Vector Algebra · JEE Main previous year questions

2023 39 questions

1. 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 ___________.

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   Answer: 3

2. 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
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   Answer: D

3. 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
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   Answer: A

4. 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 ____.

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   Answer: 2

5. 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
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   Answer: C

6. 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.$\frac{5}{\sqrt{2}}$
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   Answer: D

7. Q12023 · 25 Jan 2023 · Shift 1

   The vector $\vec{a}=-\hat{i}+2\hat{j}+\hat{k}$ is rotated through a right angle, passing through the y-axis in its way and the resulting vector is $\vec{b}$. Then the projection of $3\vec{a}+\sqrt{2}\vec{b}$ on $\vec{c}=5\hat{i}+4\hat{j}+3\hat{k}$ is :

   1. A.$\sqrt{6}$
   2. B.$2\sqrt{3}$
   3. C.$1$
   4. D.$3\sqrt{2}$
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   Answer: D

8. Q192023 · 25 Jan 2023 · Shift 1

   Let $\vec{a},\vec{b}$ and $\vec{c}$ be three non zero vectors such that $\vec{b}\cdot \vec{c}=0$ and $\vec{a}\times (\vec{b}\times \vec{c})=\frac{\vec{b}-\vec{c}}{2}$. If $\vec{d}$ be a vector such that $\vec{b}\cdot \vec{d}=\vec{a}\cdot \vec{b}$, then $(\vec{a}\times \vec{b})\cdot (\vec{c}\times \vec{d})$ is equal to

   1. A.$\frac{1}{2}$
   2. B.$-\frac{1}{4}$
   3. C.$\frac{1}{4}$
   4. D.$\frac{3}{4}$
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   Answer: C

9. Q182023 · 25 Jan 2023 · Shift 2

   Let $\vec{a}=-\hat{i}-\hat{j}+\hat{k},\vec{a}\cdot \vec{b}=1$ and $\vec{a}\times \vec{b}=\hat{i}-\hat{j}$. Then $\vec{a}-6\vec{b}$ is equal to :

   1. A.$3\left(\hat{i}+\hat{j}+\hat{k}\right)$
   2. B.$3\left(\hat{i}-\hat{j}-\hat{k}\right)$
   3. C.$3\left(\hat{i}+\hat{j}-\hat{k}\right)$
   4. D.$3\left(\hat{i}-\hat{j}+\hat{k}\right)$
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   Answer: A

10. Q192023 · 25 Jan 2023 · Shift 2

    If the four points, whose position vectors are $3\hat{i}-4\hat{j}+2\hat{k},\hat{i}+2\hat{j}-\hat{k},-2\hat{i}-\hat{j}+3\hat{k}$ and $5\hat{i}-2\alpha \hat{j}+4\hat{k}$ are coplanar, then $\alpha$ is equal to :

    1. A.$\frac{73}{17}$
    2. B.$-\frac{73}{17}$
    3. C.$-\frac{107}{17}$
    4. D.$\frac{107}{17}$
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    Answer: A

11. Q132023 · 24 Jan 2023 · Shift 1

    Let $f(x)=\{\begin{array}{ll}{x}^2\sin \left(\frac{1}{x}\right)& ,x\ne 0\\ 0& ,x=0\end{array}$ Then at $x=0$

    1. A.$f$ is continuous but $f^{\prime }$ is not continuous
    2. B.$f$ and $f^{\prime }$ both are continuous
    3. C.$f$ is continuous but not differentiable
    4. D.$f^{\prime }$ is continuous but not differentiable
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    Answer: A

12. Q302023 · 24 Jan 2023 · Shift 1· Numerical answer

    Suppose ${\sum }_{r=0}^{2023}{r}^2{}^{2023}{\mathrm{C}}_r=2023\times \alpha \times 2^{2022}$. Then the value of $\alpha$ is ____

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    Answer: 1012

13. Q42023 · 24 Jan 2023 · Shift 2

    Let $\vec{\alpha }=4\hat{i}+3\hat{j}+5\hat{k}$ and $\vec{\beta }=\hat{i}+2\hat{j}-4\hat{k}$. Let ${\vec{\beta }}_1$ be parallel to $\vec{\alpha }$ and ${\vec{\beta }}_2$ be perpendicular to $\vec{\alpha }$. If $\vec{\beta }={\vec{\beta }}_1+{\vec{\beta }}_2$, then the value of $5{\vec{\beta }}_2\cdot (\hat{i}+\hat{j}+\hat{k})$ is :

    1. A.9
    2. B.7
    3. C.6
    4. D.11
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    Answer: B

14. Q232023 · 24 Jan 2023 · Shift 2· Numerical answer

    Let $\vec{a}=\hat{i}+2\hat{j}+\lambda \hat{k},\vec{b}=3\hat{i}-5\hat{j}-\lambda \hat{k},\vec{a}\cdot \vec{c}=7,2\vec{b}\cdot \vec{c}+43=0,\vec{a}\times \vec{c}=\vec{b}\times \vec{c}$. Then $|\vec{a}\cdot \vec{b}|$ is equal to :

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    Answer: 8