Recently Active Algebra Questions

\hspace{-16}$Find a function $\mathbf{f(x)\neq x}$ such that for every $\mathbf{x\geq 0}$\\\\ $\mathbf{f\left(\frac{x}{1+x}\right)=\frac{f(x)}{1+f(x)}}$\\\\ ...

If f(x) = f(xy)+f(\frac{x}{y}) \text{for all } x \epsilon \mathbb{R^{+}} and f(1) = 0 , f ' (1) = 0 , then find the f(x). Please Help I just can't make this : f ' (1) = 0 ? ...

If Sn=(1.2)/3! + (2.22)/4! + (3.23)/5! +...............upto n terms. Then find sum of infinite terms. ...

nth term of a series can be written as ar= f(r) - f(r-1), then Sn= \sum_{r=1}^{n}{a_{r}} =f(n) - f(0) and S∞= *Image* Sn , then Value of \sum_{r=1}^{infinity}{(4r-1)5^{r}/(r^{^{2}}+r)} is?? ...

Value of k for which 4x2+4x+1=0 has exactly one point of intersection with k-|x+(1/2)| is equal to: a) 1 b) 2 c) 3/2 d) 3/4? ...

Let P(x) = x^{6}-x^{5}-x^{3}-x^{2}-x Q(x) = x^{4}-x^{3}-x^{2}-1 If a,b,c,d are roots of q(x) then find the value of P(a)+P(b)+P(d) ? ...

\left\{ \left( ^{n}C_{0}+^{n}C_{3}+...\right)-\frac{1}{2}\left(^{n}C_{1}+^{n}C_{2}+^{n}C_{4}+^{n}C_{5}+.... \right)\right\}^{2}+\\\frac{3}{4}\left(^{n}C_{1}-^{n}C_{2}+^{n}C_{4}-^{n}C_{5}+....\right)^{2} = ? ...

\hspace{-16}$Prove that\\\\ $\mathbf{\binom{2n+1}{1}.\binom{2n+1}{3}.\binom{2n+1}{5}.......\binom{2n+1}{2n-1}.\binom{2n+1}{2n+1}<\left(\frac{4^n-1}{n}\right)^n} ...

Find " m " such that (x^{2}-5x+4)^{2}+(2m+1)(x^{2}-5x+4)+3=0 has two distinct Real roots. ...

If h(x)= 10/x -2 find directly h-1(2) ...

*Image* ...

Given that the equation x^4+px^3+qx^2+rx+s=0 has four real, positive roots, prove that- (a)pr-16s\geq 0 (a)q^2-36s\geq 0 Is there any proof without using Cauchy-Schwarz? ...

Q1. We have T_{n}=(n^2+1)n! and S_{n}=T_{1}+T_{2}+....+T_{n} , If \frac{T_{n}}{S_{n}} =\frac{a}{b} where gcd(a,b)=1, , then find b-a. Q2. No of ordered pairs (a,b) such that (a+ib)^{2010}=a-ib is Q3. Let A,B,C be three subset ...

Is there any proof to (n!)2≥nn without using induction? ...

\hspace{-16}$Find Max. and Min. value of $\mathbf{f(x,y)=\frac{2x^2+7y^2-12xy}{x^2+y^2}}$\\\\ Where $\mathbf{x,y}$ are Real no. not Simultaneously Zero ...

\hspace{-16}$Find Max. of Constant $k$ such that for any positive real no. \\\\ $a\;b\;,c$ with $abc=1$ satisfy the Inequality\\\\ $a^2+b^2+c^2+3k\geq (k+1)(ab+bc+ca)$ ...

\hspace{-16}$Calculate value of \\\\ $\binom{2010}{1}-\binom{2010}{3}+\binom{2010}{5}+................-\binom{2010}{2007}+\binom{2010}{2009}=$ ...

Call a set of integers "spacy" if it contains no more than one out of any three consecutive integers. How many subsets of {1, 2, 3, ....... 12}, including the empty set, are spacy? ...

find the solutions to 2x+ 2y + 2z = 2336. here x , y and z are positive integers.. ...

\hspace{-16}$Solve for $\mathbf{x}$\\\\ $\mathbf{x^2-2x+2=\log_{\frac{2}{3}}(x^2+1)+\log_{\frac{2}{3}}(3x)}$ ...

If sec^{-1}(x-3)+tan^{-1}(\sqrt{9y^{2}-1})+sin^{-1}(x^2+y^2) =\lambda , has no solution then find the exhaustive range of \lambda . ...

$\hspace{-16}Solve the equation \\\\ $\mathbf{64^x-27=343^{x-1}+\frac{3}{7}.28^x}$ ...

Suppose that f (x) is a polynomial with integer coefficients such that f (2) = 3 and f (7) = −5. Show that f (x) has no integer roots. ...

\hspace{-16}$Find all Complex no. $\mathbf{z_{1}\;,z_{2}\;,z_{3}\in\mathbf{C}}$ in \\\\\\ $\mathbf{\begin{Vmatrix} \bold{\hspace{-70}z^3_{1}+z^3_{2}+z^3_{3}=24} \\\\ \bold{\left(z_{1}+z_{2}\right).(z_{2}+z_{3}).(z_{3}+z_{1})= ...

\hspace{-16}$If $\mathbf{a\;,b\;c\in\mathbb{R}}$ and $\mathbf{a+b+c=0\;,a^2+b^2+c^2=1}$\\\\ Then $\mathbf{a^4+b^4+c^4=}$ ...

\hspace{-16}$If $\mathbf{a\;,b\;c\geq 2\;,}$ Then find Min. value of \\\\ $\mathbf{\log_{b+c}a+\log_{c+a}b+\log_{a+b}c}$ ...

\hspace{-16}$\If $\mathbf{x=\left(8+3\sqrt{7}\right)^n\;,n\in\mathbb{N}}\;,$Then $\mathbf{x-x^2+x\left[x\right]=}$\\\\ Where $\mathbf{\left[x\right]=}$Greatest Integer Function ...

\ \boxed{\lim_{n\to\infty}\prod_{p=1}^{n}\left(1+\frac{p}{n^{2}}\right) = \sqrt{e}} ...

\hspace{-16}$Find all Integer solution $\mathbf{(x,y)}$ in $\mathbf{x^3+y^3+6xy=8}$ ...

find the number of solutions of |[x]-2x| = 4 here [.] denotes the floor function... and |.| denotes modulus.. ...