The horizontal range is four times the maximum height attained by a projectile. The angle of projection is ......... $^o$

If $a$ and $b$ are odd integers,then the roots of the equation $2ax^2 + (2a + b)x + b = 0, a \ne 0,$ will be

The curve $f(x) = e^x \sin x$ is defined in the interval $[0, 2\pi]$. The value of $x$ for which the slope of the tangent drawn to the curve at $x$ is maximum,is

Given that,$a \alpha^2+2 b \alpha+c \neq 0$ and that the system of equations
$\begin{aligned} & (a \alpha+b) x+a y+b z=0 \\ & (b \alpha+c) x+b y+c z=0 \\ & (a \alpha+b) y+(b \alpha+c) z=0\end{aligned}$
has a non-trivial solution,then $a, b$ and $c$ lie in

The solution set of the inequality $\log_{10}(x^2 - 2x - 2) \le 0$ is

$A$ body of mass $m$ hangs at one end of a string of length $l$,the other end of which is fixed. It is given a horizontal velocity so that the string just reaches a position where it makes an angle of $60^o$ with the vertical. The tension in the string at the mean position is: