The number of \(6\) digit numbers that can be formed using the digits \(0, 1, 2, 5, 7\) and \(9\) which are divisible by \(11\) and no digits is repeated, is

The point represented by \(2 + i\)  in the Argand plane moves \(1\,unit\) eastwards, then \(2\,units\) northwards and finally from there \(2\sqrt 2\,units\)  in the south-westwards direction. Then its new position in the Argand plane is at the point represented by

Let a unit vector \(\hat{\mathrm{u}}=\mathrm{x} \hat{\mathrm{i}}+\mathrm{y} \hat{\mathrm{j}}+\mathrm{z} \hat{\mathrm{k}}\) make angles \(\frac{\pi}{2}, \frac{\pi}{3}\) and \(\frac{2 \pi}{3}\) with the vectors \(\frac{1}{\sqrt{2}} \hat{\mathrm{i}}+\frac{1}{\sqrt{2}} \hat{\mathrm{k}}, \frac{1}{\sqrt{2}} \hat{\mathrm{j}}+\frac{1}{\sqrt{2}} \hat{\mathrm{k}}\) and \(\frac{1}{\sqrt{2}} \hat{\mathrm{i}}+\frac{1}{\sqrt{2}} \hat{\mathrm{j}}\) respectively. If \(\overrightarrow{\mathrm{v}}=\frac{1}{\sqrt{2}} \hat{\mathrm{i}}+\frac{1}{\sqrt{2}} \hat{\mathrm{j}}+\frac{1}{\sqrt{2}} \hat{\mathrm{k}}\), then \(|\hat{\mathrm{u}}-\overrightarrow{\mathrm{v}}|^2\) is equal to

Let \(\mathrm{f}(\mathrm{x})\) be a polynomial of degree \(5\) such that \(\mathrm{x}=\pm 1\) are its critical points. \(\mathop {\lim }\limits_{x \to 0} \left(2+\frac{f(x)}{x^{3}}\right)=4,\) then which one of the following is not true?

If the domain of the function \(f(x)=\log _e\) \(\left(\frac{2 x+3}{4 x^2+x-3}\right)+\cos ^{-1}\left(\frac{2 x-1}{x+2}\right)\) is \((\alpha, \beta]\), then the value of \(5 \beta-4 \alpha\) is equal to

Let the coefficients of three consecutive terms \(T_r, T_{r+1}\) and \(T_{r+2}\) in the binomial expansion of \((a+b)^{12}\) be in a G.P. and let \(p\) be the number of all possible values of \(r\). Let \(q\) be the sum of all rational terms in the binomial expansion of \((\sqrt[4]{3}+\sqrt[3]{4})^{12}\). Then \(\mathrm{p}+\mathrm{q}\) is equal to :

The \(8^{\text {th }}\) common term of the series \(S _1=3+7+11+15+19+\ldots . .\) ; \(S _2=1+6+11+16+21+\ldots .\) is \(.......\).

\(A\) and \(B\) alternately throw a pair of dice. \(A\) wins if he throws a sum of 5 before \(B\) throws a sum of 8 , and \(B\) wins if he throws a sum of 8 before \(A\) throws a sum of 5 . The probability, that \(A\) wins if A makes the first throw, is

If \(\mathrm{A}=\{\mathrm{x} \in {R}:|\mathrm{x}-2|>1\}, \mathrm{B}=\left\{\mathrm{x} \in {R}: \sqrt{\mathrm{x}^{2}-3}>1\right\}\), \(\mathrm{C}=\{\mathrm{x} \in {R}:|\mathrm{x}-4| \geq 2\}\) and \({Z}\) is the set of all integers, then the number of subsets of the set \((A \cap B \cap C)^{c} \cap {Z}\) is .... .

If \(\frac{{dy}}{{dx}} + y\tan x = \sin 2x\) and \(y(0)\,=1\) , then \(y(\pi)\) is equal to

If \(f: R \rightarrow R\) is a function defined by \(f(x)=[x-1] \cos \left(\frac{2 x-1}{2}\right) \pi,\) where \([.]\) denotes the greatest integer function, then \(f\) is

A light ray emerging from the point source placed at \(P( 1, 3)\) is reflected at a point \(Q\) in the axis of \(x\). If the reflected ray passes through the point \(R\) (\(6, 7)\), then the abscissa of \(Q\) is

Let \(x=x(y)\) be the solution of the differential equation \(y^2 \mathrm{~d} x+\left(x-\frac{1}{y}\right) \mathrm{d} y=0\). If \(x(1)=1\), then \(x\left(\frac{1}{2}\right)\) is :