Vector Algebra · JEE Main previous year questions

The vector $\vec{a}=-\hat{i}+2\hat{j}+\hat{k}$ is rotated through a right angle, passing through the y-axis in its way and the resulting vector is $\vec{b}$. Then the projection of $3\vec{a}+\sqrt{2}\vec{b}$ on $\vec{c}=5\hat{i}+4\hat{j}+3\hat{k}$ is :

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

Let $\vec{a},\vec{b}$ and $\vec{c}$ be three non zero vectors such that $\vec{b}\cdot \vec{c}=0$ and $\vec{a}\times (\vec{b}\times \vec{c})=\frac{\vec{b}-\vec{c}}{2}$. If $\vec{d}$ be a vector such that $\vec{b}\cdot \vec{d}=\vec{a}\cdot \vec{b}$, then $(\vec{a}\times \vec{b})\cdot (\vec{c}\times \vec{d})$ is equal to

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

Let $\vec{a}=-\hat{i}-\hat{j}+\hat{k},\vec{a}\cdot \vec{b}=1$ and $\vec{a}\times \vec{b}=\hat{i}-\hat{j}$. Then $\vec{a}-6\vec{b}$ is equal to :

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

If the four points, whose position vectors are $3\hat{i}-4\hat{j}+2\hat{k},\hat{i}+2\hat{j}-\hat{k},-2\hat{i}-\hat{j}+3\hat{k}$ and $5\hat{i}-2\alpha \hat{j}+4\hat{k}$ are coplanar, then $\alpha$ is equal to :

1. A.$\frac{73}{17}$
2. B.$-\frac{73}{17}$
3. C.$-\frac{107}{17}$
4. D.$\frac{107}{17}$

Let $f(x)=\{\begin{array}{ll}{x}^2\sin \left(\frac{1}{x}\right)& ,x\ne 0\\ 0& ,x=0\end{array}$ Then at $x=0$

1. A.$f$ is continuous but $f^{\prime }$ is not continuous
2. B.$f$ and $f^{\prime }$ both are continuous
3. C.$f$ is continuous but not differentiable
4. D.$f^{\prime }$ is continuous but not differentiable

Suppose ${\sum }_{r=0}^{2023}{r}^2{}^{2023}{\mathrm{C}}_r=2023\times \alpha \times 2^{2022}$. Then the value of $\alpha$ is ____

Let $\vec{\alpha }=4\hat{i}+3\hat{j}+5\hat{k}$ and $\vec{\beta }=\hat{i}+2\hat{j}-4\hat{k}$. Let ${\vec{\beta }}_1$ be parallel to $\vec{\alpha }$ and ${\vec{\beta }}_2$ be perpendicular to $\vec{\alpha }$. If $\vec{\beta }={\vec{\beta }}_1+{\vec{\beta }}_2$, then the value of $5{\vec{\beta }}_2\cdot (\hat{i}+\hat{j}+\hat{k})$ is :

1. A.9
2. B.7
3. C.6
4. D.11

Let $\vec{a}=\hat{i}+2\hat{j}+\lambda \hat{k},\vec{b}=3\hat{i}-5\hat{j}-\lambda \hat{k},\vec{a}\cdot \vec{c}=7,2\vec{b}\cdot \vec{c}+43=0,\vec{a}\times \vec{c}=\vec{b}\times \vec{c}$. Then $|\vec{a}\cdot \vec{b}|$ is equal to :

Let $\hat{a}$ and $\hat{b}$ be two unit vectors such that the angle between them is $\frac{\pi }{4}$. If $\theta$ is the angle between the vectors $(\hat{a}+\hat{b})$ and $(\hat{a}+2\hat{b}+2(\hat{a}\times \hat{b}))$, then the value of $164{\cos }^2\theta$ is equal to :

1. A.$90+27\sqrt{2}$
2. B.$45+18\sqrt{2}$
3. C.$90+3\sqrt{2}$
4. D.$54+90\sqrt{2}$

Let $\vec{a}=3\hat{i}+\hat{j}$ and $\vec{b}=\hat{i}+2\hat{j}+\hat{k}$. Let $\vec{c}$ be a vector satisfying $\vec{a}\times (\vec{b}\times \vec{c})=\vec{b}+\lambda \vec{c}$. If $\vec{b}$ and $\vec{c}$ are non-parallel, then the value of $\lambda$ is :

1. A.$-5$
2. B.5
3. C.1
4. D.$-1$

If $\frac{1}{2\times 3\times 4}+\frac{1}{3\times 4\times 5}+\frac{1}{4\times 5\times 6}+\dots +\frac{1}{100\times 101\times 102}=\frac{\mathrm{k}}{101}$, then $34\mathrm{k}$ is equal to ____.

Let $\vec{a},\vec{b},\vec{c}$ be three coplanar concurrent vectors such that the angle between any two of them is the same. If the product of their magnitudes is 14 and $(\vec{a}\times \vec{b})\cdot (\vec{b}\times \vec{c})+(\vec{b}\times \vec{c})\cdot (\vec{c}\times \vec{a})+(\vec{c}\times \vec{a})\cdot (\vec{a}\times \vec{b})=168$, then $|\vec{a}|+|\vec{b}|+|\vec{c}|$ is equal to :

1. A.10
2. B.14
3. C.16
4. D.18

If $(2,3,9)$, $(5,2,1)$, $(1,\lambda ,8)$ and $(\lambda ,2,3)$ are coplanar, then the product of all possible values of $\lambda$ is:

1. A.$\frac{21}{2}$
2. B.$\frac{59}{8}$
3. C.$\frac{57}{8}$
4. D.$\frac{95}{8}$

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}+\vec{b}{|}^2=|\vec{a}{|}^2+2|\vec{b}{|}^2$, $\vec{a}\cdot \vec{b}=3$ and $|\vec{a}\times \vec{b}{|}^2=75$. Then $|\vec{a}{|}^2$ is equal to ____.

Let $S=\{(x,y)\in \mathbb{N}\times \mathbb{N}:9(x-3{)}^2+16(y-4{)}^2\le 144\}$ and $T=\{(x,y)\in \mathbb{R}\times \mathbb{R}:(x-7{)}^2+(y-4{)}^2\le 36\}$. Then $n(S\cap T)$ is equal to ____.

Let the vectors $\vec{a}=(1+t)\hat{i}+(1-t)\hat{j}+\hat{k}$, $\vec{b}=(1-t)\hat{i}+(1+t)\hat{j}+2\hat{k}$ and $\vec{c}=t\hat{i}-t\hat{j}+\hat{k}$, $t\in \mathbf{R}$ be such that for $\alpha ,\beta ,\gamma \in \mathbf{R}$, $\alpha \vec{a}+\beta \vec{b}+\gamma \vec{c}=\vec{0}\Rightarrow \alpha =\beta =\gamma =0$. Then, the set of all values of $t$ is :

1. A.a non-empty finite set
2. B.equal to $\mathbf{N}$
3. C.equal to $\mathbf{R}-\{0\}$
4. D.equal to $\mathbf{R}$

Let a vector $\vec{a}$ has magnitude 9. Let a vector $\vec{b}$ be such that for every $(x,y)\in \mathbf{R}\times \mathbf{R}-\{(0,0)\}$, the vector $(x\vec{a}+y\vec{b})$ is perpendicular to the vector $(6y\vec{a}-18x\vec{b})$. Then the value of $|\vec{a}\times \vec{b}|$ is equal to :

1. A.$9\sqrt{3}$
2. B.$27\sqrt{3}$
3. C.9
4. D.81

Let S be the set of all $a\in \mathbb{R}$ for which the angle between the vectors $\vec{u}=a({\log }_{e}b)\hat{i}-6\hat{j}+3\hat{k}$ and $\vec{v}=({\log }_{e}b)\hat{i}+2\hat{j}+2a({\log }_{e}b)\hat{k}$, $(b>1)$ is acute. Then S is equal to :

1. A.$\left(-\infty ,-\frac{4}{3}\right)$
2. B.$\Phi$
3. C.$\left(-\frac{4}{3},0\right)$
4. D.$\left(\frac{12}{7},\infty \right)$

Let $\vec{a}=\alpha \hat{i}+\hat{j}+\beta \hat{k}$ and $\vec{b}=3\hat{i}-5\hat{j}+4\hat{k}$ be two vectors, such that $\vec{a}\times \vec{b}=-\hat{i}+9\hat{j}+12\hat{k}$. Then the projection of $\vec{b}-2\vec{a}$ on $\vec{b}+\vec{a}$ is equal to :

1. A.2
2. B.$\frac{39}{5}$
3. C.9
4. D.$\frac{46}{5}$

Let $\vec{a}=2\hat{i}-\hat{j}+5\hat{k}$ and $\vec{b}=\alpha \hat{i}+\beta \hat{j}+2\hat{k}$. If $((\vec{a}\times \vec{b})\times \hat{i})\cdot \hat{k}=\frac{23}{2}$, then $|\vec{b}\times 2\hat{j}|$ is equal to :

1. A.4
2. B.5
3. C.$\sqrt{21}$
4. D.$\sqrt{17}$

Let $\vec{a},\vec{b},\vec{c}$ be three non-coplanar vectors such that $\vec{a}\times \vec{b}=4\vec{c},\vec{b}\times \vec{c}=9\vec{a}$ and $\vec{c}\times \vec{a}=\alpha \vec{b},\alpha >0$. If $\left|\vec{a}\right|+\left|\vec{b}\right|+\left|\vec{c}\right|=\frac{1}{36}$, then $\alpha$ is equal to ____.

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 $-\hat{i}+2\hat{j}-2\hat{k}$ is 30, then $\alpha$ is equal to :

1. A.$\frac{15}{2}$
2. B.8
3. C.$\frac{13}{2}$
4. D.7

Let $\vec{a}=\hat{i}-\hat{j}+2\hat{k}$ and let $\vec{b}$ be a vector such that $\vec{a}\times \vec{b}=2\hat{i}-\hat{k}$ and $\vec{a}\cdot \vec{b}=3$. Then the projection of $\vec{b}$ on the vector $\vec{a}-\vec{b}$ is :

1. A.$\frac{2}{\sqrt{21}}$
2. B.$2\sqrt{\frac{3}{7}}$
3. C.$\frac{2}{3}\sqrt{\frac{7}{3}}$
4. D.$\frac{2}{3}$

Let a vector $\vec{c}$ be coplanar with the vectors $\vec{a}=-\hat{i}+\hat{j}+\hat{k}$ and $\vec{b}=2\hat{i}+\hat{j}-\hat{k}$. If the vector $\vec{c}$ also satisfies the conditions $\vec{c}\cdot \left[\left(\vec{a}+\vec{b}\right)\times \left(\vec{a}\times \vec{b}\right)\right]=-42$ and $\left(\vec{c}\times \left(\vec{a}-\vec{b}\right)\right)\cdot \hat{k}=3$, then the value of $|\vec{c}{|}^2$ is equal to :

1. A.24
2. B.29
3. C.35
4. D.42

Let $R_1=\{(a,b)\in \mathbb{N}\times \mathbb{N}:|a-b|\le 13\}$ and $R_2=\{(a,b)\in \mathbb{N}\times \mathbb{N}:|a-b|\ne 13\}$. Then on $\mathbb{N}$ :

1. A.Both $R_1$ and $R_2$ are equivalence relations
2. B.Neither $R_1$ nor $R_2$ is an equivalence relation
3. C.$R_1$ is an equivalence relation but $R_2$ is not
4. D.$R_2$ is an equivalence relation but $R_1$ is not