Q. No.A vector \({\vec{A}}\) is rotated by small angle \(\mathrm{\Delta}\mathrm{\theta}\) radians \(({\Delta}\theta<<1)\) to get a new vector \(\vec{B}.\) In that case \(|\vec{B}-\vec{A}|\) is:
1. \(\vec{A}|\Delta \theta|\)
2. \(\mathrm{|\vec{B}| \Delta \theta-|\vec{A}|}\)
3. \(|\vec{\mathrm{A}}|\left(1-\frac{\Delta \theta^2}{2}\right)\)
4. \(0\)
1. \(\vec{A}|\Delta \theta|\)
2. \(\mathrm{|\vec{B}| \Delta \theta-|\vec{A}|}\)
3. \(|\vec{\mathrm{A}}|\left(1-\frac{\Delta \theta^2}{2}\right)\)
4. \(0\)
Subtopic: Resultant of Vectors |
Level 4: Below 35%
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In a regular octagon \({ABCDEFGH},\) all sides are equal in length. The position vector of point \(A\) with respect to the center \(O\) of the octagon is given by: \(\overrightarrow{{AO}}=2 \hat{{i}}+3 \hat{{j}}-4 \hat{{k}}.\)
What is the value of the vector sum: \(\overrightarrow{{AB}}+\overrightarrow{{AC}}+\overrightarrow{{AD}}+\overrightarrow{{AE}}+\overrightarrow{{AF}}+\overrightarrow{{AG}}+\overrightarrow{{AH}} ~\text{?}\)
| 1. | \( -16 \hat{i}-24 \hat{j}+32 \hat{k} \) | 2. | \( 16 \hat{i}+24 \hat{j}-32 \hat{k} \) |
| 3. | \( 16 \hat{i}+24 \hat{j}+32 \hat{k} \) | 4. | \(16 \hat{i}-24 \hat{j}+32 \hat{k} \) |
Subtopic: Resultant of Vectors |
75%
Level 2: 60%+
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Which of the following relations is true for two unit vectors \(\hat A\) and \(\hat B\) making an angle \(\theta\) to each other?
1. \(|\hat{\mathrm{A}}+\hat{\mathrm{B}}|= |\hat{\mathrm{A}}-\hat{\mathrm{B}} \mid \tan \frac{\theta}{2} \)
2. \(|\hat{\mathrm{A}}-\hat{\mathrm{B}}|=|\hat{\mathrm{A}}+\hat{\mathrm{B}}| \tan \frac{\theta}{2} \)
3. \(|\hat{\mathrm{A}}+\hat{\mathrm{B}}|=|\hat{\mathrm{A}}-\hat{\mathrm{B}}| \cos \frac{\theta}{2} \)
4. \(|\hat{\mathrm{A}}-\hat{\mathrm{B}}|=|\hat{\mathrm{A}}+\hat{\mathrm{B}}| \cos \frac{\theta}{2}\)
1. \(|\hat{\mathrm{A}}+\hat{\mathrm{B}}|= |\hat{\mathrm{A}}-\hat{\mathrm{B}} \mid \tan \frac{\theta}{2} \)
2. \(|\hat{\mathrm{A}}-\hat{\mathrm{B}}|=|\hat{\mathrm{A}}+\hat{\mathrm{B}}| \tan \frac{\theta}{2} \)
3. \(|\hat{\mathrm{A}}+\hat{\mathrm{B}}|=|\hat{\mathrm{A}}-\hat{\mathrm{B}}| \cos \frac{\theta}{2} \)
4. \(|\hat{\mathrm{A}}-\hat{\mathrm{B}}|=|\hat{\mathrm{A}}+\hat{\mathrm{B}}| \cos \frac{\theta}{2}\)
Subtopic: Resultant of Vectors |
Level 3: 35%-60%
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Two vectors \(\vec A \) and \(\vec B\) have equal magnitudes. If the magnitude of \(\vec A + \vec B\) is equal to two times the magnitude of \(\vec A - \vec B\), then the angle between \(\vec A \) and \(\vec B\) will be:
1. \(\sin ^{-1}\left(\frac{3}{5}\right) \)
2. \(\sin ^{-1}\left(\frac{1}{3}\right) \)
3. \(\cos ^{-1}\left(\frac{3}{5}\right) \)
4. \(\cos ^{-1}\left(\frac{1}{3}\right)\)
1. \(\sin ^{-1}\left(\frac{3}{5}\right) \)
2. \(\sin ^{-1}\left(\frac{1}{3}\right) \)
3. \(\cos ^{-1}\left(\frac{3}{5}\right) \)
4. \(\cos ^{-1}\left(\frac{1}{3}\right)\)
Subtopic: Resultant of Vectors |
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Level 3: 35%-60%
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Two forces of magnitude \(A\) and \({A\over 2}\) act perpendicular to each other. The magnitude of the resultant force is equal to:
| 1. | \(\dfrac A2\) | 2. | \(\dfrac {\sqrt {5}A} { 2}\) |
| 3. | \(\dfrac {3A} {2}\) | 4. | \(\dfrac {5A} {2}\) |
Subtopic: Resultant of Vectors |
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Level 1: 80%+
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If \(\overrightarrow{\mathbf{A}}{=}{2}\hat{i}{+}{3}\hat{j}{+}{2}\hat{k}\;{and}\;\overrightarrow{\mathbf{A}}{-}\overrightarrow{\mathbf{B}}{=}{2}\hat{j}\), then find \(\left|{\overrightarrow{B}}\right|\)
1. 3
2. \(3\sqrt{3}\)
3. 2
4. \(\sqrt{3}\)
1. 3
2. \(3\sqrt{3}\)
3. 2
4. \(\sqrt{3}\)
Subtopic: Resultant of Vectors |
74%
Level 2: 60%+
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Consider a cube of side length \(a,\) as shown in the figure. What is the vector from the centre of the face \(ABOD\) to the centre of the face \(BEFO \text{?}\)
| 1. | \(\dfrac{1}{2}a\bigg(\hat k- \hat i \bigg) \) | 2. | \(\dfrac{1}{2}a\bigg(\hat i- \hat k \bigg) \) |
| 3. | \(\dfrac{1}{2}a\bigg(\hat j- \hat i \bigg) \) | 4. | \(\dfrac{1}{2}a\bigg(\hat j- \hat k \bigg) \) |
Subtopic: Resultant of Vectors |
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Level 2: 60%+
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Two vectors \(\vec {A}\) and \(\vec {B}\) have equal magnitudes. The magnitude of \((\vec {A}+\vec{B})\) is \('n’\) times the magnitude of \((\vec {A}-\vec{B}).\) The angle between \(\vec {A}\) and \(\vec {B}\) is:
1. \(\cos^{-1}\bigg[\dfrac{n^2 - 1}{n^2+1} \bigg]\)
2. \(\cos^{-1}\bigg[\dfrac{n - 1}{n+1} \bigg]\)
3. \(\sin^{-1}\bigg[\dfrac{n^2 - 1}{n^2+1} \bigg]\)
4. \(\sin^{-1}\bigg[\dfrac{n - 1}{n+1} \bigg]\)
1. \(\cos^{-1}\bigg[\dfrac{n^2 - 1}{n^2+1} \bigg]\)
2. \(\cos^{-1}\bigg[\dfrac{n - 1}{n+1} \bigg]\)
3. \(\sin^{-1}\bigg[\dfrac{n^2 - 1}{n^2+1} \bigg]\)
4. \(\sin^{-1}\bigg[\dfrac{n - 1}{n+1} \bigg]\)
Subtopic: Resultant of Vectors |
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Two vectors, \(\overrightarrow{P}\) and \(\overrightarrow{Q}\), have equal magnitudes. If the magnitude of \( \left (\overrightarrow{{P}}+\overrightarrow{{Q}} \right ) \) is \(n\) times the magnitude of \( \left (\overrightarrow{{P}}-\overrightarrow{{Q}} \right ), \) then what is the angle between \(\overrightarrow{P}\) and \(\overrightarrow{Q}\text{?}\)
| 1. | \(\sin ^{-1}\left(\dfrac{n^2-1}{n^2+1} \right) \) | 2. | \(\cos ^{-1}\left(\dfrac{n^2-1}{n^2+1} \right)\) |
| 3. | \(\sin ^{-1}\left(\dfrac{n-1}{n+1} \right)\) | 4. | \(\cos ^{-1}\left(\dfrac{n-1}{n+1} \right)\) |
Subtopic: Resultant of Vectors |
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Level 2: 60%+
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Match List I with List II.
Choose the correct answer from the options given below:
| List – I | List – II | ||
| (a) | \(\overrightarrow{{C}}-\overrightarrow{{A}}-\overrightarrow{{B}}=0\) | (i) |  |
| (b) | \(\overrightarrow{{A}}-\overrightarrow{{C}}-\overrightarrow{{B}}=0\) | (ii) |  |
| (c) | \(\overrightarrow{{B}}-\overrightarrow{{A}}-\overrightarrow{{C}}=0\) | (iii) |  |
| (d) | \(\overrightarrow{{A}}+\overrightarrow{{B}}=\overrightarrow{{-C}}\) | (iv) |  |
| 1. | \(\text { (a) } \rightarrow \text { (iv), (b) } \rightarrow \text { (i), (c) } \rightarrow \text { (iii), (d) } \rightarrow \text { (ii) }\) |
| 2. | \(\text { (a) } \rightarrow \text { (iv), (b) } \rightarrow \text { (iii), (c) } \rightarrow \text { (i), (d) } \rightarrow \text { (ii) }\) |
| 3. | \(\text { (a) } \rightarrow \text { (i), (b) } \rightarrow \text { (iv), (c) } \rightarrow \text { (ii), (d) } \rightarrow \text { (iii) }\) |
| 4. | \(\text { (a) } \rightarrow \text { (iii), (b) } \rightarrow \text { (ii), (c) } \rightarrow \text { (iv), (d) } \rightarrow \text { (i) }\) |
Subtopic: Resultant of Vectors |
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