The area (in sq. units) of the region enclosed between the parabola \(y ^{2}=2 x\) and the line \(x + y =4\) is

The common tangent to the circles \(x^2 + y^2 = 4\) and \(x^2 + y^2 + 6x + 8y - 24 = 0\) also passes through the point

A vector \(\vec n\) is inclined to \(x-\) axis at \(45^o\), to \(y-\) axis at \(60^o\) and at an acute angle to \(z-\) axis. If \(\vec n\) is a normal to a plane passing through the point \(\left( {\sqrt 2 , - 1,1} \right)\) then the equation of the plane is 

Let \(p\) and \(q\) be two positive numbers such that \(p + q =2\) and \(p ^{4}+ q ^{4}=272 .\) Then \(p\) and \(q\) are roots of the equation

\(1+3+5^2+7+9^2+\ldots\) upto 40 terms is equal to

For \(a \in C\), let \(A =\{z \in C: \operatorname{Re}( a +\overline{ z }) > \operatorname{Im}(\bar{a}+z)\}\) and \(B=\{z \in C: \operatorname{Re}(a+\bar{z}) < \operatorname{Im}(\bar{a}+z)\}\). Then among the two statements : \((S 1)\) : If \(\operatorname{Re}(A), \operatorname{Im}(A) > 0\), then the set \(A\) contains all the real numbers \((S2)\): If \(\operatorname{Re}(A), \operatorname{Im}(A) < 0\), then the set \(B\) contains all the real numbers,

A fair coin is tossed \(n\)-times such that the probability of getting at least one head is at least \(0.9 .\) Then the minimum value of \(n\) is \(....\)

Let \(k\) be a non-zero real number  If \(f(x) = {\rm{ }}\left\{ {\begin{array}{*{20}{c}}
{\frac{{\left( {{e^x} - 1} \right)^2}}{{\sin {\mkern 1mu} \left( {\frac{x}{k}} \right){\mkern 1mu} \log {\mkern 1mu} \left( {1 + \frac{x}{4}} \right)}}{\mkern 1mu} ,{\mkern 1mu} x \ne 0}\\
{{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} 12{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} ,x{\mkern 1mu} {\mkern 1mu}  = 0{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} }
\end{array}} \right.\)  is a continuous function then the value of \(k\) is

If \((1 - x^3)^{10} = \sum\limits_{r=0}^{10} a_r x^r (1-x)^{30-2r}\), then \(\dfrac{9a_9}{a_{10}}\) is equal to __________.

A scientific committee is to be formed from \(6\) Indians and \(8\) foreigners, which includes at least \(2\) Indians and double the number of foreigners as Indians. Then the number of ways, the committee can be formed, is

The mean and variance of 10 observations are 9 and 34.2, respectively. If 8 of these observations are 2, 3, 5, 10, 11, 13, 15, 21, then the mean deviation about the median of all the 10 observations is

Let \(f(\mathrm{x})=\left|2 \mathrm{x}^2+5\right| \mathrm{x}|-3|, \mathrm{x} \in \mathrm{R}\). If \(\mathrm{m}\) and \(\mathrm{n}\) denote the number of points where \(f\) is not continuous and not differentiable respectively, then \(m+n\) is equal to :

Let \(\vec{a}=3 \hat{i}+\hat{j}-\hat{k}\) and \(\overrightarrow{ c }=2 \hat{ i }-3 \hat{ j }+3 \hat{k}\). If \(\vec{b}\) is \(a\) vector such that \(\vec{a}=\vec{b} \times \vec{c}\) and \(|\vec{b}|^2=50\), then \(|72-| \vec{b}+\left.\vec{c}\right|^2 \mid\) is equal to \(..........\).