Let $p, q, r \in R^+$. If $27pqr \geq (p + q + r)^3$ and $3p + 4q + 5r = 12$,then find the value of $p^3 + q^4 + r^5$.

Let $[r]$ denote the greatest integer not exceeding $r$. The roots of the equation $3 x^2 + 6 x + 5 + \alpha (x^2 + 2 x + 2) = 0$ are complex numbers whenever $\alpha > L$ or $\alpha < M$. If $(L - M)$ is minimum,then the greatest value of $[r]$ such that $L y^2 + M y + r < 0$ for all $y \in R$ is:

Which of the following is incorrect? If $a \equiv b \pmod{m}$ and $x$ is an integer,then

Let $\tan A$ and $\tan B$, where $A, B \in (-\frac{\pi}{2}, \frac{\pi}{2})$, be the roots of the quadratic equation $x^2 - 2x - 5 = 0$. Then $20 \sin^2(\frac{A+B}{2})$ is equal to:

If $x+\frac{1}{x}=a$ and $x^2+\frac{1}{x^3}=b$,then $x^3+\frac{1}{x^2}$ is

If a polynomial $P(x)$ of degree $4$ is given by $P(x) = 2x^4 + ax^3 + bx^2 + cx + d$ such that $P(1) = 4, P(2) = 7, P(3) = 12$,and $P(4) = 19$,then find the value of $P(5)$.