Vectors · NDA previous year questions

$ABCD$ is a quadrilateral. Forces $\overrightarrow{AB},\overrightarrow{CB},\overrightarrow{CD}$ and $\overrightarrow{DA}$ act along its sides. What is their resultant?

1. A.$2\overrightarrow{CD}$
2. B.$2\overrightarrow{DA}$
3. C.$2\overrightarrow{BC}$
4. D.$2\overrightarrow{CB}$

What is the value of $b$ such that the scalar product of the vector $\hat{i}+\hat{j}+\hat{k}$ with the unit vector parallel to the sum of the vectors $2\hat{i}+4\hat{j}-5\hat{k}$ and $b\hat{i}+2\hat{j}+3\hat{k}$ is unity?

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

Consider the following statements: 1. For any three vectors $\vec{a},\vec{b},\vec{c}$; $\vec{a}\cdot \{(\vec{b}+\vec{c})\times (\vec{a}+\vec{b}+\vec{c})\}=0$ 2. For any three coplanar unit vectors $\vec{d},\vec{e},\vec{f}$; $(\vec{d}\times \vec{e})\cdot \vec{f}=1$ Which of the statements given above is/are correct?

1. A.1 only
2. B.2 only
3. C.Both 1 and 2
4. D.Neither 1 nor 2

What is the value of $\lambda$ for which the vectors $\hat{i}-\hat{j}+\hat{k}$, $2\hat{i}+\hat{j}-\hat{k}$, $\lambda \hat{i}-\hat{j}+\lambda \hat{k}$ are coplanar?

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

A vector $\vec{b}$ is collinear with the vector $\vec{a}=(2,1,-1)$ and satisfies the condition $\vec{a}\cdot \vec{b}=3$. What is $\vec{b}$ equal to?

1. A.(1, 1/2, -1/2)
2. B.(2/3, 1/3, -1/3)
3. C.(1/2, 1/4, -1/4)
4. D.(1, 1, 0)

What is the least positive integer n for which ${\left(\frac{1+i}{1-i}\right)}^n=1$?

1. A.16
2. B.12
3. C.8
4. D.4

The vectors $\vec{a}=x\hat{i}+y\hat{j}+z\hat{k}$, $\vec{b}=\hat{k}$, $\vec{c}$ are such that they form a right-handed system. What is $\vec{c}$ equal to?

1. A.$\hat{j}$
2. B.$y\hat{j}-x\hat{k}$
3. C.$y\hat{i}-x\hat{j}$
4. D.$x\hat{i}-y\hat{j}$

If $x=t^2,y=t^3$, then what is $\frac{d^{2}y}{d{x}^2}$ equal to?

1. A.1
2. B.$\frac{3}{2t}$
3. C.$\frac{3}{4t}$
4. D.$\frac{3}{2}$

What is ${\int }_{-\pi /4}^{\pi /4}{\tan }^{3}xdx$ equal to?

1. A.$\sqrt{3}$
2. B.$\frac{1}{3}$
3. C.$\frac{1}{2}$
4. D.0

If the angle between the vectors $\vec{a}$ and $\vec{b}$ is $\frac{\pi }{3}$, what is the angle between $-5\vec{a}$ and $6\vec{b}$?

1. A.$\frac{\pi }{6}$
2. B.$\frac{2\pi }{3}$
3. C.$\frac{2\pi }{5}$
4. D.$\frac{3\pi }{7}$

Let $\vec{a}$ and $\vec{b}$ be two unit vectors and $\alpha$ be the angle between them. If $(\vec{a}+\vec{b})$ is also the unit vector, then what is the value of $\alpha$?

1. A.$\frac{\pi }{4}$
2. B.$\frac{\pi }{3}$
3. C.$\frac{2\pi }{3}$
4. D.$\frac{\pi }{2}$

What is the value of $\frac{(0.101{)}_2^{(11{)}_2}+(0.011{)}_2^{(11{)}_2}}{(0.101{)}_2^{(10{)}_2}-(0.101{)}_2^{(01{)}_2}(0.011{)}_2^{(01{)}_2}+(0.011{)}_2^{(10{)}_2}}$?

1. A.$(0.001{)}_2$
2. B.$(0.01{)}_2$
3. C.$(0.1{)}_2$
4. D.$(1{)}_2$

If $\hat{a}$ and $\hat{b}$ are the unit vectors along $\bar{a}$ and $\bar{b}$ respectively, then what is the projection of $\bar{b}$ on $\bar{a}$ ?

1. A.$\bar{a}\bullet \bar{b}$
2. B.$\hat{a}\bullet \hat{b}$
3. C.$\hat{a}\bullet \bar{b}$
4. D.$|\bar{a}\times \bar{b}|$

What are the unit vectors parallel to xy-plane and perpendicular to the vector $4\hat{i}-3\hat{j}+\hat{k}$ ?

1. A.$\pm (3\hat{i}+4\hat{j})/5$
2. B.$\pm (4\hat{i}+3\hat{j})/5$
3. C.$\pm (3\hat{i}-4\hat{j})/5$
4. D.$\pm (4\hat{i}-3\hat{j})/5$

What is the vector in the $xy$-plane through origin and perpendicular to the vector $\bar{r}=a\hat{i}+b\hat{j}$ and of the same length ?

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

Given $\bar{a}=2\hat{i}-3\hat{j}+4\hat{k}$ and $\bar{b}$ is a unit vector co-directional with $\bar{a}$. If $m$ is a scalar such that $\hat{b}=m\bar{a}$, then what is the value of $m$ ?

1. A.$1/5$
2. B.$1/\sqrt{5}$
3. C.$1/29$
4. D.$1/\sqrt{29}$

The magnitude of the vectors $\bar{a}$ and $\bar{b}$ are equal and the angle between them is $60^{\circ }$. If the vectors $\lambda \bar{a}+\bar{b}$ and $\bar{a}-\lambda \bar{b}$ are perpendicular to each other, then what is the value of $\lambda$ ?

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

If $|\bar{a}|=3$, $|\bar{b}|=4$ and $|\bar{a}-\bar{b}|=7$, then what is the value of $|\bar{a}+\bar{b}|$ ?

1. A.3
2. B.2
3. C.1
4. D.0

Consider the diagonals of a quadrilateral formed by the vectors $3\hat{i}+6\hat{j}-2\hat{k}$ and $4\hat{i}-\hat{j}+3\hat{k}$. The quadrilateral must be a :

1. A.Square
2. B.Rhombus
3. C.Rectangle
4. D.None of the above

If $p=\sin (989^{\circ })\cos (991^{\circ })$, then which one of the following is correct ?

1. A.$p$ is finite and positive
2. B.$p$ is finite and negative
3. C.$p=0$
4. D.$p$ is undefined

If $A=\frac{41\pi }{12}$, then what is the value of $\frac{1-3{\tan }^{2}A}{3\tan A-{\tan }^{3}A}$ ?

1. A.$-1$
2. B.1
3. C.$1/3$
4. D.3

Which one of the following vectors of magnitude $\sqrt{51}$ makes equal angles with three vectors $\vec{a}=\frac{\hat{i}-2\hat{j}+2\hat{k}}{3}$, $\vec{b}=\frac{-4\hat{i}-3\hat{k}}{5}$ and $\vec{c}=\hat{j}$?

1. A.$5\hat{i}-\hat{j}-5\hat{k}$
2. B.$5\hat{i}+\hat{j}+5\hat{k}$
3. C.$-5\hat{i}-\hat{j}+5\hat{k}$
4. D.$5\hat{i}+5\hat{j}-\hat{k}$

If $|\vec{a}|=2$, $|\vec{b}|=5$ and $|\vec{a}\times \vec{b}|=8$, then what is the value of $\vec{a}\cdot \vec{b}$?

1. A.4
2. B.6
3. C.8
4. D.10

If $|\vec{a}+\vec{b}|=|\vec{a}-\vec{b}|$, then which one of the following is correct?

1. A.$\vec{a}$ is parallel to $\vec{b}$
2. B.$\vec{a}$ is perpendicular to $\vec{b}$
3. C.$\vec{a}=\vec{b}$
4. D.Both $\vec{a}$ and $\vec{b}$ are unit vectors

If $\vec{a}=\hat{i}-2\hat{j}+5\hat{k}$, $\vec{b}=2\hat{i}+\hat{j}-3\hat{k}$, then what is $(\vec{b}-\vec{a})\cdot (3\vec{a}+\vec{b})$ equal to?

1. A.106
2. B.$-106$
3. C.53
4. D.$-53$