${{\text{L}}_{2}}:\frac{{x-2}}{3}=\frac{{y-4}}{4}=\frac{{z-5}}{5}$ be two lines. Then which of the following points lies on the line of the shortest...

JEE Main Mathematics Medium

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