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$Evaluatiuon of $\sum_{0\leq i <j \leq n}j\binom{n}{i}$ and $\sum_{0\leq i \leq j \leq n}\binom{n}{i}\binom{n}{j}$

$If the coefficients of bi-quadratic equation are all distinct and belong $ $to the set $\{-9,-5,3,4,7\}.$ Then prove that the bi-quadratic equation has$ $at least two real roots$

$Sum of $n$ terms of series $\frac{1}{2}+\frac{1}{2!}\left(\frac{1}{2}\right)^2+\frac{1\cdot 3}{3!}\left(\frac{1}{2}\right)^3+\frac{1\cdot 3 \cdot 5}{4!}\left(\frac{1}{2}\right)^4+...$

(1)$ Find the values of $a$ for which $ax^2+(a-3)x+1<0$ for$ $at least one positive $x$ (2)$ Find the values of $a$ for which $4^t-(a-4)2^t+\frac{9}{4}a<0\forall t \in (1,2)$

$The length of latus rectum of parabola whose parametric coordinates$ $are $x=at^2+bt+c$ and $y=a't^2+b't+c'$

$If $z_{1}\;,z_{2}\;,z_{3}\;,z_{4}$ are $4$ non zero complex number such $that $\Im(z_{1}+z_{2}) = \Im(z_{3}+z_{4}) =0\;,$ Then possible value of$ \arg(\frac{z_{1}}{z_{2}})+\arg(\frac{z_{3}}{z_{4}}) = \bf{Options::} (a)\;\; 0\;\;\;\;\;\; (b)\;\; \frac{\pi}{2}\;\;\;\;\;\; (c)\;\...

$(1)\; Two point $P$ and $Q$ are taken on straight line $OA$ of length $a$ unit $ Then the probability that $PQ>b\;$ unit, Where $a>b$ $(2)\; The decimal part of logarithm of two numbers taken at random are$ $found to $7$ places , Then the probability that second number can be$...

$Evaluation of $\int_{-\infty}^{\infty}\frac{1}{(x^2+ax+a^2)(x^2+bx+b^2)}dx$ Although we have solve it using partial function Decomposition, But can we solve without using partial Decomposition. If yes then How can we solve it, Thanks

$If $f$ is a double differentiable function and satisfy the condition$ f(0)=0$ and $f(1)=0$ and $\frac{d^2}{dx^2}\left(e^{-x}f(x)-x^2\right)>0\;\forall x\in (0,1)$ $Then number of values of $x$ for which $f(x)-3=(x^2-x)e^x\;\forall x\in (0,1)$...

$If $P(x,y)$ be a variable point in $x-y$ plane such that $PA+PB=1\;,$ $Where $A(-1,0)$ and $B(1,0)\;,$Then locus of point $P$ is ................................................................................................................ $Given $PA+PB=1\;,$ Then $(PA)^2-(PB)^2=PA-PB$...

$If $f(x)$ is a twice differentiable function such that $f(x)+f''(x)=-x|\sin x|f'(x)$ $Where $x\geq 0$ and $f(0) = -3$ and $f'(0)=4\;,$ $Then maximum value of $f(x)$

$Number of permutations of $1,2,3,4,5,6,,7,8,9$ such that $1$ appear$ $somewhere to the left of $2$ and $3$ appear to the left of $4$ and $5$ $somewhere to the left of $6,$ are$ $like $815723946$ would be one such permutation$

$Calculation of $\displaystyle \frac{\tan 20^0+\tan 40^0+\tan 80^0-\tan 60^0}{\sin 40^0}$ $What i have try::, Using If $A+B+C= \pi\;,$ Then $\tan(A+B) = \tan(\pi-C)$ $So we get $\tan A+\tan B+\tan C = \tan A\cdot \tan B\cdot \tan C$ $And Using $\tan (60^0-A)\cdot \tan(A)\cdot...

$All positive integer ordered pairs $(x,y)$ for which $\displaystyle \binom{x}{y} = 2016$$

$An ellipse with length of major axis $4$ and length of minor axis $2$$ $touches the coordinate axis , Then how can we prove that locus of $ $its center is circle $x^2+y^2 = 5$$

$Prove that the number of rational points on the Circumference of a circle$ $having center $(\pi,e)$ is atmost $1$. $ Given a point $a$ and $b$ is rational ,if $a$ and $b$ both are rational number $

$Using\; Combinatorial \; argument \; How\; can\; we\; prove \; $ \displaystyle \frac{n^2!}{(n!)^2}$\; is \; an $ Integer \; Quantity\;,where \; $n\in \mathbb{N}$

$Evaluation of $\displaystyle \int_{0}^{\pi}x\cdot \cot x dx$ $I have Tried like this way.$ $Here $\displaystyle \cot x$ is $0$at $\displaystyle x = \frac{\pi}{2}$. $So we can break integral$ $ as $\displaystyle \int_{0}^{\frac{\pi}{2}}x\cdot \cot xdx +\int_{\frac{\pi}{2}}^{\pi}x\cdot \cot x...

$\displaystyle I = \int_{0}^{2}\left(3x^2-3x+1\right)\cdot \cos\left(x^3-3x^2+4x-2\right)dx$

$Evaluation\; of \; sum \; $\displaystyle \lim_{n\rightarrow \infty}\frac{1}{n^2}\sum_{k=1}^{n}\sqrt{n^2-k^2}$