If \(\alpha_1, \alpha_2, \alpha_3, \alpha_4, \alpha_5\) are the roots of \(x^5-5 x^4+9 x^3-9 x^2\) \(+5 x-1=0\) then \(\frac{1}{\alpha_1^2}+\frac{1}{\alpha_2^2}+\frac{1}{\alpha_3^2}+\frac{1}{\alpha_4^2}+\frac{1}{\alpha_5^2}=\)

If \(\alpha=\log _e(2+\sqrt{3})\), then \(\frac{\cosh \alpha}{1-\tanh \alpha}+\frac{\sinh \alpha}{1-\operatorname{coth} \alpha}=\)

A multiple choice test consists of 5 questions, each question having 4 responses. There is only one correct response and the remaining 3 are incorrect responses. If a candidate attempts all the 5 questions then the probability that he answers at least 3 questions incorrectly is

The equation of the ellipse having a vertex at \((6,1)\), a focus at \((4,1)\) and the eccentricity \(\frac{3}{5}\) is

If a parabola having its axis parallel to \(\mathrm{X}\)-axis passes through the points \((0,-1),(6,1)\) and \((-2,-3)\), then the point at which this parabola cuts the \(\mathrm{X}\)-axis is

If \(f(x)\) is a differentiable function, \(f^{\prime}(x) \geq 5 \forall x \in[2,6]\), \(f(2)=4\) and \(f(3)=15\), then a possible value of \(f(6)\)

If the points $(2,4,-1),(3,6,-1)$ and $(4,5,1)$ are three consecutive vertices of a parallelogram, then its fourth vertex is

\(\int_{\pi / 6}^{\pi / 3} \cos ^{-4} x d x=\)

When two dice are rolled, what is the probability that at least one face shows a six?

An element has a body centered cubic structure with a unit cell edge length of \(400 \mathrm{pm}\). Atomic mass of an element is \(24 \mathrm{~g} \mathrm{~mol}^{-1}\). What is the density of the element?
\[
\left(\mathrm{N}_{\mathrm{A}}=6 \times 10^{23} \mathrm{~mol}^{-1}\right)
\]

8 teachers and 4 students are sitting around a circular table at random, then the probability that no two students sit together is:

If \(\mathrm{X} \sim \mathrm{B}(9, \mathrm{p})\) is a binomial variate satisfying the equation \(\mathrm{P}(\mathrm{x}=3)=\mathrm{P}(\mathrm{x}=6)\), then \(\mathrm{P}(\mathrm{x} < 3)=\)

At \(\mathrm{T}(\mathrm{K}), \mathrm{K}_{\mathrm{c}}\) value of \(\mathrm{AO}_2(\mathrm{~g})+\mathrm{BO}_2(\mathrm{~g}) \rightleftharpoons \mathrm{AO}_3(\mathrm{~g})+\mathrm{BO}(\mathrm{g})\) is 16. In a closed 1 L flask, one mole each of \(\mathrm{AO}_2, \mathrm{BO}_2, \mathrm{AO}_3\) and BO are taken and heated to \(\mathrm{T}(\mathrm{K})\). Identify the correct statements about this equilibrium.
I) Total number of moles at equilibrium is 4
II) At equilibrium, the ratio of moles of \(\mathrm{AO}_2\) and \(\mathrm{AO}_3\) is \(1: 4\)
III) Total number of moles of \(\mathrm{AO}_2\) and \(\mathrm{BO}_2\) at equilibrium is 0.8