If \(1 + {x^4} + {x^5} = \sum\limits_{i = 0}^5 {{a_i}\,(1 + {x})^i,} \) for all \(x\) in \(R,\) then \(a_2\) is

The line \(2 x - y +1=0\) is a tangent to the circle at the point \((2,5)\) and the centre of the circle lies on \(x-2 y=4\). Then, the radius of the circle is

Let \(\vec{a}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}\) and \(\overrightarrow{\mathrm{d}}=\vec{a} \times \overrightarrow{\mathrm{b}}\). If \(\overrightarrow{\mathrm{c}}\) is a vector such that \(\vec{a} \cdot \overrightarrow{\mathrm{c}}=|\overrightarrow{\mathrm{c}}|\), \(|\overrightarrow{\mathrm{c}}-2 \vec{a}|^2=8\) and the angle between \(\overrightarrow{\mathrm{d}}\) and \(\overrightarrow{\mathrm{c}}\) is \(\frac{\pi}{4}\), then \(|10-3 \overrightarrow{\mathrm{~b}} \cdot \overrightarrow{\mathrm{c}}|+|\overrightarrow{\mathrm{d}} \times \overrightarrow{\mathrm{c}}|^2\) is equal to

The rate of growth of bacteria in a culture is proportional to the number of bacteris present and the bacteria count is \(1000\) at initial time \(t =0 .\) The number of bacteria is increased by \(20 \%\) in \(2\) hours. If the population of bacteria is \(2000\) after \(\frac{ k }{\log _{ e }\left(\frac{6}{5}\right)}\) hours, then \(\left(\frac{ k }{\log _{ c } 2}\right)^{2}\) is equal to

If \(\mathrm{A}(3,1,-1), \mathrm{B}\left(\frac{5}{3}, \frac{7}{3}, \frac{1}{3}\right), \mathrm{C}(2,2,1)\) and \(\mathrm{D}\left(\frac{10}{3}, \frac{2}{3}, \frac{-1}{3}\right)\) are the vertices of a quadrilateral \(\mathrm{ABCD}\), then its area is

Let \(\vec a = 2\hat i + {\lambda _1}\hat j + 3\hat k\), \(\vec b = 4\hat i + \left( {3 - {\lambda _2}} \right)\hat j + 6\hat k\) \(\vec c = 3\hat i + 6\hat j + \left( {{\lambda _3} - 1} \right)\hat k\) be three vectors such that \(\vec b = 2\vec a\) and \(\vec a\) is perpendicular to \(\vec c\). Then a possible value of \(\left( {{\lambda _1},{\lambda _2},{\lambda _3}} \right)\) is

Twenty metres of wire is available for fencing off a flowerbed in the form of a circular sector. Then the maximum area (in sq. m) of the flower bed is :

The area bounded by the curves \(y=|x-1|+|x-2|\) and \(y =3\) is equal to

If \(f(x)=\sin \left(\cos ^{-1}\left(\frac{1-2^{2 x}}{1+2^{2 x}}\right)\right)\) and its first derivative with respect to \(x\) is \(-\frac{ b }{ a } \log _{ e } 2\) when \(x =1,\) where \(a\) and \(b\) are integers, then the minimum value of \(\left| a ^{2}- b ^{2}\right|\) is.........

The number of solutions of the equation \(\log _{(x+1)}\left(2 x^{2}+7 x+5\right)+\log _{(2 x+5)}(x+1)^{2}-4=0, x\,>\,0\), is \(....\)

The position vectors of the vertices \(A, B\) and \(C\) of a triangle are \(2 \hat{i}-3 \hat{j}+3 \hat{k}, \quad 2 \hat{i}+2 \hat{j}+3 \hat{k} \quad\) and \(-\hat{i}+\hat{j}+3 \hat{k}\) respectively. Let \(l\) denotes the length of the angle bisector \(\mathrm{AD}\) of \(\angle \mathrm{BAC}\) where \(\mathrm{D}\) is on the line segment \(\mathrm{BC}\), then \(2 l^2\) equals :

Out of all the patients in a hospital \(89\, \%\) are found to be suffering from heart ailment and \(98\, \%\) are suffering from lungs infection. If \(\mathrm{K}\, \%\) of them are suffering from both ailments, then \(\mathrm{K}\) can not belong to the set :

Let \(f(x)=\sqrt{\lim _{r \rightarrow x}\left\{\frac{2 r^2\left[(f(r))^2-f(x) f(r)\right]}{r^2-x^2}-r^3 e^{\frac{f(r)}{r}}\right\}}\) be differentiable in \((-\infty, 0) \cup(0, \infty)\) and \(f(1)=1\). Then the value of \(ea\), such that \(f(a)=0\), is equal to ...........