The Cartesian form of the curve given by \(x=\frac{a}{2}\left(t+\frac{1}{t}\right)\), \(y=\frac{a}{2}\left(t-\frac{1}{t}\right), t\) is a parameter, is

The circle \(x^2+y^2-4 x-8 y+16=0\) rolls up along the tangent drawn to it at \((2+\sqrt{3}, 3)\) by 2 units. The equation of the circle in the new position is

If \(u=\sin \left(\frac{x}{y}\right), x=e^t\) and \(y=t^2\), then\(t^6\left(\frac{d u}{d t}\right)^2 \div e^{2 t}(t-2)^2=\)

The value of \({ }^{34} C_5+\sum_{i=0}^4(38-i) C_4\) is

An equilateral triangle is inscribed in the parabola \(y^2=8 x\), with one of its vertices is the vertex of the parabola. Then, length of the side of that triangle is

The radius of a circle is increasing at a rate of \(0.1 \mathrm{~cm} \mathrm{~s}^{-1}\). Then the rate of change of area, when its radius is \(5 \mathrm{~cm}\), is .........

The range of the function \(f(x)=\frac{x^2+x+1}{x^2-x+1}\) is

If \(a, b\) and \(c\) are the sides of \(\triangle A B C\) for which \(r_1=8, r_2=12\) and \(r_3=24\), then the ordered triad \((a, b, c)=\)

If \(f(x)=e^x ; g(x)=\ln (x)\) for all \(x \in[1, \infty)\), then \(f \circ g\) is ____

The correct statements regarding froth floatation method in metallurgy are
I. Used for the purification of sulphide ores
II. Used for the roasting of sulphide ores
III. It is based on the relative densities of gangue and ore particles
IV. It is based on the difference in the wetting properties of gangue and ore particles in frothing agent and water

A farmer owns a triangular plot in Guntur. He measures the lengths of the sides of his property as \(4 \mathrm{~cm} .5 \mathrm{~cm}\) and \(7 \mathrm{~cm}\). Then the area of land of the farmer in sq. \(\mathrm{cm}\) is

In the given logic circuit \(A=1\) and \(B=0\). The values of \(Y_1\) and \(Y_2\) are respectively.

Which of the following are disproportionation reactions?
\((\mathrm{g}=\sigma, l=(ద, \mathrm{aq}=\text { జe, } \mathrm{s}=\text { ఘ) }\)
(A) \(2 \mathrm{NO}_{2(\mathrm{~g})}+2 \mathrm{NaOH}_{(\mathrm{aq})} \rightarrow \mathrm{NaNO}_{2(\mathrm{aq})}+\mathrm{NaNO}_{3(\mathrm{aq})}+\mathrm{H}_2 \mathrm{O}_{(l)}\)
(B) \(\mathrm{Cl}_{2(\mathrm{~g})}+2 \mathrm{NaOH}_{(\mathrm{aq})} \rightarrow \mathrm{NaClO}_{(\mathrm{aq})}+\mathrm{NaCl}_{(\mathrm{aq})}+\mathrm{H}_2 \mathrm{O}_{(\mathrm{l})}\)
(C) \(3 \mathrm{ClO}^{-} \rightarrow 2 \mathrm{Cl}^{-}+\mathrm{ClO}_3^{-}\)
(D) \(3 \mathrm{Mg}_{(\mathrm{s})}+\mathrm{N}_{2(\mathrm{~g})} \rightarrow \mathrm{Mg}_3 \mathrm{~N}_{2(\mathrm{~s})}\)