Vector Algebra · JEE Main previous year questions

Let $\alpha$ and $\beta$ be the sum and the product of all the non-zero solutions of the equation $(\bar{z}{)}^2+|z|=0,z\in C$. Then $4({\alpha }^2+{\beta }^2)$ is equal to :

1. A.4
2. B.2
3. C.6
4. D.8

Let $\mathrm{ABC}$ be a triangle of area $15\sqrt{2}$ and the vectors $\overrightarrow{\mathrm{AB}}=\hat{i}+2\hat{j}-7\hat{k}$, $\overrightarrow{\mathrm{BC}}=\mathrm{a}\hat{i}+\mathrm{b}\hat{j}+\mathrm{c}\hat{k}$ and $\overrightarrow{\mathrm{AC}}=6\hat{i}+\mathrm{d}\hat{j}-2\hat{k}$, $\mathrm{d}>0$. Then the square of the length of the largest side of the triangle $\mathrm{ABC}$ is ____.

For $\lambda >0$, let $\theta$ be the angle between the vectors $\vec{a}=\hat{i}+\lambda \hat{j}-3\hat{k}$ and $\vec{b}=3\hat{i}-\hat{j}+2\hat{k}$. If the vectors $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$ are mutually perpendicular, then the value of $(14\cos \theta {)}^2$ is equal to

1. A.25
2. B.50
3. C.20
4. D.40

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k},\vec{b}=2\hat{i}+4\hat{j}-5\hat{k}$ and $\vec{c}=x\hat{i}+2\hat{j}+3\hat{k},x\in \mathbb{R}$. If $\vec{d}$ is the unit vector in the direction of $\vec{b}+\vec{c}$ such that $\vec{a}\cdot \vec{d}=1$, then $(\vec{a}\times \vec{b})\cdot \vec{c}$ is equal to

1. A.3
2. B.9
3. C.11
4. D.6

Let $\overrightarrow{\mathrm{a}}=-5\hat{i}+\hat{j}-3\hat{k},\overrightarrow{\mathrm{b}}=\hat{i}+2\hat{j}-4\hat{k}$ and $\overrightarrow{\mathrm{c}}=(((\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}})\times \hat{i})\times \hat{i})\times \hat{i}$. Then $\vec{c}\cdot (-\hat{i}+\hat{j}+\hat{k})$ is equal to :

1. A.-12
2. B.-10
3. C.-13
4. D.-15

Consider a $△ABC$ where $A(1,3,2)$, $B(-2,8,0)$ and $C(3,6,7)$. If the angle bisector of $\angle BAC$ meets the line $BC$ at $D$, then the length of the projection of the vector $\overrightarrow{AD}$ on the vector $\overrightarrow{AC}$ is :

1. A.$\frac{37}{2\sqrt{38}}$
2. B.$\sqrt{19}$
3. C.$\frac{39}{2\sqrt{38}}$
4. D.$\frac{\sqrt{38}}{2}$

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$, $\vec{b}=-\hat{i}-8\hat{j}+2\hat{k}$ and $\vec{c}=4\hat{i}+c_2\hat{j}+c_3\hat{k}$ be three vectors such that $\vec{b}\times \vec{a}=\vec{c}\times \vec{a}$. If the angle between the vector $\vec{c}$ and the vector $3\hat{i}+4\hat{j}+\hat{k}$ is $\theta$, then the greatest integer less than or equal to ${\tan }^2\theta$ is ________ .

If $f(x)=\frac{4x+3}{6x-4},x\ne \frac{2}{3}$ and $(f\circ f)(x)=g(x)$, where $g:\mathbb{R}-\left\{\frac{2}{3}\right\}\to \mathbb{R}-\left\{\frac{2}{3}\right\}$, then $(g\circ g\circ g)(4)$ is equal to

1. A.$-4$
2. B.$\frac{19}{20}$
3. C.$-\frac{19}{20}$
4. D.4

Let $\vec{a}=3\hat{i}+\hat{j}-2\hat{k}$, $\vec{b}=4\hat{i}+\hat{j}+7\hat{k}$ and $\vec{c}=\hat{i}-3\hat{j}+4\hat{k}$ be three vectors. If a vector $\vec{p}$ satisfies $\vec{p}\times \vec{b}=\vec{c}\times \vec{b}$ and $\vec{p}\cdot \vec{a}=0$, then $\vec{p}\cdot (\hat{i}-\hat{j}-\hat{k})$ is equal to

1. A.24
2. B.32
3. C.36
4. D.28

The distance of the point $Q(0,2,-2)$ from the line passing through the point $P(5,-4,3)$ and perpendicular to the lines $\vec{r}=(-3\hat{i}+2\hat{k})+\lambda (2\hat{i}+3\hat{j}+5\hat{k}),\lambda \in \mathbb{R}$ and $\vec{r}=(\hat{i}-2\hat{j}+\hat{k})+\mu (-\hat{i}+3\hat{j}+2\hat{k}),\mu \in \mathbb{R}$ is :

1. A.$\sqrt{74}$
2. B.$\sqrt{86}$
3. C.$\sqrt{54}$
4. D.$\sqrt{20}$

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 $\vec{c}=(2\vec{a}\times \vec{b})-3\vec{b}$ and the angle between $\vec{b}$ and $\vec{c}$ is $\alpha$, then $192{\sin }^2\alpha$ is equal to ____.

Let $\vec{a}=3\hat{i}+2\hat{j}+\hat{k},\vec{b}=2\hat{i}-\hat{j}+3\hat{k}$ and $\vec{c}$ be a vector such that $(\vec{a}+\vec{b})\times \vec{c}=2(\vec{a}\times \vec{b})+24\hat{j}-6\hat{k}$ and $(\vec{a}-\vec{b}+\hat{i})\cdot \vec{c}=-3$. Then $|\vec{c}{|}^2$ is equal to ____.

Let $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ and $\vec{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ be two vectors such that $|\vec{a}|=1,\vec{a}\cdot \vec{b}=2$ and $|\vec{b}|=4$. If $\vec{c}=2(\vec{a}\times \vec{b})-3\vec{b}$, then the angle between $\vec{b}$ and $\vec{c}$ is equal to:

1. A.${\cos }^{-1}\left(-\frac{1}{\sqrt{3}}\right)$
2. B.${\cos }^{-1}\left(\frac{2}{3}\right)$
3. C.${\cos }^{-1}\left(\frac{2}{\sqrt{3}}\right)$
4. D.${\cos }^{-1}\left(-\frac{\sqrt{3}}{2}\right)$

Let $\vec{a},\vec{b}$ and $\vec{c}$ be three non-zero vectors such that $\vec{b}$ and $\vec{c}$ are non-collinear. If $\vec{a}+5\vec{b}$ is collinear with $\vec{c}$, $\vec{b}+6\vec{c}$ is collinear with $\vec{a}$ and $\vec{a}+\alpha \vec{b}+\beta \vec{c}=\vec{0}$, then $\alpha +\beta$ is equal to

1. A.$30$
2. B.$-30$
3. C.$-25$
4. D.$35$

Let $O$ be the origin and the position vectors of $A$ and $B$ be $2\hat{i}+2\hat{j}+\hat{k}$ and $2\hat{i}+4\hat{j}+4\hat{k}$ respectively. If the internal bisector of $\angle \text{AOB}$ meets the line AB at C, then the length of $OC$ is

1. A.$\frac{3}{2}\sqrt{34}$
2. B.$\frac{2}{3}\sqrt{31}$
3. C.$\frac{2}{3}\sqrt{34}$
4. D.$\frac{3}{2}\sqrt{31}$

Let a unit vector $\hat{u}=x\hat{i}+y\hat{j}+z\hat{k}$ make angles $\frac{\pi }{2},\frac{\pi }{3}$ and $\frac{2\pi }{3}$ with the vectors $\frac{1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{k},\frac{1}{\sqrt{2}}\hat{j}+\frac{1}{\sqrt{2}}\hat{k}$ and $\frac{1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{j}$ respectively. If $\vec{v}=\frac{1}{\sqrt{2}}\hat{i}+\frac{1}{\sqrt{2}}\hat{j}+\frac{1}{\sqrt{2}}\hat{k}$ then $|\hat{u}-\vec{v}{|}^2$ is equal to

1. A.$\frac{11}{2}$
2. B.$\frac{5}{2}$
3. C.7
4. D.9

Let $\vec{\mathrm{a}}=\hat{i}+2\hat{j}+\hat{k}$, $\vec{\mathrm{b}}=3(\hat{i}-\hat{j}+\hat{k})$. Let $\vec{\mathrm{c}}$ be the vector such that $\vec{\mathrm{a}}\times \vec{\mathrm{c}}=\vec{\mathrm{b}}$ and $\vec{\mathrm{a}}\cdot \vec{\mathrm{c}}=3$. Then $\vec{\mathrm{a}}\cdot ((\vec{\mathrm{c}}\times \vec{\mathrm{b}})-\vec{\mathrm{b}}-\vec{\mathrm{c}})$ is equal to :

1. A.32
2. B.36
3. C.24
4. D.20

Let the position vectors of the vertices A, B and C of a triangle be $2\hat{i}+2\hat{j}+\hat{k}$, $\hat{i}+2\hat{j}+2\hat{k}$ and $2\hat{i}+\hat{j}+2\hat{k}$ respectively. Let $l_1,l_2$ and $l_3$ be the lengths of perpendiculars drawn from the orthocentre of the triangle on the sides AB, BC and CA respectively, then $l_1^2+l_2^2+l_3^2$ equals:

1. A.$\frac{1}{4}$
2. B.$\frac{1}{5}$
3. C.$\frac{1}{3}$
4. D.$\frac{1}{2}$

The position vectors of the vertices A, B and C of a triangle are $2\hat{i}-3\hat{j}+3\hat{k}$, $2\hat{i}+2\hat{j}+3\hat{k}$ and $-\hat{i}+\hat{j}+3\hat{k}$ respectively. Let $l$ denote the length of the angle bisector AD of $\angle \text{BAC}$ where D is on the line segment BC, then $2l^2$ equals :

1. A.$45$
2. B.$50$
3. C.$42$
4. D.$49$

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 $\vec{d}\times \vec{b}=\vec{c}\times \vec{b}$ and $\vec{d}\cdot \vec{a}=24$, then $|\vec{d}{|}^2$ is equal to :

1. A.$313$
2. B.$413$
3. C.$423$
4. D.$323$

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 $|\vec{b}{|}^2=50$, then $|72-|\vec{b}+\vec{c}{|}^2|$ is equal to __________.

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 equal to :

1. A.$441$
2. B.$482$
3. C.$841$
4. D.$882$

The line, that is coplanar to the line $\frac{x+3}{-3}=\frac{y-1}{1}=\frac{z-5}{5}$, is :

1. A.$\frac{x+1}{-1}=\frac{y-2}{2}=\frac{z-5}{4}$
2. B.$\frac{x+1}{-1}=\frac{y-2}{2}=\frac{z-5}{5}$
3. C.$\frac{x-1}{-1}=\frac{y-2}{2}=\frac{z-5}{5}$
4. D.$\frac{x+1}{1}=\frac{y-2}{2}=\frac{z-5}{5}$

Let for a triangle ABC, $\overrightarrow{AB}=-2\hat{i}+\hat{j}+3\hat{k}$ $\overrightarrow{CB}=\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$ $\overrightarrow{CA}=4\hat{i}+3\hat{j}+\delta \hat{k}$ If $\delta >0$ and the area of the triangle ABC is $5\sqrt{6}$, then $\overrightarrow{CB}\cdot \overrightarrow{CA}$ is equal to

1. A.$60$
2. B.$54$
3. C.$120$
4. D.$108$

Let $a,b,c$ be three distinct real numbers, none equal to one. If the vectors $a\hat{i}+\hat{j}+\hat{k}$, $\hat{i}+b\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+c\hat{k}$ are coplanar, then $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is equal to :

1. A.$-2$
2. B.$1$
3. C.$-1$
4. D.$2$