Vector Algebra · JEE Main previous year questions

2019 13 questions

1. Q152019 · 12 Apr 2019 · Shift 1

   Let $\vec{a}=3\hat{i}+2\hat{j}+2\hat{k}$ and $\vec{b}=\hat{i}+2\hat{j}-2\hat{k}$ be two vectors. If a vector perpendicular to both the vectors $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$ has the magnitude 12 then one such vector is :

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

   Answer: A

2. Q292019 · 12 Apr 2019 · Shift 1

   If the volume of parallelopiped formed by the vectors $\hat{i}+\lambda \hat{j}+\hat{k}$, $\hat{j}+\lambda \hat{k}$ and $\lambda \hat{i}+\hat{k}$ is minimum, then $\lambda$ is equal to :

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

   Answer: D

3. Q252019 · 12 Apr 2019 · Shift 2

   Let $\alpha \in \mathbb{R}$ and the three vectors $\vec{a}=\alpha \hat{i}+\hat{j}+3\hat{k}$, $\vec{b}=2\hat{i}+\hat{j}-\alpha \hat{k}$ and $\vec{c}=\alpha \hat{i}-2\hat{j}+3\hat{k}$. Then the set $S=\{\alpha :\vec{a},\vec{b}\text{ and }\vec{c}\text{ are coplanar}\}$ :

   1. A.contains exactly two numbers only one of which is positive
   2. B.is singleton
   3. C.contains exactly two positive numbers
   4. D.is empty
   Show answer

   Answer: D

4. Q252019 · 10 Apr 2019 · Shift 1

   The sum $\frac{3\times 1^3}{1^2}+\frac{5\times (1^3+2^3)}{1^2+2^2}+\frac{7\times (1^3+2^3+3^3)}{1^2+2^2+3^2}+\dots$ upto 10 terms is:

   1. A.600
   2. B.660
   3. C.680
   4. D.620
   Show answer

   Answer: B

5. Q202019 · 10 Apr 2019 · Shift 2

   The distance of the point having position vector $-\hat{i}+2\hat{j}+6\hat{k}$ from the straight line passing through the point (2, 3, - 4) and parallel to the vector, $6\hat{i}+3\hat{j}-4\hat{k}$ is :

   1. A.6
   2. B.7
   3. C.$2\sqrt{13}$
   4. D.$4\sqrt{3}$
   Show answer

   Answer: B

6. Q12019 · 9 Apr 2019 · Shift 1

   Let $\vec{\alpha }=3\hat{i}+\hat{j}$ and $\vec{\beta }=2\hat{i}-\hat{j}+3\hat{k}$. If $\vec{\beta }={\vec{\beta }}_1-{\vec{\beta }}_2$, where ${\vec{\beta }}_1$ is parallel to $\vec{\alpha }$ and ${\vec{\beta }}_2$ is perpendicular to $\vec{\alpha }$, then ${\vec{\beta }}_1\times {\vec{\beta }}_2$ is equal to

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

   Answer: B

7. Q202019 · 9 Apr 2019 · Shift 2

   If a unit vector $\vec{a}$ makes angles $\pi /3$ with $\hat{i}$, $\pi /4$ with $\hat{j}$ and $\theta \in (0,\pi )$ with $\hat{k}$, then a value of $\theta$ is :-

   1. A.$\frac{5\pi }{6}$
   2. B.$\frac{5\pi }{12}$
   3. C.$\frac{2\pi }{3}$
   4. D.$\frac{\pi }{4}$
   Show answer

   Answer: C

8. Q32019 · 8 Apr 2019 · Shift 2

   Let $\vec{a}=3\hat{i}+2\hat{j}+x\hat{k}$ and $\vec{b}=\hat{i}-\hat{j}+\hat{k}$, for some real x. Then $|\vec{a}\times \vec{b}|=r$ is possible if :

   1. A.$0<r<\sqrt{\frac{3}{2}}$
   2. B.$3\sqrt{\frac{3}{2}}<r<5\sqrt{\frac{3}{2}}$
   3. C.$r\ge 5\sqrt{\frac{3}{2}}$
   4. D.$\sqrt{\frac{3}{2}}<r\le 3\sqrt{\frac{3}{2}}$
   Show answer

   Answer: C

9. Q262019 · 8 Apr 2019 · Shift 2

   The vector equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x − y + z = 0 is :

   1. A.$\vec{r}\times (\hat{i}-\hat{k})-2=0$
   2. B.$\vec{r}\cdot (\hat{i}+\hat{k})+2=0$
   3. C.$\vec{r}\cdot (\hat{i}-\hat{k})+2=0$
   4. D.$\vec{r}\times (\hat{i}-\hat{k})+2=0$
   Show answer

   Answer: C

10. Q242019 · 12 Jan 2019 · Shift 1

    The sum of the distinct real values of $\mu$, for which the vectors, $\mu \hat{i}+\hat{j}+\hat{k}$, $\hat{i}+\mu \hat{j}+\hat{k}$, $\hat{i}+\hat{j}+\mu \hat{k}$ are co-planar, is :

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

    Answer: B

11. Q292019 · 12 Jan 2019 · Shift 2

    Let $\vec{a},\vec{b}$ and $\vec{c}$ be three unit vectors, out of which vectors $\vec{b}$ and $\vec{c}$ are non-parallel. If $\alpha$ and $\beta$ are the angles which vector $\vec{a}$ makes with vectors $\vec{b}$ and $\vec{c}$ respectively and $\vec{a}\times (\vec{b}\times \vec{c})=\frac{1}{2}\vec{b}$, then $|\alpha -\beta |$ is equal to :

    1. A.$90^{\circ }$
    2. B.$30^{\circ }$
    3. C.$45^{\circ }$
    4. D.$60^{\circ }$
    Show answer

    Answer: B

12. Q182019 · 11 Jan 2019 · Shift 2

    Let $\sqrt{3}\hat{i}+\hat{j}$, $\hat{i}+\sqrt{3}\hat{j}$ and $\beta \hat{i}+(1-\beta )\hat{j}$ respectively be the position vectors of the points A, B and C with respect to the origin O. If the distance of C from the bisector of the acute angle between OA and OB is $\frac{3}{\sqrt{2}}$, then the sum of all possible values of $\beta$ is :

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

    Answer: B

13. Q232019 · 9 Jan 2019 · Shift 1

    Let $\vec{a}=\hat{i}-\hat{j}$, $\vec{b}=\hat{i}+\hat{j}+\hat{k}$ and $\vec{c}$ be a vector such that $\vec{a}\times \vec{c}+\vec{b}=\vec{0}$ and $\vec{a}\cdot \vec{c}=4$, then $|\vec{c}{|}^2$ is equal to :

    1. A.8
    2. B.$\frac{19}{2}$
    3. C.9
    4. D.$\frac{17}{2}$
    Show answer

    Answer: B