The area bounded by the curve \(y=2 x-x^2\) and the straight line \(y=-x\) is :

If \(A=\left[\begin{array}{lll}a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a\end{array}\right], a \neq 0\) then \(|\operatorname{adj} A|\) is equal to

\(\int \frac{x+\sqrt{x}}{2 \sqrt{x}} d x=\)

The largest open interval, in which the function \(f(x)=\frac{x}{x^2+1}\) increases, is

\(\int_{-\pi}^\pi \frac{e^{\sin x}}{e^{\sin x}+e^{-\sin x}} d x\) is equal to:

\(A\) and \(B\) are two independent events. The probability that both events \(A\) and \(B\) occur is \(\frac{1}{6}\) and the probability that neither of them occur is \(\frac{1}{3}\). If \(P(A)=x, P(B)=y\) then the value of \(x+y\) is.

If \(A=\left[\begin{array}{ccc}1 & 2 & 3 \\ -4 & -5 & -6\end{array}\right], \quad B=\left[\begin{array}{cc}2 & -3 \\ 4 & -5 \\ 2 & -1\end{array}\right]\) and \(B A=\left[b_{i j}\right]\), then \(\left(b_{23}-b_{31}\right)\) is equal to

Which one of the followings is an indirect measure of the organic matter present in water?

Read the passage carefully and give the answers to the question :
Does the number of species in a community really matter to the functioning of the ecosystem? This is a question for which ecologists have not been able to give a definitive answer. For many decades, ecologists believed that communities with more species, generally, tend to be more stable than those with less species. According to the International Union for Conservation of Nature and Natural Resources (IUCN) (2004), the total number of plant and animal species described so far is more than 1.5 million.
The following are the various hypotheses proposed in explaining the greatest biological diversity in tropics except :

The equation of tangent line to \(y=2 x^2+7\) which is parallel to the line \(4 x-y+3=0\) is :

The speed of an electromagnetic wave in a medium of dielectric constant 2.25 and relative permeability 4 is:

The intensity of transmitted light is maximum when a Polaroid sheet (\( P_2 \)) is rotated between two crossed Polaroids, \( P_1 \) and \( P_3 \) when

The rate of change of the area of a circle with respect to its radius r, when r = 3 cm is :