
As shown in the figures, a uniform rod $ OO' $ of length $ l $ is hinged at the point $ O $ and held in place vertically between two walls using two massless springs of the same spring constant. The springs are connected at the midpoint and at the top-end $ (O') $ of the rod, as shown in Fig. 1, and the rod is made to oscillate by a small angular displacement. The frequency of oscillation of the rod is $ f_1 $. On the other hand, if both the springs are connected at the midpoint of the rod, as shown in Fig. 2, and the rod is made to oscillate by a small angular displacement, then the frequency of oscillation is $ f_2 $. Ignoring gravity and assuming motion only in the plane of the diagram, the value of $\frac{f_1}{f_2}$ is:
The reaction sequence given below is carried out with 16 moles of X. The yield of the major product in each step is given below the product in parentheses. The amount (in grams) of S produced is ____. 
Use: Atomic mass (in amu): H = 1, C = 12, O = 16, Br = 80
Let $ a_0, a_1, ..., a_{23} $ be real numbers such that $$ \left(1 + \frac{2}{5}x \right)^{23} = \sum_{i=0}^{23} a_i x^i $$ for every real number $ x $. Let $ a_r $ be the largest among the numbers $ a_j $ for $ 0 \leq j \leq 23 $. Then the value of $ r $ is ________.
Let $ \mathbb{R} $ denote the set of all real numbers. Then the area of the region $$ \left\{ (x, y) \in \mathbb{R} \times \mathbb{R} : x > 0, y > \frac{1}{x},\ 5x - 4y - 1 > 0,\ 4x + 4y - 17 < 0 \right\} $$ is
Diodes are used in many applications. By removing the signal component, several of them convert AC current to DC current also are known as rectifiers when used in this position. Diodes act as electric switches and can block voltage spikes, making them ideal for surgical protectors. They are hired to do digital logic. Some things are made from diodes such as power supply and voltage doubles. The sensors, as well as light on lighting devices and lasers, depend on LEDs. For electronic tuning, varactors are used, and for compressing transients in AC lines varistors are used. Zener diodes are used as voltage regulators where varactors are used for electronic tuning, and varistors are used for compressing transients in AC lines.
On the base of diodes, transistors and op-amps are built. The most common diode mode is the p-n junction. One (n) object with electrons as the charging conductor closes the second object (p) with holes (well-charged particles acted as depleted areas of electrons) as the charge carriers on this type of diode. Where they connect, the narrowing space forms, where electrons are dispersed to fill holes in the p-side. This effectively stops the flow of electrons. When a positive voltage is given to the p-side of this junction, electrons may easily move from it and the current flows into the diode to fill the holes. The shrinkage area expands as the negative bias interacts (i.e., negative voltage is given to the p-side), making the electrons harder to move across.