$A$ stone moving vertically upwards has its equation of motion $s = 490t - 4.9t^2$. The maximum height reached by the stone is

If $x = -1$ and $x = 2$ are the extreme points of the function $y = a \log|x| + bx^2 + x$,then find the values of $a$ and $b$.

The function $f(x) = \sin x (1 + \cos x)$ is maximum at $x = \dots$

The function $f(x)=x^3+a x^2+b x+c$ with $a^2 \leq 3 b$ has:

$A$ rectangular sheet of fixed perimeter with sides having their lengths in the ratio $8:15$ is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is $100$,the resulting box has maximum volume. The lengths of the sides of the rectangular sheet are:
$(A)$ $24$
$(B)$ $32$
$(C)$ $45$
$(D)$ $60$

If $x$ is real and $\alpha, \beta$ are maximum and minimum values of $\frac{x^2-x+1}{x^2+x+1}$ respectively,then $\alpha+\beta=$