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\hspace{-16}\bf{\mathbb{S}}$olve the inequality\\\\ $\bf{\mid x-1 \mid+3\mid x-3 \mid+5\mid x-5 \mid+.......+2009\mid x-2009\mid}$\\\\\\ $\bf{\geq 2 \mid x-2 \mid+4\mid x-4 \mid+6\mid x-6 \mid+.......+2008\mid x-2008\mid}$ ...

\hspace{-16}$find the last six digit of the product $\bf{(2010)\times (5)^{2014}}$ ...

\hspace{-16}\bf{\int_{0}^{\frac{\pi}{4}}\ln \left(\frac{1+\sin^2 2x}{\sin^4 x+\cos^4 x}\right)dx} ...

\hspace{-16}\bf{\int_{-1}^{1}\frac{2x^{1004}+x^{3014}+x^{2008}.\sin(x)^{2007}}{1+x^{2010}}dx} ...

\hspace{-16}$If $\bf{\mathbb{A}=\frac{1}{2\sqrt{1}}+\frac{1}{3\sqrt{2}}+\frac{1}{4\sqrt{3}}+.........+\frac{1}{100\sqrt{99}}}$\\\\\\ Then $\bf{\lfloor \mathbb{A}\rfloor =}$ ...

\hspace{-16}\mathbb{F}$ ind value of $\bf{p}$ for which the eqn. $\bf{(x^2-2x)^2-(p+3)(x^2-2x)+(p-2)=0}$\\\\\\ $\bf{(i)}\;$ has $\bf{4}$ real solutions\\\\ $\bf{(ii)}\;$has $\bf{3}$ real solutions only \\\\ $\bf{(iii)}\;$has ...

\hspace{-16}\bf{\int_{0}^{4\pi}\ln\left|13.\sin (x)+3\sqrt{3}.\cos (x)\right|dx}= ...

\hspace{-16}$If $\bf{r_{1}\;,r_{2}\;,r_{3}\;,r_{4}}$ are the roots of the equation $\bf{4x^4-ax^3+bx^2-cx+5=0}$\\\\\\ Where $\bf{r_{1}\;,r_{2}\;,r_{3}\;,r_{4}>0}$ and satisfy $\bf{\frac{r_1}{2} + \frac{r_2}{4} + \frac{r_3} ...

\hspace{-16}$No. of Integral ordered pairs $\bf{(x,y)}$ satisfying the equation\\\\\\ $\bf{\tan^{-1}\left(\frac{1}{x}\right)+\tan^{-1}\left(\frac{1}{y}\right)=\tan^{-1}\left(\frac{1}{10}\right)}$ ...

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\hspace{-16}$Calculate value of expression\\\\\\ $\bf{\sin\left(\tan^{-1}\left(\frac{1}{3}\right)+\tan^{-1}\left(\frac{1}{5}\right)+\tan^{-1}\left(\frac{1}{7}\right)+\tan^{-1}\left(\frac{1}{11}\right)+\tan^{-1}\left(\frac{1}{ ...

\hspace{-16}$If $\bf{x=\frac{\sqrt{10+\sqrt{1}}+\sqrt{10+\sqrt{2}}+\sqrt{10+\sqrt{3}}+.......+\sqrt{10+\sqrt{99}}}{\sqrt{10-\sqrt{1}}+\sqrt{10-\sqrt{2}}+\sqrt{10-\sqrt{3}}+.......+\sqrt{10-\sqrt{99}}}}$\\\\\\ Then value of $\ ...

If the roots of the equation : x4-8x3+bx2 + cx +16= 0 are positive then the root of the equation 2bx +c =0 is ???? ...

\hspace{-16}$If $\bf{f(x,y)=\frac{\sin(x)-\sin (y)}{x-y},}$ Where $\bf{x\neq y}$\\\\\\ Then $\bf{\lfloor f(x,y)\rfloor =}$\\\\\\ Where $\bf{\lfloor x \rfloor = }$ Floor function ...

\hspace{-16}$find value of $\bf{x}$ in \\\\\\ $\bf{\left|\left|\left|\left|\left|x^2-x-1\right|-2\right|-3\right|-4\right|-5\right|=x^2+x-30} ...

prove that locus of moving points such that the sum of squares of distances of any point from two fixed points is always a constant is a circle. can someone prove it without using co-ordinate geometry... i mean using plane ge ...

log |x| |x - 1| > 0 , x E R ...

sum till infinite terms of the series cot-1 3 + cot-1 7 + cot-113 + ....... is (A) pi/2 (B) cot-1 1 (C) tan-1 2 (D) none of these my approach this is something we have to find *Image* now how to approach (2) sin-1 sin 12 + co ...

\hspace{-16}$If $\bf{a,b,c\in\mathbb{R}}$ and $\bf{f(x)}$ is a Quadratic Polynomial such that\\\\ $\bf{\begin{Bmatrix} \bf{f(a)=bc} \\\\ \bf{f(b)=ca} \\\\ \bf{f(c)=ab} \end{Bmatrix}}$\\\\\\ Then $\bf{f(a+b+c)=}$ I am Getting ...

1)Find tan-1 1/x2+x+1 + tan-1 1/x2+3x+3 + tan-1 1/x2+5x+7 + tan-1 1/x2+7x+13 ..... upto n terms.... 2)If sin(cot-1(x+1))=cos(tan-1x),then x= ? 3) If \sum_{n=1}^{10}{}\sum_{m=1}^{10}{}tan^{-1}(\frac{m}{n})=k\pi\, then \: find ...

1) 2 players A and B play a series of 2n games. Each game can result in either a win or a loss for A. Find the total no. of ways in which A can win the series of these games. (All the games are to be played) Ans: 22n-1 - 1/2. ...

\hspace{-16}$Find all ordered pairs $\bf{(x,y)}$ in $\bf{x^2-y!=2001}$\\\\ Where $\bf{x,y\in \mathbb{N}}$ ...

Raindrops are falling with a speed of 5 m s -1 at an angle of 37° with the vertical. A man is climbing down a plane of inclination 30° with the horizontal, holding the umbrella vertically up to protect himself. Find the spe ...

\hspace{-16}$Determine all Real $\bf{2\times 2}$ matrix $\bf{A=\begin{pmatrix} \bf{a} & \bf{b}\\ \bf{c} & \bf{d} \end{pmatrix}}$ that satisfying \\\\ The equation $\bf{A^2+A+I=O}$ ...

N concentric thin conducting shells having radius R , R/2 , R/22 , ...... is placed . N th shell is earthed . if Q charge is given to the outer shell , find the charge on the inner most shell . ...

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\hspace{-16}$Find all Real ordered pairs $\bf{(x,y,z)}$ in $\bf{\tan^2(x)+\cot^2(y)+\csc^2(z)=\frac{1}{2}}$ ...

\hspace{-16}\bf{\int\frac{x^2.\cos^{-1}\big(x\sqrt{x}\big)}{\big(1-x^3\big)^2}dx} ...

Let f(x) be a function such that on putting any value of x we get the equation of a hyperbola which is conjugate of the hyperbola having x as eccentricity. then find, f(f(f(f(x))) ...