Let \(f : R \rightarrow R\) be a function defined by \(f ( x )=\) \(\log _{\sqrt{m}}\{\sqrt{2}(\sin x-\cos x)+m-2\}\), for some \(m\), such that the range of \(f\) is \([0,2]\). Then the value of \(m\) is \(............\)

Let \( a_{1}=1 \) and for \( n\ge1 \), \( a_{n+1}\)
= \(\frac{1}{2}a_{n}+\frac{n^{2}-2n-1}{n^{2}(n+1)^{2}} \). Then \( |\sum_{n=1}^{\infty}(a_{n}-\frac{2}{n^{2}})| \) is equal to ........... .

Consider the following two statements : Statement \(I\) : For any two non-zero complex numbers \(\mathrm{z}_1, \mathrm{z}_2\) \(\left(\left|z_1\right|+\left|z_2\right|\right)\left|\frac{z_1}{\left|z_1\right|}+\frac{z_2}{\left|z_2\right|}\right| \leq 2\left(\left|z_1\right|+\left|z_2\right|\right)\) and Statement \(II\) : If \(\mathrm{x}, \mathrm{y}, \mathrm{z}\) are three distinct complex numbers and a, b, c are three positive real numbers such that \(\frac{a}{|y-z|}=\frac{b}{|z-x|}=\frac{c}{|x-y|}\), then \(\frac{\mathrm{a}^2}{\mathrm{y}-\mathrm{z}}+\frac{\mathrm{b}^2}{\mathrm{z}-\mathrm{x}}+\frac{\mathrm{c}^2}{\mathrm{x}-\mathrm{y}}=1\) Between the above two statements,

The area, enclosed by the curves \(y=\sin x+\cos x\) and \(\mathrm{y}=|\cos \mathrm{x}-\sin \mathrm{x}|\) and the lines \(\mathrm{x}=0, \mathrm{x}=\frac{\pi}{2}\) is:

Let \(\alpha, \beta ; \alpha>\beta\), be the roots of the equation \(x^2-\sqrt{2} x-\sqrt{3}=0\). Let \(P_n=\alpha^n-\beta^n, n \in N\). Then \((11 \sqrt{3}-10 \sqrt{2}) \mathrm{P}_{10}+(11 \sqrt{2}+10) \mathrm{P}_{11}-11 \mathrm{P}_{12}\) is equal to :

The normal to the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{9}=1\) at the point \((8,3 \sqrt{3})\) on it passes through the point

If \(\lim _{n \rightarrow \infty} \frac{(n+1)^{k-1}}{n^{k+1}}[(n k+1)+(n k+2)+\ldots+\) \((n k+n)]=33 . \lim _{n \rightarrow \infty} \frac{1}{n^{k+1}} \cdot\left[1^{k}+2^{k}+3^{k}+\ldots+n^{k}\right]\), then the integral value of \(k\) is equal to \(....\)

Let the area of a \(\triangle P Q R\) with vertices \(P(5,4), Q(-2,4)\) and \(R(a, b)\) be 35 square units. If its orthocenter and centroid are \(O\left(2, \frac{14}{5}\right)\) and \(C(c, d)\) respectively, then \(c+2 d\) is equal to

A box open from top is made from a rectangular sheet of dimension \(\mathrm{a} \times \mathrm{b}\) by cutting squares each of side \(x\) from each of the four corners and folding up the flaps. If the volume of the box is maximum, then \(\mathrm{x}\) is equal to :

Let \(a_{n}\) be the \(n^{\text {th }}\) term of a G.P. of positive terms. If \(\sum\limits_{n=1}^{100} a_{2 n+1}=200\) and \(\sum\limits_{n=1}^{100} a_{2 n}=100,\) then \(\sum\limits_{n=1}^{200} a_{n}\) is equal to 

Let \(S=\left\{x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right): 9^{1-\tan ^2 x}+9^{\tan ^2 x}=10\right\}\) and \(\beta=\sum_{x \in S} \tan ^2\left(\frac{x}{3}\right)\), then \(\frac{1}{6}(\beta-14)^2\) is equal to

The sum of the cubes of all the roots of the equation \(x^{4}-3 x^{3}-2 x^{2}+3 x+1=10\) is

The number of ways in which \(21\) identical apples can be distributed among three children such that each child gets at least \(2\) apples, is