The area inside the parabola \(y^2=4 a x\), between the lines \(x=a\) and \(x=4 a\) is equal to

If \(\mathrm{f}(x)=\left\{\begin{array}{ll}\frac{\sqrt{1+\mathrm{m} x}-\sqrt{1-\mathrm{m} x}}{x} & ,-1 \leq x < 0 \ \frac{2 x+1}{x-2} & , 0 \leq x \leq 1\end{array}\right.\) is continuous in the interval \([-1,1]\), then \(\mathrm{m}\) is equal to

The distance of the point \(\mathrm{P}(3,4,4)\) from the point of intersection of the line joining the points \(\mathrm{Q}(3,-4,-5), \mathrm{R}(2,-3,1)\) and the plane \(2 x+y+z=7\) is

If \(x=a(t+\sin t), y=a(1-\cos t)\), then \(\frac{d y}{d x}=\)

With usual notations, in \(\triangle \mathrm{ABC}\), if \(\mathrm{a}=2, \mathrm{~b}=3, \mathrm{c}=5\) and \(\frac{\cos A}{a}+\frac{\cos B}{b}+\frac{\cos C}{c}=\frac{k+7}{30}\),
then \(\mathrm{k}=\)

The derivative of \(\tan ^{-1}\left(\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}\right)\) is

There are 11 points in a plane of which 5 points are collinear. Then the total number of distinct quadrilaterals with vertices at these points is

The logic gate combination circuit shown in the figure performs the logic function of

If \(u_{0}=8, u_{1}=3, u_{2}=12, u_{3}=51\), then the value of \(\Delta^{3} u_{0}\) is

If \(\bar{x}=\frac{\overline{\mathrm{b}} \times \overline{\mathrm{c}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}, \bar{y}=\frac{\overline{\mathrm{c}} \times \overline{\mathrm{a}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}\) and \(\overline{\mathrm{z}}=\frac{\overline{\mathrm{a}} \times \overline{\mathrm{b}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}\) where \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) are non-coplanar vectors, then value of \(\bar{x} \cdot(\overline{\mathrm{a}}+\overline{\mathrm{b}})+\bar{y} \cdot(\overline{\mathrm{~b}}+\overline{\mathrm{c}})+\overline{\mathrm{z}} \cdot(\overline{\mathrm{c}}+\overline{\mathrm{a}})\) is

Liver is located on _________ in the human body.

Let \(\overline{\mathrm{a}}=2 \hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}}\) and \(\overline{\mathrm{b}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}\). If \(\overline{\mathrm{c}}\) is a vector such that \(\overline{\mathrm{a}} \cdot \overline{\mathrm{c}}=|\overline{\mathrm{c}}|,|\overline{\mathrm{c}}-\overline{\mathrm{a}}|=2 \sqrt{2}\) and the angle between \((\overline{\mathrm{a}} \times \overline{\mathrm{b}})\) and \(\overline{\mathrm{c}}\) is \(30^{\circ}\), then the value of \(|(\bar{a} \times \bar{b}) \times \bar{c}|\) is equal to

Based on the graph given below, identify the INCORRECT statements.