Let \(\left(x_0, y_0\right)\) be the solution of the following equations \((2 x)^{\log 2}=(3 y)^{\log 3}\), \(3^{\log x}=2^{\log y}\), then \(x_0\) is equal to

Consider the two curves
\(C_1: y^2=4 x\)
\(C_2: x^2+y^2-6 x+1=0\), then

Match the statements of Column I with these in Column II.
[Note : Here \(z\) takes values in the complex plane and \(\operatorname{Im}(z)\) and \(\operatorname{Re}(z)\) denote respectively, the imaginary part and the real part of \(z\) ]

Column I	Column II
(A)The set of points \(z\) satisfying \(|z-i| z||=|z+i| z||\) is contained in or equal to	(p)an ellipse with eccentricity \(4 / 5\)
(B) The set of points \(z\) satisfying \(|z+4|+|z-4|=0\) is contained in or equal to	(q) the set of points \(z\) satisfying \(\operatorname{Im}(z)=0\)
(C) If \(|w|=2\), then the set of points \(z=w-\frac{1}{w}\) is contained in or equal to	(r) the set of points \(z\) satisfying \(|\operatorname{Im} z| \leq 1\)
(D) If \(|w|=1\), then the set of points \(z=w+\frac{1}{w}\) is contained in or equal to	(s) the set of points satisfying \(|\operatorname{Re} z| \leq 2\)
	(t) the set of points \(z\) satisfying \(|z| \leq 3\)

A rectangular sheet of fixed perimeter with sides having their lengths in the ratio $8:15$ is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is $100$, the resulting box has maximum volume. Then the lengths of the sides of the rectangular sheet are:

A straight line \(L\) through the point \((3,-2)\) is inclined at an angle \(60^{\circ}\) to the line \(\sqrt{3} x+y=1\). If \(L\) also intersects the \(X\)-axis, then the equation of \(L\) is

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The circle \(x^2+y^2-8 x=0\) and hyperbola \(\frac{x^2}{9}-\frac{y^2}{4}=1\) intersect at the points \(A\) and \(B\).Question:
Equation of the circle with \(A B\) as its diameter is

Let $S=\left(0,1\right)\cup \left(1, 2\right)\cup \left(3, 4\right)$ and $T=\left\{0, 1, 2, 3\right\}$. Then which of the following statements is(are) true?

Concentrated nitric acid, upon long standing, turns yellow-brown due to the formation of

When water is filled carefully in a glass, one can fill it to a height $h$ above the rim of the glass due to the surface tension of water. To calculate $h$ just before water starts flowing, model the shape of the water above the rim as a disc of thickness $h$ having semicircular edges, as shown schematically in the figure. When the pressure of water at the bottom of this disc exceeds what can be withstood due to the surface tension, the water surface breaks near the rim and water starts flowing from there. If the density of water, its surface tension and the acceleration due to gravity are ${10}^3\mathrm{kg}{\mathrm{m}}^{-3},0.07\mathrm{N}{\mathrm{m}}^{-1}$ and $10\mathrm{m}{\mathrm{s}}^{-2}$ respectively, the value of $h$(in $\mathrm{mm}$)is ________.

Liquids $\mathrm{A}$ and $\mathrm{B}$ form ideal solution for all compositions of $\mathrm{A}$ and $\mathrm{B}$ at ${25}^{\circ }C$. Two such solutions with $0.25$ and $0.50$ mole fractions of $\mathrm{A}$ have the total vapor pressures of $0.3$ and $0.4$ bar, respectively. What is the vapor pressure of pure liquid $\mathrm{B}$ in bar?

Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) and \(g: \mathbb{R} \rightarrow \mathbb{R}\) be functions defined by
\(f(x)=\left\{\begin{array}{ll}x|x| \sin \left(\frac{1}{x}\right), & x \neq 0, \\ 0, & x=0,\end{array} \quad\right.\) and \(g(x)= \begin{cases}1-2 x, & 0 \leq x \leq \frac{1}{2} \\ 0, & \text { otherwise }\end{cases}\)
Let \(a, b, c, d \in \mathbb{R}\). Define the function \(h: \mathbb{R} \rightarrow \mathbb{R}\) by
\(h(x)=a f(x)+b\left(g(x)+g\left(\frac{1}{2}-x\right)\right)+c(x-g(x))+d g(x), x \in \mathbb{R}\)
Match each entry in List-I to the correct entry in List-II.

The correct option is :

If \(\vec{a}, \vec{b}\) and \(\vec{c}\) are unit vectors satisfying \(|\vec{a}-\vec{b}|^{2}+|\vec{b}-\vec{c}|^{2}+|\vec{c}-\vec{a}|^{2}=9\), then \(|2 \vec{a}+5 \vec{b}+5 \vec{c}|\) is

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One twirls a circular ring (of mass \(M\) and radius \(R\) ) near the tip of one's finger as shown in Figure 1 . In the process the finger never loses contact with the inner rim of the ring. The finger traces out the surface of a cone, shown by the dotted line. The radius of the path traced out by the point where the ring and the finger is in contact is \(r\). The finger rotates with an angular velocity \(\omega_{0}\). The rotating ring rolls without slipping on the outside of a smaller circle described by the point where the ring and the finger is in contact (Figure 2). The coefficient of friction between the ring and the finger is \(\mu\) and the acceleration due to gravity is \(g\).








Question:

The minimum value of \(\omega_{0}\) below which the ring will drop down is