Vectors · NDA previous year questions

What is $(\vec{a}-\vec{b})\times (\vec{a}+\vec{b})$ equal to?

1. A.$\vec{0}$
2. B.$\vec{a}\times \vec{b}$
3. C.$2(\vec{a}\times \vec{b})$
4. D.$|\vec{a}{|}^2-|\vec{b}{|}^2$

A spacecraft located at $\hat{i}+2\hat{j}+3\hat{k}$ is subjected to a force $\lambda \hat{k}$ by firing a rocket. The spacecraft is subjected to a moment of magnitude

1. A.$\lambda$
2. B.$\sqrt{3}\lambda$
3. C.$\sqrt{5}\lambda$
4. D.None of the above

If $\vec{a}$ and $\vec{b}$ are vectors such that $|\vec{a}|=2,|\vec{b}|=7$ and $\vec{a}\times \vec{b}=3\hat{i}+2\hat{j}+6\hat{k}$, then what is the acute angle between $\vec{a}$ and $\vec{b}$?

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

Let $\vec{p}$ and $\vec{q}$ be the position vectors of the points P and Q respectively with respect to origin O. The points R and S divide PQ internally and externally respectively in the ratio 2 : 3. If $\overrightarrow{OR}$ and $\overrightarrow{OS}$ are perpendicular, then which one of the following is correct?

1. A.$9p^2=4q^2$
2. B.$4p^2=9q^2$
3. C.$9p=4q$
4. D.$4p=9q$

What is the moment about the point $\hat{i}+2\hat{j}-\hat{k}$ of a force represented by $3\hat{i}+\hat{k}$ acting through the point $2\hat{i}-\hat{j}+3\hat{k}$?

1. A.$-3\hat{i}+11\hat{j}+9\hat{k}$
2. B.$3\hat{i}+2\hat{j}+9\hat{k}$
3. C.$3\hat{i}+4\hat{j}+9\hat{k}$
4. D.$\hat{i}+\hat{j}+\hat{k}$

If $\vec{a}+2\vec{b}+3\vec{c}=\vec{0}$ and $\vec{a}\times \vec{b}+\vec{b}\times \vec{c}+\vec{c}\times \vec{a}=\lambda (\vec{b}\times \vec{c})$, then what is the value of $\lambda$?

1. A.$2$
2. B.$3$
3. C.$4$
4. D.$6$

If the vectors $\vec{k}$ and $\vec{A}$ are parallel to each other, then what is $k\vec{k}\times \vec{A}$ equal to?

1. A.$k^2\vec{A}$
2. B.$\vec{0}$
3. C.$-k^2\vec{A}$
4. D.$\vec{A}$

If $\alpha ,\beta$ and $\gamma$ are the angles which the vector $\overrightarrow{OP}$ ($O$ being the origin) makes with positive direction of the coordinate axes, then which of the following are correct? 1. ${\cos }^2\alpha +{\cos }^2\beta ={\sin }^2\gamma$ 2. ${\sin }^2\alpha +{\sin }^2\beta ={\cos }^2\gamma$ 3. ${\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma =2$ Select the correct answer using the code given below.

1. A.1 and 2 only
2. B.2 and 3 only
3. C.1 and 3 only
4. D.1, 2 and 3

Let $\vec{\alpha }=\hat{i}+2\hat{j}-\hat{k}$, $\vec{\beta }=2\hat{i}-\hat{j}+3\hat{k}$ and $\vec{\gamma }=2\hat{i}+\hat{j}+6\hat{k}$ be three vectors. If $\vec{\alpha }$ and $\vec{\beta }$ are both perpendicular to the vector $\vec{\delta }$ and $\vec{\delta }\cdot \vec{\gamma }=10$, then what is the magnitude of $\vec{\delta }$?

1. A.$\sqrt{3}$ units
2. B.$2\sqrt{3}$ units
3. C.$\frac{\sqrt{3}}{2}$ unit
4. D.$\frac{1}{\sqrt{3}}$ unit

If $\hat{a}$ and $\hat{b}$ are two unit vectors, then the vector $(\hat{a}+\hat{b})\times (\hat{a}\times \hat{b})$ is parallel to

1. A.$(\hat{a}-\hat{b})$
2. B.$(\hat{a}+\hat{b})$
3. C.$(2\hat{a}-\hat{b})$
4. D.$(2\hat{a}+\hat{b})$

A force $\vec{F}=\hat{i}+3\hat{j}+2\hat{k}$ acts on a particle to displace it from the point $A(\hat{i}+2\hat{j}-3\hat{k})$ to the point $B(3\hat{i}-\hat{j}+5\hat{k})$. The work done by the force will be

1. A.5 units
2. B.7 units
3. C.9 units
4. D.10 units

For any vector $\vec{a}$ $|\vec{a}\times \hat{i}{|}^2+|\vec{a}\times \hat{j}{|}^2+|\vec{a}\times \hat{k}{|}^2$ is equal to

1. A.$|\vec{a}{|}^2$
2. B.$2|\vec{a}{|}^2$
3. C.$3|\vec{a}{|}^2$
4. D.$4|\vec{a}{|}^2$

If the vectors $a\hat{i}+\hat{j}+\hat{k}$, $\hat{i}+b\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+c\hat{k}$ ($a,b,c\ne 1$) are coplanar, then the value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is equal to

1. A.0
2. B.1
3. C.$a+b+c$
4. D.$abc$

The points $P(3,2,4)$, $Q(4,5,2)$, $R(5,8,0)$ and $S(2,-1,6)$ are

1. A.vertices of a rhombus which is not a square
2. B.non-coplanar
3. C.collinear
4. D.coplanar but not collinear

If $\vec{a}=\hat{i}-\hat{j}+\hat{k}$, $\vec{b}=2\hat{i}+3\hat{j}+2\hat{k}$ and $\vec{c}=\hat{i}+m\hat{j}+n\hat{k}$ are three coplanar vectors and $|\vec{c}|=\sqrt{6}$, then which one of the following is correct?

1. A.$m=2$ and $n=\pm 1$
2. B.$m=\pm 2$ and $n=-1$
3. C.$m=2$ and $n=-1$
4. D.$m=\pm 2$ and $n=1$

Consider the following for the next two (02) items that follow: Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be three vectors such that $\vec{a}+\vec{b}+\vec{c}=\vec{0}$, and $|\vec{a}|=10$, $|\vec{b}|=6$ and $|\vec{c}|=14$. What is $\vec{a}\cdot \vec{b}+\vec{b}\cdot \vec{c}+\vec{c}\cdot \vec{a}$ equal to?

1. A.$-332$
2. B.$-166$
3. C.0
4. D.166

Consider the following for the next two (02) items that follow: Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be three vectors such that $\vec{a}+\vec{b}+\vec{c}=\vec{0}$, and $|\vec{a}|=10$, $|\vec{b}|=6$ and $|\vec{c}|=14$. What is the angle between $\vec{a}$ and $\vec{b}$?

1. A.$30^{\circ }$
2. B.$45^{\circ }$
3. C.$60^{\circ }$
4. D.$75^{\circ }$

What is a vector of unit length orthogonal to both the vectors $\hat{i}+\hat{j}+\hat{k}$ and $2\hat{i}+3\hat{j}-\hat{k}$?

1. A.$\frac{-4\hat{i}+3\hat{j}-\hat{k}}{\sqrt{26}}$
2. B.$\frac{-4\hat{i}+3\hat{j}+\hat{k}}{\sqrt{26}}$
3. C.$\frac{-3\hat{i}+2\hat{j}-\hat{k}}{\sqrt{14}}$
4. D.$\frac{-3\hat{i}+2\hat{j}+\hat{k}}{\sqrt{14}}$

If $\vec{a},\vec{b}$ and $\vec{c}$ are the position vectors of the vertices of an equilateral triangle whose orthocentre is at the origin, then which one of the following is correct?

1. A.$\vec{a}+\vec{b}+\vec{c}=\vec{0}$
2. B.$\vec{a}+\vec{b}+\vec{c}=\text{unit vector}$
3. C.$\vec{a}+\vec{b}=\vec{c}$
4. D.$\vec{a}=\vec{b}+\vec{c}$

What is the area of the parallelogram having diagonals $3\hat{i}+\hat{j}-2\hat{k}$ and $\hat{i}-3\hat{j}+4\hat{k}$?

1. A.$5\sqrt{5}$ square units
2. B.$4\sqrt{5}$ square units
3. C.$5\sqrt{3}$ square units
4. D.$15\sqrt{2}$ square units

For the next two (2) items that follow: Let $\hat{a},\hat{b}$ be two unit vectors and $\theta$ be the angle between them. What is $\cos \left(\frac{\theta }{2}\right)$ equal to?

1. A.$\frac{|\hat{a}-\hat{b}|}{2}$
2. B.$\frac{|\hat{a}+\hat{b}|}{2}$
3. C.$\frac{|\hat{a}-\hat{b}|}{4}$
4. D.$\frac{|\hat{a}+\hat{b}|}{4}$

For the next two (2) items that follow: Let $\hat{a},\hat{b}$ be two unit vectors and $\theta$ be the angle between them. What is $\sin \left(\frac{\theta }{2}\right)$ equal to?

1. A.$\frac{|\hat{a}-\hat{b}|}{2}$
2. B.$\frac{|\hat{a}+\hat{b}|}{2}$
3. C.$\frac{|\hat{a}-\hat{b}|}{4}$
4. D.$\frac{|\hat{a}+\hat{b}|}{4}$

$ABCD$ is a parallelogram and $P$ is the point of intersection of the diagonals. If $O$ is the origin, then $$\overrightarrow{OA}+\overrightarrow{OB}+\overrightarrow{OC}+\overrightarrow{OD}\text{ is equal to}$$

1. A.$4\overrightarrow{OP}$
2. B.$2\overrightarrow{OP}$
3. C.$\overrightarrow{OP}$
4. D.Null vector

If $\vec{b}$ and $\vec{c}$ are the position vectors of the points $B$ and $C$ respectively, then the position vector of the point $D$ such that $\overrightarrow{BD}=4\overrightarrow{BC}$ is

1. A.$4(\vec{c}-\vec{b})$
2. B.$-4(\vec{c}-\vec{b})$
3. C.$4\vec{c}-3\vec{b}$
4. D.$4\vec{c}+3\vec{b}$

If the position vector $\vec{a}$ of the point $(5,n)$ is such that $|\vec{a}|=13$, then the value/values of $n$ can be

1. A.$\pm 8$
2. B.$\pm 12$
3. C.8 only
4. D.12 only