The maximum area of a right-angled triangle with hypotenuse $h$ is

$y = 2x^3 - 8x^2 + 10x - 4$ is a function defined on $[1, 2]$. If the tangent drawn at a point $(a, b)$ on the graph of this function is parallel to the $X$-axis and $a \in (1, 2)$,then $a =$

For the function $f(x) = x \cos \frac{1}{x}, \quad x \geq 1$,consider the following statements:
$(A)$ For at least one $x$ in the interval $[1, \infty), f(x+2)-f(x) < 2$
$(B)$ $\lim _{x \rightarrow \infty} f^{\prime}(x) = 1$
$(C)$ For all $x$ in the interval $[1, \infty), f(x+2)-f(x) > 2$
$(D)$ $f^{\prime}(x)$ is strictly decreasing in the interval $[1, \infty)$
Which of the following combinations of statements is correct?

The minimum value of $2x^2+x-1$ is:

An enemy Apache helicopter is flying along the curve given by $y = x^{2} + 7$. $A$ soldier,placed at $(3, 7)$,wants to shoot down the helicopter when it is nearest to him. Find the nearest distance.

Let $f(x) = x^{4} - 4x^{3} + 4x^{2} + c$,where $c \in R$. Then,