Vector Algebra · JEE Main previous year questions

2022 27 questions

1. Q92022 · 29 Jul 2022 · Shift 1

   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.$54+90\sqrt{2}$
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   Answer: A

2. Q182022 · 29 Jul 2022 · Shift 1

   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.$-1$
   Show answer

   Answer: A

3. Q302022 · 29 Jul 2022 · Shift 1· Numerical answer

   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 ____.

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   Answer: 286

4. Q132022 · 29 Jul 2022 · Shift 2

   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
   Show answer

   Answer: C

5. Q192022 · 29 Jul 2022 · Shift 2

   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.$\frac{95}{8}$
   Show answer

   Answer: D

6. Q262022 · 29 Jul 2022 · Shift 2· Numerical answer

   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 ____.

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   Answer: 14

7. Q272022 · 29 Jul 2022 · Shift 2· Numerical answer

   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 ____.

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   Answer: 27

8. Q32022 · 28 Jul 2022 · Shift 1

   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 $\mathbf{R}$
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   Answer: C

9. Q52022 · 28 Jul 2022 · Shift 1

   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
   Show answer

   Answer: B

10. Q132022 · 28 Jul 2022 · Shift 2

    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.$\left(\frac{12}{7},\infty \right)$
    Show answer

    Answer: B

11. Q112022 · 27 Jul 2022 · Shift 1

    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.$\frac{46}{5}$
    Show answer

    Answer: D

12. Q172022 · 27 Jul 2022 · Shift 1

    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.$\sqrt{17}$
    Show answer

    Answer: B

13. Q272022 · 27 Jul 2022 · Shift 2· Numerical answer

    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 ____.

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    Answer: 36

14. Q122022 · 26 Jul 2022 · Shift 1

    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
    Show answer

    Answer: D

15. Q122022 · 25 Jul 2022 · Shift 2

    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.$\frac{2}{3}$
    Show answer

    Answer: A

16. Q182022 · 30 Jun 2022 · Shift 1

    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
    Show answer

    Answer: C

17. Q12022 · 28 Jun 2022 · Shift 2

    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
    Show answer

    Answer: B

18. Q142022 · 28 Jun 2022 · Shift 2

    Let $\vec{a}=\alpha \hat{i}+2\hat{j}-\hat{k}$ and $\vec{b}=-2\hat{i}+\alpha \hat{j}+\hat{k}$, where $\alpha \in \mathbb{R}$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\vec{a}$ and $\vec{b}$ is $\sqrt{15({\alpha }^2+4)}$, then the value of $2{\left|\vec{a}\right|}^2+\left(\vec{a}\cdot \vec{b}\right){\left|\vec{b}\right|}^2$ is equal to :

    1. A.$10$
    2. B.$7$
    3. C.$9$
    4. D.$14$
    Show answer

    Answer: D

19. Q182022 · 28 Jun 2022 · Shift 2

    Let $\vec{a}$ be a vector which is perpendicular to the vector $3\hat{i}+\frac{1}{2}\hat{j}+2\hat{k}$. If $\vec{a}\times (2\hat{i}+\hat{k})=2\hat{i}-13\hat{j}-4\hat{k}$, then the projection of the vector $\vec{a}$ on the vector $2\hat{i}+2\hat{j}+\hat{k}$ is :

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

    Answer: C

20. Q132022 · 27 Jun 2022 · Shift 1

    Let $\vec{a}=\hat{i}+\hat{j}-\hat{k}$ and $\vec{c}=2\hat{i}-3\hat{j}+2\hat{k}$. Then the number of vectors $\vec{b}$ such that $\vec{b}\times \vec{c}=\vec{a}$ and $|\vec{b}|\in \{1,2,.......,10\}$ is :

    1. A.0
    2. B.1
    3. C.2
    4. D.3
    Show answer

    Answer: A

21. Q252022 · 27 Jun 2022 · Shift 1· Numerical answer

    Let $A_1=\{(x,y):|x|\le y^2,|x|+2y\le 8\}$ and $A_2=\{(x,y):|x|+|y|\le k\}$. If $27(\text{Area }{A}_1)=5(\text{Area }{A}_2)$, then $k$ is equal to :

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    Answer: 6

22. Q102022 · 27 Jun 2022 · Shift 2

    Let $\vec{a}$ and $\vec{b}$ be the vectors along the diagonals of a parallelogram having area $2\sqrt{2}$. Let the angle between $\vec{a}$ and $\vec{b}$ be acute, $|\vec{a}|=1$, and $|\vec{a}\cdot \vec{b}|=|\vec{a}\times \vec{b}|$. If $\vec{c}=2\sqrt{2}\left(\vec{a}\times \vec{b}\right)-2\vec{b}$, then an angle between $\vec{b}$ and $\vec{c}$ is :

    1. A.$\frac{\pi }{4}$
    2. B.$-\frac{\pi }{4}$
    3. C.$\frac{5\pi }{6}$
    4. D.$\frac{3\pi }{4}$
    Show answer

    Answer: D

23. Q112022 · 26 Jun 2022 · Shift 2

    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 of $\vec{a}$ and $\vec{b}$ whose projection on $\vec{c}$ is $\frac{2}{\sqrt{3}}$. If $\vec{v}\cdot \hat{j}=7$, then $\vec{v}\cdot (\hat{i}+\hat{k})$ is equal to :

    1. A.6
    2. B.7
    3. C.8
    4. D.9
    Show answer

    Answer: D

24. Q112022 · 25 Jun 2022 · Shift 1

    Let $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ $a_i>0$, $i=1,2,3$ be a vector which makes equal angles with the coordinate axes $\mathrm{OX}$, $\mathrm{OY}$ and $\mathrm{OZ}$. Also, let the projection of $\vec{a}$ on the vector $3\hat{i}+4\hat{j}$ be $7$. Let $\vec{b}$ be a vector obtained by rotating $\vec{a}$ with $90^{\circ }$. If $\vec{a},\vec{b}$ and x-axis are coplanar, then projection of a vector $\vec{b}$ on $3\hat{i}+4\hat{j}$ is equal to:

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

    Answer: B

25. Q222022 · 25 Jun 2022 · Shift 1· Numerical answer

    Let $\theta$ be the angle between the vectors $\vec{a}$ and $\vec{b}$, where $|\vec{a}|=4$, $|\vec{b}|=3$ and $\theta \in \left(\frac{\pi }{4},\frac{\pi }{3}\right)$. Then ${\left|(\vec{a}-\vec{b})\times (\vec{a}+\vec{b})\right|}^2+4{\left(\vec{a}\cdot \vec{b}\right)}^2$ is equal to $\underline{}$.

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    Answer: 576