Vectors · NDA previous year questions

If $\vec{a}+3\vec{b}=3\hat{i}-\hat{j}$ and $2\vec{a}+\vec{b}=\hat{i}-2\hat{j}$, then what is the angle between $\vec{a}$ and $\vec{b}$

1. A.0
2. B.$\frac{\pi }{6}$
3. C.$\frac{\pi }{3}$
4. D.$\frac{\pi }{2}$

If $(\vec{a}+\vec{b})$ is perpendicular to $\vec{a}$ and magnitude of $\vec{b}$ is twice that of $\vec{a}$, then what is the value of $(4\vec{a}+\vec{b})\cdot \vec{b}$ equal to?

1. A.0
2. B.1
3. C.$8|\vec{a}{|}^2$
4. D.$8|\vec{b}{|}^2$

Let $\vec{a},\vec{b}$ and $\vec{c}$ be three vectors such that $\vec{a},\vec{b}$ and $\vec{c}$ are co-planar. Which of the following is/are correct? 1. $(\vec{a}\times \vec{b})\times \vec{c}$ is co-planar with $\vec{a}$ and $\vec{b}$ 2. $(\vec{a}\times \vec{b})\times \vec{c}$ is perpendicular to $\vec{a}$ and $\vec{b}$ Select the correct answer using the code given below.

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

If the position vectors of A and B are $(\sqrt{2}-1)\hat{i}-\hat{j}$ and $\hat{i}+(\sqrt{2}+1)\hat{j}$ respectively, then what is the magnitude of $\overrightarrow{AB}$?

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

If $y=(1+x)(1+x^2)(1+x^4)(1+x^8)(1+x^{16})$ then what is $\frac{\mathrm{d}y}{\mathrm{d}x}$ at $x=0$ equal to?

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

A vector $\vec{r}=a\hat{i}+b\hat{j}$ is equally inclined to both x and y axes. If the magnitude of the vector is 2 units, then what are the values of a and b respectively?

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

Consider the following statements in respect of a vector $\vec{c}=\vec{a}+\vec{b}$, where $|\vec{a}|=|\vec{b}|\ne 0$: 1. $\vec{c}$ is perpendicular to $(\vec{a}-\vec{b})$. 2. $\vec{c}$ is perpendicular to $\vec{a}\times \vec{b}$. Which of the above statement is/are correct?

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

If $\vec{a}$ and $\vec{b}$ are two vectors such that $|\vec{a}+\vec{b}|=|\vec{a}-\vec{b}|=4$, then which one of the following is correct?

1. A.$\vec{a}$ and $\vec{b}$ must be unit vectors.
2. B.$\vec{a}$ must be parallel to $\vec{b}$.
3. C.$\vec{a}$ must be perpendicular to $\vec{b}$.
4. D.$\vec{a}$ must be equal to $\vec{b}$.

If $\vec{a},\vec{b}$ and $\vec{c}$ are coplanar, then what is $(2\vec{a}\times 3\vec{b})\cdot 4\vec{c}+(5\vec{b}\times 3\vec{c})\cdot 6\vec{a}$ equal to?

1. A.114
2. B.66
3. C.0
4. D.-66

Consider the following statements: 1. The cross product of two unit vectors is always a unit vector. 2. The dot product of two unit vectors is always unity. 3. The magnitude of sum of two unit vectors is always greater than the magnitude of their difference. Which of the above statements are not correct?

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

If $\hat{a}$ is a unit vector in the $xy$-plane making an angle $30^{\circ }$ with the positive $x$-axis, then what is $\hat{a}$ equal to?

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

Two adjacent sides of a parallelogram are $2\hat{i}-4\hat{j}+5\hat{k}$ and $\hat{i}-2\hat{j}-3\hat{k}$. What is the magnitude of dot product of vectors which represent its diagonals?

1. A.21
2. B.25
3. C.31
4. D.36

If $|\vec{a}\times \vec{b}{|}^2+|\vec{a}\cdot \vec{b}{|}^2=144$ and $|\vec{a}|=4$, then what is $|\vec{b}|$ equal to?

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

If the vectors $\vec{a}=2\hat{i}-3\hat{j}+\hat{k}$, $\vec{b}=\hat{i}+2\hat{j}-3\hat{k}$ and $\vec{c}=\hat{j}+p\hat{k}$ are coplanar, then what is the value of $p$?

1. A.1
2. B.$-1$
3. C.5
4. D.$-5$

What is ${\lim }_{x\to 1}\frac{x+x^2+x^3-3}{x-1}$ equal to?

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

If $\vec{a}$ and $\vec{b}$ are unit vectors and $\theta$ is the angle between them, then what is ${\sin }^2\left(\frac{\theta }{2}\right)$ equal to?

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

Consider the following equations for two vectors $\vec{a}$ and $\vec{b}$: 1. $(\vec{a}+\vec{b})\cdot (\vec{a}-\vec{b})=|\vec{a}{|}^2-|\vec{b}{|}^2$ 2. $(|\vec{a}+\vec{b}|)(|\vec{a}-\vec{b}|)=|\vec{a}{|}^2-|\vec{b}{|}^2$ 3. $|\vec{a}\cdot \vec{b}{|}^2+|\vec{a}\times \vec{b}{|}^2=|\vec{a}{|}^2|\vec{b}{|}^2$ Which of the above statements are correct?

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

If the magnitude of the sum of two non-zero vectors is equal to the magnitude of their difference, then which one of the following is correct?

1. A.The vectors are parallel
2. B.The vectors are perpendicular
3. C.The vectors are anti-parallel
4. D.The vectors must be unit vectors

What is the scalar projection of $\vec{a}=\hat{i}-2\hat{j}+\hat{k}$ on $\vec{b}=4\hat{i}-4\hat{j}+7\hat{k}$?

1. A.$\frac{\sqrt{6}}{9}$
2. B.$\frac{19}{9}$
3. C.$\frac{9}{19}$
4. D.$\frac{\sqrt{6}}{19}$

If the position vectors of points A and B are $3\hat{i}-2\hat{j}+\hat{k}$ and $2\hat{i}+4\hat{j}-3\hat{k}$ respectively, then what is the length of $\overrightarrow{AB}$?

1. A.$\sqrt{14}$
2. B.$\sqrt{29}$
3. C.$\sqrt{43}$
4. D.$\sqrt{53}$

The sine of the angle between vectors $\vec{a}=2\hat{i}-6\hat{j}-3\hat{k}$ and $\vec{b}=4\hat{i}+3\hat{j}-\hat{k}$ is

1. A.$\frac{1}{\sqrt{26}}$
2. B.$\frac{5}{\sqrt{26}}$
3. C.$\frac{5}{26}$
4. D.$\frac{1}{26}$

What is the value of $\lambda$ for which the vectors $3\hat{i}+4\hat{j}-\hat{k}$ and $-2\hat{i}+\lambda \hat{j}+10\hat{k}$ are perpendicular ?

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

A unit vector perpendicular to each of the vectors $2\hat{i}-\hat{j}+\hat{k}$ and $3\hat{i}-4\hat{j}-\hat{k}$ is

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

If $|\vec{a}|=3,|\vec{b}|=4$ and $|\vec{a}-\vec{b}|=5$, then what is the value of $|\vec{a}+\vec{b}|$?

1. A.$8$
2. B.$6$
3. C.$5\sqrt{2}$
4. D.$5$

Let $\vec{a},\vec{b}$ and $\vec{c}$ be three mutually perpendicular vectors each of unit magnitude. If $\vec{A}=\vec{a}+\vec{b}+\vec{c},\vec{B}=\vec{a}-\vec{b}+\vec{c}$ and $\vec{C}=\vec{a}-\vec{b}-\vec{c}$, then which one of the following is correct?

1. A.$|\vec{A}|>|\vec{B}|>|\vec{C}|$
2. B.$|\vec{A}|=|\vec{B}|\ne |\vec{C}|$
3. C.$|\vec{A}|=|\vec{B}|=|\vec{C}|$
4. D.$|\vec{A}|\ne |\vec{B}|\ne |\vec{C}|$