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Find the values of\;a\; for which the in equation x^2 + ax + a^2 + 6a < 0\forall x\in (1,2)

Total no. of 7 digit no. formed by using the digit 1,2,3,4,5,6, 7,8,9 which is Divisible by 8 When (i) No Repetition (II) Repetition of Digit

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1) The Letter of the word "ZENITH" are written in all possible orders. How many words are possible if all these written out as a Dictonary What is the rank of the word "ZENITH" 2) The Letter of the word "SURITI" are written in all possible orders. How many words are possible if all these...

In how many ways 4 letters can be posted in 5 letter boxes. ans: = each of the 4 letters can be posted in any one of Letter boxes in 5 ways . So Using Principle of Multiplication Total no. of ways is = 5*5*5*5 = 5^4 But I have a confusion Can we not post all letters in one Letter Box...

How many ways there are to arrange the letter of the word "GARDEN" with the vowels in alphabet order.

\bf{If \; -5 \leq \frac{x^2+ax+b}{x^2+2x+3}\leq 4\;\; and \;\; a,b\in \mathbb{N}. \; Then \; a^2+b^2\; = }

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\text{The\; no.\; of \; times \; The \; digit \; 5 \; can \; be \; written\; when\; listing \; Integer\; from \; 1\; to \; 1000}

\text{Find \; Sum \; of \; n \; terms \; of \; following\; series} 1+\frac{x}{b_{1}}+\frac{x.(x+b_{1})}{b_{1}.b_{2}}+\frac{x.(x+b_{1})(x+b_{2})}{b_{1}.b_{2}.b_{3}}+................+\frac{x.(x+b_{1})..........(x+b_{n-1})}{b_{1}.b_{2}.........b_{n}}

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(1)\;\; \int \frac{1-\cot^{n-2}x}{\left(\tan x+\cot^{n-1}x\right)}dx (2)\;\; \int\frac{1}{x^6+1}dx

\bf{If \; a^2+bc\;,b^2+ca\;,c^2+ab\; are \; in \; arithmetic \; progression \; (A.P)\;. Then \; prove \; that \; \frac{1}{a}\;,\frac{1}{b} \;, \frac{1}{c}\; are \; in \; A.P }

\bf{find \;\; all \;\; Complex \;\; number \;\; z \;\; which \;\; satisfy\;\; \mid z \mid = \frac{1}{\mid z \mid} = \mid z-1 \mid }

\displaystyle \bf{\int_{1}^{3}\frac{\ln(2x+2)}{x^2+2x+15}dx} \bf{My \;\; Solution::} \bf{Let \;\; (x+1) = t\Leftrightarrow dx = dt\;\; and \;\; changing \;\; Limit\;,\; We\;\; Get} \bf{\int_{2}^{4}\frac{\ln(2t)}{t^2+\left(\sqrt{14}\right)^2}dx } \bf{Using\;\; I.B.P\;,We\;\; Get}...

\hspace{-16}\bf{\int%20\frac{1}{x^4+1}dx=\frac{1}{2}.\int\frac{2}{x^4+1}dx}$\\\\\\%20$\bf{=\frac{1}{2}\int\frac{\left((x^2+1)-(x^2-1)\right)}{x^4+1}dx}$\\\\\\%20$\bf{=\frac{1}{2}\int\frac{x^2+1}{x^4+1}dx-\frac{1}{2}\int\frac{x^2-1}{x^4+1}dx}$\\\\\\%20Now%20Let%20$\bf{\mathbb{I}=\frac{1}{2}\int\fr...

\hspace{-16}$If%20$\bf{a,b,c%3E0}$,%20Then%20$\bf{\mathbb{A.M}\geq%20\mathbb{G.M}\geq%20\mathbb{H.M}}$\\\\\\%20Means%20$\bf{\frac{a+b+c}{3}\geq%20\left(a.b.c\right)^{\frac{1}{3}}\geq%20\frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}}$\\\\\\%20Now%20Here%20We%20have%20to%20prove%20that%20$\bf{\frac{...

\displaystyle \int_{0}^{1}\frac{\sqrt{1-x^2}}{1-x^2.\sin^2\;x}dx