If $y=y\left(x\right)$ satisfies the differential equation $8\sqrt{x}\left(\sqrt{9+\sqrt{x}}\right)dy={\left(\sqrt{4+\sqrt{9+\sqrt{x}}}\right)}^{-1}dx$ and $x>0, y\left(0\right)=\sqrt{7},$ then $y\left(256\right)=$

Let \(f(x)=(1-x)^{2} \sin ^{2} x+x^{2}\) for all \(x \in I R\) and let \(g(x)=\int_{1}^{x}\left(\frac{2(t-1)}{t+1}-\ln t\right) f(t) d t\) for all \(x \in(1, \infty)\).

Question: Consider the statements:

\(P\) : There exists some \(x \in \mathrm{R}\) such that \(f(x)+2 x\)

\(=2\left(1+x^{2}\right)\)

\(Q\) : There exists some \(x \in \mathrm{R}\) such that \(2 f(x)+1\) \(=2 x(1+x)\)

Then

One Indian and four American men and their wives are to be seated randomly around a circular table. Then, the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife, is

The product of all positive real values of $x$ satisfying the equation $x^{\left(16{\left({\log }_{5}x\right)}^3-68{\log }_{5}x\right)}=5^{-16}$ is

Let $f\mathbb{ }:\mathbb{R}\to \left(0, \infty \right)$ and $g\mathbb{ }:\mathbb{R}\to \mathbb{R}$ be twice differentiable functions such that $f^{″}$ and $g^{″}$ are continuous functions on $\mathbb{R}$ . Suppose $f^{\prime }\left(2\right)=g\left(2\right)=0, f^{″}\left(2\right)\ne 0$ and $g^{\prime }\left(2\right) \ne 0.\underset{x\to 2}{\lim }\frac{f\left(x\right) g\left(x\right)}{f^{\prime } \left(x\right) g^{\prime }\left(x\right)}=1,$ then

Let \(g(x)=\frac{(x-1)^n}{\log \cos ^m(x-1)} ; 0 < x < 2, m\) and \(n\) are integers, \(m \neq 0, n>0\) and let \(p\) be the left hand derivative of \(|x-1|\) at \(x=1\). If \(\lim _{x \rightarrow 1^{+}} g(x)=p\), then

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

When two identical batteries of internal resistance \(1 \Omega\) each are connected in series across a resistor \(R\), the rate of heat produced in \(R\) is \(J_1\). When the same batteries are connected in parallel across \(R\), the rate is \(J_2\). If \(J_1=2.25 J_2\) then the value of \(R\) in \(\Omega\) is

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The concentration of potassium ions inside a biological cell is at least twenty times higher than the outside. The resulting potential difference across the cell is important in several processes such as transmission of nerve impulses and maintaining the ion balance. A simple model for such a concentration cell involving a metal \(\mathrm{M}\) is :
\(
M(s) \mid M^{+}(\alpha q ; 0.05 \text { molar }) \| M^{+}(a q ; 1 \text { molar })\) \(\mid M(s)
\)
For the above electrolytic cell the magnitude of the cell potential \(\left|E_{\text {cell }}\right|=70 \mathrm{mV}\).
Question:
If the \(0.05\) molar solution of \(M^{+}\)is replaced by a \(0.0025\) molar \(M^{+}\) solution, then the magnitude of the cell potential would be

A block of mass ${\mathrm{m}}_1=1\mathrm{}\mathrm{k}\mathrm{g}$ another mass ${\mathrm{m}}_2=2\mathrm{}\mathrm{k}\mathrm{g}$ , are placed together (see figure) on an inclined plane with angle of inclination $\mathrm{\theta }$ . Various values of $\mathrm{\theta }$ are given in List I. The coefficient of friction between the block ${\mathrm{m}}_1$ and the plane is always zero. The coefficient of static and dynamic friction between the block ${\mathrm{m}}_2$ and the plane are equal to $\mathrm{\mu }=0.3$ . In List II expressions for the friction on the block ${\mathrm{m}}_2$ are given. Match the correct expression of the friction in List II with the angles given in List I, and choose the correct option. The acceleration due to gravity is denoted by g.
[Useful information: $\tan \left({5.5}^{\mathrm{o}}\right)\approx 0.1;\tan \left({11.5}^{\mathrm{o}}\right)\approx 0.2;\tan \left({16.5}^{\mathrm{o}}\right)\approx 0.3$ ]

	List I		List II
A.	$\mathrm{\theta }=5^{\mathrm{o}}$	$\mathrm{P}.$	${\mathrm{m}}_2\mathrm{g}\sin \mathrm{\theta }$
B.	$\mathrm{\theta }={10}^{\mathrm{o}}$	$\mathrm{Q}.$	$\left({\mathrm{m}}_1+{\mathrm{m}}_2\right)\mathrm{g}\sin \mathrm{\theta }$
C.	$\mathrm{\theta }={15}^{\mathrm{o}}$	$\mathrm{R}.$	$\mathrm{\mu }{\mathrm{m}}_2\mathrm{}\mathrm{g}\cos \mathrm{\theta }$
D.	$\mathrm{\theta }={20}^{\mathrm{o}}$	$\mathrm{S}.$	$\mathrm{\mu }\left({\mathrm{m}}_1+{\mathrm{m}}_2\right)\mathrm{g}\cos \mathrm{\theta }$

A double star system consists of two stars \(A\) and \(B\) which have time period \(T_A\) and \(T_B\). Radius \(R_A\) and \(R_B\) and mass \(M_A\) and \(M_B\). Choose the correct option.

In a neutral or faintly alkaline solution, $8$ moles of permanganate anion quantitatively oxidize thiosulphate anions to produce $\mathrm{X}$ moles of a sulphur containing product. The magnitude of $\mathrm{X}$ is:

Consider an electron in the \(n=3\) orbit of a hydrogen-like atom with atomic number \(Z\). At absolute temperature \(T\), a neutron having thermal energy \(k_{\mathrm{B}} T\) has the same de Broglie wavelength as that of this electron. If this temperature is given by \(T=\frac{Z^2 h^2}{\alpha \pi^2 a_0^2 m_N k_B}\), (where \(h\) is the Planck's constant, \(k_B\) is the Boltzmann constant, \(m_{\mathrm{N}}\) is the mass of the neutron and \(a_0\) is the first Bohr radius of hydrogen atom) then the value of \(\alpha\) is ________