In a single throw of two dice, find
Question: In a single throw of two dice, find P (a total of 10) Solution: We know that, Probability of occurrence of an event $=\frac{\text { Total no.of Desired outcomes }}{\text { Total no.of outcomes }}$ Total outcomes are $(1,1),(1,2),(1,3),(1,4),(1,5),(1,6)$, $(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)$, $(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)$, $(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)$, $(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)$, $(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)$ Desired outcomes are $(4,6),(5,5),(6,4)$ ...
Read More →Four identical monochromatic sources A, B, C, D
Question: Four identical monochromatic sources A, B, C, D as shown in the figure produce waves of the same wavelength and are coherent. Two receiver R1 and R2 are at great but equal distances from B. (i) Which of the two receivers picks up the larger signal? (ii) Which of the two receivers picks up the larger signal when B is turned off? (iii) Which of the two receivers picks up the larger signal when D is turned off? (iv) Which of the two receivers can distinguish which of the sources B or D ha...
Read More →In a single throw of two dice, find
Question: In a single throw of two dice, find P (a number greater than 3 on each die) Solution: We know that, Probability of occurrence of an event $=\frac{\text { Total no.of Desired outcomes }}{\text { Total no.of outcomes }}$ Total outcomes are $(1,1),(1,2),(1,3),(1,4),(1,5),(1,6)$, $(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)$, $(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)$, $(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)$, $(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)$, $(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)$ Desired outcomes ar...
Read More →AC = CO = D, S1 C = S2 C = d << D A small transparent slab containing material
Question: AC = CO = D, S1 C = S2 C = d D A small transparent slab containing material of =1.5 is placed along AS2. What will be the distance from O of the principal maxima and of the first minima on either side of the principal maxima obtained in the absence of the glass slab? Solution: ∆x = 2d sin +(-1)L sin 0 = -1/16 From central maxima, OP = -D/16 $\sin \theta_{1}=\frac{\pm \lambda / 2-d / 8}{2 d}$ On the positive side, $\sin \theta_{1}^{+}=3 / 16$ On the negative side, $\sin \theta_{1}^{-}=-...
Read More →In a single throw of two dice, find
Question: In a single throw of two dice, find P (an odd number on the first die and a 6 on the second) Solution: We know that, Probability of occurrence of an event $=\frac{\text { Total no.of Desired outcomes }}{\text { Total no.of outcomes }}$ Total outcomes are $(1,1),(1,2),(1,3),(1,4),(1,5),(1,6)$ $(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)$, $(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)$, $(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)$, $(5,1),(5,2),(5,3),(5,4),(5,5),(5,6)$, $(6,1),(6,2),(6,3),(6,4),(6,5),(6,6)$ Des...
Read More →The figure shown a two-slit arrangement
Question: The figure shown a two-slit arrangement with a source which emits unpolarised light. P is a polariser with axis whose direction is not given. If I0is the intensity of the principal maxima when no polarizer is present, calculate in the present case, the intensity of the principal maxima as well as of the first minima. Solution: Amplitude of the wave in perpendicular polarisation $A_{\perp}=A_{\perp}^{0}(\sin (k x-\omega t)+\sin (k x-\omega t+\phi))$ Amplitude of the wave in parallel pol...
Read More →In a single throw of two dice, find the probability of
Question: In a single throw of two dice, find the probability of (i) getting a sum less than 6 (ii) getting a doublet of odd numbers (iii) getting the sum as a prime number Solution: (i) We know that, Probability of occurrence of an event $=\frac{\text { Total no. of Desired outcomes }}{\text { Total no.of outcomes }}$ Outcomes are $(1,1),(1,2),(1,3),(1,4),(1,5),(1,6)$, $(2,1),(2,2),(2,3),(2,4),(2,5),(2,6)$, $(3,1),(3,2),(3,3),(3,4),(3,5),(3,6)$, $(4,1),(4,2),(4,3),(4,4),(4,5),(4,6)$, $(5,1),(5,...
Read More →Consider a two-slit interference arrangements
Question: Consider a two-slit interference arrangements such that the distance of the screen from the slits is half the distance between the slits. Obtain the value of D in terms of such that the first minima on the screen falls at a distance D from the centre O. Solution: The minima will occur when ∆x = S2P S1P = (2n-1/2) S1P = D2+ (D x)2 S2P = D2+ (D + x)2 T2P = D + x T1P = D x [D2+ (D+x)2]-1/2 [D2+(D-x)2]1/2= /2 D = 0.404...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\left(1-x^{2}\right)}{x(1-2 x)} d x$ Solution: Given $I=\int \frac{1-x^{2}}{(1-2 x) x} d x$ Rewriting, we get $\int \frac{x^{2}-1}{x(2 x-1)} d x$ Expressing the integral $\int \frac{\mathrm{P}(\mathrm{x})}{\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}} \mathrm{dx}=\int \mathrm{Q}(\mathrm{x}) \mathrm{dx}+\int \frac{\mathrm{R}(\mathrm{x})}{\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}} \mathrm{dx}$ $\Rightarrow \int \frac{x^{2}-1}{x(2 x-1)} d x=\int\left(\...
Read More →For the same objective,
Question: For the same objective, find the ratio of the least separation between two points to be distinguished by a microscope for light of 5000 Aoand electrons accelerated through 100V used as the illuminating substance. Solution: 5000 Ao = 5000 10-10m 1/d = 2 sin /1.22 dmin = 1.22 / 2 sin d = 1.22/10 10-10m When the 100V light is used, dmin = 1.22 d/ 2 sin dmin = 1.22 1.22 10-10/2 sin The required ratio = dmin/dmin = 1.22/5000 = 0.244 10-3...
Read More →Can reflection result in plane-polarized light
Question: Can reflection result in plane-polarized light if the light is incident on the interface from the side with higher refractive index? Solution: When the angle of incidence is equal to the Brewsters angle such that the transmitted light is polarized and reflected light is plane-polarized. Following is the equation: $\operatorname{tani}_{B}=\mu_{2}^{1}=\frac{\mu_{2}}{\mu_{1}}$ Such that 2 1 When the light travels in a medium, the critical angle is given as $\operatorname{sini}_{C}=\frac{\...
Read More →A die is thrown. Find the probability of
Question: A die is thrown. Find the probability of getting a number between 3 and 6 Solution: We know that, Probability of occurrence of an event $=\frac{\text { Total no.of Desired outcomes }}{\text { Total no.of outcomes }}$ As 4,5 are two numbers between 3 and, so the desired outcomes are 3,6, and total outcomes are $1,2,3,4,5,6$ Therefore, total no.of outcomes are 6, and total no. of desired outcomes are 2 Probability of getting a number between 3 and 6 $=\frac{2}{6}$ $=\frac{1}{3}$ Conclusi...
Read More →A polaroid
Question: A polaroid (I) is placed in front of a monochromatic source. Another polaroid (II) is placed in front of this polaroid (I) and rotated till no light passes. A third polaroid (III) is now placed in between (I) and (II). In this case, will light emerge from (II)? Explain. Solution: A polaroid (I) is placed in front of a monochromatic source and polaroid (II) is placed in front of polaroid (I). When the light passes through the polaroid (II), the light is unaffected. Polaroid (II) is rota...
Read More →A die is thrown. Find the probability of
Question: A die is thrown. Find the probability of getting a multiple of 3 Solution: We know that, Probability of occurrence of an event $=\frac{\text { Total no. of Desired outcomes }}{\text { Total no. of outcomes }}$ As 3,6 are multiples up to 6, so the desired outcomes are 3,6, and total outcomes are $1,2,3,4,5,6$ Therefore, total no. of outcomes are 6, and total no. of desired outcomes are 2 Probability of getting multiple of 3 $=\frac{2}{6}$ $=\frac{1}{3}$ Conclusion: Probability of gettin...
Read More →The human eye has an approximate angular
Question: The human eye has an approximate angular resolution of = 5.8 104rad and a typical photo printer prints a minimum of 300 dpi (dots per inch, 1 inch = 2.54 cm). At what minimal distance z should a printed page be held so that one does not see the individual dots. Solution: Angular separation, = 5.8 104rad The average distance between the two dots = 2.54/300 = 0.85 10-2cm At distance z cm, the angle subtended = 0.85 10-2/z Resolution angle for human = 0.85 10-2/z Maximum distance between ...
Read More →A die is thrown. Find the probability of
Question: A die is thrown. Find the probability of getting a prime number Solution: We know that, Probability of occurrence of an event $=\frac{\text { Total no.of Desired outcomes }}{\text { Total no.of outcomes }}$ As $2,3,5$ are prime numbers up to 6 , so the desired outcomes are $2,3,5$, and total outcomes are $1,2,3,4,5,6$ Therefore, total no. of outcomes are 6 , and total no. of desired outcomes are 3 Probability of getting a prime number $=\frac{3}{6}$ $=\frac{1}{2}$ Conclusion: Probabili...
Read More →Why is the diffraction of sound waves
Question: Why is the diffraction of sound waves more evident in daily experience than that of lightwave? Solution: The frequency of sound waves varies from 20 Hz to 20,000 Hz with a wavelength of 15 m to 15 mm respectively. The diffraction of sound takes place when the size of the slit is comparable to the wavelength of the sound. The wavelength of visible light is 0.4 to 0.7 micron. The diffraction of the light doesnt take place when the size of the slits is different from the wavelength of the...
Read More →What is the shape of the wavefront
Question: What is the shape of the wavefront on earth for sunlight? Solution: We know that the distance between the sun and the earth is very large. We also know that the sun is spherical and can be considered as a point source of light which is present at a very large distance. When the point source is kept at a distance, the wavefront formed will be almost plane....
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{x^{2}+x-1}{x^{2}+x-6} d x$ Solution: Consider $I=\int \frac{x^{2}+x-1}{x^{2}+x-6} d x$ Expressing the integral $\int \frac{\mathrm{P}(\mathrm{x})}{\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}} \mathrm{dx}=\int \mathrm{Q}(\mathrm{x}) \mathrm{dx}+\int \frac{\mathrm{R}(\mathrm{x})}{\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}} \mathrm{dx}$ Let $x^{2}+x-1=x^{2}+x-6+5$ $\Rightarrow \int \frac{x^{2}+x-1}{x^{2}+x-6} d x=\int\left(\frac{x^{2}+x-6}{x^{2}+x-6}+\...
Read More →Consider a point at the focal point of a convergent lens.
Question: Consider a point at the focal point of a convergent lens. Another convergent lens of short focal length is placed on the other side. What is the nature of the wavefronts emerging from the final image? Solution: The orientation of wavefront and the ray are perpendicular to each other. In the above figure, L1 is the source of parallel rays, forming I2 at the focal length of the lens. The image formed acts as L2 through which the light rays get converged into L2 and the final image I is f...
Read More →A die is thrown. Find the probability of
Question: A die is thrown. Find the probability of getting an odd number Solution: We know that, Probability of occurrence of an event $=\frac{\text { Total no.of Desired outcomes }}{\text { Total no.of outcomes }}$ As $1,3,5$ are odd numbers up to 6 , so the desired outcomes are $1,3,5$, and total outcomes are $1,2,3,4,5,6$ Therefore, total no. of outcomes are 6 , and total no. of desired outcomes are 3 Probability of getting an odd number $=\frac{3}{6}$ $=\frac{1}{2}$ Conclusion: Probability o...
Read More →Is Huygen’s principle valid for longitudinal
Question: Is Huygens principle valid for longitudinal sound waves? Solution: Yes, Huygens principle is valid for longitudinal sound waves....
Read More →For light diverging from a point source
Question: For light diverging from a point source (a) the wavefront is spherical (b) the intensity decreases in proportion to the distance squared (c) the wavefront is parabolic (d) the intensity at the wavefront does not depend on the distance Solution: The correct answer is (a) the wavefront is spherical (b) the intensity decreases in proportion to the distance squared...
Read More →Consider the diffraction pattern for a small pinhole.
Question: Consider the diffraction pattern for a small pinhole. As the size of the hole is increased (a) the size decreases (b) the intensity increases (c) the size increases (d) the intensity decreases Solution: The correct answer is (a) the size decreases (b) the intensity increases...
Read More →A die is thrown. Find the probability of
Question: A die is thrown. Find the probability of getting a 2 or a 3 Solution: We know that, Probability of occurrence of an event $=\frac{\text { Total no.of Desired outcomes }}{\text { Total no.of outcomes }}$ Total outcomes are $1,2,3,4,5,6$, and the desired outcomes are 2,3 Therefore, total no.of outcomes are 6 , and total no.of desired outcomes are 2 Probability of getting a 2 or 3 $=\frac{2}{6}$ $=\frac{1}{3}$ Conclusion: Probability of getting 2 or 3 when a die is thrown is $\frac{1}{3}$...
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