Let the median and the mean deviation about the median of \(7\) observation \(170,125,230,190,210\), \(a, b\) be 1\(70\) and \(\frac{205}{7}\) respectively. Then the mean deviation about the mean of these \(7\) observations is :

For \(k \in N\), let \(\frac{1}{\alpha(\alpha+1)(\alpha+2) \ldots(\alpha+20)}=\sum_{k=0}^{20} \frac{A_{k}}{a+k}\), where \(a\,>\,0\). Then the value of \(100\left(\frac{A_{14}+A_{15}}{A_{13}}\right)^{2}\) is equal to \(....\)

The number of elements in the set \(S=\left\{x \in R : 2 \cos \left(\frac{x^{2}+x}{6}\right)=4^{x}+4^{-x}\right\}\) is\(.....\)

Let the centre of the circle \(x^2 + y^2 + 2gx + 2fy + 25 = 0\) be in the first quadrant and lie on the line \(2x - y = 4\). Let the area of an equilateral triangle inscribed in the circle be \(27\sqrt{3}\). Then the square of the length of the chord of the circle on the line \(x = 1\) is _______.

If \(0 < a , b < 1,\) and \(\tan ^{-1} a +\tan ^{-1} b =\frac{\pi}{4},\) then the value of \((a+b)-\left(\frac{a^{2}+b^{2}}{2}\right)+\left(\frac{a^{3}+b^{3}}{3}\right)-\left(\frac{a^{4}+b^{4}}{4}\right)+\ldots\) is ..... .

Let \(L_1: \overrightarrow{\mathrm{r}}=(\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}})+\lambda(\hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}), \lambda \in \mathrm{R}\) \(L_2: \overrightarrow{\mathrm{r}}=(\hat{\mathrm{j}}-\hat{\mathrm{k}})+\mu(3 \hat{\mathrm{i}}+\hat{\mathrm{j}}+\mathrm{p} \hat{\mathrm{k}}), \mu \in \mathrm{R}\) and  \(L_3: \overrightarrow{\mathrm{r}}=\delta(\ell \hat{\mathrm{i}}+\mathrm{m} \hat{\mathrm{j}}+\mathrm{n} \hat{\mathrm{k}}) \delta \in \mathrm{R}\). Be three lines such that \(\mathrm{L}_1\) is perpendicular to \(\mathrm{L}_2\) and \(L_3\) is perpendicular to both \(L_1\) and \(L_2\). Then the point which lies on \(\mathrm{L}_3\) is

Let \(\mathrm{ABCD}\) and \(\mathrm{AEFG}\) be squares of side \(4\) and \(2\) units, respectively. The point \(\mathrm{E}\) is on the line segment \(\mathrm{AB}\) and the point \(\mathrm{F}\) is on the diagonal \(\mathrm{AC}\). Then the radius \(r\) of the circle passing through the point \(\mathrm{F}\) and touching the line segments \(\mathrm{BC}\) and \(\mathrm{CD}\) satisfies :

If the \(10^{\text {th }}\) term of an A.P. is \(\frac{1}{20}\) and its \(20^{\text {th }}\) term is \(\frac{1}{10},\) then the sum of its first \(200\) terms is

The value \(\sum \limits_{ r =0}^{22}{ }^{22} C _{ r }{ }^{23} C _{ r }\) is \(.......\)

If in a regular polygon the number of diagonals is \(54\), then the number of sides of this polygon is

Let a unit vector which makes an angle of \(60^{\circ}\) with \(2 \hat{i}+2 \hat{j}-\hat{k}\) and an angle of \(45^{\circ}\) with \(\hat{i}-\hat{k}\) be \(\overrightarrow{\mathrm{C}}\). Then \(\overrightarrow{\mathrm{C}}+\left(-\frac{1}{2} \hat{\mathrm{i}}+\frac{1}{3 \sqrt{2}} \hat{\mathrm{j}}-\frac{\sqrt{2}}{3} \hat{\mathrm{k}}\right)\) is :

Let \(A =\{1,2,3,4,5,6,7\}\). Then the relation \(R =\) \(\{( x , y ) \in A \times A : x + y =7\}\) is

The number of integral values of \('k'\) for which the equation \(3 \sin x+4 \cos x=k+1\) has a solution, \(k\) \(\in R\) is