Motion in plane practice · Hard

A man can swim with a speed of $4\,\,km{h^{ - 1}}$ in still water. He crosses a river 1 km wider that flows steadily at $3\,\,km{h^{ - 1}}$ If he makes his strokes normal to the river current how far down the river does he go when he reaches the other bank?

A vector $\mathbf{a}$ is turned without a change in its length through a small angles $d \theta$. The value of $|\Delta \mathbf{a}|$ and $\Delta a$ are, respectively.

If the equation of motion of a projectile is $y = 3x - \frac{1}{8}{x^2}$ the range and maximum height are respectively ($y$ and $x$ are in metres)

A boat travels 4 hr upstream and 2 hr down the stream for covering the same distance. The ratio of velocity of boat to the water in river

If $\vec A = 3\hat i + 4\hat j,\,\vec B = 7\hat i + 24\hat j$ then the vector having the same magnitude as $\vec B$ and parallel to $\vec A$ is

If $\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}+\hat{\mathbf{j}}, \mathbf{c}=\hat{\mathbf{i}}$ and $(\mathbf{a} \times \mathbf{b}) \times \mathbf{c}=\lambda \mathbf{a}+\mu \mathrm{b}$, then $\lambda+\mu$ is equal to