Let \(f: R \rightarrow R\) be defined as \(f(x)=5 x^2\). Choose the correct answer :

If \(x^2+2 x y+y^3=10\), then the value of \(\frac{d y}{d x}\) is:

Which of the following is not the specification of the Sinking Fund?

The probability that in a year of the 22nd century chosen at random, there will be 53 Sundays is:

The corner points of the feasible region determined by the system of linear constraints are \((0, 0), (0, 40), (20, 40), (60, 20), (60, 0)\). The objective function is \(Z = 4x + 3y\).
Compare the quantity in Column A and Column B
Column A : Column B :
Maximum of Z 325

The shortest distance between the following lines :
\(\vec{r}=(\hat{i}+\hat{j}-\hat{k})+s(2 \hat{i}+\hat{j}+\hat{k})\)
\(\vec{r}=(\hat{i}+\hat{j}+2 \hat{k})+t(4 \hat{i}+2 \hat{j}+2 \hat{k})\), where \(s\) and \(t\) are scalars, is :

If A and B are two distinct events such that P(A|B) = P(B|A), then which of the following is/are possible?
(A) A = B
(B) P(A) = P(B)
(C) \(A \subset B\) but \(A \neq B\)
(D) \(A \cap B=\phi\)
Choose the correct answer from the options given below:

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.
The bottled juices are clarified by the combination of -

In which of the following cases, the value of impact parameter (for alpha particle scattering experiment by Geiger-Marsden) will be minimum ?

How many episodes of mass extinctions of species have occurred on earth since the origin and diversification of life?

Read the passage carefully and answer the questions
The carbonyl group of aldehydes and ketones is highly polar. The positively charged carbon is readily attacked by electron-rich nucleophile and the negatively charged oxygen atom is attacked by an electro-deficient electrophile. Aldehydes and ketones undergo nucleophilic addition reactions in the presence of acid or a base. The mechanism involves attack of a nucleophile on the electron deficient carbonyl carbon to form a tetrahedral intermediate. A new bond is formed between the carbon and oxygen. Protonation of the anion furnished by the reaction medium yields a neutral addition product. The addition product is same whether the reaction is catalyzed by acid or a base. For example in the presence of base aldehydes and ketones react with hydrogen cyanide to form cyanohydrin.
The source of nucleophile in the base catalyzed cyanohydrin formation is:

Decomposers break down complex organic matter into -

A simple random sample of 100 items from a population with \(\sigma=8\) has a standard error of mean :