Functions

If for x, y <!-- [if gte mso 9]><xml> </xml><![endif]-->  $\in$ R, x < 0, y = $lo{g}_{10}x+lo{g}_{10}{x}^{1/3}+lo{g}_{10}{x}^{1/9}+.....upto\infty$ <!-- [if gte mso 9]><xml> </xml><![endif]--> terms and $\frac{2+4+6+....+2y}{3+6+9+....+3y}=\frac{4}{lo{g}_{10}x},$ <!-- [if gte mso 9]><xml> </xml><![endif]--> then the ordered pair (x, y) is equal to

Let f(x) = <!-- [if gte mso 9]><xml> </xml><![endif]-->  $cos\left(2ta{n}^{-1}sin\left(co{t}^{-1}\sqrt[]{\frac{1-x}{x}}\right)\right),0<x<1.Then:$

For any real number x, let [x] denote the largest integer less than equal to x. Let f be a real valued function defined on the interval [-10, 10] by f (x) = $\left\{\begin{array}{cc}\hfill x-\left[x\right],if\left[x\right]\hfill & \hfill 1+\left[x\right]-x,if\left[x\right]\hfill \end{array}\right\}\begin{array}{cc}\hfill isodd\hfill & \hfill iseven\hfill \end{array}.$ <!-- [if gte mso 9]><xml> </xml><![endif]--> Then the value of <!-- [if gte mso 9]><xml> </xml><![endif]-->  $\frac{{\pi }^2}{10}\underset{-10}{\overset{10}{{\displaystyle \int f\left(x\right)}}}cos\pi xdxis:$

Let a, b and $\gamma$ <!-- [if gte mso 9]><xml> </xml><![endif]-->be three positive real number. Let f(x) = ax⁵ + bx³ + $\gamma x,x\in R$ <!-- [if gte mso 9]><xml> </xml><![endif]--> and g : R ® R be such that $g\left(f\left(x\right)\right)=x$ <!-- [if gte mso 9]><xml> </xml><![endif]--> for all x $\in R.$ <!-- [if gte mso 9]><xml> </xml><![endif]--> If $a_1,a_2,a_3.....,a_n$ <!-- [if gte mso 9]><xml> </xml><![endif]--> be in arithmetic progression with mean zero then the value of $f\left(g\left(\frac{1}{n}{\displaystyle \sum _{i=1}^{n}f\left(a_i\right)}\right)\right)$ <!-- [if gte mso 9]><xml> </xml><![endif]--> is equal to:

Let f(x) = $min\left\{\left[x-1\right],\left[x-2\right],....,\left[x-10\right]\right\}$ <!-- [if gte mso 9]><xml> </xml><![endif]--> where [t] denotes the greatest integer $\le$ <!-- [if gte mso 9]><xml> </xml><![endif]--> t. Then ${\displaystyle \underset{0}{\overset{10}{\int }}f\left(x\right)dx+{\displaystyle \underset{0}{\overset{10}{\int }}{\left(f\left(x\right)\right)}^{2}dx+{\displaystyle \underset{0}{\overset{10}{\int }}\left|f\left(x\right)\right|dx}}}$ <!-- [if gte mso 9]><xml> </xml><![endif]--> is equal to……….

For p, q $\in R,$ <!-- [if gte mso 9]><xml> </xml><![endif]-->consider the real valued function f(x) = (x – p)² – q, $x\in R$ <!-- [if gte mso 9]><xml> </xml><![endif]--> and q > 0. Let a₁, a₂, a₃ and a₄ be in an arithmetic progression with mean p and positive common difference. If $\left|f\left(a_i\right)\right|=500$ <!-- [if gte mso 9]><xml> </xml><![endif]-->for all I = 1, 2, 3, 4, then the absolute difference between the roots of f(x) = 0 is………….

Let f : N -> R be a function such that f(x + y) = 2f(x) f(y) for natural number x and y. If f(1) = 2, then the value of a for which ${\displaystyle \sum _{k=1}^{10}f\left(\alpha +k\right)}=\frac{512}{3}\left(2^{20}-1\right)$ <!-- [if gte mso 9]><xml> </xml><![endif]--> holds is:

Let [t] denote the greatest integer $\le t$ and {t} denote the fractional part of t. The integral value of for which the left hand limit of the function.

$f\left(x\right)=\left[1+x\right]+\frac{{\alpha }^{2\left[x\right]+\left\{x\right\}}+\left[x\right]-1}{2\left[x\right]+\left\{x\right\}}atx=0$ is equal to $\alpha -\frac{4}{3},$ is.