The general solution of the differential equation \(\frac{d y}{d x}=-4 x y^2\) is given by

\(\int\left(\frac{1}{\log _e x}-\frac{1}{\left(\log _e x\right)^2}\right) d x\) is equal to

The area of the parallelogram of which \(\hat{i}\) and \(\hat{i}+\hat{j}\) are adjacent sides is :

The angle between the vectors \(\hat{i}-2 \hat{j}+\hat{k}\) and \(3 \hat{i}+2 \hat{j}+\hat{k}\) is:

The area of the region bounded by \(y^2=9 x, x=2, x=4\) and the \(x\)-axis in the first quadrant, is

Which of the following is not correct?

If \(A\) is a \(3 \times 3\) matrix such that \(|\operatorname{adj} A|=9\) and \(\left|k A^{-1}\right|=9\), then the value of \(k\) are :

An electric field in \(x-y\) plane is given by the equation \(\vec{E}=E_0 \hat{i} N / C\). A positively charged particle is given a velocity of \(5 \hat{j} m / s\) at point \(x=3, y=0\) in \(x-y\) plane. The particle will experience force :

Plants can capture only 2-10% of the PAR and this small amount of energy sustains the entire world. Here PAR stands for:

The system of equations
\(x+y+z=7\)
\(x+2 y+3 z=5\)
\(x+3 y+\lambda z=\mu\)
has a unique solution, if

The large holes in 'Swiss Cheese' are due to :

If \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\), then \(\frac{d^2 y}{d x^2}\) is equal to

Read the following passage carefully and answer the given questions. During anaerobic sludge digestion, bacteria produce a mixture of gases such as methane, hydrogen sulphide and carbon dioxide. These gases form biogas and can be used as source of energy as it is inflammable. The bioactive molecule, cyclosporin A, that is used as an immunosuppressive agent in organ-transplant patients, is produced by the fungus Trichoderma polysporum. You must have noticed that bottled fruit juices bought from the market are clearer as compared to those made at home. This is because the bottled juices are clarified by the use of pectinases and proteases. Statins produced by the yeast Monascus purpureus have been commercialised as blood-cholesterol lowering agents. It acts by competitively inhibiting the enzyme responsible for synthesis of cholesterol. The BOD test measures the rate of uptake of oxygen by microorganisms in a sample of water and thus, indirectly, BOD is a measure of the organic matter present in the water.
During anaerobic sludge digestion, bacteria produce the biogas. The composition of this biogas is -