Vectors · NDA previous year questions

Consider the following in respect of moment of a force: 1. The moment of force about a point is independent of point of application of force. 2. The moment of a force about a line is a vector quantity. 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

For any vector $\vec{r}$, what is $(\vec{r}\cdot \hat{i})\left(\vec{r}\times \hat{i}\right)+(\vec{r}\cdot \hat{j})\left(\vec{r}\times \hat{j}\right)+(\vec{r}\cdot \hat{k})\left(\vec{r}\times \hat{k}\right)$ equal to?

1. A.$\vec{0}$
2. B.$\vec{r}$
3. C.$2\vec{r}$
4. D.$3\vec{r}$

Let $\vec{a}$ and $\vec{b}$ be two vectors of magnitude 4 inclined at an angle $\frac{\pi }{3}$, then what is the angle between $\vec{a}$ and $\vec{a}-\vec{b}$?

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

What is the length of projection of the vector $\hat{i}+2\hat{j}+3\hat{k}$ on the vector $2\hat{i}+3\hat{j}-2\hat{k}$ ?

1. A.$\frac{1}{\sqrt{17}}$
2. B.$\frac{2}{\sqrt{17}}$
3. C.$\frac{3}{\sqrt{17}}$
4. D.$\frac{2}{\sqrt{14}}$

If $(\vec{a}\times \vec{b}{)}^2+(\vec{a}\cdot \vec{b}{)}^2=144$ and $|\vec{b}|=4$, then what is the value of $|\vec{a}|$ ?

1. A.3
2. B.4
3. C.5
4. D.6

If $\theta$ is the angle between vectors $\vec{a}$ and $\vec{b}$ such that $\vec{a}\cdot \vec{b}\ge 0$, then which one of the following is correct?

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

The vectors $60\hat{i}+3\hat{j}$, $40\hat{i}-8\hat{j}$ and $\beta \hat{i}-52\hat{j}$ are collinear if :

1. A.$\beta =20$
2. B.$\beta =40$
3. C.$\beta =-40$
4. D.$\beta =26$

Consider the following in respect of the vectors $\vec{a}=(0,1,1)$ and $\vec{b}=(1,0,1)$: 1. The number of unit vectors perpendicular to both $\vec{a}$ and $\vec{b}$ is only one. 2. The angle between the vectors is $\frac{\pi }{3}$. 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

1. A.$\frac{1}{3}$
2. B.$\frac{4}{9}$
3. C.$\frac{5}{9}$
4. D.$\frac{2}{3}$

Consider the following for the next two (02) items that follow : Let a vector $\vec{a}=4\hat{i}-8\hat{j}+\hat{k}$ make angles $\alpha ,\beta ,\gamma$ with the positive directions of x, y, z axes respectively. What is $\cos 2\beta +\cos 2\gamma$ equal to ?

1. A.$-\frac{32}{81}$
2. B.$-\frac{16}{81}$
3. C.$\frac{16}{81}$
4. D.$\frac{32}{81}$

Which of the above points lie on the line joining A and B ?

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

1. A.2
2. B.3
3. C.√6
4. D.√3

Consider the following for the next three (03) items that follow : Let $f(x)=P{e}^x+Q{e}^{2x}+R{e}^{3x}$ where P, Q, R are real numbers. Further $f(0)=6$, $f^{\prime }(\ln 3)=282$ and ${\int }_0^{\ln 2}f(x)\mathrm{dx}=11$ ‘What is the value of Q ?

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

What is the value of R ?

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

What is f'(0) equal to ?

1. A.18
2. B.16
3. C.15
4. D.14

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

Let $\vec{a},\vec{b}$ and $\vec{c}$ be unit vectors lying on the same plane. What is $\{(3\vec{a}+2\vec{b})\times (5\vec{a}-4\vec{c})\}\cdot (\vec{b}+2\vec{c})$ equal to ?

1. A.-8
2. B.-32
3. C.8
4. D.0

What are the values of x for which the angle between the vectors $2x^2\hat{i}+3x\hat{j}+\hat{k}$ and $\hat{i}-2\hat{j}+x^2\hat{k}$ is obtuse ?

1. A.$0<x<2$
2. B.$x<0$
3. C.$x>2$
4. D.$0\le x\le 2$

The position vectors of vertices A, B and C of triangle ABC are respectively $\hat{j}+\hat{k},3\hat{i}+\hat{j}+5\hat{k}$ and $3\hat{j}+3\hat{k}$. What is angle C equal to?

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

If $4\hat{i}+\hat{j}-3\hat{k}$ and $p\hat{i}+q\hat{j}-2\hat{k}$ are collinear vectors, then what are the possible values of p and q respectively?

1. A.4, 1
2. B.1, 4
3. C.$\frac{8}{3},\frac{2}{3}$
4. D.$\frac{2}{3},\frac{8}{3}$

If $\vec{a},\vec{b},\vec{c}$, are the position vectors of the vertices A, B, C respectively of a triangle ABC and G is the centroid of the triangle, then what is $\overrightarrow{AG}$ equal to ? .

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

Consider the following statements : 1. Dot product over vector addition is distributive 2. Cross product over vector addition is distributive 3. Cross product of vectors is associative Which of the above statements is/are correct ?

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

Let $\vec{a},\vec{b},\vec{c}$ be three non-zero vectors such that $\vec{a}\times \vec{b}=\vec{c}$. Consider the following statements: 1. $\vec{a}$ is unique if $\vec{b}$ and $\vec{c}$ are given 2. $\vec{c}$ is unique if $\vec{a}$ and $\vec{b}$ are given Which of the above statements is/are correct?

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

Let $\vec{a}$ and $\vec{b}$ be two unit vectors such that $|\vec{a}-\vec{b}|<2$. If $2\theta$ is the angle between $\vec{a}$ and $\vec{b}$, then which one of the following is correct ?

1. A.$0\le \sin \theta <1$ only
2. B.$-\frac{1}{2}<\sin \theta <\frac{1}{2}$ only
3. C.$-1<\sin \theta <0$ only
4. D.$-1<\sin \theta <1$

Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be unit vectors such that $\vec{a}\times \vec{b}$ is perpendicular to $\vec{c}$. If $\theta$ is the angle between $\vec{a}$ and $\vec{b}$, then which of the following is/are correct? 1. $\vec{a}\times \vec{b}=\sin \theta \vec{c}$ 2. $\vec{a}\cdot (\vec{b}\times \vec{c})=0$ 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