Vector Algebra · JEE Main previous year questions

2020 13 questions

1. Q212020 · 6 Sept 2020 · Shift 1· Numerical answer

   If $\vec{a}$ and $\vec{b}$ are unit vectors, then the greatest value of $$\sqrt{3}|\vec{a}+\vec{b}|+|\vec{a}-\vec{b}|$$ is ______ .

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   Answer: 4

2. Q242020 · 6 Sept 2020 · Shift 2· Numerical answer

   If $\vec{x}$ and $\vec{y}$ be two non-zero vectors such that $|\vec{x}+\vec{y}|=|\vec{x}|$ and $2\vec{x}+\lambda \vec{y}$ is perpendicular to $\vec{y}$, then the value of $\lambda$ is ______ .

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   Answer: 1

3. Q22020 · 3 Sept 2020 · Shift 1

   The lines $\vec{r}=(\hat{i}-\hat{j})+l(2\hat{i}+\hat{k})$ and $\vec{r}=(2\hat{i}-\hat{j})+m(\hat{i}+\hat{j}+\hat{k})$

   1. A.do not intersect for any values of $l$ and m
   2. B.intersect for all values of $l$ and m
   3. C.intersect when $l=2$ and $\mathrm{m}=\frac{1}{2}$
   4. D.intersect when $l=1$ and $\mathrm{m}=2$
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   Answer: A

4. Q212020 · 3 Sept 2020 · Shift 1· Numerical answer

   Let $A=\left[\begin{array}{cc}x& 1\\ 1& 0\end{array}\right],x\in \mathbb{R}$ and $A^4=[a_{ij}]$. If $a_{11}=109$, then $a_{22}$ is equal to _______.

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   Answer: 10

5. Q42020 · 3 Sept 2020 · Shift 2

   Let $a,b,c\in \mathbb{R}$ be such that $a^2+b^2+c^2=1$. If $a\cos \theta =b\cos \left(\theta +\frac{2\pi }{3}\right)=c\cos \left(\theta +\frac{4\pi }{3}\right)$, where $\theta =\frac{\pi }{9}$, then the angle between the vectors $a\hat{i}+b\hat{j}+c\hat{k}$ and $b\hat{i}+c\hat{j}+a\hat{k}$ is :

   1. A.0
   2. B.$\frac{\pi }{9}$
   3. C.$\frac{2\pi }{3}$
   4. D.$\frac{\pi }{2}$
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   Answer: D

6. Q222020 · 2 Sept 2020 · Shift 1· Numerical answer

   Let $\vec{a},\vec{b}$ and $\vec{c}$ be three unit vectors such that ${\left|\vec{a}-\vec{b}\right|}^2+{\left|\vec{a}-\vec{c}\right|}^2=8$. Then ${\left|\vec{a}+2\vec{b}\right|}^2+{\left|\vec{a}+2\vec{c}\right|}^2$ is equal to _______.

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   Answer: 2

7. Q232020 · 2 Sept 2020 · Shift 2· Numerical answer

   Let the position vectors of points 'A' and 'B' be $\hat{i}+\hat{j}+\hat{k}$ and $2\hat{i}+\hat{j}+3\hat{k}$, respectively. A point 'P' divides the line segment AB internally in the ratio $\lambda :1(\lambda >0)$. If O is the origin and ${\left|\overrightarrow{OB}\cdot \overrightarrow{OP}-3\left|\overrightarrow{OA}\times \overrightarrow{OP}\right|\right|}^2=6$, then $\lambda$ is equal to _______.

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   Answer: 0.8

8. Q242020 · 9 Jan 2020 · Shift 1· Numerical answer

   The projection of the line segment joining the points (1, –1, 3) and (2, –4, 11) on the line joining the points (–1, 2, 3) and (3, –2, 10) is ____________.

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   Answer: 8

9. Q252020 · 9 Jan 2020 · Shift 1· Numerical answer

   If the vectors, $\vec{p}=(a+1)\hat{i}+a\hat{j}+a\hat{k}$, $\vec{q}=a\hat{i}+(a+1)\hat{j}+a\hat{k}$ and $\vec{r}=a\hat{i}+a\hat{j}+(a+1)\hat{k}(a\in \mathbb{R})$ are coplanar and $3(\vec{p}\cdot \vec{q}{)}^2-\lambda |\vec{r}\times \vec{q}{|}^2=0$, then the value of $\lambda$ is _______.

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   Answer: 1

10. Q222020 · 9 Jan 2020 · Shift 2· Numerical answer

    Let $\vec{a},\vec{b}$ and $\vec{c}$ be three vectors such that $|\vec{a}|=\sqrt{3}$, $|\vec{b}|=5$, $\vec{b}\cdot \vec{c}=10$ and the angle between $\vec{b}$ and $\vec{c}$ is $\frac{\pi }{3}$. If $\vec{a}$ is perpendicular to the vector $\vec{b}\times \vec{c}$, then $\left|\vec{a}\times \left(\vec{b}\times \vec{c}\right)\right|$ is equal to _______.

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    Answer: 30

11. Q12020 · 8 Jan 2020 · Shift 2

    Let $\vec{a}=\hat{i}-2\hat{j}+\hat{k}$ and $\vec{b}=\hat{i}-\hat{j}+\hat{k}$ be two vectors. If $\vec{c}$ is a vector such that $\vec{b}\times \vec{c}=\vec{b}\times \vec{a}$ and $\vec{c}\cdot \vec{a}=0$, then $\vec{c}\cdot \vec{b}$ is equal to

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

12. Q42020 · 7 Jan 2020 · Shift 1

    A vector $\vec{a}=\alpha \hat{i}+2\hat{j}+\beta \hat{k}$ $(\alpha ,\beta \in \mathbb{R})$ lies in the plane of the vectors, $\vec{b}=\hat{i}+\hat{j}$ and $\vec{c}=\hat{i}-\hat{j}+4\hat{k}$. If $\vec{a}$ bisects the angle between $\vec{b}$ and $\vec{c}$, then:

    1. A.$\vec{a}\cdot \hat{i}+3=0$
    2. B.$\vec{a}\cdot \hat{k}-4=0$
    3. C.$\vec{a}\cdot \hat{i}+1=0$
    4. D.$\vec{a}\cdot \hat{k}+2=0$
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    Answer: B

13. Q62020 · 7 Jan 2020 · Shift 2

    Let $\vec{a},\vec{b}$ and $\vec{c}$ be three unit vectors such that $\vec{a}+\vec{b}+\vec{c}=\vec{0}$. If $\lambda =\vec{a}\cdot \vec{b}+\vec{b}\cdot \vec{c}+\vec{c}\cdot \vec{a}$ and $\vec{d}=\vec{a}\times \vec{b}+\vec{b}\times \vec{c}+\vec{c}\times \vec{a}$, then the ordered pair, $(\lambda ,\vec{d})$ is equal to :

    1. A.$\left(\frac{3}{2},3\vec{a}\times \vec{c}\right)$
    2. B.$\left(-\frac{3}{2},3\vec{c}\times \vec{b}\right)$
    3. C.$\left(-\frac{3}{2},3\vec{a}\times \vec{b}\right)$
    4. D.$\left(\frac{3}{2},3\vec{b}\times \vec{c}\right)$
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    Answer: C