A fair coin is tossed 99 times. If X is the number of times head occur then \(\mathrm{P}[\mathrm{X}=\mathrm{r}]\) is maximum when \(\mathrm{r}=\)

The curve \(x^4-2 x y^2+y^2+3 x-3 y=0\) cuts the X -axis at \((0,0)\) at an angle of

The angle between the tangents drawn from the origin to the circle \((x-7)^2+(y+1)^2=25\) is

If \(\alpha+\beta=\frac{\pi}{2}\) and \(\beta+\gamma=\alpha\), then \(\tan \alpha\) equals

If inverse of \(\left[\begin{array}{ccc}1 & 2 & x \\ 4 & -1 & 7 \\ 2 & 4 & -6\end{array}\right]\) does not exist, then \(x=\)

The bacteria increases at the rate proportional to the number of bacteria present. If
the original number ' \(\mathrm{N}^{\prime}\) doubles in 4 hours then the number of bacteria in 12 hours
will be

\(\lim _{x \rightarrow 0} \frac{27^x-9^x-3^x+1}{\sqrt{5}-\sqrt{4+\cos x}}=\)

If \(\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}\) are non-coplanar vectors and \(\overline{\mathrm{p}}=\frac{\overline{\mathrm{b}} \times \overline{\mathrm{c}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}, \overline{\mathrm{q}}=\frac{\overline{\mathrm{c}} \times \overline{\mathrm{a}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}, \overline{\mathrm{r}}=\frac{\overline{\mathrm{a}} \times \overline{\mathrm{b}}}{[\overline{\mathrm{a}} \overline{\mathrm{b}} \overline{\mathrm{c}}]}, \quad\) then \(2 \overline{\mathrm{a}} \cdot \overline{\mathrm{p}}+\overline{\mathrm{b}} \cdot \overline{\mathrm{q}}+\overline{\mathrm{c}} \cdot \overline{\mathrm{r}}=\)

A beam of unpolarized light passes through a tourmaline crystal A and then it passes through a second tourmaline crystal B oriented so that its principal plane is parallel to that of A. The intensity of emergent light is \(\mathrm{I}_0\). Now \(B\) is rotated by \(45^{\circ}\) about the ray. The emergent light will have intensity \(\left(\cos 45^{\circ}=\frac{1}{\sqrt{2}}\right)\)

A vehicle without passengers is moving on a frictionless horizontal road with velocity ' \(u\) ' can be stopped in a distance ' \(d\) '. Now \(40 \%\) of it's weight is added. If the retardation remains same the stopping distance at velocity ' \(u\) ' is

The capacitance of a parallel plate capacitor is \(2.5 \mu \mathrm{F}\). When it is half filled with a dielectric as shown in figure, its capacitance becomes \(5 \mu \mathrm{F}\). The dielectric constant of the dielectric is

The area of triangle with vertices \((1,2,0),(1,0, a)\) and \((0,3,1)\) is \(\sqrt{6}\) sq. units, then the values of ' \(a\) ' are

Phenol on reaction with aqueous solution of bromine gives