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Let \(S=S_{1} \cap S_{2} \cap S_{3}\), where

\(S_{1}=\{z \in \mathbb{C}:|z|<4\}, \quad S_{2}=\left\{z \in \mathbb{C}: \operatorname{Im}\left[\frac{z-1+\sqrt{3} i}{1-\sqrt{3} i}\right]>0\right\}\) and \(S_{3}=\{z \in \mathbb{C}: \operatorname{Re} z>0\}\).


Question:

Area of \(S=\)

Let \(S\) denote the locus of the point of intersection of the pair of lines
\(4 x-3 y=12 \alpha\)
\(4 \alpha x+3 \alpha y=12\)
where \(\alpha\) varies over the set of non-zero real numbers. Let \(T\) be the tangent to \(S\) passing through the points \((p, 0)\) and \((0, q), q>0\), and parallel to the line \(4 x-\frac{3}{\sqrt{2}} y=0\).
Then the value of \(p q\) is

Let $\left[x\right]$ be the greatest integer less than or equals to $x$. Then, at which of the following point(s) the function $f\left(x\right)=x\cos \left(\pi \left(x+\left[x\right]\right)\right)$ is discontinuous?

Let $a_1, a_2, a_3, .........$ be a sequence of positive integers in arithmetic progression with common difference $2$. Also, let $b_1, b_2, b_3, ........$ be a sequence of positive integers in geometric progression with common ratio $2$. If $a_1=b_1=c$, then the number of all possible values of $c$, for which the equality $2 \left(a_1+a_2+........+a_n\right)=b_1+b_2+.....+b_n$ holds for some positive integer $n$, is _______

Let $\overline{z}$ denote the complex conjugate of a complex number $z$ and let $i=\sqrt{-1}$. In the set of complex numbers, the number of distinct roots of the equation $\overline{z}-z^2=i\left(\overline{z}+z^2\right)$ is _______.

For $\text{a}\in \text{R}$ (the set of all real numbers), $\text{a}\ne -1$, $\underset{\text{n}\to \infty }{\text{lim}}\frac{\left(1^{\text{a}}+2^{\text{a}}+\dots +{\text{n}}^{\text{a}}\right)}{{\left(\text{n}+1\right)}^{\text{a}-1}\left[\left(\text{na}+1\right)+\left(\text{na}+2\right)+\dots +\left(\text{na}+\text{n}\right)\right]}=\frac{1}{60}$ Then a =

The number of seven digit integers, with sum of the digits equal to 10 and formed by using the digits 1,2 and 3 only, is

Let RS be the diameter of the circle $x^2+y^2=1$ where, $S$ is the point$(1, 0)$. Let $P$be a variable point (other than $R \& S$) on the circle and tangents to the circle at $S \& P$meet at the point$Q$. The normal to the circle at P intersects a line drawn through$Q$ parallel to $RS$ at point $E$ . Then, the locus of $E$ passes through the point (s):

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The capacitor of capacitance \(C\) can be charged (with the help of a resistance \(R\) ) by a voltage source \(V\), by closing switch \(S_1\) while keeping switch \(S_2\) open. The capacitor can be connected in series with an inductor \(L\) by closing switch \(S_2\) and opening \(S_1\).

Question :
After the capacitor gets fully charged, \(S_1\) is opened and \(S_2\) is closed so that the inductor is connected in series with the capacitor. Then,

A field line is shown in the figure. This field cannot represent

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A thermal power plant produces electric power of \(600 \mathrm{~kW}\) at \(4000 \mathrm{~V}\), which is to be transported to a place \(20 \mathrm{~km}\) away from the power plant for consumers' usage. It can be transported either directly with a cable of large current carrying capacity or by using a combination of step-up and step-down transformers at the two ends. The drawback of the direct transmission is the large energy dissipation. In the method using transformers, the dissipation is much smaller. In this method, a step-up transformer is used at the plant side so that the current is reduced to a smaller value. At the consumers' end, a step-down transformer is used to supply power to the consumers at the specified lower voltage. It is reasonable to assume that the power cable is purely resistive and the transformers are ideal with a power factor unity. All the currents and voltages mentioned are rms values.

Question :

In the method using the transformers, assume that the ratio of the number of turns in the primary to that in the secondary in the step-up transformer is \(1: 10\). If the power to the consumers has to be supplied at \(200 \mathrm{~V}\), the ratio of the number of turns in the primary to that in the secondary in the step-down transformer is

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A thermally insulating cylinder has a thermally insulating and frictionless movable partition in the middle, as shown in the figure below. On each side of the partition, there is one mole of an ideal gas, with specific heat at constant volume, \(C_{V}=2 R\). Here, \(R\) is the gas constant. Initially, each side has a volume \(V_{0}\) and temperature \(T_{0}\). The left side has an electric heater, which is turned on at very low power to transfer heat \(Q\) to the gas on the left side. As a result the partition moves slowly towards the right reducing the right side volume to \(V_{0} / 2\). Consequently, the gas temperatures on the left and the right sides become \(T_{L}\) and \(T_{R}\), respectively. Ignore the changes in the temperatures of the cylinder, heater and the partition.

Question :
The value of $\frac{T_R}{T_0}$ is:

Statement I Band gap in germanium is small.
Statement II The energy spread of each germanium atomic energy level is infinitesimally small.