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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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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 \(\mathrm{(D)}\) |
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For a real angle \(\theta,\) the expression \(k=5\sin\theta-12\cos\theta\) is given. What is the maximum possible value of \(k\)?
1. \(12\)
2. \(5\)
3. \(13\)
4. \(17\)
1. \(12\)
2. \(5\)
3. \(13\)
4. \(17\)
Subtopic: Trigonometry |
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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 |
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Level 1: 80%+
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\(\overrightarrow A\) and \(\overrightarrow {B}\) are two vectors given by \(\overrightarrow {A}= 2\hat i + 3\hat j\) and \(\overrightarrow {B}= \hat i + \hat j\). The component of \(\overrightarrow A\) parallel to \(\overrightarrow B\) is:
1. \(\frac{(2\hat i -\hat j)}{2}\)
2. \(\frac{5}{2}(\hat i - \hat j)\)
3. \(\frac{5}{2}(\hat i + \hat j)\)
4. \(\frac{(3\hat i -2\hat j)}{2}\)
Subtopic: Scalar Product |
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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 |
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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 |
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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 |
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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. \(\dfrac{4}{\sqrt3}~\text{ms}^{-2}\)
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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 |
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