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A particle moves along x-axis as $\mathrm{x}=4\left(\mathrm{t}-2\right)+\mathrm{a}{\left(\mathrm{t}-2\right)}^2$
Which of the following is true?
1.  The initial velocity of the particle is 4.
2.  The acceleration of the particle is 2a.
3.  The particle is at the origin at t=0.
4.  None of these

If \(f \left(x\right) = x^{2} - 2 x + 4\), then \(f(x)\) has:

A particle is moving along x-axis. The velocity v of particle varies with its position x as $\mathrm{v}=\frac{1}{\mathrm{x}}$. Find velocity of particle as a function of time t given that at t=0, x=1 .
1.  $\mathrm{v}=\sqrt{2\mathrm{t}+1}$
2.  $\mathrm{v}=\frac{1}{\sqrt{2\mathrm{t}+1}}$
3.  $\mathrm{v}=\frac{1}{\mathrm{t}}$
4.  None of these

Temperature of a body varies with time as $\mathrm{T}=\left({\mathrm{T}}_0+{\mathrm{at}}^2+\mathrm{b} \mathrm{sint}\right)\mathrm{K}$, where ${\mathrm{T}}_0$ is the temperature in Kelvin at $\mathrm{t}=0 \sec \& \mathrm{a}=2/\mathrm{\pi } \mathrm{K}/{\mathrm{s}}^2 \& \mathrm{b}=-\mathrm{4 }\mathrm{K}$, then the rate of change of temperature $\left(\frac{\mathrm{dT}}{\mathrm{dt}}\right)$ at $\mathrm{t}=\mathrm{\pi } \sec$ is:
1. \(8~\text{K}\)
2. \(80~\text{K}\)
3. \(8~\text{K/sec}\)
4. \(80~\text{K/sec}\)

The velocity of a particle moving on the x-axis is given by $\mathrm{v}={\mathrm{x}}^2+\mathrm{x}$ where v is in m/s and x is in m. Find its acceleration in $\mathrm{m}/{\mathrm{s}}^2$ when passing through the point x=2m.
1.  0
2.  5
3.  11
4.  30

The displacement of the particle is zero at \(t=0\) and at \(t=t\) it is \(x\). It starts moving in the \(x\)-direction with a velocity that varies as \(v = k \sqrt{x}\), where \(k\) is constant. The velocity will: (Here, \(v=\frac{dx}{dt}\))

Let y = 50t - t${}^2$, the value of y is maximum, when t =
1. 10
2. 5
3. 0
4. -5

If $\mathrm{y} = {\log }_{\mathrm{e}} {\mathrm{x}}^2$, then the value of $\frac{\mathrm{dy}}{\mathrm{dx}}$ is
$1. \frac{2}{\mathrm{x}}$
$2. \frac{1}{\mathrm{x}}$
$3. \frac{1}{2\mathrm{x}}$
$4. \frac{1}{{\mathrm{x}}^2}$

$\mathrm{If} \mathrm{y} = \sin \left(4\mathrm{x}+ \frac{2\mathrm{\pi }}{3}\right), \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} \frac{\mathrm{dy}}{\mathrm{dx}} \mathrm{is}$
$1. \cos \left(4\mathrm{x}+ \frac{2\mathrm{\pi }}{3}\right)$
$2. 4\cos \left(4\mathrm{x}+ \frac{2\mathrm{\pi }}{3}\right)$
$3. -\cos \left(4\mathrm{x}+ \frac{2\mathrm{\pi }}{3}\right)$
$4. -4\cos \left(4\mathrm{x}+ \frac{2\mathrm{\pi }}{3}\right)$

If x = $\frac{{\mathrm{t}}^3}{3}-\frac{5}{2}{t}^2 + 6t + 1$, then the value $\frac{{\mathrm{d}}^2\mathrm{x}}{{\mathrm{dt}}^2}$, when $\frac{\mathrm{dx}}{\mathrm{dt}}$ is zero is
$1. \pm 1$
$2. \pm 2$
$3. \pm 3$
$4. \pm 4$

If y = xsinx, then find $\frac{\mathrm{dy}}{\mathrm{dx}}$ 
1. xsinx + cosx
2. sinx
3. xcosx
4. xcosx + sinx

If $\mathrm{I} = \int \frac{1}{\sqrt{\mathrm{x}}}\mathrm{dx}$, then the value of I is-
$1. \sqrt{\mathrm{x}} +\hspace{0.17em}\mathrm{c}$
$2. 2\sqrt{\mathrm{x}} +\hspace{0.17em}\mathrm{c}$
$3. \frac{1}{\sqrt{\mathrm{x}}} + \mathrm{c}$
$4. \frac{1}{\mathrm{x}} + \mathrm{c}$

If $\mathrm{l} = \int \left({\sec }^{2}2\mathrm{x}\right)\mathrm{dx}$, then the value of l is
1. tanx + c
2. tan2x + c
3. $\frac{\tan 2\mathrm{x}}{2} +\hspace{0.17em}\mathrm{c}$
4. cot2x + c

Using integration, find the area enclosed by the curve y = x - x${}^2$ and positive x-axis is
$1. \frac{1}{2}$
$2. \frac{1}{3}$
$3. \frac{1}{4}$
$4. \frac{1}{6}$

$\int (3t^2 +\hspace{0.17em}5) \mathrm{dt}$
$1. {\mathrm{t}}^3 + 5\mathrm{t}$
$2. 6\mathrm{t}$
$3. 6\mathrm{t} + \mathrm{c}$
$4. {\mathrm{t}}^3 + 5\mathrm{t} + \mathrm{c}$

Evaluate the integral, $\mathrm{x} = \underset{0}{\overset{\mathrm{\pi }/\mathrm{\omega }}{\int }} (1 - \sin \mathrm{\omega t})\mathrm{dt}$
$1. \frac{\mathrm{\pi }-1}{\mathrm{\omega }}$
$2. \frac{\mathrm{\pi }-2}{\mathrm{\omega }}$
$3. \frac{\mathrm{\pi }+1}{\mathrm{\omega }}$
$4. \frac{\mathrm{\pi }+2}{\mathrm{\omega }}$
 
 

If y = x${}^2$sinx, then the value of $\frac{\mathrm{dy}}{\mathrm{dx}}$ is
$1. {\mathrm{x}}^2\mathrm{cosx} + 2\mathrm{xsinx}$
$2. \mathrm{xcosx} + {\mathrm{x}}^2\mathrm{sinx}$
$3. \mathrm{xcosx} + 2{\mathrm{x}}^2\mathrm{sinx}$
$4. {\mathrm{x}}^2\mathrm{cosx} + \mathrm{xsinx}$

$\int \frac{dx}{3x+5}=$
1. Not defined
2. ln (3x + 5) + c
3. $\frac{\ln (3\mathrm{x}+ 5)}{3} +\hspace{0.17em}\mathrm{c}$
4. 3ln (3x + 5) + c

Let $\mathrm{y}={\mathrm{x}}^2+\mathrm{x}$, the minimum value of y is
1. 1
2. 0
3. $\frac{1}{4}$
4. $-\frac{1}{4}$