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Q. No.For \(x \ll 1,\) the approximate value of \((1+x)^2\) is:
1. \(1+x^2\)
2. \(1+x\)
3. \(1+2x\)
4. \(1+2x+x^2\)
1. \(1+x^2\)
2. \(1+x\)
3. \(1+2x\)
4. \(1+2x+x^2\)
Subtopic: Differentiation |
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Level 1: 80%+
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A particle having a velocity \(v=v_0\) at \(t=0\) is decelerated at the rate \(|a|=\alpha\sqrt v,\) where \(\alpha\) is a positive constant.
Choose the correct option from the options given below:
| \(\mathrm{(A)}\) | The particle comes to rest at \(t=\dfrac{2\sqrt{v_0}}{\alpha} \) |
| \(\mathrm{(B)}\) | The particle will come to rest at infinity. |
| \(\mathrm{(C)}\) | The distance travelled by the particle before coming to rest is \(\dfrac{2v_0^{3/2}}{\alpha}\) |
| \(\mathrm{(D)}\) | The distance travelled by the particle before coming to rest is \(\dfrac{2v_0^{3/2}}{3\alpha}\) |
| 1. | \(\mathrm{(A)}\) and \(\mathrm{(B)}\) | 2. | \(\mathrm{(B)}\) and \(\mathrm{(C)}\) |
| 3. | \(\mathrm{(C)}\) and \(\mathrm{(D)}\) | 4. | \(\mathrm{(A)}\) and \(\mathrm{(D)}\) |
Level 3: 35%-60%
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If two vectors \(\overrightarrow{P}=\hat{i}+2 m \hat{j}+m \hat{k}\) and \(\overrightarrow{Q}=4 \hat{i}-2 \hat{j}+m \hat{k}\) are perpendicular to each other. Then, the value of \(m\) will be:
1. \(-1\)
2. \(2\)
3. \(1\)
4. \(3\)
1. \(-1\)
2. \(2\)
3. \(1\)
4. \(3\)
Subtopic: Vector Product |
82%
Level 1: 80%+
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The displacement \(x\) of a particle varies with time as \(x=ae^{\alpha t}+be^{\beta t}\) where \(a,b,\alpha,\beta\) are constant and are positives. The velocity of the particle will:
| 1. | drop to zero when \(\alpha=\beta\) |
| 2. | be independent of \(\alpha\) and \(\beta\) |
| 3. | go on increasing with time |
| 4. | go on decreasing with time |
Subtopic: Differentiation |
76%
Level 2: 60%+
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If \(x \ll 1,\) then \((1+x)^{-1}\) is approximately equal to:
1. \(1+x\)
2. \(1-x\)
3. \(1+x^2\)
4. \(1-x^2\)
1. \(1+x\)
2. \(1-x\)
3. \(1+x^2\)
4. \(1-x^2\)
Subtopic: Differentiation |
91%
Level 1: 80%+
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A fox runs along the straight line \(y=x,\) while a rabbit moves along the circular path \(x^2+y^2=8.\) Which of the following coordinates represents a point where the fox can catch the rabbit?
1. \((2,2)\)
2. \((2,-2)\)
3. \((-2,2)\)
4. \((-1,-1)\)
1. \((2,2)\)
2. \((2,-2)\)
3. \((-2,2)\)
4. \((-1,-1)\)
Subtopic: Co-ordinate geometry |
80%
Level 1: 80%+
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A particle is moving along a straight line whose velocity-displacement graph is as shown in the figure. What is the magnitude of acceleration when displacement is \(3~\text m?\)
1. \(4\sqrt3~\text{ms}^{-2}\)
2. \(3\sqrt3~\text{ms}^{-2}\)
3. \(\sqrt3~\text{ms}^{-2}\)
4. \(\dfrac{4}{\sqrt3}~\text{ms}^{-2}\)
1. \(4\sqrt3~\text{ms}^{-2}\)
2. \(3\sqrt3~\text{ms}^{-2}\)
3. \(\sqrt3~\text{ms}^{-2}\)
4. \(\dfrac{4}{\sqrt3}~\text{ms}^{-2}\)
62%
Level 2: 60%+
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All of the following describe the magnitude and direction of a vector EXCEPT:
| 1. | \(10~\text{m/s} \) west |
| 2. | \(10~\text{m/s} \) in a circle |
| 3. | \(20~\text{m} \) to the left |
| 4. | \(20~\text{m} \) straight up |
Subtopic: Scalars & Vectors |
86%
Level 1: 80%+
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The magnitudes of forces \(\vec F_A\) and \(\vec F_B\) are \(400~\text{N}\) and \(300~\text{N},\) respectively, as shown in the diagram. What is the magnitude of the net force in each of the three given cases?
| 1. | In Case \(1\), the net force is \(100~\text{N}\); Case \(2\), \(700~\text{N}\); and Case \(3\), \(500~\text{N}.\) |
| 2. | In Case \(1\), the net force is \(700~\text{N}\); Case \(2\), \(100~\text{N}\); and Case \(3\), \(500~\text{N}.\) |
| 3. | In Case \(1\), the net force is \(350~\text{N}\); Case \(2\), \(50~\text{N}\); and Case \(3\), \(450~\text{N}.\) |
| 4. | In Case \(1\), the net force is \(350~\text{N}\); Case \(2\), \(50~\text{N}\), and Case \(3\), \(550~\text{N}.\) |
Subtopic: Resultant of Vectors |
91%
Level 1: 80%+
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In the figure,\(P,O,\) and \(Q\) lie on a straight line. The value of \(x\) is:
| 1. | \(20^\circ\) | 2. | \(25^\circ\) |
| 3. | \(30^\circ\) | 4. | \(35^\circ\) |
Subtopic: Geometry |
91%
Level 1: 80%+
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