The solution of the differential equation \(\frac{d r}{d t}=-r t, r(0)=r_0\) is

By using the determinants the area of the triangle with vertices \((0,0),(0,1)\) and \((1,0)\) is :

If \(x=a t^2\) and \(y=a^3 t^3\), then \(\frac{d^2 y}{d x^2}\) will be :

The probability that A speaks the truth is \(4 / 5\). A coin is tossed and A reports 'Heads'. The probability that it actually showed 'Heads' is :

If \(\left|\begin{array}{ccc}-1 & 1 & 0 \\ a & b & 1 \\ 1 & 2 & 1\end{array}\right|\) is a singular matrix, then the relation between a and b is:

The value of \(\int\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)^2 d x\) is:

A container is full of pure milk. From this container, one third of milk is taken out and then an equal amount of water is poured into the container to fill it. This operation is repeated four times. The final ratio of milk and water in the container is :

\(\tan ^{-1} \frac{1}{\sqrt{3}}-\sin ^{-1} \frac{1}{2}\) is equal to :

The self inductance of an inductor does not depend on the :

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.
Statins are produced by:

Answer the questions on the basis of passage given below :
As aldehydes ketones possess the carbonyl functional group, they show similar reactions. They exhibit nucleophilic addition reactions, reduction, oxidations, reactions due to alpha-Hydrogen atom, and many other reactions, like Cannizzaro reaction to electrophilic substitution reactions (in case of aromatic aldehydes ketones).
These reactions help in the preparation of a number of important compounds, like acetic acid, vinyl acetate, polymers and in perfume and dye industries. Nucleophilic addition reactions form cyanohydrins, acetals, alcohols and N -substituted derivatives like imine, oxime, semicarbazone, phenylhydrazone Schiff's base. Reactions due to alpha-hydrogen atoms help in formation of aldols and alpha, beta-unsaturated aldehydes ketones.
Which of the following does not give iodoform test?


Choose the correct answer from the options given below :

Copying genetic information from one strand of DNA into mRNA is:

Choose the correct statement(s) for intermolecular dehydration of alcohols to form ethers
(A) The formation of ether may involve a nucleophilic bimolecular \(S_N2\) reaction or \(S_N1\) reaction
(B) The method is successful with secondary and tertiary alcohols.
(C) The reaction requires protic acids
(D) The reaction is carried out at low temperature.
Choose the correct answer from the options given below: