Paragraph:
Let \(V_r\) denotes the sum of the first \(r\) terms of an arithmetic progression \((A P)\) whose first term is \(r\) and the common difference is \((2 r-1)\). Let \(T_r=V_{r+1}-V_r-2\) and \(Q_r=T_{r+1}-T_r\) for \(r=1,2, \ldots\)
Question:
\(T_r\) is always

Let \(g_{i}:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow \mathbb{R}, i=1,2\), and \(f:\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right] \rightarrow \mathbb{R}\) be functions such that \(g_{1}(x)=1, g_{2}(x)=|4 x-\pi|\) and \(f(x)=\sin ^{2} x\), for all \(x \in\left[\frac{\pi}{8}, \frac{3 \pi}{8}\right]\)

Define

\[

S_{i}=\int_{\frac{\pi}{8}}^{\frac{3 \pi}{8}} f(x) \cdot g_{i}(x) d x, \quad i=1,2

\]

The value of $\frac{16S_1}{\pi }$ is ___.

Match List-I with List - II and select the correct answer using the code given below the lists 
 

	List-I		List-II
A.	Volume of parallelepiped determined by vectors  $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ is $2$. Then the volume of the parallelepiped determined by vectors $2\left(\overrightarrow{a}\times \overrightarrow{b}\right),3\left(\overrightarrow{b}\times \overrightarrow{c}\right)$ and $\left(\overrightarrow{c}\times \overrightarrow{a}\right)$ is	P.	$100$
B.	Volume of parallel piped determined by vectors  $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ is $5$. Then the volume of the parallelepiped determined by vectors $3\left(\overrightarrow{a}+\overrightarrow{b}\right),\left(\overrightarrow{b}+\overrightarrow{c}\right)$ and $2\left(\overrightarrow{c}+\overrightarrow{a}\right)$ is	Q.	$30$
C	Area of a triangle with adjacent sides determined by vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $20$.Then the area of the triangle with adjacent sides determined by vectors $\left(2\overrightarrow{a}+3\overrightarrow{b}\right)$ and $\left(\overrightarrow{a}-\overrightarrow{b}\right)$  is	R.	$24$
D	Area of a parallelogram with adjacent sides determined by vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $30$. Then the area of the parallelogram with adjacent sides determined by vectors $\left(\overrightarrow{a}+\overrightarrow{b}\right)$ and $\overrightarrow{a}$ is	S.	$60$

Let the function $f:R\to R$ be defined by $f\left(x\right)=x^3-x^2+\left(x-1\right)\sin x$ and let $g:R\to R$ be an arbitrary function. Let $fg:R\to R$ be the product function defined by $\left(fg\right)\left(x\right)=f\left(x\right)g\left(x\right)$. Then which of the following statements is/are TRUE?

Let \(f:(0,1) \rightarrow R\) be defined by \(f(x)=\frac{b-x}{1-b x}\), where \(b\) is a constant such that \(0 < b < 1\). Then,

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Let \(V_r\) denotes the sum of the first \(r\) terms of an arithmetic progression \((A P)\) whose first term is \(r\) and the common difference is \((2 r-1)\). Let \(T_r=V_{r+1}-V_r-2\) and \(Q_r=T_{r+1}-T_r\) for \(r=1,2, \ldots\)
Question:
Which one of the following is a correct statement?

If the distance between the plane \(A x-2 y+z=d\) and the plane containing the lines \(\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4} \quad\) and \(\frac{x-2}{3}=\frac{y-3}{4}=\frac{z-4}{5}\) is \(\sqrt{6}\), then \(|d|\) is

A vernier calipers has \(1 \mathrm{~mm}\) marks on the main scale. It has 20 equal divisions on the vernier scale which match with 16 main scale divisions. For this vernier calipers, the least count is

For a real number $\alpha ,$ if the system $\left[\begin{array}{ccc}1& \alpha & {\alpha }^2\\ \alpha & 1& \alpha \\ {\alpha }^2& \alpha & 1\end{array}\right] \left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}1\\ -1\\ 1\end{array}\right]$ of linear equations, has infinitely many solutions, then $1+\alpha +{\alpha }^2=$

Paragraph:
Properties such as boiling point, freezing point and vapour pressure of a pure solvent change when solute molecules are added to get homogeneous solution. These are called colligative properties. Applications of colligative properties are very useful in day-to-day life. One of its examples is the use of ethylene glycol and water mixture as anti-freezing liquid in the radiator automobiles.
A solution \(M\) is prepared by mixing ethanol and water. The mole fraction of ethanol in the mixtrue is \(0.9\)
Given Freezing point depression constant of water \(\left(k^{\text {water }}\right)=1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}\)
Freezing point depression constant of ethanol \(\left(k_f^{\text {ethanol }}\right)=2.0 \mathrm{Kkg} \mathrm{mol}^{-1}\)
Boiling point elevation constant of water \(\left(k_b^{\text {water }}\right)=0.52 \mathrm{Kkg} \mathrm{mol}^{-1}\)
Boiling point elevation constant of ethanol \(\left(k_b^{\text {ethanol }}\right)=1.2 \mathrm{Kkg} \mathrm{mol}^{-1}\)
Standard freezing point of water \(=273 \mathrm{~K}\)
Standard freezing point of ethanol \(=155.7 \mathrm{~K}\)
Standard boiling point of water \(=373 \mathrm{~K}\)
Standard boiling point of ethanol \(=351.5 \mathrm{~K}\)
Vapour pressure of pure water \(=328 \mathrm{~mm}\) of \(\mathrm{Hg}\)
Vapour pressure of pure ethanol \(=40 \mathrm{~mm}\) of \(\mathrm{Hg}\)
Molecular weight of water \(=18 \mathrm{~g} \mathrm{~mol}^{-1}\)
Molecular weight of ethanol \(=46 \mathrm{~g} \mathrm{~mol}^{-1}\)
In answering the following questions, consider the solutions to be ideal dilute solutions and solutes to be non-volatile and non-dissociative.

Question:
The freezing point of the solution \(M\) is

Two large circular discs separated by a distance of $0.01\mathrm{m}$ are connected to a battery via a switch as shown in the figure. Charged oil drops of density $900\mathrm{kg}{\mathrm{m}}^{-3}$ are released through a tiny hole at the centre of the top disc. Once some oil drops achieve terminal velocity, the switch is closed to apply a voltage of $200\mathrm{V}$ across the discs. As a result, an oil drop of radius $8\times {10}^{-7}\mathrm{m}$ stops moving vertically and floats between the discs. The number of electrons present in this oil drop is __________. (neglect the buoyancy force, take acceleration due to gravity $=10\mathrm{m}{\mathrm{s}}^{-2}$ and charge on an electron (e) $=1.6\times {10}^{-19}\mathrm{C}$)

Column I gives some devices and Column II gives some processes on which the functioning of these devices depend. Match the devices in Column I with the processes in Column II and indicate your answer by darkening appropriate bubbles in the \(4 \times 4\) matrix given in the ORS.

The correct option(s) related to the extraction of iron from its ore in the blast furnace operating in the temperature range $900-1500 \mathrm{K}$ is(are)