Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{1}{5-4 \sin x} d x$ Solution: Given $I=\int \frac{1}{5-4 \sin x} d x$ We know that $\sin x=\frac{2 \tan \frac{x}{2}}{1+\tan \frac{2}{2}}$ $\Rightarrow \int \frac{1}{5-4 \sin x} d x=\int \frac{1}{5-4\left(\frac{2 \tan \frac{x}{2}}{1+\tan \frac{2 x}{2}}\right)} d x$ $=\int \frac{1+\tan ^{2} \frac{x}{2}}{5\left(1+\tan ^{2} \frac{x}{2}\right)-4\left(2 \tan \frac{x}{2}\right)} d x$ Replacing $1+\tan ^{2} x / 2$ in numerator by $\sec ^{2} x / 2$,...
Read More →Iodine molecules are held in
Question: Iodine molecules are held in the crystals lattice by ____________. (i) london forces (ii) dipole-dipole interactions (iii) covalent bonds (iv) coulombic forces Solution: Option (i) is the answer....
Read More →The sharp melting point of crystalline solids
Question: The sharp melting point of crystalline solids is due to ___________. (i) a regular arrangement of constituent particles observed over a short distance in the crystal lattice. (ii) a regular arrangement of constituent particles observed over a long distance in the crystal lattice. (iii) the same arrangement of constituent particles in different directions. (iv) a different arrangement of constituent particles in different directions. Solution: Option (ii)a regular arrangement of constit...
Read More →Which of the following statement is
Question: Which of the following statement is not true about amorphous solids? (i) On heating, they may become crystalline at a certain temperature. (ii) They may become crystalline on keeping for a long time. (iii) Amorphous solids can be moulded by heating. (iv) They are anisotropic. Solution: Option (iv)They are anisotropic is the answer....
Read More →Which of the following is true about
Question: Which of the following is true about the value of the refractive index of quartz glass? (i) Same in all directions (ii) Different in different directions (iii) Cannot be measured (iv) Always zero Solution: Option (i)Same in all directions is the answer...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{1}{5+4 \cos x} d x$ Solution: Given $I=\int \frac{1}{5+4 \cos x} d x$ We know that $\cos x=\frac{1-\tan ^{2} \frac{x}{2}}{1+\tan ^{2} \frac{x}{2}}$ $\Rightarrow \int \frac{1}{5+4 \cos x} d x=\int \frac{1}{5+4\left(\frac{1-\tan ^{2} \frac{x}{2}}{1+\tan ^{2} \frac{x}{2}}\right)} d x$ $=\int \frac{1+\tan ^{2} \frac{x}{2}}{5\left(1+\tan ^{2} \frac{x}{2}\right)+4\left(1-\tan ^{2} \frac{x}{2}\right)} d x$ Replacing $1+\tan ^{2} x / 2$ in numerato...
Read More →A die is thrown twice.
Question: A die is thrown twice. What is the probability that at least one of the two throws comes up with the number 4? Solution: Given : A die is thrown twice To find : Probability that at least one of the two throws comes up with the number 4 The formula used : Probability = A die is numbered from 1 to 6 When a die is thrown twice, total number of outcomes $=6^{2}=36$ Favourable outcomes = $\{(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(1,4),(2,4),(3,4),(5,4),(6,4)\}$ Favourable number of outcomes = ...
Read More →Which of the following arrangements
Question: Which of the following arrangements shows the schematic alignment of magnetic moments of antiferromagnetic substances? Solution: Option (iv) is the answer....
Read More →Which of the following is an amorphous solid?
Question: Which of the following is an amorphous solid? (i) Graphite (C) (ii) Quartz glass (SiO2) (iii) Chrome alum (iv) Silicon carbide (SiC) Solution: Option (ii)Quartz glass (SiO2) is the answer....
Read More →Which of the following is not a characteristic
Question: Which of the following is not a characteristic of a crystalline solid? (i) Definite and characteristic heat of fusion. (ii) Isotropic nature. (iii) A regular periodically repeated pattern of arrangement of constituent particles in the entire crystal. (iv) A true solid Solution: Option (ii)Isotropic nature.is the answer....
Read More →A number is chosen from the numbers 1 to 100.
Question: A number is chosen from the numbers 1 to 100. Find the probability of its being divisible by 4 or 6. Solution: let A denote the event that the number is divisible by 4 and B denote the event that the number is divisible by 4. To find : Probability that the number is both divisible by 4 or $6=P(A$ or $B)$ The formula used : Probability = $P(A$ or $B)=P(A)+P(B)-P(A$ and $B)$ Numbers from 1 to 100 divisible by 4 are $4,8,12,16,20,24,28,32,36,40,44,48,52$, $56,60,64,68,72,76,80,84,88,92,96...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{1}{\cos 2 x+3 \sin ^{2} x} d x$ Solution: Given $I=\int \frac{1}{\cos 2 x+3 \sin ^{2} x} d x$ We know that $\cos 2 x=1-2 \sin ^{2} x$ $\Rightarrow \int \frac{1}{\cos 2 x+3 \sin ^{2} x} d x=\int \frac{1}{1-2 \sin ^{2} x+3 \sin ^{2} x} d x$ $=\int \frac{1}{1+\sin ^{2} x} d x$ Dividing numerator and denominator by $\cos ^{2} x$, $\Rightarrow \int \frac{1}{1+\sin ^{2} x} d x=\int \frac{\sec ^{2} x}{\sec ^{2} x+\tan ^{2} x} d x$ Replacing $\sec ...
Read More →Which of the following conditions favours
Question: Which of the following conditions favours the existence of a substance in the solid state? (i) High temperature (ii) Low temperature (iii) High thermal energy (iv) Weak cohesive forces Solution: Option (ii)Low temperature is the answer....
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{1}{\sin ^{2} x+\sin 2 x} d x$ Solution: Given $I=\int \frac{1}{\sin ^{2} x+\sin 2 x} d x$ We know that $\sin 2 x=2 \sin x \cos x$ $\Rightarrow I=\int \frac{1}{\sin ^{2} x+2 \sin x \cos x} d x$ Dividing numerator and denominator by $\cos ^{2} x$, $\Rightarrow \int \frac{1}{\sin ^{2} x+2 \sin x \cos x} d x=\int \frac{\sec ^{2} x}{\tan ^{2} x+2 \tan x} d x$ Putting $\tan x=t$ so that $\sec ^{2} x d x=d t$, $\Rightarrow \int \frac{\sec ^{2} x}{...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $w \int \frac{1}{\cos x(\sin x+2 \cos x)} d x$ Solution: Given I $=\int \frac{1}{\cos x(\sin x+2 \cos x)} d x$ $\Rightarrow I=\int \frac{1}{\cos x(\sin x+2 \cos x)} d x=\int \frac{1}{\cos x \sin x+2 \cos ^{2} x} d x$ Dividing the numerator and denominator by $\cos ^{2} x$, $\Rightarrow \int \frac{1}{\cos x \sin x+2 \cos ^{2} x} d x=\int \frac{\sec ^{2} x}{\tan x+2} d x$ Putting $\tan x+2=\mathrm{t}$ so that $\sec ^{2} x \mathrm{dx}=\mathrm{dt}$, $\Righ...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $w \int \frac{1}{\cos x(\sin x+2 \cos x)} d x$ Solution: Given I $=\int \frac{1}{\cos x(\sin x+2 \cos x)} d x$ $\Rightarrow I=\int \frac{1}{\cos x(\sin x+2 \cos x)} d x=\int \frac{1}{\cos x \sin x+2 \cos ^{2} x} d x$ Dividing the numerator and denominator by $\cos ^{2} x$, $\Rightarrow \int \frac{1}{\cos x \sin x+2 \cos ^{2} x} d x=\int \frac{\sec ^{2} x}{\tan x+2} d x$ Putting $\tan x+2=\mathrm{t}$ so that $\sec ^{2} x \mathrm{dx}=\mathrm{dt}$, $\Righ...
Read More →An audio signal is modulated by a carrier wave of 20MHz
Question: An audio signal is modulated by a carrier wave of 20MHz such that the bandwidth required for modulation is 3kHz. Could this wave be demodulated by a diode detector which has the values of R and C as (i) R = 1 kΩ, C = 0.01F (ii) R = 10 kΩ, C = 0.01F (iii) R = 10 kΩ, C = 0.1F Solution: Carrier wave frequency, fc 20 MHz = 20 106Hz Bandwidth = 2fm = 3 103Hz fm = 1.5 103Hz 1/fc = 0.5 10-7 1/fm = 0.7 10-3 (i) 1/fc RC 1/fm Therefore, it can be demodulated (ii) 1/fc RC 1/fm Therefore, it can b...
Read More →(i) Draw the plot of amplitude versus ‘ω’
Question: (i) Draw the plot of amplitude versus for an amplitude modulated wave whose carrier wave (c ) is carrying two modulating signals, 1 and 2 (2 1). [Hint: Follow derivation from Eq 15.6 of NCERT Textbook of XII] (ii) Is the plot symmetrical about c? Comment especially about plot in region c (iii) Extrapolate and predict the problems one can expect if more waves are to be modulated (iv) Suggest solutions to the above problem. In the process can one understand another advantage of modulatio...
Read More →An amplitude modulated wave is as shown in the figure.
Question: An amplitude modulated wave is as shown in the figure. Calculate (i) the percentage modulation (ii) peak carrier voltage and (iii) peak value of information voltage Solution: Maximum voltage, Vmax = 100/2 = 50 V Minimum voltage, Vmin = 20/2 = 10 V (i) Percentage modulation = 66.67% (ii) Peak carrier voltage, Vc = 30 V (iii) Peak value of information voltage = 20 V...
Read More →A 50 MHz sky wave takes 4.04 ms to reach
Question: A 50 MHz sky wave takes 4.04 ms to reach a receiver via re-transmission from a satellite 600 km above earths surface. Assuming re-transmission time by satellite negligible, find the distance between source and receiver. If communication between the two was to be done by Line of Sight (LOS) method, what should size and placement of receiving and transmitting antenna be? Solution: Velocity of waves = 3 108m/s Time to reach a receiver = 4.04 10-3s Height of satellite, h = 600 km Radius of...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\sin 2 x}{\sin ^{4} x+\cos ^{4} x} d x$ Solution: Given $I=\int \frac{\sin 2 x}{\sin ^{4} x+\cos ^{4} x} d x$ Dividing the numerator and denominator by $\cos ^{4} x$, $\Rightarrow \int \frac{\sin 2 x}{\sin ^{4} x+\cos ^{4} x} d x=\int \frac{2 \tan x \sec ^{2} x}{\tan ^{4} x+1} d x$ Putting $\tan ^{2} x=t$ so that $2 \tan x \sec ^{2} x d x=d t$ $\Rightarrow \int \frac{2 \tan x \sec ^{2} x}{\tan ^{4} x+1} d x=\int \frac{d t}{t^{2}+1}$ We know...
Read More →(i) The intensity of a light pulse travelling along a communication
Question: (i) The intensity of a light pulse travelling along a communication channel decreases exponentially with distance x according to the relation I = Ioex, where I o is the intensity at x = 0 and is the attenuation constant. Show that the intensity reduces by 75 per cent after a distance of (ln 4/) (ii) Attenuation of a signal can be expressed in decibel (dB) according to the relation dB =10 log10 (I/I0). What is the attenuation in dB/km for an optical fibre in which the intensity falls by...
Read More →A card is drawn at random from a well-shuffled deck of 52 cards.
Question: A card is drawn at random from a well-shuffled deck of 52 cards. Find the probability of its being a spade or a king. Solution: let A denote the event that the card drawn is spade and B denote the event that card drawn is king. In a pack of 52 cards, there are 13 spade cards and 4 king cards Given : $P(A)=\frac{13}{52}, P(B)=\frac{4}{52}$ To find : Probability that card drawn is either a queen or heart $=P(A$ or $B)$ The formula used : Probability = $P(A$ or $B)=P(A)+P(B)-P(A$ and $B)$...
Read More →On radiating (sending out) an AM modulated signal,
Question: On radiating (sending out) an AM modulated signal, the total radiated power is due to energy carried by c, c m c + m. Suggest ways to minimise the cost of radiation without compromising on information Solution: Total radiated power due to energy carried by c, (c m) and (c + m). Sideband frequencies are generally close to the carrier frequency. Only the sideband frequencies carry information in an amplitude modulated signal....
Read More →The maximum frequency for reflection of sky waves
Question: The maximum frequency for reflection of sky waves from a certain layer of the ionosphere is found to be f max = 9(Nmax) 1/2, where Nmax is the maximum electron density at that layer of the ionosphere. On a certain day, it is observed that signals of frequencies higher than 5MHz are not received by reflection from the F1 layer of the ionosphere while signals of frequencies higher than 8MHz are not received by reflection from the F2 layer of the ionosphere. Estimate the maximum electron ...
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