Vector Algebra · JEE Main previous year questions

2025 17 questions

1. Q182025 · 8 Apr 2025 · Shift 2

   Let $\vec{a}=\hat{i}+2\hat{j}+\hat{k}$ and $\vec{b}=2\hat{i}+\hat{j}-\hat{k}$. Let $\hat{c}$ be a unit vector in the plane of the vectors $\vec{a}$ and $\vec{b}$ and be perpendicular to $\vec{a}$. Then such a vector $\hat{c}$ is:

   1. A.$\frac{1}{\sqrt{2}}(-\hat{i}+\hat{k})$
   2. B.$\frac{1}{\sqrt{5}}(\hat{j}-2\hat{k})$
   3. C.$\frac{1}{\sqrt{3}}(\hat{i}-\hat{j}+\hat{k})$
   4. D.$\frac{1}{\sqrt{3}}(-\hat{i}+\hat{j}-\hat{k})$
   Show answer

   Answer: A

2. Q132025 · 4 Apr 2025 · Shift 1

   Consider two vectors $\vec{u}=3\hat{i}-\hat{j}$ and $\vec{v}=2\hat{i}+\hat{j}-\lambda \hat{k},\lambda >0$. The angle between them is given by ${\cos }^{-1}\left(\frac{\sqrt{5}}{2\sqrt{7}}\right)$. Let $\vec{v}={\vec{v}}_1+{\vec{v}}_2$, where ${\vec{v}}_1$ is parallel to $\vec{u}$ and ${\vec{v}}_2$ is perpendicular to $\vec{u}$. Then the value of ${\left|{\vec{v}}_1\right|}^2+{\left|{\vec{v}}_2\right|}^2$ is equal to

   1. A.$\frac{23}{2}$
   2. B.$\frac{25}{2}$
   3. C.$10$
   4. D.$14$
   Show answer

   Answer: D

3. Q232025 · 4 Apr 2025 · Shift 2· Numerical answer

   Let the three sides of a triangle ABC be given by the vectors $2\hat{i}-\hat{j}+\hat{k},\hat{i}-3\hat{j}-5\hat{k}$ and $3\hat{i}-4\hat{j}-4\hat{k}$. Let $G$ be the centroid of the triangle $ABC$. Then $6\left(|\overrightarrow{\mathrm{AG}}{|}^2+|\overrightarrow{\mathrm{BG}}{|}^2+|\overrightarrow{\mathrm{CG}}{|}^2\right)$ is equal to ____ .

   Show answer

   Answer: 164

4. Q222025 · 3 Apr 2025 · Shift 1· Numerical answer

   Let $\vec{a}=\hat{i}+\hat{j}+\hat{k},\vec{b}=3\hat{i}+2\hat{j}-\hat{k},\vec{c}=\lambda \hat{j}+\mu \hat{k}$ and $\hat{d}$ be a unit vector such that $\vec{a}\times \hat{d}=\vec{b}\times \hat{d}$ and $\vec{c}\cdot \hat{d}=1$. If $\vec{c}$ is perpendicular to $\vec{a}$, then $|3\lambda \hat{d}+\mu \vec{c}{|}^2$ is equal to____

   Show answer

   Answer: 5

5. Q102025 · 3 Apr 2025 · Shift 2

   Let the equation $x(x+2)(12-k)=2$ have equal roots. Then the distance of the point $\left(k,\frac{k}{2}\right)$ from the line $3x+4y+5=0$ is

   1. A.$15$
   2. B.$12$
   3. C.$5\sqrt{3}$
   4. D.$15\sqrt{5}$
   Show answer

   Answer: A

6. Q242025 · 3 Apr 2025 · Shift 2· Numerical answer

   Let $\vec{a}=\hat{i}+2\hat{j}+\hat{k},\vec{b}=3\hat{i}-3\hat{j}+3\hat{k},\vec{c}=2\hat{i}-\hat{j}+2\hat{k}$ and $\vec{d}$ be a vector such that $\vec{b}\times \vec{d}=\vec{c}\times \vec{d}$ and $\vec{a}\cdot \vec{d}=4$. Then $|(\vec{a}\times \vec{d}){|}^2$ is equal to__________.

   Show answer

   Answer: 128

7. Q132025 · 2 Apr 2025 · Shift 1

   If $\vec{a}$ is a nonzero vector such that its projections on the vectors $2\hat{i}-\hat{j}+2\hat{k},\hat{i}+2\hat{j}-2\hat{k}$ and $\hat{k}$ are equal, then a unit vector along $\vec{a}$ is :

   1. A.$\frac{1}{\sqrt{155}}(-7\hat{i}+9\hat{j}+5\hat{k})$
   2. B.$\frac{1}{\sqrt{155}}(-7\hat{i}+9\hat{j}-5\hat{k})$
   3. C.$\frac{1}{\sqrt{155}}(7\hat{i}+9\hat{j}-5\hat{k})$
   4. D.$\frac{1}{\sqrt{155}}(7\hat{i}+9\hat{j}+5\hat{k})$
   Show answer

   Answer: D

8. Q192025 · 2 Apr 2025 · Shift 2

   Let $\vec{a}=2\hat{i}-3\hat{j}+\hat{k}$, $\vec{b}=3\hat{i}+2\hat{j}+5\hat{k}$ and a vector $\vec{c}$ be such that $(\vec{a}-\vec{c})\times \vec{b}=-18\hat{i}-3\hat{j}+12\hat{k}$ and $\vec{a}\cdot \vec{c}=3$. If $\vec{b}\times \vec{c}=\vec{d}$, then $|\vec{a}\cdot \vec{d}|$ is equal to :

   1. A.15
   2. B.18
   3. C.12
   4. D.9
   Show answer

   Answer: A

9. Q52025 · 29 Jan 2025 · Shift 2

   Let $\hat{a}$ be a unit vector perpendicular to the vectors $\vec{b}=\hat{i}-2\hat{j}+3\hat{k}$ and $\vec{c}=2\hat{i}+3\hat{j}-\hat{k}$, and $\hat{a}$ makes an angle of ${\cos }^{-1}\left(-\frac{1}{3}\right)$ with the vector $\hat{i}+\hat{j}+\hat{k}$. If $\hat{a}$ makes an angle of $\frac{\pi }{3}$ with the vector $\hat{i}+\alpha \hat{j}+\hat{k}$, then the value of $\alpha$ is:

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

   Answer: C

10. Q82025 · 29 Jan 2025 · Shift 2

    Let the line $x+y=1$ meet the axes of $x$ and $y$ at $A$ and $B$, respectively. A right angled triangle $AMN$ is inscribed in the triangle $OAB$, where $O$ is the origin and the points $M$ and $N$ lie on the lines $OB$ and $AB$, respectively. If the area of the triangle $AMN$ is $\frac{4}{9}$ of the area of the triangle $OAB$ and $AN:NB=\lambda :1$, then the sum of all possible value(s) of $\lambda$ is:

    1. A.$\frac{1}{2}$
    2. B.$\frac{5}{2}$
    3. C.$2$
    4. D.$\frac{13}{6}$
    Show answer

    Answer: C

11. Q252025 · 28 Jan 2025 · Shift 1· Numerical answer

    Let $\vec{a}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}},\vec{\mathrm{b}}=2\hat{\mathrm{i}}+2\hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\vec{\mathrm{d}}=\vec{\mathrm{a}}\times \vec{\mathrm{b}}$. If $\vec{\mathrm{c}}$ is a vector such that $\vec{\mathrm{a}}\cdot \vec{\mathrm{c}}=|\vec{\mathrm{c}}|$, $|\vec{\mathrm{c}}-2\vec{\mathrm{a}}{|}^2=8$ and the angle between $\vec{\mathrm{d}}$ and $\vec{\mathrm{c}}$ is $\frac{\pi }{4}$, then $|10-3\vec{\mathrm{b}}\cdot \vec{\mathrm{c}}|+|\vec{\mathrm{d}}\times \vec{\mathrm{c}}{|}^2$ is equal to ____.

    Show answer

    Answer: 6

12. Q132025 · 24 Jan 2025 · Shift 1

    Let $\vec{a}=\hat{i}+2\hat{j}+3\hat{k},\vec{b}=3\hat{i}+\hat{j}-\hat{k}$ and $\vec{c}$ be three vectors such that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}$. If the vector $\vec{c}$ is perpendicular to $\vec{b}$ and $\vec{a}\cdot \vec{c}=5$, then $|\vec{c}|$ is equal to

    1. A.$\sqrt{\frac{11}{6}}$
    2. B.$\frac{1}{3\sqrt{2}}$
    3. C.$18$
    4. D.$16$
    Show answer

    Answer: A

13. Q102025 · 24 Jan 2025 · Shift 2

    Let the position vectors of three vertices of a triangle be $4\vec{p}+\vec{q}-3\vec{r}$, $-5\vec{p}+\vec{q}+2\vec{r}$ and $2\vec{p}-\vec{q}+2\vec{r}$. If the position vectors of the orthocenter and the circumcenter of the triangle are $\frac{\vec{p}+\vec{q}+\vec{r}}{4}$ and $\alpha \vec{p}+\beta \vec{q}+\gamma \vec{r}$ respectively, then $\alpha +2\beta +5\gamma$ is equal to :

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

    Answer: B

14. Q132025 · 24 Jan 2025 · Shift 2

    Let $\vec{a}=3\hat{i}-\hat{j}+2\hat{k}$, $\vec{b}=\vec{a}\times (\hat{i}-2\hat{k})$ and $\vec{c}=\vec{b}\times \hat{k}$. Then the projection of $\vec{c}-2\hat{j}$ on $\vec{a}$ is :

    1. A.$2\sqrt{7}$
    2. B.$3\sqrt{7}$
    3. C.$\sqrt{14}$
    4. D.$2\sqrt{14}$
    Show answer

    Answer: D

15. Q42025 · 23 Jan 2025 · Shift 2

    Let the point $\mathrm{A}$ divide the line segment joining the points $\mathrm{P}(-1,-1,2)$ and $\mathrm{Q}(5,5,10)$ internally in the ratio $r:1(r>0)$. If $\mathrm{O}$ is the origin and $(\overrightarrow{\mathrm{OQ}}\cdot \overrightarrow{\mathrm{OA}})-\frac{1}{5}|\overrightarrow{\mathrm{OP}}\times \overrightarrow{\mathrm{OA}}{|}^2=10$, then the value of $r$ is :

    1. A.$\sqrt{7}$
    2. B.14
    3. C.7
    4. D.3
    Show answer

    Answer: C

16. Q142025 · 23 Jan 2025 · Shift 2

    Let $A=\{(x,y)\in \mathbf{R}\times \mathbf{R}:|x+y|\ge 3\}$ and $B=\{(x,y)\in \mathbf{R}\times \mathbf{R}:|x|+|y|\le 3\}$. If $C=\{(x,y)\in A\cap B:x=0\text{ or }y=0\}$, then ${\sum }_{(x,y)\in C}|x+y|$ is :

    1. A.18
    2. B.24
    3. C.15
    4. D.12
    Show answer

    Answer: D

17. Q222025 · 22 Jan 2025 · Shift 1· Numerical answer

    Let $\vec{c}$ be the projection vector of $\vec{b}=\lambda \hat{i}+4\hat{k},\lambda >0$, on the vector $\vec{a}=\hat{i}+2\hat{j}+2\hat{k}$. If $|\vec{a}+\vec{c}|=7$, then the area of the parallelogram formed by the vectors $\vec{b}$ and $\vec{c}$ is _________.

    Show answer

    Answer: 16