In the diagram, the lines QR and ST are parallel to each other. The shortest distance between these two lines is half the shortest distance between the point P and the line QR. What is the ratio of the area of the triangle PST to the area of the trapezium SQRT?
Note: The figure shown is representative
In the adjoining figure, $\triangle CAB$ is a right triangle, right angled at A and $AD \perp BC$. Prove that $\triangle ADB \sim \triangle CDA$. Further, if $BC = 10$ cm and $CD = 2$ cm, find the length of AD.
The relationship between two variables \( x \) and \( y \) is given by \( x + py + q = 0 \) and is shown in the figure. Find the values of \( p \) and \( q \). Note: The figure shown is representative.
An ideal low pass filter has frequency response given by \[ H(j\omega) = \begin{cases} 1, & |\omega| \leq 200\pi \\ 0, & \text{otherwise} \end{cases} \] Let \( h(t) \) be its time domain representation. Then h(0) = _________ (round off to the nearest integer).
A controller \( D(s) \) of the form \( (1 + K_D s) \) is to be designed for the plant \[ G(s) = \frac{1000\sqrt{2}}{s(s+10)^2} \] as shown in the figure. The value of \( K_D \) that yields a phase margin of \(45^\circ\) at the gain cross-over frequency of 10 rad/sec is __________ (round off to one decimal place).