When a metal rod of length L placed normal to a uniform magnetic field B is moved at a velocity of v perpendicular to the field, the induced emf across its ends is

In a parallel plate capacitor of capacitance C, a metal sheet is inserted between the plates parallel to them. If the thickness of the sheet is one third of the separation between the plates, the capacitance of the capacitor will be

The potential energy of an electron in the first excited state of hydrogen atom is -6.8 eV. The kinetic energy of the electron in its ground state will be :

Two parallel conductors each carrying a current of 10 mA in opposite directions are placed at a distance 10 cm apart. The force per unit length between two conductors is

Two charged particles, placed at a distance d apart in vacuum, exert a force F on each other. Now, each of the charges is doubled. To keep the force unchanged, the distance between the charges should be changed to :

Which of the following statements are correct?
(A) The electrostatic repulsive force between the protons can be greater than the nuclear force to bind the nucleons together inside a nucleus.
(B) The repulsive electrostatic force between protons in smaller nuclei is much smaller than the nuclear force between nucleons inside a nucleus.
(C) The gravitational force between nucleons is much smaller than the nuclear force between the nucleons inside a nucleus.
(D) The binding energy per nucleon between nucleons is almost constant because the nuclear force is a long range force.
Choose the correct answer from the options given below:

A series LCR circuit is connected to a 230 V, 50 rad/s source. Given L = 5 H, C = 80 \( \mu \)F and R = 40 \( \Omega \). Impedance of the circuit is

Which of the following gives the correct order of stability constant in solution?

If \((x+1) e^y=1\), then :

A proton and an alpha particle enter a uniform magnetic field with the same velocity and move along circular paths. The time period of the alpha particle will be:

We use microbes and microbially derived products almost every day. Bacteria called lactic acid bacteria (LAB) grow in milk to convert it into curd. The dough, which is used to make bread, is fermented by yeast called Saccharomyces cerevisiae. Microbes are used to produce industrial products like lactic acid, acetic acid and alcohol, which are used in a variety of processes in the industry. However, certain bacteria, which grow anaerobically on cellulosic material, produce large amount of methane along with CO2 and H2. These bacteria are collectively called methanogens, and one such common bacterium is Methanobacterium. These bacteria are commonly found in the anaerobic sludge during sewage treatment. Microbes are also used for commercial and industrial production of certain chemicals like organic acids, alcohols and enzymes.
Examples of acid producers are Aspergillus niger (a fungus) of citric acid. You are familiar with the commonly used antibiotic Penicillin. Do you know that Penicillin was the first antibiotic to be discovered, and it was a chance discovery? 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.
Citric acid is commercially obtained by Aspergillus niger, which is a:

Arrange the following alkyl halides in order of increasing reactivity towards \(S_N^1\) reaction.
(A) \(\left( CH _3\right)_3 C - I\)
(B) \(\left( CH _3\right)_3 C - F\)
(C) \(\left( CH _3\right)_3 C - Br\)
(D) \(\left( CH _3\right)_3 C - Cl\)
Choose the correct answer from the options given below:

The general solution of the differential equation \(\frac{d y}{d x}=1+x+y+x y\) is given by: