$...

Let $\alpha$ and $\beta$ be the roots of $x^2-3 x+p=0$ and $\gamma$ and $\delta$ be the roots of $x^2-6 x+q=0$. If $\alpha, \beta, \gamma, \delta$...

https://competishun.com/let-s-be-the-set-of-all-integer-solutions-x-y-z-of-the-system-of-equationsbr-beginaligned-x-2-y5-z0-2-x4-yz0-7-x14-y9-z0-endaligned-such-that-15-leq-x2y2z2-leq-150-then-the-number-of-elements-in-the-s/

Let $\alpha$ and $\beta$ the roots of the quadratic equation $x^2 \sin \theta-x(\sin \theta \cos \theta+1)+\cos \theta=0\left(0<\theta<45^{\circ}\right)$, and $\alpha<\beta$. Then $\sum_{n=0}^{\infty}\left(\alpha^n+\frac{(-1)^n}{\beta^n}\right)$ is equal to :

The sum, $\sum_{n=1}^7 \frac{n(n+1)(2 n+1)}{4}$ is equal toThe sum, $\sum_{n=1}^7 \frac{n(n+1)(2 n+1)}{4}$ is equal to

If 19 th term of a non-zero A.P. is zero, then its $\left(49^{\text {th }}\right.$ term) $(29$ th term $)$ is

Let $\mathrm{a}_{\mathrm{n}}$ be the $\mathrm{n}^{\text {th }}$ term of a G.P. of positive terms. If $\sum_{n=1}^{100} a_{2 n+1}=200$ and $\sum_{n=1}^{100} a_{2 n}=100$, then $...

If the $10^{\text {th }}$ term of an A.P. is $\frac{1}{20}$ and its $20^{\text {th }}$ term is $\frac{1}{10}$, then the sum of its first...

Let $x, y$ be positive real numbers and $m, n$ positive integers. The maximum value of the expression $\frac{x^m y^n}{\left(1+x^{2 m}\right)\left(1+y^{2 n}\right)}$ is

If the sum of the series $20+19 \frac{3}{5}+19 \frac{1}{5}+18 \frac{4}{5}+\ldots .$. upto $n$th term is 488 and the $n$th term is negative, then