Vector Algebra · JEE Main previous year questions

2024 31 questions

1. Q22024 · 9 Apr 2024 · Shift 1

   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
   Show answer

   Answer: A

2. Q122024 · 9 Apr 2024 · Shift 2

   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.
   Show answer

   Answer: D

3. Q42024 · 8 Apr 2024 · Shift 1

   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.$(-\frac{4}{3},1)$
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   Answer: B

4. Q232024 · 8 Apr 2024 · Shift 1· Numerical answer

   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 ________.

   Show answer

   Answer: 569

5. Q52024 · 8 Apr 2024 · Shift 2

   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
   Show answer

   Answer: B

6. Q142024 · 8 Apr 2024 · Shift 2

   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
   Show answer

   Answer: B

7. Q82024 · 6 Apr 2024 · Shift 2

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

   Answer: C

8. Q92024 · 6 Apr 2024 · Shift 2

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

   Answer: C

9. Q72024 · 5 Apr 2024 · Shift 2

   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
   Show answer

   Answer: C

10. Q192024 · 5 Apr 2024 · Shift 2

    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
    Show answer

    Answer: A

11. Q12024 · 4 Apr 2024 · Shift 1

    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
    Show answer

    Answer: A

12. Q82024 · 4 Apr 2024 · Shift 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.$\frac{\sqrt{2}}{3}\hat{i}+\frac{1}{3\sqrt{2}}\hat{j}-\frac{1}{2}\hat{k}$
    Show answer

    Answer: C

13. Q192024 · 4 Apr 2024 · Shift 1

    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
    Show answer

    Answer: A

14. Q292024 · 4 Apr 2024 · Shift 1· Numerical answer

    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 ____.

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    Answer: 54

15. Q102024 · 4 Apr 2024 · Shift 2

    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
    Show answer

    Answer: A

16. Q202024 · 4 Apr 2024 · Shift 2

    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
    Show answer

    Answer: C

17. Q82024 · 1 Feb 2024 · Shift 1

    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
    Show answer

    Answer: A

18. Q32024 · 1 Feb 2024 · Shift 2

    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.$\frac{\sqrt{38}}{2}$
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    Answer: A

19. Q282024 · 1 Feb 2024 · Shift 2· Numerical answer

    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 ________ .

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    Answer: 38

20. Q62024 · 31 Jan 2024 · Shift 1

    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
    Show answer

    Answer: D

21. Q92024 · 31 Jan 2024 · Shift 1

    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
    Show answer

    Answer: B

22. Q122024 · 31 Jan 2024 · Shift 1

    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.$\sqrt{20}$
    Show answer

    Answer: A

23. Q232024 · 31 Jan 2024 · Shift 1· Numerical answer

    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 ____.

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    Answer: 48

24. Q212024 · 31 Jan 2024 · Shift 2· Numerical answer

    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 ____.

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    Answer: 38

25. Q42024 · 30 Jan 2024 · Shift 1

    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.${\cos }^{-1}\left(-\frac{\sqrt{3}}{2}\right)$
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    Answer: D