What amount of conc. \(\mathrm{H}_2 \mathrm{SO}_4\) solution should be used to prepare \(500 \mathrm{~mL}\) of \(0.5 \mathrm{M} \mathrm{H}_2 \mathrm{SO}_4\) ? (The concentration of \(\mathrm{H}_2 \mathrm{SO}_4\) solution being used is \(90 \%\) and molecular mass of \(\mathrm{H}_2 \mathrm{SO}_4=98.079 \mathrm{~g} \mathrm{~mol}^{-1}\) )

Which of the following complex ions is most stable?

Statement I Despite having aldehyde group, glucose does not give Schiff test.
Statement II Glucose exists in \(\alpha\) and \(\beta\) crystalline forms.

Which of the following orders are correct against the stated property?
I) \(\mathrm{NaO}_2 < \mathrm{KO}_2 < \mathrm{RbO}_2 < \mathrm{CsO}_2\) - stability
II) \(\mathrm{Mg}(\mathrm{OH})_2 < \mathrm{Ca}(\mathrm{OH})_2 < \mathrm{Sr}(\mathrm{OH})_2\) - basic strength
III) \(\mathrm{MgCO}_3 < \mathrm{CaCO}_3 < \mathrm{SrCO}_3\) - thermal stability

The radius of the first orbit of \(\mathrm{Li}^{2+}\) is \(\mathrm{X} Å\). The radius of the third orbit of \(\mathrm{He}^{+}\)(in \(Å\) ) is

Strongest conjugate base among the following is

\(\mathrm{KMnO}_4\) oxidises hydrogen sulphide in acidic medium. The number of moles of \(\mathrm{KMnO}_4\) which react with one mole of hydrogen sulphide is

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) \(3 \mathrm{Mg}_{(\mathrm{s})}+\mathrm{N}_{2(\mathrm{~g})} \rightarrow \mathrm{Mg}_3 \mathrm{~N}_{2(\mathrm{~s})}\)

If \(7 \hat{i}-4 \hat{j}+5 \hat{k}\) is the position vector of the vertex \(A\) of a tetrahedron \(A B C D\) and \(-\hat{i}+4 \hat{j}-3 \hat{k}\) is the position vector of the centroid of the triangle \(\mathrm{BCD}\), then the position vector of the centroid of the tetrahedron \(A B C D\) is

If \(a, b\) and \(c\) are three distinct real numbers and \(\lim _{x \rightarrow \infty} \frac{(b-c) x^2+(c-a) x+(a-b)}{(a-b) x^2+(b-c) x+(c-a)}=\frac{1}{2}\), then \(a+2 c=\)

The maximum value of \(y=x(\log x)^2\) is

\(\int_0^1 \frac{x^4+1}{x^6+1} d x=\)

In a right angled triangle, if the position vector of the vertex having the right angle is \(-3 \mathrm{i}+5 \mathrm{j}+2 \mathrm{k}\) and the position vector of the midpoint of its hypotenuse is \(6 \mathrm{i}+2 \mathrm{j}+5 \mathrm{k}\), then the position vector of its centroid is