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Q. No.If \(t=\sqrt{x}+4,\) then \(\left(\dfrac{{dx}}{{dt}}\right)_{{t}=4}\) is:
1. \(4\)
2. \(0\)
3. \(8\)
4. \(16\)
Subtopic: Differentiation |
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Level 2: 60%+
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\(\vec A\) is a vector quantity such that \(| \vec A|\) = non-zero constant. Which of the following expressions is true for \(\vec A\)?
1. \(\overrightarrow{\mathrm{A}} \cdot \overrightarrow{\mathrm{A}}=0 \)
2. \(\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{A}}<0 \)
3. \(\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{A}}=0 \)
4. \(\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{A}}>0\)
1. \(\overrightarrow{\mathrm{A}} \cdot \overrightarrow{\mathrm{A}}=0 \)
2. \(\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{A}}<0 \)
3. \(\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{A}}=0 \)
4. \(\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{A}}>0\)
Subtopic: Scalar Product |
69%
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. \(|\hat{{A}}+\hat{{B}}|= |\hat{{A}}-\hat{{B}} \mid \tan \dfrac{\theta}{2} \)
2. \(|\hat{{A}}-\hat{{B}}|=|\hat{{A}}+\hat{{B}}| \tan \dfrac{\theta}{2} \)
3. \(|\hat{{A}}+\hat{{B}}|=|\hat{{A}}-\hat{{B}}| \cos \dfrac{\theta}{2} \)
4. \(|\hat{{A}}-\hat{{B}}|=|\hat{{A}}+\hat{{B}}| \cos \dfrac{\theta}{2}\)
1. \(|\hat{{A}}+\hat{{B}}|= |\hat{{A}}-\hat{{B}} \mid \tan \dfrac{\theta}{2} \)
2. \(|\hat{{A}}-\hat{{B}}|=|\hat{{A}}+\hat{{B}}| \tan \dfrac{\theta}{2} \)
3. \(|\hat{{A}}+\hat{{B}}|=|\hat{{A}}-\hat{{B}}| \cos \dfrac{\theta}{2} \)
4. \(|\hat{{A}}-\hat{{B}}|=|\hat{{A}}+\hat{{B}}| \cos \dfrac{\theta}{2}\)
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 |
56%
Level 3: 35%-60%
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The scalar projection of the vector \(\vec{A}=2\vec{i}+4\vec{j}-2\vec{k}\) on the vector \(\vec{B}=\vec{i}+2\vec{j}+\alpha\vec{k}\) is zero. The value of \(\alpha\) is:
1. \(3\)
2. \(5\)
3. \(7\)
4. \(9\)
1. \(3\)
2. \(5\)
3. \(7\)
4. \(9\)
Subtopic: Scalar Product |
81%
Level 1: 80%+
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If \(\overrightarrow{{A}}=(2 \hat{{i}}+3 \hat{{j}}-\hat{\mathrm{k}}) ~\text {m}\) and \(\overrightarrow{\mathrm{B}}=( \hat{{i}}+2\hat{{j}}+2\hat{{k}}) ~\text{m}\). The magnitude of the component of the vector \(\vec A\) along the vector \(\vec B\) will be:
1. \(5\)
2. \(4\)
3. \(3\)
4. \(2\)
1. \(5\)
2. \(4\)
3. \(3\)
4. \(2\)
Subtopic: Scalar Product |
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Level 1: 80%+
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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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A force vector \(\vec{F}\) lies in the \({xy} \)-plane and makes an angle of \(30^\circ\) with the positive \({y}\text-\)axis, as shown in the figure. If the \({y}\text-\)component of the force is given as \(2\sqrt{3}~\text{N},\) what is the corresponding \({x}\text-\)component of the force?
| 1. | \(2\sqrt{3}~\text{N} \) | 2. | \(2~\text{N}\) |
| 3. | \(3~\text{N}\) | 4. | \(3\sqrt{2}~\text{N} \) |
Subtopic: Resolution of Vectors |
74%
Level 2: 60%+
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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 |
79%
Level 2: 60%+
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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 |
76%
Level 2: 60%+
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