$A$ particle moves in the $xy$ plane according to the law $x = a \sin \omega t$ and $y = a(1 - \cos \omega t)$,where $a$ and $\omega$ are constants. The particle traces:

The ranges and heights for two projectiles projected with the same initial velocity at angles $42^{\circ}$ and $48^{\circ}$ with the horizontal are $R_{1}, R_{2}$ and $H_{1}, H_{2}$ respectively. Choose the correct option:

Fill in the blanks given below:
$(a)$ At an angle of .......... the horizontal range of a projectile is maximum.
$(b)$ The angle between the instantaneous velocity and instantaneous acceleration of a particle moving in a circular path with constant speed is ..........
$(c)$ If $\overrightarrow{A} = 4\widehat{i} + 3\widehat{j}$,then $|\overrightarrow{A}| = ..........$

$A$ particle moving in a circle of radius $R$ with a uniform speed takes a time $T$ to complete one revolution. If this particle were projected with the same speed at an angle $\theta$ to the horizontal,the maximum height attained by it equals $4R$. The angle of projection,$\theta$,is then given by

$A$ projectile is launched from the origin in the $xy$ plane ($x$ is the horizontal and $y$ is the vertically up direction) making an angle $\alpha$ from the $x$-axis. If its distance $r = \sqrt{x^2 + y^2}$ from the origin is plotted against $x$,the resulting curves show different behaviors for launch angles $\alpha_1$ and $\alpha_2$ as shown in the figure. For $\alpha_1$,$r(x)$ keeps increasing with $x$,while for $\alpha_2$,$r(x)$ increases and reaches a maximum,then decreases and goes through a minimum before increasing again. The switch between these two cases takes place at a critical angle $\alpha_c$ (where $\alpha_1 < \alpha_c < \alpha_2$). The value of $\alpha_c$ is (where $v_0$ is the initial speed of the projectile and $g$ is the acceleration due to gravity).

In a conical pendulum,the bob is rotated with different angular velocities and the tension in the string is calculated for different values of $\omega$. Which of the following is the correct graph between $T$ and $\omega$?