Vector Algebra · JEE Main previous year questions

Let the equation $x(x+2)(12-k)=2$ have equal roots. Then the distance of the point $\left(k,\frac{k}{2}\right)$ from the line $3x+4y+5=0$ is

1. A.$15$
2. B.$12$
3. C.$5\sqrt{3}$
4. D.$15\sqrt{5}$

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 $\vec{b}\times \vec{d}=\vec{c}\times \vec{d}$ and $\vec{a}\cdot \vec{d}=4$. Then $|(\vec{a}\times \vec{d}){|}^2$ is equal to__________.

If $\vec{a}$ is a nonzero vector such that its projections on the vectors $2\hat{i}-\hat{j}+2\hat{k},\hat{i}+2\hat{j}-2\hat{k}$ and $\hat{k}$ are equal, then a unit vector along $\vec{a}$ is :

1. A.$\frac{1}{\sqrt{155}}(-7\hat{i}+9\hat{j}+5\hat{k})$
2. B.$\frac{1}{\sqrt{155}}(-7\hat{i}+9\hat{j}-5\hat{k})$
3. C.$\frac{1}{\sqrt{155}}(7\hat{i}+9\hat{j}-5\hat{k})$
4. D.$\frac{1}{\sqrt{155}}(7\hat{i}+9\hat{j}+5\hat{k})$

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 $(\vec{a}-\vec{c})\times \vec{b}=-18\hat{i}-3\hat{j}+12\hat{k}$ and $\vec{a}\cdot \vec{c}=3$. If $\vec{b}\times \vec{c}=\vec{d}$, then $|\vec{a}\cdot \vec{d}|$ is equal to :

1. A.15
2. B.18
3. C.12
4. D.9

Let $\hat{a}$ be a unit vector perpendicular to the vectors $\vec{b}=\hat{i}-2\hat{j}+3\hat{k}$ and $\vec{c}=2\hat{i}+3\hat{j}-\hat{k}$, and $\hat{a}$ makes an angle of ${\cos }^{-1}\left(-\frac{1}{3}\right)$ with the vector $\hat{i}+\hat{j}+\hat{k}$. If $\hat{a}$ makes an angle of $\frac{\pi }{3}$ with the vector $\hat{i}+\alpha \hat{j}+\hat{k}$, then the value of $\alpha$ is:

1. A.$\sqrt{3}$
2. B.$\sqrt{6}$
3. C.$-\sqrt{6}$
4. D.$-\sqrt{3}$

Let the line $x+y=1$ meet the axes of $x$ and $y$ at $A$ and $B$, respectively. A right angled triangle $AMN$ is inscribed in the triangle $OAB$, where $O$ is the origin and the points $M$ and $N$ lie on the lines $OB$ and $AB$, respectively. If the area of the triangle $AMN$ is $\frac{4}{9}$ of the area of the triangle $OAB$ and $AN:NB=\lambda :1$, then the sum of all possible value(s) of $\lambda$ is:

1. A.$\frac{1}{2}$
2. B.$\frac{5}{2}$
3. C.$2$
4. D.$\frac{13}{6}$

Let $\vec{a}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}},\vec{\mathrm{b}}=2\hat{\mathrm{i}}+2\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\vec{\mathrm{d}}=\vec{\mathrm{a}}\times \vec{\mathrm{b}}$. If $\vec{\mathrm{c}}$ is a vector such that $\vec{\mathrm{a}}\cdot \vec{\mathrm{c}}=|\vec{\mathrm{c}}|$, $|\vec{\mathrm{c}}-2\vec{\mathrm{a}}{|}^2=8$ and the angle between $\vec{\mathrm{d}}$ and $\vec{\mathrm{c}}$ is $\frac{\pi }{4}$, then $|10-3\vec{\mathrm{b}}\cdot \vec{\mathrm{c}}|+|\vec{\mathrm{d}}\times \vec{\mathrm{c}}{|}^2$ is equal to ____.

Let $\vec{a}=\hat{i}+2\hat{j}+3\hat{k},\vec{b}=3\hat{i}+\hat{j}-\hat{k}$ and $\vec{c}$ be three vectors such that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}$. If the vector $\vec{c}$ is perpendicular to $\vec{b}$ and $\vec{a}\cdot \vec{c}=5$, then $|\vec{c}|$ is equal to

1. A.$\sqrt{\frac{11}{6}}$
2. B.$\frac{1}{3\sqrt{2}}$
3. C.$18$
4. D.$16$

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 $2\vec{p}-\vec{q}+2\vec{r}$. If the position vectors of the orthocenter and the circumcenter of the triangle are $\frac{\vec{p}+\vec{q}+\vec{r}}{4}$ and $\alpha \vec{p}+\beta \vec{q}+\gamma \vec{r}$ respectively, then $\alpha +2\beta +5\gamma$ is equal to :

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

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

1. A.$2\sqrt{7}$
2. B.$3\sqrt{7}$
3. C.$\sqrt{14}$
4. D.$2\sqrt{14}$

Let the point $\mathrm{A}$ divide the line segment joining the points $\mathrm{P}(-1,-1,2)$ and $\mathrm{Q}(5,5,10)$ internally in the ratio $r:1(r>0)$. If $\mathrm{O}$ is the origin and $(\overrightarrow{\mathrm{OQ}}\cdot \overrightarrow{\mathrm{OA}})-\frac{1}{5}|\overrightarrow{\mathrm{OP}}\times \overrightarrow{\mathrm{OA}}{|}^2=10$, then the value of $r$ is :

1. A.$\sqrt{7}$
2. B.14
3. C.7
4. D.3

Let $A=\{(x,y)\in \mathbf{R}\times \mathbf{R}:|x+y|\ge 3\}$ and $B=\{(x,y)\in \mathbf{R}\times \mathbf{R}:|x|+|y|\le 3\}$. If $C=\{(x,y)\in A\cap B:x=0\text{ or }y=0\}$, then ${\sum }_{(x,y)\in C}|x+y|$ is :

1. A.18
2. B.24
3. C.15
4. D.12

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 the area of the parallelogram formed by the vectors $\vec{b}$ and $\vec{c}$ is _________.

Let three vectors $\vec{a}=\alpha \hat{i}+4\hat{j}+2\hat{k}$, $\vec{b}=5\hat{i}+3\hat{j}+4\hat{k}$, $\vec{c}=x\hat{i}+y\hat{j}+z\hat{k}$ form a triangle such that $\vec{c}=\vec{a}-\vec{b}$ and the area of the triangle is $5\sqrt{6}$. If $\alpha$ is a positive real number, then $|\vec{c}{|}^2$ is equal to:

1. A.14
2. B.12
3. C.16
4. D.10

Between the following two statements: Statement I : Let $\vec{a}=\hat{i}+2\hat{j}-3\hat{k}$ and $\vec{b}=2\hat{i}+\hat{j}-\hat{k}$. Then the vector $\vec{r}$ satisfying $\vec{a}\times \vec{r}=\vec{a}\times \vec{b}$ and $\vec{a}\cdot \vec{r}=0$ is of magnitude $\sqrt{10}$. Statement II : In a triangle $ABC$, $\cos 2A+\cos 2B+\cos 2C\ge -\frac{3}{2}$.

1. A.Both Statement I and Statement II are correct.
2. B.Both Statement I and Statement II are incorrect.
3. C.Statement I is correct but Statement II is incorrect.
4. D.Statement I is incorrect but Statement II is correct.

The set of all $\alpha$, for which the vectors $\vec{a}=\alpha t\hat{i}+6\hat{j}-3\hat{k}$ and $\vec{b}=t\hat{i}-2\hat{j}-2\alpha t\hat{k}$ are inclined at an obtuse angle for all $t\in \mathbb{R}$, is

1. A.$[0,1)$
2. B.$(-\frac{4}{3},0]$
3. C.$(-2,0]$
4. D.$(-\frac{4}{3},1)$

Let $\vec{a}=9\hat{i}-13\hat{j}+25\hat{k},\vec{b}=3\hat{i}+7\hat{j}-13\hat{k}$ and $\vec{c}=17\hat{i}-2\hat{j}+\hat{k}$ be three given vectors. If $\vec{r}$ is a vector such that $\vec{r}\times \vec{a}=(\vec{b}+\vec{c})\times \vec{a}$ and $\vec{r}\cdot (\vec{b}-\vec{c})=0$, then $\frac{|593\vec{r}+67\vec{a}{|}^2}{(593{)}^2}$ is equal to ________.

Let $\vec{a}=4\hat{i}-\hat{j}+\hat{k},\vec{b}=11\hat{i}-\hat{j}+\hat{k}$ and $\vec{c}$ be a vector such that $(\vec{a}+\vec{b})\times \vec{c}=\vec{c}\times (-2\vec{a}+3\vec{b})$. If $(2\vec{a}+3\vec{b})\cdot \vec{c}=1670$, then $|\vec{c}{|}^2$ is equal to:

1. A.1600
2. B.1618
3. C.1627
4. D.1609

Let $\vec{a}=\hat{i}+2\hat{j}+3\hat{k},\vec{b}=2\hat{i}+3\hat{j}-5\hat{k}$ and $\vec{c}=3\hat{i}-\hat{j}+\lambda \hat{k}$ be three vectors. Let $\vec{r}$ be a unit vector along $\vec{b}+\vec{c}$. If $\vec{r}\cdot \vec{a}=3$, then $3\lambda$ is equal to:

1. A.21
2. B.25
3. C.27
4. D.30

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

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

Let $\vec{a}=6\hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=\hat{i}+\hat{j}$. If $\vec{c}$ is a vector such that $|\vec{c}|\ge 6,\vec{a}\cdot \vec{c}=6|\vec{c}|,|\vec{c}-\vec{a}|=2\sqrt{2}$ and the angle between $\vec{a}\times \vec{b}$ and $\vec{c}$ is $60^{\circ }$, then $|(\vec{a}\times \vec{b})\times \vec{c}|$ is equal to:

1. A.$\frac{3}{2}\sqrt{6}$
2. B.$\frac{9}{2}(6-\sqrt{6})$
3. C.$\frac{9}{2}(6+\sqrt{6})$
4. D.$\frac{3}{2}\sqrt{3}$

Let $\vec{a}=2\hat{i}+5\hat{j}-\hat{k},\vec{b}=2\hat{i}-2\hat{j}+2\hat{k}$ and $\vec{c}$ be three vectors such that $(\vec{c}+\hat{i})\times (\vec{a}+\vec{b}+\hat{i})=\vec{a}\times (\vec{c}+\hat{i})$. If $\vec{a}\cdot \vec{c}=-29$, then $\vec{c}\cdot (-2\hat{i}+\hat{j}+\hat{k})$ is equal to:

1. A.15
2. B.10
3. C.5
4. D.12

Consider three vectors $\vec{a},\vec{b},\vec{c}$. Let $|\vec{a}|=2,|\vec{b}|=3$ and $\vec{a}=\vec{b}\times \vec{c}$. If $\alpha \in \left[0,\frac{\pi }{3}\right]$ is the angle between the vectors $\vec{b}$ and $\vec{c}$, then the minimum value of $27|\vec{c}-\vec{a}{|}^2$ is equal to:

1. A.124
2. B.110
3. C.121
4. D.105

Let $f(x)=\{\begin{array}{ll}-2,& -2\le x\le 0\\ x-2,& 0<x\le 2\end{array}$ and $h(x)=f(|x|)+|f(x)|$. Then ${\int }_{-2}^{2}h(x)dx$ is equal to:

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

Let a unit vector which makes an angle of $60^{\circ }$ with $2\hat{i}+2\hat{j}-\hat{k}$ and an angle of $45^{\circ }$ with $\hat{i}-\hat{k}$ be $\vec{C}$. Then $\vec{C}+\left(-\frac{1}{2}\hat{i}+\frac{1}{3\sqrt{2}}\hat{j}-\frac{\sqrt{2}}{3}\hat{k}\right)$ is:

1. A.$-\frac{\sqrt{2}}{3}\hat{i}+\frac{\sqrt{2}}{3}\hat{j}+\left(\frac{1}{2}+\frac{2\sqrt{2}}{3}\right)\hat{k}$
2. B.$\left(\frac{1}{\sqrt{3}}+\frac{1}{2}\right)\hat{i}+\left(\frac{1}{\sqrt{3}}-\frac{1}{3\sqrt{2}}\right)\hat{j}+\left(\frac{1}{\sqrt{3}}+\frac{\sqrt{2}}{3}\right)\hat{k}$
3. C.$\frac{\sqrt{2}}{3}\hat{i}-\frac{1}{2}\hat{k}$
4. D.$\frac{\sqrt{2}}{3}\hat{i}+\frac{1}{3\sqrt{2}}\hat{j}-\frac{1}{2}\hat{k}$