The radius of spherical balloon is decreasing at the rate of \(0.1 cm / sec\), the rate at which its volume is decreasing, when its radius is 0.5 cm is:

If \(x+\frac{1}{x}=2\), the principal value of \(\sin ^{-1} x\) is :

If \(\left[\begin{array}{cc}2 a+b & a-2 b \\ 5 c-d & 4 c+3 d\end{array}\right]=\left[\begin{array}{cc}4 & -3 \\ 11 & 24\end{array}\right]\), then the value of \(a+2 b-3 c+4 d\) is equal to

\(\frac{d}{d x}\left(e^{2 \log _e x^3}\right)\) equals

The value of the determinant \(\left|\begin{array}{ccc}a \cos \theta & b \sin \theta & 0 \\ -b \sin \theta & a \cos \theta & 0 \\ 0 & 0 & c\end{array}\right|\) is :

A machine has been producing steel tubes of mean inner diameter of 2 cm.
A sample of 10 tubes gives an inner diameter of 2.01 cm and a standard deviation of 0.063 cm.
Test the hypothesis that the machine is working in the proper order at 5% level of significance and answer, which of the following statements are correct?
[Given that \(t_{0.05}=2.262\)]
(A) The value of the test statistic is t = 0.476
(B) The null hypothesis is rejected at \(5 \%\) level of significance.
(C) The null hypothesis is accepted at \(5 \%\) level of significance.
(D) The degree of freedom is 10
Choose the correct answer from the options given below:

The integrating factor of the differential equation \(x \frac{d y}{d x}+2 y=x^2\) is:

A point charge is kept at the centre of a metallic insulated spherical shell. Then :

Read the following passage carefully and answer the given questions.
Bioactive molecule, cyclosporin A, that is used as an immunosuppressive agent in organ-transplant patients, is produced by the fungus Trichoderma polysporum. 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. Baculoviruses are pathogens that attack insects and other arthropods. The majority of baculoviruses used as biological control agents are in the genus Nucleopolyhedrovirus. These viruses are excellent candidates for species-specific, narrow spectrum insecticidal applications. They have been shown to have no negative impacts on plants, mammals, birds, fish or even on non-target insects. BOD refers to the amount of the oxygen that would be consumed if all the organic matter in one liter of water were oxidised by bacteria. The sewage water is treated till the BOD is reduced. Cheese, is one of the oldest food items in which microbes were used. Different varieties of cheese are known by their characteristic texture, flavour and taste. For example, the large holes in ‘Swiss cheese’ are due to production of a large amount of CO₂ by a bacterium named Propionibacterium sharmanii.
The greater the BOD of waste water-

Identify the regions, where sacred groves are not found?

In which ecosystem is the biomass of primary consumers greater than producers?

Identify " \(Y\) " in the following reaction:
\(R-NH_2+HNO_2 \xrightarrow{NaNO_2+HCl} X\xrightarrow{H_2 O} Y+N_2+HCl\)

From the following statement regarding the order of reaction, the statement that is not true is