Integration

Q. A streetcar moves rectilinearly from station $A$ to the next station $B$ with an acceleration varying according to the law $f=a-bx$, where $a$ and $b$ are constant, and $x$ is the distance from station $A$. The distance between the two stations and maximum velocity are

1. $x=\frac{2a}{b},v_{max}=\frac{a}{\sqrt{b}}$
2. $x=\frac{b}{2a},v_{max}=\frac{a}{b}$
3. $x=\frac{a}{2b},v_{max}=\frac{b}{\sqrt{a}}$
4. $x=\frac{a}{b},v_{max}=\frac{\sqrt{a}}{b}$

Q.

What is the antiderivative of ${\sec }^2\left(x\right)$?

Q. The flux passing through the surface $S_5$ will be

1. $-0.135N{m}^2{C}^{-1}$
2. $-0.054N{m}^2{C}^{-1}$
3. $-0.081N{m}^2{C}^{-1}$
4. $-0.054N{m}^2{C}^{-1}$

Q. The velocity of a particle is at any time related to the distance travelled by the particle by the relation $v=ax+b$, where a is positive and b $\le \frac{a}{2}.$ Which of the following statements will be true for this motion? (given x = 0 when t = 0)

1. The displacement of the particle at time t is $x=\frac{b}{a}(e^{at}-1)$
2. The particle will experience a retardation if b < 0
3. The particle will be at rest at time t = 0
4. The motion of the particle is under constant acceleration.