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\hspace{-16}(1)\;\; \bf{\int_{-\pi}^{\pi}\left(\sum_{k=1}^{2013}\sin (kx)\right)^2dx}$\\\\\\ $(2)\;\; \bf{\int_{-\pi}^{\pi}\left(\sum_{k=1}^{2013}\cos (kx)\right)^2dx}$\\\\\\ ...
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\hspace{-16}\bf{\prod_{m=1}^{n-1}\sin \left(\frac{m\cdot \pi}{n}\right)=} ...
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\hspace{-16}$Find all real polynomials $\bf{p(x)}$ such that $\bf{p(x)\cdot p(x+1)=p(x^2)\;\forall x\in \mathbb{Z}}$ ...
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\hspace{-16}\bf{(1)\;\; \int\sqrt{a+\sqrt{b+\sqrt{x}}}\; dx}$\\\\\\ $\bf{(2)\;\;\int \sqrt{1+2\sqrt{x-x^2}}\;dx}$ ...
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\hspace{-16}$In how many ways can the selection of $\bf{8}$ letters be done frm $\bf{24}$ letters\\\\ of which $\bf{8}$ are $\bf{'a'}$ and $\bf{8}$ are $\bf{'b'}$ and rest are unlike. ...
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\hspace{-16}\bf{(1)\;\;}$ Total no. of real solution in $\bf{2^x = x^2}$\\\\\\ $\bf{(2)\;\;}$ Total no. of real solution in $\bf{2^x = 1+x^2}$\\\\\\ $\bf{(3)\;\;}$ Total no. of real solution in $\bf{2^x+3^x+4^x+5^x = 10x+4}$\ ...
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\hspace{-16}\bf{(1)\;\;}$ Total Integer ordered pair,s of $\bf{(x,y)}$ in $\bf{x^2-y! = 2001}$\\\\\\ $\bf{(2)\;\;}$ Total Integer ordered pair,s of $\bf{(x,y)}$ in $\bf{x^2-7y! = 2011}$\\\\\\ $\bf{(3)\;\;}$ Total Integer orde ...
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\hspace{-16}\bf{(1)\;\;}$ The sum of $\bf{5}$ digit no. that can be formed using the digits $\bf{0,0,1,2,3,4}$\\\\\\ $\bf{(2)\;\;}$ The sum of $\bf{5}$ digit no. that can be formed using the digits $\bf{0,0,1,1,2,3}$ ...
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\hspace{-16}\bf{(1)\;\; \int\frac{1}{(1+x^4)^{\frac{1}{4}}}dx}$\\\\\\ $\bf{(2)\;\; \int\frac{1}{(1-x^4)^{\frac{1}{4}}}dx}$\\\\\\ $\bf{(3)\;\;\int\frac{1}{(1+x^4)}dx}$\\\\\\ $\bf{(4)\;\;\int\frac{1}{(1+x^6)}dx}$ ...
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\hspace{-18}$Integer values of $\bf{x}$ for which $\bf{x^4+x^3+x^2+x+1}$ is a perfect square. ...
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\hspace{-18}$All positive Integer ordered pairs $\bf{(x,y)}$ for which $\bf{\binom{x}{y} = 120}$ ...
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\hspace{-18}$(1) The number of four digits having only two consecutive digits identical is\\\\\\ (2) The number of four digits having only three consecutive digits are\\\\ identical is ...
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\hspace{-16}$If $\bf{34! = 295232799039604140847618609643520000000}$.Then $\bf{(a,b,c,d)}$\\\\ ...
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\hspace{-16}$Solution for $\bf{a\;,b\;,c}$ in \\\\ $\bf{a[a]+c\{c\}-b\{b\}=0.16}$\\\\ $\bf{b+a\{a\}-c\{c\} = 0.25}$\\\\ $\bf{c[c]+b\{b\}-a\{a\} = 0.49}$\\\\ Where $\bf{[x] = }$ Integer part of $\bf{x}$\\\\ and $\bf{\{x\} = }$ ...
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\hspace{-16}$If $\bf{\int_{0}^{\infty}\frac{\sin x}{x} = \frac{\pi}{2}\;\; .}$ Then value of $\bf{\int_{0}^{\infty}\frac{\sin^3 x}{x^3} = }$ ...
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\hspace{-16}$Determine all pairs $\bf{(a, b)}$ of natural numbers, for which the number\\\\ of $ \bf{a ^ 3 + 6ab + 1} $ and $ \bf{b ^ 3 + 6ab + 1}$ are cubes of natural numbers. ...
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\hspace{-16}$factors of $\bf{a(b-c)^3+b(c-a)^3+c(a-b)^3}$ ...
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\hspace{-16}$Calculation of real values of $\bf{(a,b,c)}$ such that $\bf{x^3-ax^2+bx-c =0}$\\\\ has a roots $\bf{a\;,b}$ and $\bf{c.}$ ...
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\bf{\int\frac{1}{(x^2-x+1)\sqrt{x^2+x+1}}dx} ...
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\hspace{-16}$Solution for real $\bf{\left(a\;,b\;,c\right)}$ in \\\\ $\bf{[a]+c\{c\}-b\{b\}=0.16}$\\\\ $\bf{b+a\{a\}-c\{c\} = 0.25}$\\\\ $\bf{c[c]+b\{b\}-a\{a\} = 0.49}$\\\\ Where $\bf{[x] =}$ Integer part of $\bf{x}$\\\\ and ...
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\hspace{-16} $ Minimum value of $\bf{\left|z-1-i \right| + \left |z+2-3i \right| + \left |z+3+2i \right|}$\\\\\\ where $\bf{z = x+iy}$ and $\bf{i = \sqrt{-1}}$ ...
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\hspace{-16}\bf{\int_{0}^{1}x^{2013}.(1-x)^{2014}dx} ...
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\hspace{-16}$(i) $\bf{\int\frac{1}{1+\sin^4 x}dx}$\\\\\\ (ii) $\bf{\int\frac{1}{1+\cos^4 x}dx}$ ...
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\hspace{-16}$Total Real solution of the equations in Diff. cases\\\\\\ (i) $\bf{2^x = 1+x^2}$\\\\\\ (ii) $\bf{e^x = x^2}$ ...
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\hspace{-16}$ Prove that there exists a power of the number $\bf{2}$ such that the last\\\\ $\bf{1000}$ digits in its decimal representation are all $\bf{1}$ and $\bf{2}$ ...
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\hspace{-16}$If $\bf{\mathbb{S} = \sum_{r=4}^{1000000}\frac{1}{r^{\frac{1}{3}}}}.$ Then value of $\bf{\left[\mathbb{S}\right] = }$\\\\\\ where $\bf{[x] = }$ Greatest Integer function ...
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\hspace{-16}\bf{(1)\;\;}$ In how many ways can the letters of the word $"\bf{PERMUTATIONS}"$\\\\ be arranged so that there is always exactly $\bf{4}$ letters between $\bf{P}$ and $\bf{S}$, is\\\\\\ $\bf{(2)\;\;}$ Calculate To ...
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\hspace{-16}$Calculate total no. of real solution in $\bf{e ^x = x^n}$\\\\ where $\bf{n\in \mathbb{N}}$ ...
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\hspace{-16}$Calculate total no. of real solution in each case\\\\\\ $\bf{(1)\;\;2^x = 1+x^2}$\\\\\\ $\bf{(2)\;\; 2^x+3^x+4^x = x^2}$\\\\\\ $\bf{(3)\;\; 3^x+4^x+5^x = 1+x^2}$ ...
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\hspace{-16}$Calculate total no. of real solution in each case\\\\\\ $\bf{(1)\;\;2^x = 1+x^2}$\\\\\\ $\bf{(2)\;\; 2^x+3^x+4^x = x^2}$\\\\\\ $\bf{(3)\;\; 3^x+4^x+5^x = 1+x^2}$ ...
