Let \(\mathbb{R}\) denote the set of all real numbers. Let \(z_1=1+2 i\) and \(z_2=3 i\) be two complex numbers, where \(i=\sqrt{-1}\).
Let \(S=\{(x, y) \in \mathbb{R} \times \mathbb{R}:\left|x+i y-z_1\right|=2\)\(\left|x+i y-z_2\right|\}\).
Then which of the following statements is (are) TRUE?

The normal at a point \(P\) on the ellipse \(x^2+4 y^2=16\) meets the \(x\)-axis at \(Q\). If \(M\) is the mid-point of the line segment \(P Q\), then the locus of \(M\) intersects the latusrectum of the given ellipse at the points

Let \(P(6,3)\) be a point on the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\). If the normal at the point \(P\) intersects the \(X\)-axis at \((9,0)\), then the eccentricity of the hyperbola is

Consider three sets $E_1=\{1,2,3\},F_1=\{1,3,4\}$ and $G_1=\{2,3,4,5\}$. Two elements are chosen at random, without replacement, from the set $E_1$, and let $S_1$ denote the set of these chosen elements. Let $E_2=E_1-S_1$ and $F_2=F_1\cup S_1$. Now two elements are chosen at random, without replacement, from the set $F_2$ and let $S_2$ denote the set of these chosen elements. Let $G_2=G_1\cup S_2$. Finally, two elements are chosen at random, without replacement, from the set $G_2$ and let $S_3$ denote the set of these chosen elements.
Let $E_3=E_2\cup S_3$. Given that $E_1=E_3$, let $p$ be the conditional probability of the event $S_1=\{1,2\}$. Then the value of $p$ is

Let $a, b, x$ and $y$ be real numbers such that $a-b=1$ and $y\ne 0$ . If the complex number $z=x+iy$ satisfies $Im\left(\frac{az+b}{z+1}\right)=y,$ then which of the following is (are) possible value (s) of $x?$

The function \(f:[0,3] \rightarrow[1,29]\), defined by \(f(x)=2 x^{3}-15 x^{2}+36 x+1\), is

Let $\mathrm{X}\mathrm{ }\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{ }\mathrm{Y}$ be two arbitrary, $3\mathrm{ }\times 3$ , non - zero, skew - symmetric matrices and Z be an arbitrary $3\mathrm{ }\times 3$ , non - zero, symmetric matrix. Then which of the following matrices is (are) skew symmetric ?

Two moles of ideal helium gas are in a rubber balloon at \(30^{\circ} \mathrm{C}\). The balloon is fully expandable and can be assumed to require no energy in its expansion. The temperature of the gas in the balloon is slowly changed to \(35^{\circ} \mathrm{C}\). The amount of heat required in raising the temperature is nearly (take \(R=8.31 \mathrm{~J} / \mathrm{mol} . \mathrm{K}\) )

Consider the given data with frequency distribution $\begin{array}{lllllll}{x}_i& 3& 8& 11& 10& 5& 4\\ f_i& 5& 2& 3& 2& 4& 4\end{array}$ Match each entry in List-I to the correct entries in List-II.

	List-I		List-II
$\left(\mathrm{P}\right)$	The mean of the above data is	$\left(1\right)$	$2.5$
$\left(\mathrm{Q}\right)$	The median of the above data is	$\left(2\right)$	$5$
$\left(\mathrm{R}\right)$	The mean deviation about the mean of the above data is	$\left(3\right)$	$6$
$\left(\mathrm{S}\right)$	The mean deviation about the median of the above data is	$\left(4\right)$	$2.7$
		$\left(5\right)$	$2.4$

The correct option is

Let \(p(x)\) be a real polynomial of least degree which has a local maximum at \(x=1\) and a local minimum at \(x=3\). If \(p(1)=6\) and \(p(3)=2\), then \(p^{\prime}(0)\) is

Paragraph:
p-amino-N, N-dimethylaniline is added to a strongly acidic solution of \(X\). The resulting solution is treated with a few drops of aqueous solution of \(Y\) to yield blue colouration due to the formation of methylene blue. Treatment of the aqueous solution of \(Y\) with the reagent potassium hexacyanoferrate(II) leads to the formation of an intense blue precipitate. The precipitate dissolves an excess addition of the reagent. Similarly, treatment of the solution of \(Y\) with the solution of potassium hexacyanoferrate(III) leads to a brown colouration due to the formation of \(Z\).
Question:
The compound \(X\), is

Let $z$ be a complex number with non-zero imaginary part. If $\frac{2+3z+4z^2}{2-3z+4z^2}$ is a real number, then the value of ${\left|z\right|}^2$ is ______.

A tiny spherical oil drop carrying a net charge \(q\) is balanced in still air with a vertical uniform electric field of strength \(\frac{81 \pi}{7} \times 10^5 \mathrm{Vm}^{-1}\). When the field is switched off, the drop is observed to fall with terminal velocity \(2 \times 10^{-3} \mathrm{~ms}^{-1}\). Given \(g=9.8 \mathrm{~ms}^{-2}\), viscosity of the air \(=1.8 \times 10^{-5} \mathrm{Ns} \mathrm{m}^{-2}\) and the density of oil \(=900 \mathrm{~kg} \mathrm{~m}^{-3}\), the magnitude of \(q\) is