The area (in sq. units) of the region bounded by the curve \(y=x^5\), the \(x\)-axis and the ordinates \(x=-1\) and \(x=1\) is equal to:

If A and B are independent events, then which of the following is not true?

The value of integral \(\int \frac{\cos 2 x-\cos 2 \alpha}{\cos x-\cos \alpha} d x\), where \(\alpha\) is a constant is :

If \(\int \frac{2 x}{x^2+3 x+2} d x=m \log |x+2|+n \log |x+1|+C\), then the values of m and n are:

The vector equation of the line passing through the point \((-1,5,4)\) and perpendicular to the plane \(z=0\) is :

\(\int_0^2|x-1| d x=\)

An annuity in which the periodic payment begin on a fixed date and continue forever is called:

Amoebic dysentery is caused by:

If \(A=\left[\begin{array}{cc}\alpha & \beta \\ \gamma & -\alpha\end{array}\right]\) is such that \(A^2=I\), then:

Patients with diabetes have :

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 micro-organisms in a sample of water and thus, indirectly, BOD is a measure of the organic matter present in the water.
BOD is a measure of-

If \(y=\log _2\left(\log _2 x\right)\), then \(\frac{d y}{d x}\) is equal to :

Two capacitors of capacitances of \( 6 \, \mu F \) and \( 12 \, \mu F \) are connected in series with a 12 V battery. The total energy stored in the combination is