In the following \([x]\) denotes the greatest integer less than or equal to \(x\). Match the functions in Column I with the properties Column II.

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) be a function such that \(f(x+y)=f(x)+f(y)\) for all \(x, y \in \mathbb{R}\), and \(g: \mathbb{R} \rightarrow(0, \infty)\) be a function such that \(g(x+y)=g(x) g(y)\) for all \(x, y \in \mathbb{R}\). If \(f\left(\frac{-3}{5}\right)=12\) and \(g\left(\frac{-1}{3}\right)=2\), then the value of \(\left(f\left(\frac{1}{4}\right)+g(-2)-8\right) g(0)\) is . ________.

Let $E, F$ and $G$ be three events having probabilities $P\left(E\right)=\frac{1}{8},P\left(F\right)=\frac{1}{6}$ and $P\left(G\right)=\frac{1}{4},$ and let $P(E\cap F\cap G)=\frac{1}{10}$
For any event $H$, if $H^c$ denotes its complement, then which of the following statements is (are) TRUE ?

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Tangents are drawn from the point \(P(3,4)\) to the ellipse \(\frac{x^2}{9}+\frac{y^2}{4}=1\) touching the ellipse at points \(A\) and \(B\).Question:
The coordinates of \(A\) and \(B\) are

The letters of the word \(\mathrm{COCHIN}\) are permuted and all the permutations are arranged in an alphabetical order as in an English dictionary. The number of words that appear before the word \(\mathrm{COCHIN}\) is

Let \(\mathbf{a}=-\hat{\mathbf{i}}-\hat{\mathbf{k}}, \mathbf{b}=-\hat{\mathbf{i}}+\hat{\mathbf{j}}\) and \(\mathbf{c}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}\) be three given vectors. If \(\mathbf{r}\) is a vector such that \(\mathbf{r} \times \mathbf{b}=\mathbf{c} \times \mathbf{b}\) and \(\mathbf{r} \cdot \mathbf{a}=0\), then the value of \(\mathbf{r} \cdot \mathbf{b}\) is

Consider the following frequency distribution :

Value	4	5	8	9	6	12	11
Frequency	5	\(f_1\)	\(f_2\)	2	1	1	3

Suppose that the sum of the frequencies is 19 and the median of this frequency distribution is 6. For the given frequency distribution, let \(\alpha\) denote the mean deviation about the mean, \(\beta\) denote the mean deviation about the median, and \(\sigma^2\) denote the variance.
Match each entry in List-I to the correct entry in List-II and choose the correct option.
\(\begin{array}{|l|l|l|l|} \hline & \text{LIST - I} & & \text{LIST - II} \\ \hline (P) & 7 f_1+9 f_2 \text{ is equal to} & (1) & 146 \\ \hline (Q) & 19 \alpha \text{ is equal to} & (2) & 47 \\ \hline (R) & 19 \beta \text{ is equal to} & (3) & 48 \\ \hline (S) & 19 \sigma^2 \text{ is equal to} & (4) & 145 \\ \hline & & (5) & 55 \\ \hline \end{array}\)

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In hexagonal system of crystals, a frequently encountered arrangement of atoms is described as a hexagonal prism. Here, the top and bottom of the cell are regular hexagons and three atoms are sandwiched in between them. A space-filling model of this structure, called hexagonal close-packed (HCP), is constituted of a sphere on a flat surface surrounded in the same plane by six identical spheres as closely as possible. Three spheres are then placed over the first layer so that they touch each other and represent the second layer. Each one of the three spheres touches three spheres of the bottom layer. Finally, the second layer is covered with a third layer that is identical to the bottom layer in relative position. Assume radius of every sphere to be ' \(r\) '.
Question:
The number of atoms in this HCP unit cell is

In ${\mathrm{R}}^3$ , let L be a straight line passing through the origin. Suppose that all the points on L are at a constant distance from the two planes ${\mathrm{P}}_1\mathrm{ }:\mathrm{x}+2\mathrm{y}-\mathrm{z}+1=0\mathrm{ }$ and ${\mathrm{P}}_2\mathrm{ }:2\mathrm{x}-\mathrm{y}+\mathrm{z}-1=0.$ Let M be the locus of the feet of the perpendiculars drawn from the points on L to the plane ${\mathrm{P}}_1$ . Which of the following points lie (s) on M ?

The value of \(b\) for which the equations \(x^2+b x-1=0, x^2+x+b=0\) have one root in common is

To an evacuated vessel with movable piston under external pressure of \(1 \mathrm{~atm}\), \(0.1\) mole of He and \(1.0\) mole of an unknown compound (vapour pressure \(0.68 mathrm{~atm}\) at \(0^{\circ} \mathrm{C}\) ) are introduced. Considering the ideal gas behaviour, the total volume (in litre) of the gases at \(0^{\circ} \mathrm{C}\) is close to

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Copper is the most noble of the first row transition metals and occurs in small deposits in several countries. Ores of copper include chalcanthite \(\left(\mathrm{CuSO}_4 \cdot 5 \mathrm{H}_2 \mathrm{O}\right)\), atacamite \(\left(\mathrm{Cu}_2 \mathrm{Cl}(\mathrm{OH})_3\right)\), cuprite \(\left(\mathrm{Cu}_2 \mathrm{O}\right)\), copper glance \(\left(\mathrm{Cu}_2 \mathrm{~S}\right)\) and malachite \(\left(\mathrm{Cu}_2(\mathrm{OH})_2 \mathrm{CO}_3\right)\). However, 80\% of the world copper production comes from the ore chalcopyrite \(\left(\mathrm{CuFeS}_2\right)\). The extraction of copper from chalcopyrite involves partial roasting, removal of iron and self-reduction.
Question:
Iron is removed from chalcopyrite as

Column I gives a list of possible set of parameters measured in some experiments. The variations of the parameters in the form of graphs are shown in Column II. Match the set of parameters given in Column I with the graphs given in Column II. Indicate your answer by darkening the appropriate bubble of the \(4 \times 4\) matrix given in the ORS.