Let $f:\left(0, 1\right)\to \mathrm{ℝ}$ be the function defined as $f\left(x\right)=\sqrt{n}$ if $x\in [\frac{1}{n+1}, \frac{1}{n})$ where $n\in \mathrm{\mathbb{N}}$. Let $g:\left(0, 1\right)\to \mathrm{ℝ}$ be a function such that ${\int }_{x^2}^x\sqrt{\frac{1-t}{t}}dt<g\left(x\right)<2\sqrt{x}$ for all $x\in \left(0, 1\right)$. Then $\underset{x\to 0}{\lim }f\left(x\right) g\left(x\right)$

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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 equation of the locus of the point whose distance from the point \(P\) and the line \(A B\) are equal, is

Let \(a\) and \(b\) be two nonzero real numbers. If the coefficient of \(x^5\) in the expansion of \(\left(a x^2+\frac{70}{27 b x}\right)^4\) is equal to the coefficient of \(x^{-5}\) in the expansion of \(\left(a x-\frac{1}{b x^2}\right)^7\), then the value of \(2 b\) is

Lines \(L_1: y-x=0\) and \(L_2: 2 x+y=0\) intersect the line \(L_3: y+2=0\) at \(P\) and \(Q\), respectively. The bisector of the acute angle between \(L_1\) and \(L_2\) intersects \(L_3\) at \(R\).
Statement I The ratio \(P R: R Q\) equals \(2 \sqrt{2}: \sqrt{5}\).
Statement II In any triangle, bisector of an angle divides the triangle into two similar triangles.

Let \(O(0,0), P(3,4)\) and \(Q(6,0)\) be the vertices of the \(\triangle O P Q\). The point \(R\) inside the \(\triangle O P Q\) is such that the \(\triangle O P R, \triangle P Q R, \triangle O Q R\) are of equal area. The coordinates of \(R\) are

Consider \(L_1: 2 x+3 y+p-3=0 ; L_2: 2 x+3 y+p+3=0\) where \(p\) is a real number and \(C: x^2+y^2+6 x-10 y+30=0\).
Statement 1 If line \(L_1\) is a chord of circle \(C\), then line \(L_2\) is not always a diameter of circle \(C\).
Statement 2 If line \(L_1\) is a diameter of circle \(C\), then line \(L_2\) is not a chord of circle \(C\).

The value of $\sum _{k=1}^{13}\frac{1}{\sin \left(\frac{\pi }{4}+\frac{\left(k-1\right)\pi }{6}\right)\sin \left(\frac{\pi }{4}+\frac{k\pi }{6}\right)}$ is equal to

A current carrying wire heats a metal rod. The wire provides a constant power $\left(\mathrm{P}\right)$ to the rod. The metal rod is enclosed in an insulated container. It is observed that the temperature $\left(\mathrm{T}\right)$ in the metal rod changes with time $\left(\mathrm{t}\right)$ as:
$\mathrm{T}\left(\mathrm{t}\right)={\mathrm{T}}_0\left(1+\mathrm{\beta }{\mathrm{t}}^{1/4}\right)$
where $\mathrm{\beta }$ is a constant with appropriate dimension while ${\mathrm{T}}_0$ is a constant with dimension of temperature. The heat capacity of the metal is:

A solution of colourless salt \(\mathrm{H}\) on boiling with excess \(\mathrm{NaOH}\) produces a non-flammable gas. The gas evolution ceases after sometime. Upon addition of \(\mathrm{Zn}\) dust to the same solution, the gas evolution restarts. The colourless salt (s) \(\mathrm{H}\) is (are)

Aqueous solutions of \(\mathrm{HNO}_3, \mathrm{KOH}\), \(\mathrm{CH}_3 \mathrm{COOH}\), and \(\mathrm{CH}_3 \mathrm{COONa}\) of identical concentrations are provided. The pair(s) of solutions which form a buffer upon mixing is(are)

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 electrochemical cell shown below is a concentration cell. \(\mathrm{M} \mid \mathrm{M}^{2+}\) (saturated solution of a sparingly soluble salt, \(\left.\mathrm{MX}_{2}\right) \| \mathrm{M}^{2 \perp}\left(0.001 \mathrm{~mol} \mathrm{dm}^{-3}\right) \mid \mathrm{M}\).
The emf of the cell depends on the difference in concentrations of \(\mathrm{M}^{2+}\) ions at the two electrodes. The emf of the cell at \(298 \mathrm{~K}\) is \(0.059 \mathrm{~V}\).
Question:
The value of \(\Delta \mathrm{G}\left(\mathrm{kJ} \mathrm{mol}^{-1}\right)\) for the given cell is (take 1 \(\mathrm{F}=96500 \mathrm{Cmol}^{-1}\) )

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Chemical reactions involve interaction of atoms and molecules. A large number of atoms/molecules (approximately \(6.023 \times 10^{23}\) ) are present in a few grams of any chemical compound varying with their atomic/molecular masses. To handle such large numbers conveniently, the mole concept was introduced. This concept has implications in diverse areas such as analytical chemistry, biochemistry, electrochemistry and radiochemistry. The following example illustrates a typical case, involving chemical/electrochemical reaction, which requires a clear understanding of the mole concept.
A \(4.0\) molar aqueous solution of \(\mathrm{NaCl}\) is prepared and \(500 \mathrm{~mL}\) of this solution is electrolysed. This leads to the evolution of chlorine gas at one of electrodes (atomic mass: \(\mathrm{Na}=23, \mathrm{Hg}=200 ; 1\) faraday \(=96500\) coulombs).
Question:
If the cathode is a Hg electrode, the maximum weight \((g)\) of amalgam formed from this solution is