A charged particle would continue to move
Question: A charged particle would continue to move with a constant velocity in a region wherein, (a) E = 0, B 0 (b) E 0, B 0 (c) E 0, B = 0 (d) E = 0, B = 0 Solution: (a) E = 0, B 0 (b) E 0, B 0 (d) E = 0, B = 0...
Read More →write each of the following using ‘if and only if’ :
Question: write each of the following using if and only if : (i) In order to get A grade, it is necessary and sufficient that you do all the homework regularly. (ii) If you watch television, then your mind is free, and if your mind is free, then you watch television. Solution: (i) You get an A grade if and only if you do all your homework regularly. (ii) You watch television if and only if your mind is free....
Read More →A cubical region of space is filled with some uniform electric
Question: A cubical region of space is filled with some uniform electric and magnetic fields. An electron enters the cube across one of its faces with velocity v and a positron enters via opposite face with velocity v. At this instant, (a) the electric forces on both the particles cause identical acceleration (b) the magnetic forces on both the particles cause equal accelerations (c) both particles gain or lose energy at the same rate (d) the motion of the centre of mass (CM) is determined by B ...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{1}{\sqrt{(x-\alpha)(\beta-x)}} d x,(\beta\alpha)$ Solution: let $\mathrm{I}=\int \frac{1}{\sqrt{(\mathrm{x}-\mathrm{\alpha})(\beta-\mathrm{x})}} \mathrm{dx},(\operatorname{as} \beta\alpha)$ $=\int \frac{1}{\sqrt{-x^{2}-x(\alpha+\beta)-\alpha \beta}} d x$ $=\int \frac{1}{\sqrt{-\left[\mathrm{x}^{2}-2 \mathrm{x}\left(\frac{\alpha+\beta}{2}\right)+\left(\frac{\alpha+\beta}{2}\right)^{2}-\left(\frac{\alpha+\beta}{2}\right)^{2}+\alpha \beta\righ...
Read More →Given below are some pairs of statements. Combine each pair using if and only if :
Question: Given below are some pairs of statements. Combine each pair using if and only if : (i) $p$ : If a quadrilateral is equiangular, then it is a rectangle. $q$ : If a quadrilateral is a rectangle, then it is equiangular. (ii) $p$ : If the sum of the digits of a number is divisible by 3 , then the number is divisible by 3 . $q$ : If a number is divisible by 3 , then the sum of its digits is divisible by $3 .$ (iii) $\mathbf{p}: \mathbf{A}$ quadrilateral is a parallelogram if its diagonals b...
Read More →Two identical current-carrying coaxial loops,
Question: Two identical current-carrying coaxial loops, carry current I in an opposite sense. A simple amperian loop passes through both of them once. Calling the loop as C, (a) $\oint_{c} B . d l=\mp 2 \mu_{0} I$ (b) the value of $\oint_{c} B \cdot d l$ is independent of sense of $c$ (c) there may be a point on C where B and dl are perpendicular (d) B vanishes everywhere on C Solution: (b) the value of $\oint_{c} B \cdot d l$ is independent of sense of $C$ (c) there may be a point on C where B ...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{1}{\sqrt{3 x^{2}+5 x+7}} d x$ Solution: let $I=\int \frac{1}{\sqrt{3 x^{2}+5 x+7}} d x$ $=\frac{1}{\sqrt{3}} \int \frac{1}{\sqrt{x^{2}+\frac{5}{3} x+\frac{7}{3}}} d x$ $=\frac{1}{\sqrt{3}} \int \frac{1}{\sqrt{x^{2}+2 x\left(\frac{5}{6}\right)+\left(\frac{5}{6}\right)^{2}-\left(\frac{5}{6}\right)^{2}+\frac{7}{3}}} d x$ $=\frac{1}{\sqrt{3}} \int \frac{1}{\sqrt{\left(x+\frac{5}{6}\right)^{2}-\frac{59}{36}}} d x$ $\operatorname{let}\left(x+\fra...
Read More →Consider a wire carrying a steady current,
Question: Consider a wire carrying a steady current, I placed un a uniform magnetic field B perpendicular to its length. Consider the charges inside the wire. It is known that magnetic forces do not work. This implies that (a) motion of charges inside the conductor is unaffected by B since they do not absorb energy (b) some charges inside the wire move to the surface as a result of B (c) if the wire moves under the influence of B, no work is done by the force (d) if the wire moves under the infl...
Read More →The gyro-magnetic ratio of an electron in an H-atom,
Question: The gyro-magnetic ratio of an electron in an H-atom, according to Bohr model is (a) independent of which orbit it is in (b) negative (c) positive (d) increases with the quantum number n Solution: (a) independent of which orbit it is in (b) negative...
Read More →Write the converse and contrapositive of each of the following :
Question: Write the converse and contrapositive of each of the following : (i) If $x$ is a prime number, then $x$ is odd. (ii) If a positive integer $\mathrm{n}$ is divisible by 9 , then the sum of its digits is divisible by $9 .$ (iii) If the two lines are parallel, then they do not intersect in the same plane. (iv) If the diagonal of a quadrilateral bisect each other, then it is a parallelogram. (v) If $A$ and $B$ are subsets of $X$ such that $A \subseteq B$, then $(X-B) \subseteq(X-A)$ (vi) I...
Read More →A circular current loop of magnetic moment M
Question: A circular current loop of magnetic moment M is in an arbitrary orientation in an external magnetic field B. The work done to rotate the loop by 30o about an axis perpendicular to its plane is (a) MB (b) 3 MB/2 (c) MB/2 (d) zero Solution: (d) zero...
Read More →In a cyclotron, a charged particle
Question: In a cyclotron, a charged particle (a) undergoes acceleration all the time (b) speeds up between the dees because of the magnetic field (c) speeds up in a dee (d) slows down within a dee and speeds up between dees Solution: (a) undergoes acceleration all the time...
Read More →An electron is projected with uniform velocity along
Question: An electron is projected with uniform velocity along the axis of a current-carrying long solenoid. Which of the following is true? (a) the electron will be accelerated along the axis (b) the electron path will be circular about the axis (c) the electron will experience a force at 45oto the axis and hence execute a helical path (d) the electron will continue to move with uniform velocity along the axis of the solenoid Solution: (d) the electron will continue to move with uniform velocit...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{1}{\sqrt{5-4 x-2 x^{2}}} d x$ Solution: Let $I=\int \frac{1}{\sqrt{5-4 x-2 x^{2}}} d x$ $=\int \frac{1}{\sqrt{-2\left[x^{2}+2 x-\frac{5}{2}\right]}} d x$ $=\frac{1}{\sqrt{2}} \int \frac{1}{\sqrt{-\left[x^{2}+2 x+(1)^{2}-(1)^{2}-\frac{5}{2}\right]}} d x$ $=\frac{1}{\sqrt{2}} \int \frac{1}{\sqrt{-\left[(x+1)^{2}-\frac{7}{2}\right]}} d x$ $=\frac{1}{\sqrt{2}} \int \frac{1}{\sqrt{\frac{7}{2}-(x+1)^{2}}} d x$ Let $(x+1)=t$ Differentiating both s...
Read More →A current circular loop of radius R is placed in
Question: A current circular loop of radius R is placed in the x-y plane with centre at the origin. Half of the lop with x 0 is now bent so that it now lies in the y-z plane. (a) the magnitude of magnetic moment now diminishes (b) the magnetic moment does not change (c) the magnitude of B at (0,0,z),z R increases (d) the magnitude of B at (0,0,z),z R is unchanged Solution: (a) the magnitude of magnetic moment now diminishes...
Read More →Biot-Savart law indicates that the moving electrons
Question: Biot-Savart law indicates that the moving electrons produce a magnetic field B such that (a) B ┴ v (b) B ‖ v (c) it obeys inverse cube law (d) it is along the line joining the electrons and point of observation Solution: (a) B ┴ v...
Read More →Two charged particles traverse identical helical
Question: Two charged particles traverse identical helical paths in an opposite sense in a uniform magnetic field $B=B_{0} \hat{k}$ (a) they have equal z-components of momenta (b) they must have equal charges (c) they necessarily represent a particle-antiparticle pair (d) the charge to mass ratio satisfy: (e/m)1 + (e/m)2 = 0 Solution: (d) the charge to mass ratio satisfy: (e/m)1 + (e/m)2 = 0...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{1}{\sqrt{8+3 x-x^{2}}} d x$ Solution: $8+3 x-x 2$ can be written as $8-\left(x^{2}-3 x+\frac{9}{4}-\frac{9}{4}\right)$ Therefore $8-\left(x^{2}-3 x+\frac{9}{4}-\frac{9}{4}\right)$ $=\frac{41}{4}-\left(x-\frac{3}{2}\right)^{2}$ $\int \frac{1}{\sqrt{8+3 x-x^{2}}} d x=\int \frac{1}{\sqrt{\frac{41}{4}-\left(x-\frac{3}{2}\right)^{2}}} d x$ Let $x-3 / 2=t$ $d x=d t$ $\int \frac{1}{\sqrt{\frac{41}{4}-\left(x-\frac{3}{2}\right)^{2}}} d x=\int \frac...
Read More →Write each of the following statements in the form ‘if …. then’ :
Question: Write each of the following statements in the form if . then : (i) A rhombus is a square only if each of its angles measures $90^{\circ}$. (ii) When a number is a multiple of 9 , it is necessarily a multiple of $3 .$ (iii) You get a job implies that your credentials are good. (iv) Atmospheric humidity increase only if it rains (v) If a number is not a multiple of 3 , then it is not a multiple of 6 . Solution: (i) If each of the angles of a rhombus measures 90, then the rhombus is a squ...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{1}{\sqrt{2 x-x^{2}}} d x$ Solution: let $I=\int \frac{1}{\sqrt{2 x-x^{2}}} d x$ $=\int \frac{1}{\sqrt{-\left(x^{2}-2 x\right)}} d x$ $=\int \frac{1}{\sqrt{-\left[x^{2}-2 x(1)+1^{2}-1^{2}\right]}} d x$ $=\int \frac{1}{\sqrt{-\left[(x-1)^{2}-1\right]}} d x$ $=\int \frac{1}{\sqrt{1-(x-1)^{2}}} d x$ let $(x-1)=t$ $d x=d t$ SO, I $=\int \frac{1}{\sqrt{1-t^{2}}} d t$ $=\sin ^{-1} \mathrm{t}+\mathrm{c}\left[\right.$ since $\left.\int \frac{1}{\sqr...
Read More →Rewrite the following statement in five different ways conveying the same meaning.
Question: Rewrite the following statement in five different ways conveying the same meaning. If a given number is a multiple of 6, then it is a multiple of 3. Solution: I. A given number is a multiple of 6; it implies that it is a multiple of 3. II. A given number is a multiple of 6 only if it is a multiple of 3 . III. For a given number to be a multiple of 6 , it is necessary that it is a multiple of $3 .$ IV. For a given number to be a multiple of 3 , it is sufficient that the number is multip...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{1}{\cos x+\cos e c x} d x$ Solution: let $I=\frac{1}{\cos x+\operatorname{cosec} x} d x$ Multiply and divide by $\sin x$ $I=\frac{\frac{1}{\sin x}}{\frac{\cos x}{\sin x}+\frac{\operatorname{cosecx}}{\sin x}} d x$ $=\frac{\operatorname{cosec} x}{\cot x+\operatorname{cosec}^{2} x} d x$ $=\frac{\operatorname{cosec} x}{\cot x+1+\cot ^{2} x} d x$ $=\frac{\operatorname{cosec} x}{\cot ^{2} x+\cot x+1} d x$ Let $\cot x=t$ $-\operatorname{cosec} x d...
Read More →(a) Consider the circuit in the figure. How much energy is absorbed
Question: (a) Consider the circuit in the figure. How much energy is absorbed by electrons from the initial state of no current to the state of drift velocity? (b) Electrons give up energy at the rate of RI2per second to the thermal energy. What time scale would one associate with energy in problem a) n = no of electron/volume = 1029/m3, length of circuit = 10 cm, cross-section = A = 1mm2 Solution: (a) Current is given as I = V/R from the Ohms law Therefore, I = 1A But, I = neAvd vd = I/neA When...
Read More →Let A = [2, 3, 5, 7]. Examine whether the statements given below are true or false.
Question: Let A = [2, 3, 5, 7]. Examine whether the statements given below are true or false. (i) $\exists x \in A$ such that $x+39$. (ii) $\exists x \in A$ such that $x$ is even. (iii) $\exists x \in A$ such that $x+2=6$. (iv) $\forall x \in A, x$ is prime. (v) $\forall x \in A, x+210$. (vi) $\forall x \in A, x+4 \geq 11$ Solution: $A=[2,3,5,7]$ (given in the question). The given statement is: $\exists x \in A$ such that $x+39$. So, we need to see whether there exists x which belongs to A, such...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{e^{x}}{\left(1+e^{x}\right)\left(2+e^{x}\right)} d x$ Solution: To evaluate the following integral following steps: Let $e^{x}=t \ldots .(i)$ $\Rightarrow \mathrm{e}^{\mathrm{x}} \mathrm{d} \mathrm{x}=\mathrm{dt}$ Now $\int \frac{e^{x}}{\left(1+e^{x}\right)\left(2+e^{x}\right)} d x=\int \frac{1}{(1+t)(2+t)} d t$ $=\int \frac{1}{(1+t)} d t-\int \frac{1}{(2+t)} d t$ $=\log |(1+t)|-\log |(2+t)|+c$ $=\log \left|\frac{1+t}{2+t}\right|+c\left[\lo...
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