The cubic equation whose roots are thrice to each of the roots of \(x^3+2 x^2-4 x+1=0\) is

In a \(\triangle A B C\), let \(\angle C=\frac{\pi}{2}\). If \(r\) and \(R\) are respectively inradius and circumradius of \(A B C\), then \(R+r=\)

If \((\alpha, \beta)\) is the external centre of similitude of the circles \(x^2+y^2=3\) and \(x^2+y^2-2 x+4 y+4=0\), then \(\frac{\beta}{\alpha}=\)

If the pair of straight lines \(6 x^2-5 x y+y^2=0\) makes angles \(\alpha\) and \(\beta\) with the \(X\)-axis, then \(\tan (\alpha-\beta)=\)

If $a=2, b=3,c=4$ in a triangle $ABC$, then $\cos C=$

\(\lim _{n \rightarrow \infty}\left[\begin{array}{l}\frac{1}{n}+\frac{n^2}{(n+1)^3}+\frac{n^2}{(n+2)^3}+\frac{n^2}{(n+3)^3} \\ +\ldots+\frac{1}{125 n}\end{array}\right]=\)

A unit vector orthogonal to the vector \(3 \bar{i}+4 \bar{j}+5 \bar{k}\) and coplanar with the vectors \(\bar{i}+\bar{j}+\bar{k}\) and \(\bar{i}-\bar{j}+\bar{k}\) is

If \(x^2+y^2-4 x-2 y+5=0\) and \(x^2+y^2-6 x-4 y-3=\) 0 are members of a coaxial system of circles, then the centre of a point circle in the system is

A molecule \((X)\) has (i) four sigma bonds formed by the overlap of \(s p^2\) and \(s\) orbitals (ii) one sigma bond formed by \(s p^2\) and \(s p^2\) orbitals and (iii) one \(\pi\) bond formed by \(p_x\) and \(p_z\) orbitals. Which of the following is \(X\) ?

The standard reduction potentials of \(\mathrm{Zn}^{2+}\left|\mathrm{Zn}, \mathrm{Cu}^{2+}\right| \mathrm{Cu}\) and \(\mathrm{Ag}^{+} \mid \mathrm{Cu}\) and \(\mathrm{Ag}^{+} \mid \mathrm{Ag}\) are respectively \(-0.76,0.34\) and \(0.8 \mathrm{~V}\). The following cells were constructed :
(1) \(\mathrm{Zn}\left|\mathrm{Zn}^{2+}\right|\left|\mathrm{Cu}^{2+}\right| \mathrm{Cu}\)
(2) \(\mathrm{Zn}\left|\mathrm{Zn}^{2+}\right|\left|\mathrm{Ag}^{+}\right| \mathrm{Ag}\)
(3) \(\mathrm{Cu}\left|\mathrm{Cu}^{2+} \| \mathrm{Ag}^{+}\right| \mathrm{Ag}\)
What is the correct order of \(E_{\text {cell }}^{\circ}\) of these cells?

The equation of pair of straight lines parallel to \(X\)-axis and touching the circle \(x^2+y^2-6 x-4 y-12=0\)

When an electron placed in a uniform magnetic field is accelerated from rest through a potential difference \(\mathrm{V}_1\), it experiences a force F . If the potential difference is changed to \(\mathrm{V}_2\), the force experienced by the electron in same magnetic field is 2 F , then the ratio of potential differences \(\frac{\mathrm{V}_2}{\mathrm{~V}_1}\) is

The ratio of the displacements of a freely falling body during second and fifth seconds of its motion is