Vectors · NDA previous year questions

Passage: Let $\vec{a},\vec{b},\vec{c}$ be unit vectors. Further, $\vec{a}$ is perpendicular to $\vec{b}$; $\vec{c}$ makes an angle $\frac{\pi }{3}$ with both $\vec{a}$ and $\vec{b}$; and $\vec{c}=p\vec{a}+q\vec{b}+r(\vec{a}\times \vec{b})$. Question: What is the value of $r^2$ ?

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

Passage: Let $\vec{a},\vec{b},\vec{c}$ be unit vectors. Further, $\vec{a}$ is perpendicular to $\vec{b}$; $\vec{c}$ makes an angle $\frac{\pi }{3}$ with both $\vec{a}$ and $\vec{b}$; and $\vec{c}=p\vec{a}+q\vec{b}+r(\vec{a}\times \vec{b})$. Question: What is the value of $(p+q)$ ?

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

Passage: Let $\vec{a}\times \vec{b}=\vec{c}$ and $\vec{b}\times \vec{c}=\vec{a}$. Question: Consider the following statements : I. $(\vec{a}\times \vec{b})\cdot \vec{c}+(\vec{b}\times \vec{c})\cdot \vec{a}=(\vec{c}\times \vec{a})\cdot \vec{b}$ II. $\{(\vec{a}\times \vec{b})\times (\vec{b}\times \vec{c})\}\cdot \vec{b}=1$ Which of the statements given above is/are correct ?

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

Passage: Let $\vec{a}\times \vec{b}=\vec{c}and\vec{b}\times \vec{c}=\vec{a}$. Question: Consider the following statements : I. $\vec{a},\vec{b},\vec{c}$ are orthogonal in pairs. II. $\vec{a},\vec{b},\vec{c}$ are unit vectors. Which of the statements given above is/are correct ?

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

Passage: Let $\vec{a},\vec{b},\vec{c},\vec{a}+\vec{b},\vec{b}+\vec{c},\vec{a}+\vec{b}+\vec{c}$ be unit vectors. Question: What is the angle between $\vec{a}$ and $\vec{c}$ ?

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

Passage: Let $\vec{a},\vec{b},\vec{c},\vec{a}+\vec{b},\vec{b}+\vec{c},\vec{a}+\vec{b}+\vec{c}$ be unit vectors. Question: What is the angle between $\vec{a}$ and $\vec{b}$ ?

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

How many of the following can be a vector perpendicular to both the vectors $2\hat{i}-\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+3\hat{k}$ ? I. $4\hat{i}+5\hat{j}-3\hat{k}$ II. $-8\hat{i}-10\hat{j}+6\hat{k}$ III. $\frac{1}{50}(-4\hat{i}-5\hat{j}+3\hat{k})$ Select the correct answer.

1. A.None
2. B.One
3. C.Two
4. D.All three

What is the area of the parallelogram whose sides are represented by the vectors $\hat{i}+2\hat{j}+3\hat{k}$ and $2\hat{i}+\hat{j}+2\hat{k}$ ?

1. A.$\frac{1}{2}\sqrt{26}$ square units
2. B.$\frac{1}{2}\sqrt{27}$ square units
3. C.$\sqrt{26}$ square units
4. D.$\sqrt{27}$ square units

The position vectors of the vertices A, B, Cand D of a quadrilateral ABCD are given by $3\hat{i}+4\hat{j}-2\hat{k}$ , $4\hat{i}-4\hat{j}-3\hat{k},2\hat{i}-3\hat{j}+2\hat{k}$ and $6\hat{i}-2\hat{j}+\hat{k}$ respectively. What is the angle between the diagonals AC and BD of the quadrilateral?

1. A.$90^{\circ }$
2. B.$75^{\circ }$
3. C.$60^{\circ }$
4. D.$45^{\circ }$

A force $\vec{F}=2\hat{i}-\lambda \hat{j}+5\hat{k}$ is applied at the point A(1, 2, 5) If its moment about the point B(- 1, - 2, 3) is $16\hat{i}-6\hat{j}+2\lambda \hat{k}$, then what is the value of $\lambda$ ?

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

For the following two (02) items: Let $$f(x)=\{\begin{array}{ll}\frac{1-\cos 2x}{x^2}& ,x<0\\ 9& ,x=0\\ \frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}& ,x>0\end{array}$$ What is ${\lim }_{x\to 0^{-}}f(x)$ equal to?

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

For the following two (02) items: Let $$f(x)=\{\begin{array}{ll}\frac{1-\cos 2x}{x^2}& ,x<0\\ 9& ,x=0\\ \frac{\sqrt{x}}{\sqrt{16+\sqrt{x}}-4}& ,x>0\end{array}$$ What is ${\lim }_{x\to 0^{+}}f(x)$ equal to?

1. A.6
2. B.7
3. C.8
4. D.9

Consider the following statements with respect to a vector d = (a × b) × c: I. d is coplanar with a and b. II. d is perpendicular to c. Which of the statements given above is/are correct?

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

The position vectors of three points A, B and C are $\vec{a}$, $\vec{b}$ and $\vec{c}$ respectively such that $3\vec{a}-4\vec{b}+\vec{c}=\vec{0}$. What is AB:BC equal to?

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

The position vectors of three points A, B and C respectively, where $\vec{a},\vec{b}$ and $\vec{c}$ respectively, where $\vec{c}=({\cos }^2\theta )\vec{a}+({\sin }^2\theta )\vec{b}$. What is $(\vec{a}\times \vec{b})+(\vec{b}\times \vec{c})+(\vec{c}\times \vec{a})$ equal to?

1. A.$\vec{0}$
2. B.$2\vec{c}$
3. C.$3\vec{c}$
4. D.Unit vector

Let $\vec{a},\vec{b},(\vec{a}\times \vec{b})$ be unit vectors. What is $\vec{a}.\vec{b}$ equal to?

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

Let $x=\sec \theta -\cos \theta$ and $y={\sec }^4\theta -{\cos }^4\theta$ What is ${\left(\frac{dy}{dx}\right)}^2$ equal to?

1. A.$\frac{4(y^2+4)}{(x^2+4)}$
2. B.$\frac{4(y^2-4)}{(x^2-4)}$
3. C.$\frac{16(y^2+4)}{(x^2+4)}$
4. D.$\frac{16(y^2-4)}{(x^2-4)}$

Let $x=\sec \theta -\cos \theta$ and $y={\sec }^4\theta -{\cos }^4\theta$ What $\left[\frac{x^2+4}{y^2+4}\frac{dy}{dx}\left(x^2+4\frac{d^{2}y}{d{x}^2}-16y\right)\right]$ equal to?

1. A.$16x$
2. B.$16y$
3. C.$-16x$
4. D.$-16y$

What is 3α + 2β equal to if (2î + 6ĵ + 27k̂) × (î + αĵ + βk̂) is a null vector?

1. A.36
2. B.33
3. C.30
4. D.27

For what value of the angle between the vectors $\vec{a}$ and $\vec{b}$ is the quantity $|\vec{a}\times \vec{b}|+\sqrt{3}|\vec{a}\cdot \vec{b}|$ maximum?

1. A.$0^{\circ }$
2. B.$30^{\circ }$
3. C.$45^{\circ }$
4. D.$60^{\circ }$

Let $\theta$ be the angle between two unit vectors $\vec{a}$ and $\vec{b}$. if $\vec{a}+2\vec{b}$ is perpendicular to $5\vec{a}-4\vec{b}$ then what is $\cos \theta +\cos 2\theta$ equal to?

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

Let ABCDEF be a regular hexagon. If $\overrightarrow{\mathrm{AD}}=m\overrightarrow{\mathrm{BC}}$ and $\overrightarrow{\mathrm{CF}}=n\overrightarrow{\mathrm{AB}}$ then what is mn equal to

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

The vectors $\vec{a}$, $\vec{b}$ and $\vec{c}$ are of the same length. If taken pairwise they form equal angles. If $\vec{a}=\hat{i}+\hat{j}$ and $\vec{b}=\hat{j}+\hat{k}$, then what can $\vec{c}$ be equal to? I. $\hat{i}+\hat{k}$ II. $\frac{-\hat{i}+4\hat{j}-\hat{k}}{3}$ Select the correct answer using the code given below.

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

The diagonals of a quadrilateral ABCD are along the lines x - 2y = 1 and 4x + 2y = 3. The quadrilateral ABCD may be a

1. A.rectangle
2. B.cyclic quadrilateral
3. C.parallelogram
4. D.rhombus

If a vector of magnitude 2 units makes an angle $\frac{\pi }{3}$ with $2\hat{i}$, $\frac{\pi }{4}$ with $3\hat{j}$ and an acute angle $\theta$ with $4\hat{k}$, then what are the components of the vector?

1. A.$(1,\sqrt{2},1)$
2. B.$(1,-\sqrt{2},1)$
3. C.$(1,-\sqrt{2},-1)$
4. D.$(1,\sqrt{2},-1)$