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Let \(A, B, C\) be three sets of complex numbers as defined below.
\(A=\{z ; \operatorname{lm} z \geq 1\} ; B=\{z:|z-2-i|=3\} ;\) \(C=\{z: \operatorname{Re}((1-i) z)=\sqrt{2}\}\).
Question:
The number of elements in the set \(A \cap B \cap C\) is

Tangents are drawn to the hyperbola \(\frac{x^{2}}{9}-\frac{y^{2}}{4}=1\), parallel to the straight line \(2 x-y=1\). The points of contact of the tangents on the hyperbola are

Suppose $\det \left[\begin{array}{cc}\sum _{k=0}^{n}k& \sum _{k=0}^n{}_{\mathrm{ }}{}^n{C}_k{k}^2\\ \sum _{k=0}^n{}_{\mathrm{ }}{}^n{C}_{k}k& \sum _{k=0}^n{}_{\mathrm{ }}{}^n{C}_k{3}^k\end{array}\right]=0,$ holds for some positive integer $n.$ Then $\sum _{k=0}^n\frac{{}_{ }{}^n{C}_k}{k+1}$ equals

A debate club consists of 6 girls and 4 boys. A team of 4 members is to be selected from this club including the selection of a captain (from among these 4 members) for the team. If the team has to include at most one boy, then the number of ways of selecting the team is

Consider the hyperbola $H :x^2-y^2=1$ and a circle S with center $N\left(x_2, 0\right).$ Suppose that H and S touch each other at point $P\left(x_1, y_1\right)$ with $x_1>1 and y_1>0.$ The common tangent to H and S at P intersects the x - axis at point M. If $(l, m)$ is the centroid of the triangle $\mathrm{\Delta }PMN,$ then the correct expression(s) is(are)

Let $PQRS$ be a quadrilateral in a plane, where $QR=1,\angle PQR=\angle QRS=70°,\angle PQS=15°$ and $\angle PRS=40°$. If $\angle RPS=\theta °,PQ=\alpha$ and $PS=\beta$, then the interval(s) that contain(s) the value of $4\alpha \beta \sin \theta °$ is/are

Let \(\int(x)=\left\{\begin{array}{r}x^{2}\left|\cos \frac{\pi}{x}\right|, \quad x \neq 0 \\ 0, \quad x=0\end{array}, x \in R\right.\) then \(\int\) is

The structure of a peptide is given below.
If the absolute values of the net charge of the peptide at $\mathrm{pH}=2,\mathrm{pH}=6,$ and $\mathrm{pH}=11$ are $\left|{\mathrm{z}}_1\right|,\left|{\mathrm{z}}_2\right|,$ and $\left|{\mathrm{z}}_3\right|,$ respectively, then what is $\left|{\mathrm{z}}_1\right|+\left|{\mathrm{z}}_2\right|+\left|{\mathrm{z}}_3\right|$?

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A small spherical monoatomic ideal gas bubble \(\left(\gamma=\frac{5}{3}\right)\) is trapped inside a liquid of density \(\rho_1\) (see figure). Assume that the bubble does not exchange any heat with the liquid. The bubble contains \(n\) moles of gas. The temperature of the gas when the bubble is at the bottom is \(T_0\), the height of the liquid is \(H\) and the atmospheric pressure is \(p_0\) (Neglect surface tension).

Question:
The buoyancy force acting on the gas bubble is (Assume \(R\) is the universal gas constant)

One mole of an ideal gas at $300 \mathrm{K}$ in thermal contact with its surroundings expands isothermally from $1.0 \mathrm{L}$ to $2.0 \mathrm{L}$ against a constant pressure of $3.0 \mathrm{atm}$. In this process, the change in entropy of the surroundings $\left(∆{\mathrm{S}}_{\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{r}}\right)$ in $\mathrm{J}\mathrm{ }{\mathrm{K}}^{-1}$ is:
$\left(1 \mathrm{L} \mathrm{atm} = 101.3 \mathrm{J}\right)$

The periodic table consists of 18 groups. An isotope of copper, on bombardment with protons, undergoes a nuclear reaction yielding element \(X\) as shown below. To which group, element \(X\) belongs in the periodic table?

\({ }_{29}^{63} \mathrm{Cu}+{ }_{1}^{1} \mathrm{H} \rightarrow 6_{0}^{1} n+{ }_{2}^{4} \alpha+2{ }_{1}^{1} \mathrm{H}+\mathrm{X}\)

Based on VSEPR model, match the xenon compounds given in List-I with the corresponding geometries and the number of lone pairs on xenon given in List-II and choose the correct option.

List - I	List - II
(P) \(XeF _2\)	(1) Trigonal bipyramidal and two lone pair of electrons
(Q) \(XeF _4\)	(2) Tetrahedral and one lone pair of electrons
(R) \(XeO _3\)	(3) Octahedral and two lone pair of electrons
(S) \(XeO _3 F_2\)	(4) Trigonal bipyramidal and no lone pair of electrons
	(5) Trigonal bipyramidal and three lone pair of electrons

A particle of mass \(m\) is under the influence of the gravitational field of a body of mass \(M(\gg m)\). The particle is moving in a circular orbit of radius \(r_0\) with time period \(T_0\) around the mass \(M\). Then, the particle is subjected to an additional central force, corresponding to the potential energy \(V_{\mathrm{c}}(r)=m \alpha / r^3\), where \(\alpha\) is a positive constant of suitable dimensions and \(r\) is the distance from the center of the orbit. If the particle moves in the same circular orbit of radius \(r_0\) in the combined gravitational potential due to \(M\) and \(V_{\mathrm{c}}(r)\), but with a new time period \(T_1\), then \(\left(T_1^2-T_0^2\right) / T_1^2\) is given by
[ \(G\) is the gravitational constant.]