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Solve the differential equation:
(1 + y2) dx = (tan−1 y − x) dy
Concept: General and Particular Solutions of a Differential Equation
Find the shortest distance between the lines `vecr = (4hati - hatj) + lambda(hati+2hatj-3hatk)` and `vecr = (hati - hatj + 2hatk) + mu(2hati + 4hatj - 5hatk)`
Concept: Shortest Distance Between Two Lines
A die, whose faces are marked 1, 2, 3 in red and 4, 5, 6 in green is tossed. Let A be the event "number obtained is even" and B be the event "number obtained is red". Find if A and B are independent events.
Concept: Independent Events
Suppose a girl throws a die. If she gets 1 or 2 she tosses a coin three times and notes the number of tails. If she gets 3,4,5 or 6, she tosses a coin once and notes whether a ‘head’ or ‘tail’ is obtained. If she obtained exactly one ‘tail’, what is the probability that she threw 3,4,5 or 6 with the die ?
Concept: Bayes’ Theorem
"The outward electric flux due to charge +Q is independent of the shape and size of the surface which encloses is." Give two reasons to justify this statement.
Concept: Electric Flux
Find the equivalent capacitance of the network shown in the figure, when each capacitor is of 1 μF. When the ends X and Y are connected to a 6 V battery, find out (i) the charge and (ii) the energy stored in the network.
Concept: Capacitors and Capacitance
Derive an expression for drift velocity of free electrons.
Concept: Drift of Electrons and the Origin of Resistivity
Distinguish between emf and terminal voltage of a cell.
Concept: Cells, EMF, and Internal Resistance
Explain the term ‘drift velocity’ of electrons in conductor. Hence obtain the expression for the current through a conductor in terms of ‘drift velocity’.
Concept: Drift of Electrons and the Origin of Resistivity
Answer the following question.
A cell of emf E and internal resistance r is connected across a variable resistor R. Plot the shape of graphs showing a variation of terminal voltage V with (i) R and (ii) circuit current I.
Concept: Cells, EMF, and Internal Resistance
Obtain the expression for current sensitivity of moving coil galvanometer.
Concept: Moving Coil Galvanometer
Two infinitely long straight parallel wires, '1' and '2', carrying steady currents I1 and I2 in the same direction are separated by a distance d. Obtain the expression for the magnetic field `vecB`due to the wire '1' acting on wire '2'. Hence find out, with the help of a suitable diagram, the magnitude and direction of this force per unit length on wire '2' due to wire '1'. How does the nature of this force changes if the currents are in opposite direction? Use this expression to define the S.I. unit of current.
Concept: Force Between Two Parallel Currents (Ampere’s Law)
Increasing the current sensitivity of a galvanometer may not necessarily increase its voltage sensitivity. Explain, giving reason.
Concept: Moving Coil Galvanometer
Define self-inductance of a coil.
Concept: Solenoid
State how a moving coil galvanometer can be converted into an ammeter.
Concept: Moving Coil Galvanometer
How is current sensitivity increased?
Concept: Moving Coil Galvanometer
In what way is Gauss's law in magnetism different from that used in electrostatics ? Explain briefly.
Concept: Magnetism and Gauss’s Law
State Lenz's law. Illustrate, by giving an example, how this law helps in predicting the direction of the current in a loop in the presence of a changing magnetic flux.
Concept: Lenz’s Law and Conservation of Energy
In a given coil of self-inductance of 5 mH, current changes from 4 A to 1 A in 30 ms. Calculate the emf induced in the coil.
Concept: Inductance >> Self Inductance
Derive the expression for the self-inductance of a long solenoid of cross sectional area A and length l, having n turns per unit length.
Concept: Inductance >> Self Inductance
