. If \(\alpha, \beta\) are the roots of the equation \(x^2-2 x+4=0\), then \(\alpha^n+\beta^n=\ldots \ldots\) \(x \cos \left(\frac{n \pi}{3}\right)\) for any \(n \in \mathbf{N}\)

Let ' \(\mathrm{O}\) ' be the origin, \(\mathrm{A}\) and \(\mathrm{B}\) be two points with position vectors \(-3 \hat{i}-3 \hat{j}+4 \hat{k}\) and \(4 \hat{i}-4 \hat{j}-3 \hat{k}\) respectively. Let \(P\) be a point such that the line drawn through \(P\) parallel to \(\overrightarrow{\mathrm{OB}}\) meets \(\mathrm{OA}\) in \(\mathrm{L}\) and another line through \(\mathrm{P}\) parallel to \(\overrightarrow{\mathrm{OA}}\) meets \(\mathrm{OB}\) in \(\mathrm{M}\). If \(\mathrm{L}\) divides \(\mathrm{OA}\) in the ratio \(2: 3\) and \(\mathrm{M}\) divides \(\mathrm{OB}\) in the ratio \(3: 2\), then the distance from 0 to \(\mathrm{P}\) is

The lines \(a x^2+2 h x y+b y^2=0\) are at right angles if

If the sum of the slopes of the lines given by \(4 x^2+2 \lambda x y-7 y^2=0\) is equal to the product of the slopes, then \(\lambda\) is equal to

If \(\alpha, \beta\) and \(\gamma\) are the roots of the equation \(x^3+a^2+b x\) \(+c=0\), then the roots of the equation \(x^3+\left(2 b-a^2\right) x^2+\) \(\left(b^2-2 a c\right) x-c^2=0\) are

Let \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) and \(\mathrm{d}\) be non-zero numbers. If the point of intersection of the lines \(4 a x+2 a y+c=0\) and \(5 b x+2 b y\) \(+\mathrm{d}=0\) lies in the fourth quadrant and is equidistant from the two coordinate axes, then

If \(\int \frac{d x}{(1+\sqrt{x}) \sqrt{x-x^2}}=\frac{A \sqrt{x}}{\sqrt{1-x}}+\frac{B}{\sqrt{1-x}}+C\), where \(C\) is real constant, then \(A+B\) equals to

A unit vector coplanar with \(\mathbf{i}+\mathbf{j}+3 \mathbf{k}\) and \(\mathbf{i}+3 \mathbf{j}+\mathbf{k}\) and perpendicular to \(\mathbf{i}+\mathbf{j}+\mathbf{k}\) is

A student writes an examination which contains eight true or false questions. If he answers six or more questions correctly, he passes the examination. If the student answers all the questions, then the probability that he fails in the examination is

\(\mathrm{KMnO}_4\) reacts with \(\mathrm{KI}\), in basic medium to form \(\mathrm{I}_2\) and \(\mathrm{MnO}_2\). When \(250 \mathrm{~mL}\) of \(0.1 \mathrm{M} \mathrm{KI}\) solution is mixed with \(250 \mathrm{~mL}\) of \(0.02 \mathrm{M}\) \(\mathrm{KMnO}_4\) in basic medium, what is the number of moles of \(\mathrm{I}_2\) formed?

If the first ionisation enthalpy of \(\mathrm{Li}, \mathrm{Be}\) and C respectively are \(520,899,1086 \mathrm{~kJ} \mathrm{~mol}^{-1}\), the first ionisation enthalpy (in \(\mathrm{kJ} \mathrm{mol}^{-1}\) ) of B will be

Let \([\mathrm{x}]\) represent the greatest integer less than or equal to \(\mathrm{x},\{\mathrm{x}\}=\mathrm{x}-[\mathrm{x}]\), \(\sqrt{2}=1.414\) and \(\sqrt{3}=1.732\). If \(f(x)=\left\{x+\left[\frac{x}{1+x^2}\right]\right\}\) is a real valued function, then \(\mathrm{f}(\sqrt{2})+\mathrm{f}(-\sqrt{3})=\)

Which one of the following is used in the preparation of cellulose nitrate?