Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{x^{2}}{x^{6}-a^{6}} d x$ Solution: let $I=\int \frac{x^{2}}{x^{6}-a^{6}} d x$ $=\int \frac{x^{2}}{\left(x^{3}\right)^{2}-\left(a^{3}\right)^{2}} d x$ Let $\mathrm{x}^{3}=\mathrm{t} \ldots . .$ (i) $\Rightarrow 3 x^{2} d x=d t$ $I=\frac{1}{3} \int \frac{1}{t^{2}-\left(a^{3}\right)^{2}} d t$ $I=\frac{1}{3} \times \frac{1}{2 \times a^{3}} \log \left|\frac{t-a^{3}}{t+a^{3}}\right|+c$ [since, $\left.\int \frac{1}{x^{2}-(a)^{2}} d x=\frac{1}{2 \t...
Read More →Is the momentum conserved
Question: Is the momentum conserved when charge crosses a junction in an electric circuit? Why or why not? Solution: The momentum is not conserved when the charge crosses a junction in an electric circuit. This is because the drift velocity is proportional to the electric field....
Read More →In a meter bridge, the point D is a neutral point.
Question: In a meter bridge, the point D is a neutral point. (a) the meter bridge can have no other neutral point for this set of resistances (b) when the jockey contacts a point on meter wire left of D, current flows to B from the wire (c) when the jockey contacts a point on a meter wire to the right of D, current flows from B to the wire through the galvanometer (d) when R is increased, the neutral point shifts to left Solution: The correct answer is (a) the meter bridge can have no other neut...
Read More →The measurement of an unknown resistance R
Question: The measurement of an unknown resistance R is to be carried out using Wheatstone bridge. Two students perform an experiment in two ways. The first student take R2 = 10Ω and R1 = 5Ω. The other student takes R2 = 1000 Ω and R1 = 500 Ω. In the standard arm, both take R3 = 5 Ω. Both find R = R2/R1 R3 = 10 Ω within errors. (a) the errors of measurement of the two students are the same (b) errors of measurement do depend on the accuracy with which R2 and R1 can be measured (c) if the student...
Read More →Write the negation of each of the following statements:
Question: Write the negation of each of the following statements: (i) Every natural number is greater than $0 .$ (ii) Both the diagonals of a rectangle are equal. (iii) The sum of 4 and 5 is 8 . (iv) The number 6 is greater than $4 .$ (v) Every natural number is an integer. (vi) The number $-5$ is a rational number (vii) All cats scratch. (viii) There exists a rational number $x$ such that $x^{2}=3$. (ix) All students study mathematics at the elementary level. (x) Every student has paid the fees...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{3 x^{5}}{1+x^{12}} d x$ Solution: let $I=\int \frac{3 x^{5}}{1+x^{12}} d x$ $=\int \frac{3 x^{5}}{1+\left(x^{6}\right)^{2}} d x$ Let $\mathrm{x}^{6}=\mathrm{t} \ldots . .$ (i) $\Rightarrow 6 x^{5} d x=d t$ $I=\frac{3}{6} \int \frac{1}{(t)^{2}+1} d t$ $I=\frac{1}{2} \tan ^{-1} t+c$ [since, $\left.\int \frac{1}{1+(\mathrm{x})^{2}} \mathrm{dx}=\tan ^{-1} \mathrm{x}+\mathrm{c}\right]$ $I=\frac{1}{2} \tan ^{-1}\left(x^{6}\right)+c$ [using (i)]...
Read More →Temperature dependence of resistivity ρ(T)
Question: Temperature dependence of resistivity (T) of semiconductors, insulators, and metals is significantly based on the following factors: (a) number of charge carriers can change with temperature T (b) time interval between two successive collisions can depend on T (c) length of material can be a function of T (d) mass of carriers is a function of T Solution: The correct answer is (a) number of charge carriers can change with temperature T (b) time interval between two successive collisions...
Read More →Consider a simple circuit in the figure.
Question: Consider a simple circuit in the figure. stands for a variable resistance R. R can vary from R0 to infinity. r is internal resistance of the battery (a) potential drop across AB is nearly constant as R is varied (b) current through R is nearly a constant as R is varied (c) current I depends sensitively on R (d) I V/r+R always Solution: The correct answer is (a) potential drop across AB is nearly constant as R is varied (d) I V/r+R always...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{x}{x^{4}+2 x^{2}+3} d x$ Solution: Let $I=\int \frac{x}{x^{4}+2 x^{2}+3} d x$ Let $\mathrm{x}^{2}=\mathrm{t} \ldots \ldots \cdots(\mathrm{i})$ $\Rightarrow 2 x d x=d t$ $I=\frac{1}{2} \int \frac{1}{t^{2}+2 t+3} d t$ $=\frac{1}{2} \int \frac{1}{t^{2}+2 t+1-1+3} d t$ $=\frac{1}{2} \int \frac{1}{(\mathrm{t}+1)^{2}+2} \mathrm{dt}$ Put $t+1=u$ .......(ii) $\Rightarrow \mathrm{dt}=\mathrm{du}$ $I=\frac{1}{2} \int \frac{1}{(u)^{2}+(\sqrt{2})^{2}} ...
Read More →Kirchhoff’s junction rule is a reflection of
Question: Kirchhoffs junction rule is a reflection of (a) conservation of current density vector (b) conservation of charge (c) the fact that the momentum with which a charged particle approaches a junction is unchanged as the charged particle leaves the junction (d) the fact that there is no accumulation of charges at a junction Solution: The correct answer is (b) conservation of charge (d) the fact that there is no accumulation of charges at a junction...
Read More →Which of the following characteristics
Question: Which of the following characteristics of electrons determines the current in a conductor? (a) drift velocity alone (b) thermal velocity alone (c) both drift velocity and thermal velocity (d) neither drift nor thermal velocity Solution: The correct answer is (a) drift velocity alone...
Read More →Which of the following statements are true and which are false?
Question: Which of the following statements are true and which are false? In each case give a valid reason for your answer. (i) p: $\sqrt{11}$ is an irrational number (ii) q: Circle is a particular case of an ellipse. (iii) r: Each radius of a circle is a chord of the circle (iv) S: The center of a circle bisects each chord of the circle (v) $\mathrm{t}$ : If $\mathrm{a}$ and $\mathrm{b}$ are integers such that $\mathrm{a}\mathrm{b}$, then $-\mathrm{a}-\mathrm{b}$. (vi) $y$ : The quadratic equat...
Read More →A metal rod of length 10 cm and a rectangular cross-section
Question: A metal rod of length 10 cm and a rectangular cross-section of 1 cm 1/2 cm is connected to battery across opposite faces. The resistance will be (a) maximum when the battery is connected across 1 cm 1/2 cm faces (b) maximum when the battery is connected across 10 cm 1 cm faces (c) maximum when the battery is connected across 10 cm 1/2 cm faces (d) same irrespective of the three faces Solution: The correct answer is (a) maximum when the battery is connected across 1 cm 1/2 cm faces...
Read More →Two cells of emf’s approximately 5V and 10V are to be accurately
Question: Two cells of emfs approximately 5V and 10V are to be accurately compared using a potentiometer of length 400 cm. (a) the battery that runs the potentiometer should have voltage of 8V (b) the battery of potentiometer can have a voltage of 15V and R adjusted so that the potential drop across the wire slightly exceeds 10V (c) the first portion of 50 cm of wire itself should have a potential drop of 10V (d) potentiometer is usually used for comparing resistances and not voltages Solution: ...
Read More →A resistance R is to be measured using a meter bridge.
Question: A resistance R is to be measured using a meter bridge. Student chooses the standard resistance S to be 100Ω. He finds the null point at l1 = 2.9 cm . He is told to attempt to improve the accuracy. Which of the following is a useful way? (a) he should measure l1 more accurately (b) he should change S to 1000 Ω and repeat the experiment (c) he should change S to 3 Ω and repeat the experiment (d) he should give up hope of a more accurate measurement with a meter bridge Solution: The corre...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{1}{e^{x}+e^{-x}} d x$ Solution: let I $=\int \frac{1}{\mathrm{e}^{\mathrm{x}}+\mathrm{e}^{-\mathrm{x}}} \mathrm{dx}$ $=\int \frac{1}{e^{x}+\frac{1}{e^{x}}} d x$ $=\int \frac{e^{x}}{\left(e^{x}\right)^{2}+1} d x$ $\Rightarrow e^{x} d x=d t$ $I=\int \frac{1}{(t)^{2}+1} d t$ $I=\tan ^{-1} t+c$ $\left[\right.$ since, $\left.\int \frac{1}{1+(\mathrm{x})^{2}} \mathrm{dx}=\tan ^{-1} \mathrm{x}+\mathrm{c}\right]$ $I=\tan ^{-1}\left(e^{x}\right)+c[$...
Read More →Two batteries of emf ε1 and ε2 and internal
Question: Two batteries of emf 1 and 2 and internal resistances r1 and r2 respectively are connected in parallel as shown in the figure. (a) the equivalent emf eq of the two cells is between 1 and 2 that is 1 eq 2 (b) the equivalent emf eq is smaller than 1 (c) the eq is given by eq = 1 + 2 always (d) eq is independent of internal resistances r1 and r2 Solution: The correct answer is (a) the equivalent emf eq of the two cells is between 1 and 2 that is 1 eq 2...
Read More →Consider a current-carrying wire in the shape of a circle.
Question: Consider a current-carrying wire in the shape of a circle. Note that as the current progresses along the wire, the direction of j changes in an exact manner, while the current I remain unaffected. The agent that is essentially responsible for is (a) source of emf (b) electric field produced by charges accumulated on the surface of wire (c) the charges just behind a given segment of wire which push them just the right way by repulsion (d) the charges ahead Solution: The correct answer i...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{e^{3 x}}{4 e^{6 x}-9} d x$ Solution: let $I=\int \frac{e^{2 x}}{4 e^{6 x_{-9}}} d x$ Let $\mathrm{e}^{3 \mathrm{x}}=\mathrm{t} \ldots \ldots(\mathrm{i})$ $\Rightarrow 3 e^{3 x} d x=d t$ $I=\frac{1}{3} \int \frac{1}{4 t^{2}-9} d t$ $=\frac{1}{12} \int \frac{1}{t^{2}-\frac{9}{4}} d t$ $I=\frac{1}{12} \int \frac{1}{t^{2}-\left(\frac{3}{2}\right)^{2}} d t$ $I=\frac{1}{36} \log \left|\frac{t-\frac{3}{2}}{t+\frac{3}{2}}\right|+c$ [since, $\left.\...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{e^{x}}{e^{2 x}+5 e^{x}+6} d x$ Solution: let $I=\int \frac{e^{x}}{e^{2 x}+5 e^{x}+6} d x$ Let $\mathrm{e}^{\mathrm{x}}=\mathrm{t} \ldots \ldots$ (i) $\Rightarrow \mathrm{e}^{\mathrm{x}} \mathrm{dx}=\mathrm{dt}$ $=\int \frac{1}{t^{2}+5 t+6} d t$ $=\int \frac{1}{t^{2}+2 t \times \frac{5}{2}+\left(\frac{5}{2}\right)^{2}-\left(\frac{5}{2}\right)^{2}+6} d t$ $=\int \frac{1}{\left(t+\frac{5}{2}\right)^{2}-\frac{1}{4}} d t$ Let $\mathrm{t}+\frac{5...
Read More →Which of the following sentences are statements?
Question: Which of the following sentences are statements? In case of a statement, mention whether it is true or false. (i) Paris is in France. (ii) Each prime number has exactly two factors. (iii) The equation $x^{2}+5|x|+6=0$ has no real roots. (iv) $(2+\sqrt{3})$ is a complex number. (v) Is 6 a positive integer? (vi) The product of $-3$ and $-2$ is $-6$. (vii) The angles opposite the equal sides of an isosceles triangle are equal. (viii) Oh! It is too hot. (ix) Monika is a beautiful girl. (x)...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\cos x}{\sin ^{2} x+4 \sin x+5} d x$ Solution: Let $I=\int \frac{\cos x}{\sin ^{2} x+4 \sin x+5} d x$ Let $\sin x=t \ldots . .(i)$ $\Rightarrow \cos x d x=d t$ $S O, I=\int \frac{d t}{t^{2}+4 t+5}$ $=\int \frac{d t}{t^{2}+(2 t)(2)+2^{2}-2^{2}+5}$ $\int \frac{d t}{(t+2)^{2}+1}$ Again, let $t+2=u \ldots$ (ii) $\Rightarrow \mathrm{dt}=\mathrm{du}$ $I=\int \frac{d u}{u^{2}+1}$ $=\tan ^{-1} u+c$ [since, $\left.\int \frac{1}{1+(\mathrm{x})^{2}} \...
Read More →Two charges –q each are separated by distance 2d.
Question: Two charges q each are separated by distance 2d. A third charge +q is kept at midpoint O. Find potential energy of +q as a function of small distance x from O due to q charges. Sketch PE versus x and convince yourself that the charge at O is in an unstable equilibrium. Solution: In the above figure, +q is the charge that got displaced from O towards (-d,0). This is written as $U=q\left(V_{1}+V_{2}\right)=q \frac{1}{4 \pi \epsilon_{0}} \frac{-q}{(d-x)}+\frac{-q}{d+x}$ $U=\frac{1}{2 \pi ...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{e^{x}}{1+e^{2 x}} d x$ Solution: : let $\mathrm{I}=\int \frac{\mathrm{e}^{\mathrm{x}}}{1+\mathrm{e}^{2 \mathrm{x}}} \mathrm{dx}$ Let $e^{x}=t \ldots \ldots$ (i) $\Rightarrow e^{x} d x=d t$ SO, $I=\int \frac{d t}{(1)^{2}+t^{2}}$ $I=\tan ^{-1} t+c$ [since, $\left.\int \frac{1}{1+(\mathrm{x})^{2}} \mathrm{~d} \mathrm{x}=\tan ^{-1} \mathrm{x}+\mathrm{c}\right]$ $I=\tan ^{-1}\left(e^{x}\right)+c$ [using(i)]...
Read More →Two charges q1 and q2 are placed at (0, 0, d)
Question: Two charges q1and q2are placed at (0, 0, d) and (0, 0, -d) respectively. Find the locus of points where the potential a zero. Solution: We know that the potential at point P is V = Vi Where Vi= qi/40, riis the magnitude of the position vector P V = 1/4 0 qi/rpi When (x,y,z) plane is considered, the two charges lie on the z-axis and is separated by 2d. The potential is given as $\frac{q_{1}}{\sqrt{x^{2}+y^{2}+(z-d)^{2}}}+\frac{q_{2}}{\sqrt{x^{2}+y^{2}+(z+d)^{2}}}=0$ Squaring the equatio...
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