If \(\vec{a}, \vec{b}\) and \(\vec{c}\) are three unit vectors such that \(\vec{a}+2 \vec{b}-3 \vec{c}=\overrightarrow{0}\), then the value of \(2 \vec{a} \cdot \vec{b}-6 \vec{b} \cdot \vec{c}-3 \vec{c} \cdot \vec{a}\) is

The unit digit of \(12^{12}\) is :

The system of equations:
\(x+y+z=5\)
\(x+2 y+3 z=9\)
\(x+3 y+\lambda z=\mu\) has a unique solution, if

In a 10 km race, \(A, B\) and \(C\) each run at uniform speed and get the first, second and third positions respectively.
If \(A\) beats \(B\) by 1 km and \(B\) beats \(C\) by 1 km , then by how much distance does \(A\) beat \(C\) ?

(A) f(x) = |x| is continuous function on R (the set of real numbers)
(B) f(x) = [x] is continuous function on R
(C) f(x) = sin x is continuous function on R
(D) f(x) = tanx is continuous function on R
(E) f(x) = sinx + cosx is continuous function on R

If \(A\) and \(B\) are two invertible matrices, then which of the following statements are correct?
(A) \(\left|A^{-1}\right|=|A|^{-1}\)
(B) \(\operatorname{adj} A=|A| A^{-1}\)
(C) \((A B)^{-1}=A^{-1} B^{-1}\)
(D) \((A+B)^{-1}=A^{-1}+B^{-1}\)
Choose the correct answer from the options given below:

If \(y=x^2 \log x\) then \(\frac{d^2 y}{d x^2}=\)

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.
Statins, used as blood-cholesterol lowering agents are produced by -

If \(f(x)=\left\{\begin{array}{ll}\frac{\sin \pi x}{5 x}, & x \neq 0 \\ k, & x=0\end{array}\right.\) is continuous at \(x=0\), then \(k\) is :

If the length of a simple pendulum is decreased by \(3 \backslash \%\), the percentage change in its period T is
[Note : \(T=2 \pi \sqrt{\frac{L}{g}}\) where \(L\) is the length of pendulum and \(g\) is constant]

When a slab of insulating material 4 mm thick is introduced between the plates of a parallel plate capacitor of separation 4 mm, it is found that the distance between the plates has to be increased by 3.2 mm to restore the capacity to its original value. The dielectric constant of the material is :

The external diameter of a 5 m long hollow copper tube is 10 cm and the thickness of its wall is 5 mm. If the specific resistance of copper be \( 1.7 \times 10^{-8} \, \Omega\text{-m} \), then its resistance would be approximately

Answer the question on the basis of passage given below:
Proteins are the polymers of about twenty different a-amino acids which are linked by peptide bonds. Proteins perform various structural and dynamic functions in the organisms. Proteins which contain only a-amino acids are called simple proteins. The secondary or tertiary structure of proteins get disturbed on change of pH or temperature and they are not able to perform their functions. This is called denaturation of proteins. Enzymes are biocatalysts which speed up the reactions in biosystems. Vitamins are accessory food factors required in the diet. They are classified as fat soluble (A, D, E and K) and water soluble (B group and C). Deficiency of vitamins leads to many diseases. Nucleic acids are the polymers of nucleotides which in turn consist of a base, a pentose sugar and phosphate moiety. Nucleic acids are responsible for the transfer of characters from parents to offsprings. There are two types of nucleic acids DNA and RNA. DNA contains a five carbon sugar molecule called 2-deoxyribose whereas RNA contains ribose.
RNA and DNA differ due to presence/absence of oxygen at which carbon of sugar moiety?