If \({ }^n \mathrm{C}_0+\frac{1}{2}{ }^n \mathrm{C}_1+\frac{1}{3}{ }^n \mathrm{C}_2+\ldots+\frac{1}{n}{ }^n \mathrm{C}_{n-1}+\frac{1}{n+1}{ }^n \mathrm{C}_n=\frac{1023}{10}\), then \(\mathrm{n}=\)
If \(3 \sin 2 \theta=2 \sin 3 \theta\) and \(0<\theta<\pi\), then the value of \(\sin \theta\) is equal to
If in triangle ABC , with usual notations \(\sin \frac{\mathrm{A}}{2} \cdot \sin \frac{\mathrm{C}}{2}=\sin \frac{\mathrm{B}}{2}\) and 2 s is the perimeter of the triangle, then the value of \(s\) is
\[ \int\limits_0^1 \frac{1}{2+\sqrt{x}} d x= \]
Let M and N be foots of the perpendiculars drawn from the point \(\mathrm{P}(\mathrm{a}, \mathrm{a}, \mathrm{a})\) on the lines \(x-y=0, \mathrm{z}=1\) and \(x+y=0, \mathrm{z}=-1\) respectively and if \(\angle \mathrm{MPN}=90^{\circ}\) then \(\mathrm{a}^2=\)
If \(\bar{a}, \bar{b}, \bar{c}\) are non coplanar unit vectors such that \(\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})=\frac{\overline{\mathrm{b}}+\overline{\mathrm{c}}}{\sqrt{2}}\) then the angle between \(\overline{\mathrm{a}}\) and \(\overline{\mathrm{b}}\) is
The joint equation of the bisector of the angle between the lines \(2 x^2+11 x y+3 y^2=0\) is
If a random variable X has the following probability distribution of X
\[ \begin{array}{|l|c|c|c|c|c|c|c|c|} \hline \mathrm{X}=x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline \mathrm{P}(\mathrm{X}=x) & 0 & \mathrm{k} & 2 \mathrm{k} & 2 \mathrm{k} & 3 \mathrm{k} & \mathrm{k}^2 & 2 \mathrm{k}^2 & 7 \mathrm{k}^2+\mathrm{k} \\ \hline \end{array} \]
Then \(P(x \geq 6)=\)
If \(y=\log _3\left(\log _3 x\right)\) then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) at \(x=3\) is \(\ldots \ldots\)
A pair of tangents are drawn to the circle \(x^2+y^2+6 x-4 y-12=0\) from a point \(\mathrm{P}(-4,-5)\), then the area enclosed between these tangents and the area of the circle is
A family has 3 children. The probability that all the three children are girls, given that at least one of them is a girl is
