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 $\text{P}=\left[\begin{array}{ccc}3& -1& -2\\ 2& 0& \text{α}\\ 3& -5& 0\end{array}\right]$ , where $\text{α}\in \text{R},$ Suppose $\text{Q}=\left[{\text{q}}_{\text{i}\text{j}}\right]$ is a matrix such that$PQ = kI$, where $\text{k}\in \text{R},\text{k}\ne 0$ and$I$ is the identity matrix of order 3. If ${\text{q}}_{23}=-\frac{\text{k}}{8}$ and $\text{det}\left(\text{Q}\right)=\frac{{\text{k}}^2}{2}$ then 

For how many values of $p$, the circle $x^2+y^2+2x+4y-p=0$ and the coordinate axes have exactly three common points?

Let \(\mathbf{A}\) be vector parallel to line of intersection of planes \(P_1\) and \(P_2\) through origin. \(P_1\) is parallel to the vectors \(2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) and \(4 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}\) and \(P_2\) is parallel to \(\hat{\mathbf{j}}-\hat{\mathbf{k}}\) and \(3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}\), then the angle between vector \(\mathbf{A}\) and \(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}\) is

Which of the following values of $\mathrm{\alpha }$ satisfy the equation $\left|\begin{array}{ccc}{\left(1+\mathrm{\alpha }\right)}^2& {\left(1+2\mathrm{\alpha }\right)}^2& {\left(1+3\mathrm{\alpha }\right)}^2\\ {\left(2+\mathrm{\alpha }\right)}^2& {\left(2+2\mathrm{\alpha }\right)}^2& {\left(2+3\mathrm{\alpha }\right)}^2\\ {\left(3+\mathrm{\alpha }\right)}^2& {\left(3+2\mathrm{\alpha }\right)}^2& {\left(3+3\mathrm{\alpha }\right)}^2\end{array}\right|=\mathrm{ }-648\mathrm{ }\mathrm{\alpha }$ ?

For any $y\in \mathrm{ℝ}$, let ${\cot }^{-1}\left(y\right)\in \left(0,\pi \right)$ and ${\tan }^{-1}\left(y\right)\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$. Then the sum of all the solutions of the equation ${\tan }^{-1}\left(\frac{6y}{9-y^2}\right)+{\cot }^{-1}\left(\frac{9-y^2}{6y}\right)=\frac{2\pi }{3}$ for $0<\left|y\right|<3$, is equal to

A hyperbola, having the transverse axis of length \(2 \sin \theta\), is confocal with the ellipse \(3 x^2+4 y^2=12\). Then, its equation is

Match the compounds in Column I with their characteristic test(s)/reactions(s) given in Column II. Indicate your answer by darkening the appropriate bubbles of the \(4 \times 4\) matrix given in the ORS.

Match the anionic species given in Column I that are present in the ores(s) given in Column II.

Column I	Column II
A. Carbonate	P. Siderite
B. Sulphide	Q. Malachite
C. Hydroxide	R. Bauxite
D. Oxide	S. Calamine
	T. Argentite

Paragraph I: An organic compound \(\mathrm{P}\) with molecular formula \(\mathrm{C}_5 \mathrm{H}_{18} \mathrm{O}_2\) decolorizes bromine water and also shows positive iodoform test. \(\mathrm{P}\) on ozonolysis followed by treatment with \(\mathrm{H}_2 \mathrm{O}_2\) gives \(\mathrm{Q}\) and \(\mathrm{R}\). While compound \(\mathrm{Q}\) shows positive iodoform test, compound \(\mathrm{R}\) does not give positive iodoform test. \(\mathrm{Q}\) and \(\mathrm{R}\) on oxidation with pyridinium chlorochromate (PCC) followed by heating give \(\mathrm{S}\) and \(\mathrm{T}\), respectively. Both \(\mathrm{S}\) and \(\mathrm{T}\) show positive iodoform test.
Complete copolymerization of 500 moles of \(\mathrm{Q}\) and 500 moles of \(\mathrm{R}\) gives one mole of a single acyclic copolymer \(\mathrm{U}\).
[Given, atomic mass: \(\mathrm{H}=1, \mathrm{C}=12, \mathrm{O}=16\)]
Question: The molecular weight of \(\mathbf{U}\) is

Calamine, malachite, magnetite and cryolite, respectively are:

A glass tube of uniform internal radius \((r)\) has a valve separating the two identical ends. Initially, the valve is in a tightly closed position. End 1 has a hemispherical soap bubble of radius \(r\). End 2 has sub-hemispherical soap bubble as shown in figure. Just after opening the valve,

Three concentric metallic spherical shells of radii \(R, 2 R, 3 R\) are given charges \(Q_1, Q_2, Q_3\), respectively. It is found that the surface charge densities on the outer surfaces of the shells are equal. Then, the ratio of the charges given to the shells, \(Q_1: Q_2: Q_3\), is