Vector Algebra · JEE Main previous year questions

2021 24 questions

1. Q242021 · 1 Sept 2021 · Shift 2· Numerical answer

   Let the points of intersections of the lines $x-y+1=0$, $x-2y+3=0$ and $2x-5y+11=0$ are the mid points of the sides of a triangle $\Delta \mathrm{ABC}$. Then, the area of the $\Delta \mathrm{ABC}$ is ___________.

   Show answer

   Answer: 6

2. Q272021 · 1 Sept 2021 · Shift 2· Numerical answer

   Let $\vec{a}=2\hat{i}-\hat{j}+2\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}-\hat{k}$. Let a vector $\vec{v}$ be in the plane containing $\vec{a}$ and $\vec{b}$. If $\vec{v}$ is perpendicular to the vector $3\hat{i}+2\hat{j}-\hat{k}$ and its projection on $\vec{a}$ is 19 units, then $|2\vec{v}{|}^2$ is equal to ___________.

   Show answer

   Answer: 1494

3. Q12021 · 26 Aug 2021 · Shift 2

   Let $[t]$ denote the greatest integer less than or equal to t. Let $f(x)=x-[x]$, $g(x)=1-x+[x]$, and $h(x)=\min \{f(x),g(x)\}$, $x\in [-2,2]$. Then h is :

   1. A.continuous in $[-2,2]$ but not differentiable at more than four points in $(-2,2)$
   2. B.not continuous at exactly three points in $[-2,2]$
   3. C.continuous in $[-2,2]$ but not differentiable at exactly three points in $(-2,2)$
   4. D.not continuous at exactly four points in $[-2,2]$
   Show answer

   Answer: A

4. Q132021 · 26 Aug 2021 · Shift 2
   
   1. A.$5$
   2. B.$2\sqrt{10}$
   3. C.$5\sqrt{3}$
   4. D.$5\sqrt{2}$
   Show answer

   Answer: D

5. Q232021 · 26 Aug 2021 · Shift 2· Numerical answer

   If the projection of the vector $\hat{i}+2\hat{j}+\hat{k}$ on the sum of the two vectors $2\hat{i}+4\hat{j}-5\hat{k}$ and $-\lambda \hat{i}+2\hat{j}+3\hat{k}$ is 1, then $\lambda$ is equal to ________.

   Show answer

   Answer: 5

6. Q192021 · 27 Jul 2021 · Shift 2

   Let $\vec{a},\vec{b}$ and $\vec{c}$ be three vectors such that $\vec{a}=\vec{b}\times (\vec{b}\times \vec{c})$. If magnitudes of the vectors $\vec{a},\vec{b}$ and $\vec{c}$ are $\sqrt{2}$, 1 and 2 respectively and the angle between $\vec{b}$ and $\vec{c}$ is $\theta \left(0<\theta <\frac{\pi }{2}\right)$, then the value of $1+\tan \theta$ is equal to :

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

   Answer: B

7. Q282021 · 27 Jul 2021 · Shift 2· Numerical answer

   Let $\vec{a}=\hat{i}-\alpha \hat{j}+\beta \hat{k},\vec{b}=3\hat{i}+\beta \hat{j}-\alpha \hat{k}$ and $\vec{c}=-\alpha \hat{i}-2\hat{j}+\hat{k}$, where $\alpha$ and $\beta$ are integers. If $\vec{a}\cdot \vec{b}=-1$ and $\vec{b}\cdot \vec{c}=10$, then $\left(\vec{a}\times \vec{b}\right)\cdot \vec{c}$ is equal to ____.

   Show answer

   Answer: 9

8. Q162021 · 25 Jul 2021 · Shift 1

   Let the vectors $$(2+a+b)\hat{i}+(a+2b+c)\hat{j}-(b+c)\hat{k},(1+b)\hat{i}+2b\hat{j}-b\hat{k}\text{ and }(2+b)\hat{i}+2b\hat{j}+(1-b)\hat{k},a,b,c\in R$$ be co-planar. Then which of the following is true?

   1. A.2b = a + c
   2. B.3c = a + b
   3. C.a = b + 2c
   4. D.2a = b + c
   Show answer

   Answer: A

9. Q232021 · 25 Jul 2021 · Shift 1· Numerical answer

   Let $\vec{p}=2\hat{i}+3\hat{j}+\hat{k}$ and $\vec{q}=\hat{i}+2\hat{j}+\hat{k}$ be two vectors. If a vector $\vec{r}=(\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k})$ is perpendicular to each of the vectors $((\vec{p}+\vec{q})$ and $(\vec{p}-\vec{q})$, and $\left|\vec{r}\right|=\sqrt{3}$, then $|\alpha |+|\beta |+|\gamma |$ is equal to ____.

   Show answer

   Answer: 3

10. Q82021 · 25 Jul 2021 · Shift 2

    Let a, b and c be distinct positive numbers. If the vectors $a\hat{i}+a\hat{j}+c\hat{k},\hat{i}+\hat{k}$ and $c\hat{i}+c\hat{j}+b\hat{k}$ are co-planar, then c is equal to :

    1. A.$\frac{2}{\frac{1}{a}+\frac{1}{b}}$
    2. B.$\frac{a+b}{2}$
    3. C.$\frac{1}{a}+\frac{1}{b}$
    4. D.$\sqrt{ab}$
    Show answer

    Answer: D

11. Q162021 · 22 Jul 2021 · Shift 2

    Let a vector $\vec{a}$ be coplanar with vectors $\vec{b}=2\hat{i}+\hat{j}+\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}+\hat{k}$. If $\vec{a}$ is perpendicular to $\vec{d}=3\hat{i}+2\hat{j}+6\hat{k}$, and $|\vec{a}|=\sqrt{10}$. Then a possible value of $[\vec{a}\vec{b}\vec{c}]+[\vec{a}\vec{b}\vec{d}]+[\vec{a}\vec{c}\vec{d}]$ is equal to :

    1. A.$-42$
    2. B.$-40$
    3. C.$-29$
    4. D.$-38$
    Show answer

    Answer: A

12. Q172021 · 22 Jul 2021 · Shift 2

    Let three vectors $\vec{a},\vec{b}$ and $\vec{c}$ be such that $\vec{a}\times \vec{b}=\vec{c}$, $\vec{b}\times \vec{c}=\vec{a}$ and $|\vec{a}|=2$. Then which one of the following is not true?

    1. A.$\vec{a}\times ((\vec{b}+\vec{c})\times (\vec{b}-\vec{c}))=\vec{0}$
    2. B.$\text{Projection of }\vec{a}\text{ on }(\vec{b}\times \vec{c})\text{ is }2$
    3. C.$[\vec{a}\vec{b}\vec{c}]+[\vec{c}\vec{a}\vec{b}]=8$
    4. D.$|3\vec{a}+\vec{b}-2\vec{c}{|}^2=51$
    Show answer

    Answer: D

13. Q192021 · 20 Jul 2021 · Shift 1

    Let $\vec{a}=2\hat{i}+\hat{j}-2\hat{k}$ and $\vec{b}=\hat{i}+\hat{j}$. If $\vec{c}$ is a vector such that $\vec{a}\cdot \vec{c}=|\vec{c}|$, $|\vec{c}-\vec{a}|=2\sqrt{2}$ and the angle between $(\vec{a}\times \vec{b})$ and $\vec{c}$ is $\frac{\pi }{6}$, then the value of $\left|(\vec{a}\times \vec{b})\times \vec{c}\right|$ is :

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

    Answer: D

14. Q212021 · 20 Jul 2021 · Shift 1· Numerical answer

    Let $\vec{a},\vec{b},\vec{c}$ be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle $\theta$, with the vector $\vec{a}+\vec{b}+\vec{c}$. Then $36{\cos }^{2}2\theta$ is equal to ________.

    Show answer

    Answer: 4

15. Q292021 · 20 Jul 2021 · Shift 1· Numerical answer

    Let P be a plane passing through the points $(1,0,1)$, $(1,-2,1)$ and $(0,1,-2)$. Let a vector $\vec{a}=\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$ be such that $\vec{a}$ is parallel to the plane P, perpendicular to $(\hat{i}+2\hat{j}+3\hat{k})$ and $\vec{a}\cdot (\hat{i}+\hat{j}+2\hat{k})=2$, then $(\alpha -\beta +\gamma {)}^2$ equals ________.

    Show answer

    Answer: 81

16. Q72021 · 18 Mar 2021 · Shift 1

    A vector $\vec{a}$ has components $3p$ and $1$ with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the counter clockwise sense. If, with respect to new system, $\vec{a}$ has components $p+1$ and $\sqrt{10}$, then the value of $p$ is equal to :

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

    Answer: D

17. Q52021 · 18 Mar 2021 · Shift 2

    Let $\vec{a}$ and $\vec{b}$ be two non-zero vectors perpendicular to each other and $|\vec{a}|=|\vec{b}|$. If $|\vec{a}\times \vec{b}|=|\vec{a}|$, then the angle between the vectors $\left(\vec{a}+\vec{b}+\left(\vec{a}\times \vec{b}\right)\right)$ and $\vec{a}$ is equal to :

    1. A.${\sin }^{-1}\left(\frac{1}{\sqrt{6}}\right)$
    2. B.${\cos }^{-1}\left(\frac{1}{\sqrt{2}}\right)$
    3. C.${\sin }^{-1}\left(\frac{1}{\sqrt{3}}\right)$
    4. D.${\cos }^{-1}\left(\frac{1}{\sqrt{3}}\right)$
    Show answer

    Answer: D

18. Q102021 · 18 Mar 2021 · Shift 2

    In a triangle ABC, if $|\overrightarrow{BC}|=8$, $|\overrightarrow{CA}|=7$, $|\overrightarrow{AB}|=10$, then the projection of the vector $\overrightarrow{AB}$ on $\overrightarrow{AC}$ is equal to :

    1. A.$\frac{25}{4}$
    2. B.$\frac{127}{20}$
    3. C.$\frac{85}{14}$
    4. D.$\frac{115}{16}$
    Show answer

    Answer: C

19. Q12021 · 16 Mar 2021 · Shift 1

    Let a vector $\alpha \hat{i}+\beta \hat{j}$ be obtained by rotating the vector $\sqrt{3}\hat{i}+\hat{j}$ by an angle $45^{\circ }$ about the origin in counterclockwise direction in the first quadrant. Then the area of triangle having vertices $(\alpha ,\beta )$, $(0,\beta )$ and $(0,0)$ is equal to :

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

    Answer: B

20. Q32021 · 16 Mar 2021 · Shift 1

    Let the position vectors of two points P and Q be $3\hat{i}-\hat{j}+2\hat{k}$ and $\hat{i}+2\hat{j}-4\hat{k}$, respectively. Let R and S be two points such that the direction ratios of lines PR and QS are $(4,-1,2)$ and $(-2,1,-2)$, respectively. Let lines PR and QS intersect at T. If the vector $\overrightarrow{TA}$ is perpendicular to both $\overrightarrow{PR}$ and $\overrightarrow{QS}$ and the length of vector $\overrightarrow{TA}$ is $\sqrt{5}$ units, then the modulus of a position vector of A is :

    1. A.$\sqrt{171}$
    2. B.$\sqrt{227}$
    3. C.$\sqrt{482}$
    4. D.$\sqrt{5}$
    Show answer

    Answer: A

21. Q162021 · 26 Feb 2021 · Shift 1

    If $\vec{a}$ and $\vec{b}$ are perpendicular, then $\vec{a}\times \left(\vec{a}\times \left(\vec{a}\times \left(\vec{a}\times \vec{b}\right)\right)\right)$ is equal to :

    1. A.$\frac{1}{2}|\vec{a}{|}^4\vec{b}$
    2. B.$\vec{0}$
    3. C.$\vec{a}\times \vec{b}$
    4. D.$|\vec{a}{|}^4\vec{b}$
    Show answer

    Answer: D

22. Q192021 · 26 Feb 2021 · Shift 1

    If $(1,5,35)$, $(7,5,5)$, $(1,\lambda ,7)$ and $(2\lambda ,1,2)$ are coplanar, then the sum of all possible values of $\lambda$ is :

    1. A.$-\frac{44}{5}$
    2. B.$-\frac{39}{5}$
    3. C.$\frac{44}{5}$
    4. D.$\frac{39}{5}$
    Show answer

    Answer: C

23. Q132021 · 26 Feb 2021 · Shift 2

    If vectors $\overrightarrow{a_1}=x\hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{a_2}=\hat{i}+y\hat{j}+z\hat{k}$ are collinear, then a possible unit vector parallel to the vector $x\hat{i}+y\hat{j}+z\hat{k}$ is :

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

    Answer: A

24. Q262021 · 25 Feb 2021 · Shift 2· Numerical answer

    Let $\vec{a}=\hat{i}+\alpha \hat{j}+3\hat{k}$ and $\vec{b}=3\hat{i}-\alpha \hat{j}+\hat{k}$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\vec{a}$ and $\vec{b}$ is $8\sqrt{3}$ square units, then $\vec{a}\cdot \vec{b}$ is equal to ______ .

    Show answer

    Answer: 2