Vector Algebra · JEE Main previous year questions

2023 39 questions

1. Q172023 · 12 Apr 2023 · Shift 1

   Let $\lambda \in \mathbb{Z}$, $\vec{a}=\lambda \hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=3\hat{i}-\hat{j}+2\hat{k}$. Let $\vec{c}$ be a vector such that $(\vec{a}+\vec{b}+\vec{c})\times \vec{c}=\vec{0}$, $\vec{a}\cdot \vec{c}=-17$ and $\vec{b}\cdot \vec{c}=-20$. Then $|\vec{c}\times (\lambda \hat{i}+\hat{j}+\hat{k}){|}^2$ is equal to :

   1. A.$53$
   2. B.$62$
   3. C.$49$
   4. D.$46$
   Show answer

   Answer: D

2. Q82023 · 11 Apr 2023 · Shift 1

   For any vector $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$, with $10|a_i|<1,i=1,2,3$, consider the following statements : (A): $\max \{|a_1|,|a_2|,|a_3|\}\le |\vec{a}|$ (B) : $|\vec{a}|\le 3\max \{|a_1|,|a_2|,|a_3|\}$

   1. A.Only (B) is true
   2. B.Only (A) is true
   3. C.Neither (A) nor (B) is true
   4. D.Both (A) and (B) are true
   Show answer

   Answer: D

3. Q132023 · 11 Apr 2023 · Shift 1

   Let $\vec{a}$ be a non-zero vector parallel to the line of intersection of the two planes described by $\hat{i}+\hat{j},\hat{i}+\hat{k}$ and $\hat{i}-\hat{j},\hat{j}-\hat{k}$. If $\theta$ is the angle between the vector $\vec{a}$ and the vector $\vec{b}=2\hat{i}-2\hat{j}+\hat{k}$ and $\vec{a}\cdot \vec{b}=6$, then the ordered pair $(\theta ,|\vec{a}\times \vec{b}|)$ is equal to :

   1. A.$\left(\frac{\pi }{3},3\sqrt{6}\right)$
   2. B.$\left(\frac{\pi }{3},6\right)$
   3. C.$\left(\frac{\pi }{4},3\sqrt{6}\right)$
   4. D.$\left(\frac{\pi }{4},6\right)$
   Show answer

   Answer: D

4. Q252023 · 11 Apr 2023 · Shift 1· Numerical answer

   Let a line $l$ pass through the origin and be perpendicular to the lines $l_1:\vec{r}=(\hat{i}-11\hat{j}-7\hat{k})+\lambda (\hat{i}+2\hat{j}+3\hat{k}),\lambda \in \mathbb{R}$ and $l_2:\vec{r}=(-\hat{i}+\hat{k})+\mu (2\hat{i}+2\hat{j}+\hat{k}),\mu \in \mathbb{R}$. If P is the point of intersection of $l$ and $l_1$, and $Q(\alpha ,\beta ,\gamma )$ is the foot of perpendicular from P on $l_2$, then $9(\alpha +\beta +\gamma )$ is equal to _________.

   Show answer

   Answer: 5

5. Q172023 · 11 Apr 2023 · Shift 2

   If four distinct points with position vectors $\vec{a},\vec{b},\vec{c}$ and $\vec{d}$ are coplanar, then $[\vec{a}\vec{b}\vec{c}]$ is equal to :

   1. A.$[\vec{d}\vec{b}\vec{a}]+[\vec{a}\vec{c}\vec{d}]+[\vec{d}\vec{b}\vec{c}]$
   2. B.$[\vec{b}\vec{c}\vec{d}]+[\vec{d}\vec{a}\vec{c}]+[\vec{d}\vec{b}\vec{a}]$
   3. C.$[\vec{a}\vec{d}\vec{b}]+[\vec{d}\vec{c}\vec{a}]+[\vec{d}\vec{b}\vec{c}]$
   4. D.$[\vec{d}\vec{c}\vec{a}]+[\vec{b}\vec{d}\vec{a}]+[\vec{c}\vec{d}\vec{b}]$
   Show answer

   Answer: D

6. Q212023 · 11 Apr 2023 · Shift 2· Numerical answer

   Let $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$ and $\vec{b}=\hat{i}+\hat{j}-\hat{k}$. If $\vec{c}$ is a vector such that $\vec{a}\cdot \vec{c}=11$, $\vec{b}\cdot (\vec{a}\times \vec{c})=27$ and $\vec{b}\cdot \vec{c}=-\sqrt{3}|\vec{b}|$, then $|\vec{a}\times \vec{c}{|}^2$ is equal to _________.

   Show answer

   Answer: 285

7. Q42023 · 10 Apr 2023 · Shift 1

   Let O be the origin and the position vector of the point P be $-\hat{i}-2\hat{j}+3\hat{k}$. If the position vectors of the points A, B and C are $-2\hat{i}+\hat{j}-3\hat{k}$, $2\hat{i}+4\hat{j}-2\hat{k}$ and $-4\hat{i}+2\hat{j}-\hat{k}$ respectively, then the projection of the vector $\overrightarrow{OP}$ on a vector perpendicular to the vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$ is :

   1. A.$\frac{7}{3}$
   2. B.3
   3. C.$\frac{10}{3}$
   4. D.$\frac{8}{3}$
   Show answer

   Answer: B

8. Q92023 · 10 Apr 2023 · Shift 2

   Let $\vec{a}=2\hat{i}+7\hat{j}-\hat{k}$, $\vec{b}=3\hat{i}+5\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}+2\hat{k}$. Let $\vec{d}$ be a vector which is perpendicular to both $\vec{a}$ and $\vec{b}$, and $\vec{c}\cdot \vec{d}=12$. Then $(-\hat{i}+\hat{j}-\hat{k})\cdot (\vec{c}\times \vec{d})$ is equal to :

   1. A.24
   2. B.42
   3. C.44
   4. D.48
   Show answer

   Answer: C

9. Q162023 · 8 Apr 2023 · Shift 1

   If the points with position vectors $\alpha \hat{i}+10\hat{j}+13\hat{k},6\hat{i}+11\hat{j}+11\hat{k},\frac{9}{2}\hat{i}+\beta \hat{j}-8\hat{k}$ are collinear, then $(19\alpha -6\beta {)}^2$ is equal to :

   1. A.$16$
   2. B.$49$
   3. C.$36$
   4. D.$25$
   Show answer

   Answer: C

10. Q252023 · 8 Apr 2023 · Shift 1· Numerical answer

    Let $\vec{a}=6\hat{i}+9\hat{j}+12\hat{k},\vec{b}=\alpha \hat{i}+11\hat{j}-2\hat{k}$ and $\vec{c}$ be vectors such that $\vec{a}\times \vec{c}=\vec{a}\times \vec{b}$. If $\vec{a}\cdot \vec{c}=-12,\vec{c}\cdot (\hat{i}-2\hat{j}+\hat{k})=5$, then $\vec{c}\cdot (\hat{i}+\hat{j}+\hat{k})$ is equal to ___________.

    Show answer

    Answer: 11

11. Q162023 · 8 Apr 2023 · Shift 2

    Let the vectors ${\vec{u}}_1=\hat{i}+\hat{j}+a\hat{k},{\vec{u}}_2=\hat{i}+b\hat{j}+\hat{k}$ and ${\vec{u}}_3=c\hat{i}+\hat{j}+\hat{k}$ be coplanar. If the vectors ${\vec{v}}_1=(a+b)\hat{i}+c\hat{j}+c\hat{k},{\vec{v}}_2=a\hat{i}+(b+c)\hat{j}+a\hat{k}$ and ${\vec{v}}_3=b\hat{i}+b\hat{j}+(c+a)\hat{k}$ are also coplanar, then $6(a+b+c)$ is equal to :

    1. A.$12$
    2. B.$6$
    3. C.$0$
    4. D.$4$
    Show answer

    Answer: A

12. Q92023 · 6 Apr 2023 · Shift 1

    Let $\vec{a}=2\hat{i}+3\hat{j}+4\hat{k},\vec{b}=\hat{i}-2\hat{j}-2\hat{k}$ and $\vec{c}=-\hat{i}+4\hat{j}+3\hat{k}$. If $\vec{d}$ is a vector perpendicular to both $\vec{b}$ and $\vec{c}$, and $\vec{a}\cdot \vec{d}=18$, then $|\vec{a}\times \vec{d}{|}^2$ is equal to :

    1. A.680
    2. B.720
    3. C.760
    4. D.640
    Show answer

    Answer: B

13. Q172023 · 6 Apr 2023 · Shift 1

    Let the position vectors of the points A, B, C and D be $5\hat{i}+5\hat{j}+2\lambda \hat{k},\hat{i}+2\hat{j}+3\hat{k},-2\hat{i}+\lambda \hat{j}+4\hat{k}$ and $-\hat{i}+5\hat{j}+6\hat{k}$. Let the set $S=\{\lambda \in \mathbb{R}:\text{the points A, B, C and D are coplanar}\}$. Then ${\sum }_{\lambda \in S}(\lambda +2{)}^2$ is equal to :

    1. A.$\frac{37}{2}$
    2. B.25
    3. C.13
    4. D.41
    Show answer

    Answer: D

14. Q162023 · 6 Apr 2023 · Shift 2

    Let the vectors $\vec{a},\vec{b},\vec{c}$ represent three coterminous edges of a parallelopiped of volume V. Then the volume of the parallelopiped, whose coterminous edges are represented by $\vec{a},\vec{b}+\vec{c}$ and $\vec{a}+2\vec{b}+3\vec{c}$ is equal to :

    1. A.3 V
    2. B.2 V
    3. C.6 V
    4. D.V
    Show answer

    Answer: D

15. Q202023 · 6 Apr 2023 · Shift 2

    The sum of all values of $\alpha$, for which the points whose position vectors are $\hat{i}-2\hat{j}+3\hat{k},2\hat{i}-3\hat{j}+4\hat{k},(\alpha +1)\hat{i}+2\hat{k}$ and $9\hat{i}+(\alpha -8)\hat{j}+6\hat{k}$ are coplanar, is equal to :

    1. A.6
    2. B.4
    3. C.-2
    4. D.2
    Show answer

    Answer: D

16. Q302023 · 1 Feb 2023 · Shift 1· Numerical answer

    Let $\vec{v}=\alpha \hat{i}+2\hat{j}-3\hat{k},\vec{w}=2\alpha \hat{i}+\hat{j}-\hat{k}$ and $\vec{u}$ be a vector such that $|\vec{u}|=\alpha >0$. If the minimum value of the scalar triple product $\left[\vec{u}\vec{v}\vec{w}\right]$ is $-\alpha \sqrt{3401}$, and $|\vec{u}\cdot \hat{i}{|}^2=\frac{m}{n}$ where $m$ and $n$ are coprime natural numbers, then $m+n$ is equal to ________.

    Show answer

    Answer: 3501

17. Q112023 · 31 Jan 2023 · Shift 1

    Let $\vec{a}=2\hat{i}+\hat{j}+\hat{k}$, and $\vec{b}$ and $\vec{c}$ be two nonzero vectors such that $|\vec{a}+\vec{b}+\vec{c}|=|\vec{a}+\vec{b}-\vec{c}|$ and $\vec{b}\cdot \vec{c}=0$. Consider the following two statements: (A) $|\vec{a}+\lambda \vec{c}|\ge |\vec{a}|$ for all $\lambda \in \mathbb{R}$. (B) $\vec{a}$ and $\vec{c}$ are always parallel. Then,

    1. A.only (B) is correct
    2. B.both (A) and (B) are correct
    3. C.only (A) is correct
    4. D.neither (A) nor (B) is correct
    Show answer

    Answer: C

18. Q262023 · 31 Jan 2023 · Shift 1· Numerical answer

    Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}|=\sqrt{14}$, $|\vec{b}|=\sqrt{6}$ and $|\vec{a}\times \vec{b}|=\sqrt{48}$. Then $(\vec{a}\cdot \vec{b}{)}^2$ is equal to ___________.

    Show answer

    Answer: 36

19. Q12023 · 31 Jan 2023 · Shift 2

    Let $\vec{a}=\hat{i}+2\hat{j}+3\hat{k},\vec{b}=\hat{i}-\hat{j}+2\hat{k}$ and $\vec{c}=5\hat{i}-3\hat{j}+3\hat{k}$ be three vectors. If $\vec{r}$ is a vector such that, $\vec{r}\times \vec{b}=\vec{c}\times \vec{b}$ and $\vec{r}\cdot \vec{a}=0$, then $25|\vec{r}{|}^2$ is equal to :

    1. A.336
    2. B.449
    3. C.339
    4. D.560
    Show answer

    Answer: C

20. Q252023 · 31 Jan 2023 · Shift 2· Numerical answer

    Let $\vec{a},\vec{b},\vec{c}$ be three vectors such that $|\vec{a}|=\sqrt{31},4|\vec{b}|=|\vec{c}|=2$ and $2(\vec{a}\times \vec{b})=3(\vec{c}\times \vec{a})$. If the angle between $\vec{b}$ and $\vec{c}$ is $\frac{2\pi }{3}$, then ${\left(\frac{\vec{a}\times \vec{c}}{\vec{a}\cdot \vec{b}}\right)}^2$ is equal to ___________.

    Show answer

    Answer: 3

21. Q152023 · 30 Jan 2023 · Shift 1

    If $\vec{a},\vec{b},\vec{c}$ are three non-zero vectors and $\hat{n}$ is a unit vector perpendicular to $\vec{c}$ such that $\vec{a}=\alpha \vec{b}-\hat{n}$, $(\alpha \ne 0)$ and $\vec{b}\cdot \vec{c}=12$, then $\left|\vec{c}\times (\vec{a}\times \vec{b})\right|$ is equal to :

    1. A.15
    2. B.9
    3. C.6
    4. D.12
    Show answer

    Answer: D

22. Q182023 · 29 Jan 2023 · Shift 1

    If the vectors $\vec{a}=\lambda \hat{i}+\mu \hat{j}+4\hat{k},\vec{b}=-2\hat{i}+4\hat{j}-2\hat{k}$ and $\vec{c}=2\hat{i}+3\hat{j}+\hat{k}$ are coplanar and the projection of $\vec{a}$ on the vector $\vec{b}$ is $\sqrt{54}$ units, then the sum of all possible values of $\lambda +\mu$ is equal to :

    1. A.24
    2. B.0
    3. C.18
    4. D.6
    Show answer

    Answer: A

23. Q302023 · 29 Jan 2023 · Shift 1· Numerical answer

    Let $\vec{a},\vec{b}$ and $\vec{c}$ be three non-zero non-coplanar vectors. Let the position vectors of four points $A,B,C$ and $D$ be $\vec{a}-\vec{b}+\vec{c},\lambda \vec{a}-3\vec{b}+4\vec{c},-\vec{a}+2\vec{b}-3\vec{c}$ and $2\vec{a}-4\vec{b}+6\vec{c}$ respectively. If $\overrightarrow{AB},\overrightarrow{AC}$ and $\overrightarrow{AD}$ are coplanar, then $\lambda$ is equal to ____.

    Show answer

    Answer: 2

24. Q42023 · 29 Jan 2023 · Shift 2

    If $\vec{a}=\hat{i}+2\hat{k},\vec{b}=\hat{i}+\hat{j}+\hat{k},\vec{c}=7\hat{i}-3\hat{j}+4\hat{k},\vec{r}\times \vec{b}+\vec{b}\times \vec{c}=\vec{0}$ and $\vec{r}\cdot \vec{a}=0$. Then $\vec{r}\cdot \vec{c}$ is equal to :

    1. A.36
    2. B.30
    3. C.34
    4. D.32
    Show answer

    Answer: C

25. Q102023 · 29 Jan 2023 · Shift 2

    Let $\vec{a}=4\hat{i}+3\hat{j}$ and $\vec{b}=3\hat{i}-4\hat{j}+5\hat{k}$. If $\vec{c}$ is a vector such that $\vec{c}\cdot \left(\vec{a}\times \vec{b}\right)+25=0,\vec{c}\cdot (\hat{i}+\hat{j}+\hat{k})=4$, and projection of $\vec{c}$ on $\vec{a}$ is 1, then the projection of $\vec{c}$ on $\vec{b}$ equals :

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

    Answer: D