If 500 mL of a 5M solution is diluted
Question: If 500 mL of a 5M solution is diluted to 1500 mL, what will be the molarity of the solution obtained? (i) 1.5 M (ii) 1.66 M (iii) 0.017 M (iv) 1.59 M Solution: Option (ii) is the answer....
Read More →What will be the molarity of a solution,
Question: What will be the molarity of a solution, which contains 5.85 g of NaCl(s) per 500 mL? (i) 4 mol L-1 (ii) 20 molL-1 (iii) 0.2 molL-1 (iv) 2molL-1 Solution: Option (iii) is the answer....
Read More →A measured temperature on the Fahrenheit
Question: A measured temperature on the Fahrenheit scale is 200 F. What will this reading be on a Celsius scale? (i) 40 C (ii) 94 C (iii) 93.3 C (iv) 30 C Solution: Option (iii) is the answer....
Read More →Two students performed the same experiment
Question: Two students performed the same experiment separately and each one of them recorded two readings of mass which are given below. The correct reading of mass is 3.0 g. Based on given data, mark the correct option out of the following statements. Student Readings (i) Results of both the students are neither accurate nor precise. (ii) Results of student A are both precise and accurate. (iii) Results of student B are neither precise nor accurate. (iv) Results of student B are both precise a...
Read More →Find the general solution of each of the following equations:
Question: Find the general solution of each of the following equations: (i) $\sin x=\frac{\sqrt{3}}{2}$ (ii) $\cos x=1$ (iii) $\sec \mathrm{x}=\sqrt{2}$ Solution: To Find: General solution. (i) Given: $\sin x=\frac{\sqrt{3}}{2}$ Formula used: $\sin \theta=\sin \alpha \Rightarrow \theta=\mathrm{n} \pi+(-1)^{\mathrm{n}} \alpha, \mathrm{n} \in$ ред By using above formula, we have $\sin x=\frac{\sqrt{3}}{2}=\sin \frac{\pi}{3} \Longrightarrow x=n \pi+(-1)^{n} \cdot \frac{\pi}{3}$ So general solution is...
Read More →Find the point on the curve
Question: Find the point on the curve $2 a^{2} y=x^{3}-3 a x^{2}$ where the tangent is parallel to the $x$-axis. Solution: Given: The curve is $2 a^{2} y=x^{3}-3 a x^{2}$ Differentiating the above w.r.t $\mathrm{x}$ $\Rightarrow 2 \mathrm{a}^{2} \times \frac{\mathrm{dy}}{\mathrm{dx}}=3 \mathrm{x}^{3}-1-3 \times 2 \mathrm{ax}^{2-1}$ $\Rightarrow 2 \mathrm{a}^{2} \frac{\mathrm{dy}}{\mathrm{dx}}=3 \mathrm{x}^{2}-6 \mathrm{ax}$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{3 \mathrm{x}^{2}-6 \m...
Read More →Find the general solution of each of the following equations:
Question: Find the general solution of each of the following equations: (i) $\sin 3 x=0$ (ii) $\sin \frac{3 \mathrm{x}}{2}=0$ (iii) $\sin \left(x+\frac{\pi}{5}\right)=0$ (iv) $\cos 2 x=0$ (v) $\cos \frac{5 x}{2}=0$ (vi) $\cos \left(x+\frac{\pi}{10}\right)=0$ (vii) $\tan 2 x=0$ (viii) $\tan \left(3 \mathrm{x}+\frac{\pi}{6}\right)=0$ (ix) $\tan \left(2 \mathrm{x}-\frac{\pi}{4}\right)=0$ Solution: To Find: General solution. [NOTE: A solution of a trigonometry equation generalized by means of period...
Read More →Find the point on the curve
Question: Find the point on the curve $x^{2}+y^{2}=13$, the tangent at each one of which is parallel to the line $2 x+3 y=7$. Solution: Given: The curve $x^{2}+y^{2}=13$ and the line $2 x+3 y=7$ $x^{2}+y^{2}=13$ Differentiating the above w.r.t $x$ $\Rightarrow 2 x^{2}-1+2 y^{2}-1 \frac{d y}{d x}=0$ $\Rightarrow 2 x+2 y \frac{d y}{d x}=0$ $\Rightarrow 2\left(x+y \frac{d y}{d x}\right)=0$ $\Rightarrow\left(x+y \frac{d y}{d x}\right)=0$ $\Rightarrow y \frac{d y}{d x}=-x$ $\Rightarrow \frac{d y}{d x...
Read More →Find a point on the curve
Question: Find a point on the curve $y=3 x^{2}+4$ at which the tangent is perpendicular to the line whose slope is $-\frac{1}{6}$. Solution: Given: The curve $y=3 x^{2}+4$ and the Slope of the tangent is $\frac{-1}{6}$ $y=3 x^{2}+4$ Differentiating the above w.r.t $x$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=2 \times 3 \mathrm{x}^{2-1}+0$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=6 \times \ldots(1)$ Since, tangent is perpendicular to the line, $\therefore$ The Slope of the normal $=\frac{...
Read More →For the harmonic travelling
Question: For the harmonic travelling wave y = 2 cos 2 (10t-0.0080x+3.5) where x and y are in cm and t is second. What is the phase difference between the oscillatory motion at two points separated by a distance of (a) 4 m (b) 0.5 m (c) /2 (d) 3 /4 (e) what is the phase difference between the oscillation of a particle located at x = 100 cm at t = Ts and t = 5s? Solution: Given, y = 2 cos 2 (10t-0.0080x+3.5) a = 2 = 20 k = 0.016 = 7 (a) path difference, p = 4m = 400 cm Substituting the values, we...
Read More →In the given progressive waves y = 5
Question: In the given progressive waves y = 5 sin (100 t 0.4 x) where y and x are in m, t is in s. What is the (a) amplitude (b) wavelength (c) frequency (d) wave velocity (e) particle velocity amplitude Solution: Given, The wave is travelling in +x direction The equation is y = 5 sin (100 t 0.4 x) (a) amplitude, a = 5 m (b) wavelength, = 2/ = 5 m (c) frequency, v = 50 Hz (d) wave velocity, v = 250 m/s (e) particle velocity amplitude = 500 m/s...
Read More →At what points on the curve
Question: At what points on the curve $y=2 x^{2}-x+1$ is the tangent parallel to the line $y=3 x+4 ?$ Solution: Given: The curve is $y=2 x^{2}-x+1$ and the line $y=3 x+4$ First, we will find The Slope of tangent $y=2 x^{2}-x+1$ $\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(2 \mathrm{x}^{2}\right)-\frac{\mathrm{d}}{\mathrm{dx}}(\mathrm{x})+\frac{\mathrm{d}}{\mathrm{dx}}(1)$ $\Rightarrow \frac{\mathrm{d} \mathrm{y}}{\mathrm{dx}}=4 \mathrm{x}-1 \ldots(1)$ $y=3 x+4...
Read More →Given below are some functions of x and t to
Question: Given below are some functions of x and t to represent the displacement of an elastic wave. (a) y = 5 cos (4x) sin (20t) (b) y = 4 sin (5x-t/2) + 3 cos (5x-t/2) (c) y = 10 cos [(252-250)t] cos [(252+250) t] (d) y = 100 cos (100 t + 0.5x) State which of these represent (a) a travelling wave along -x direction (b) a stationary wave (c) beats (d) a travelling wave along +x direction Give reasons for your answers Solution: (a) When a wave travels in (-x) direction, it must have +kx which i...
Read More →If c is r.m.s speed of molecules in a gas
Question: If c is r.m.s speed of molecules in a gas and v is the speed of sound waves in the gas, show that c/v is constant and independent of temperature for all diatomic gases. Solution: We know following is the equation for molecules: $c=\sqrt{\frac{3 P}{\rho}}$ $c=\sqrt{\frac{3 R T}{M}}$ p/ = PT/M Where, M is the molar mass of the gas $v=\sqrt{\frac{\gamma P}{\rho}}=\sqrt{\frac{\gamma R T}{M}}$ c/v is given as: $\frac{c}{v}=\frac{\sqrt{\frac{3 R T}{M}}}{\sqrt{\frac{\gamma R T}{M}}}=\sqrt{\fr...
Read More →A train standing at the outer signal of a railway
Question: A train standing at the outer signal of a railway station blows a whistle of frequency 400 Hz still air. The train beings to move with a speed of 10 m/s towards the platform. What is the frequency of the sound for an observer standing on the platform? Solution: v0= 400 Hz vz= 10 m/s Velocity of sound in air, va= 330 m/s v is the frequency heard by the observer standing on the platform v = (330)(400)/320 = 412.5 Hz...
Read More →A steel wire has a length of 12 m and a mass of 2.10 kg.
Question: A steel wire has a length of 12 m and a mass of 2.10 kg. What will be the speed of a transverse wave on this wire when a tension of 2.06 104N is applied? Solution: Length, l = 12 m Total mass, M = 2.10 kg m = M/l = 2.1/12 Tension, T = 2.06 104N Therefore, v is given as $v=\sqrt{\frac{T}{m}}=\frac{2.06 \times 10^{4} \times 12}{2.10}=3.43 \times 10^{2}$ v = 343.0 m/s...
Read More →When two waves of almost equal frequencies
Question: When two waves of almost equal frequencies n1and n2reach at a point simultaneously, what is the time interval between successive maxima? Solution: It is given that the two frequencies are almost equal, that is n1= n2. For the formation of the beats, the frequencies must be n2 n1. The no.of frequencies per second in maxima = n = n2 n1 Therefore, the time period of the maxima = 1/n = 1/n2 n1second....
Read More →Find the principal solutions of each of the following equations :
Question: Find the principal solutions of each of the following equations : (i) $\sin x=\frac{-1}{2}$ (ii) $\sqrt{2} \cos x+1=0$ (iii) $\tan x=-1$ (iv) $\sqrt{3} \operatorname{cosec} x+2=0$ (v) $\tan \mathrm{x}=-\sqrt{3}$ (vi) $\sqrt{3} \sec x+2=0$ Solution: To Find: Principal solution. (i) Given: $\sin x=\frac{-1}{2}$ Formula used: $\sin \theta=\sin \alpha \Rightarrow \theta=n \pi+(-1)^{n} \alpha, n \in 1$ By using above formula, we have $\sin x=\frac{-1}{2}=-\sin \frac{\pi}{6}=\sin \left(\pi+\...
Read More →At what temperatures will the speed
Question: At what temperatures will the speed of sound in air be 3 times its value at 0oC? Solution: We know that, $v \propto \sqrt{T}$ Given that, vT= 3v0 $\frac{3 v_{0}}{v_{0}}=\sqrt{\frac{T}{273+0}} \Rightarrow \sqrt{T}=3 \sqrt{273}$ T = 2457 273 = 2184oC...
Read More →A sitar wire is replaced by another wire
Question: A sitar wire is replaced by another wire of same length and material but of three times earlier radius. If the tension in the wire remains the same, by what factor will the frequency change? Solution: The frequency of the stretched wire is given as: $v=\frac{n}{2 L} \sqrt{\frac{T}{m}}$ Given that, No.of harmonic n, length L, and tension T are the same in both the cases. Therefore, $v \propto \frac{1}{\sqrt{m}} \Rightarrow \frac{v_{1}}{v_{2}}=\frac{\sqrt{m_{2}}}{\sqrt{m_{1}}}$ Substitut...
Read More →Find a point on the curve
Question: Find a point on the curve $y=3 x^{2}-9 x+8$ at which the tangents are equally inclined with the axes. Solution: Given: The curve is $y=3 x^{2}-9 x+8$ Differentiating the above w.r.t $x$ $\Rightarrow y=3 x^{2}-9 x+8$ $\Rightarrow \frac{d y}{d x}=2 \times 3 x^{2}-1-9+0$ $\Rightarrow \frac{d y}{d x}=6 x-9 \ldots(1)$ Since, the tangent are equally inclined with axes i.e, $\theta=\frac{\pi}{4}$ or $\theta=\frac{-\pi}{4}$ $\therefore \frac{d y}{d x}=$ The Slope of the tangent $=\tan \theta$ ...
Read More →The displacement of an elastic wave is given
Question: The displacement of an elastic wave is given by the function y = 3 sin t + 4 cos t where y is in cm and t is in second. Calculate the resultant amplitude. Solution: Given, y = 3 sin t + 4 cos t Lets consider 3 = a cos 4 = a sin Then y = a cos t + a sin t y = a sin (t + ) tan = 4/3 = tan-14/3 When the above equation is squared, we het a2= 25 a = 5 Therefore, the new amplitude is 5 cm....
Read More →A tuning fork A, marked 512 Hz, produces 5 beats per second,
Question: A tuning fork A, marked 512 Hz, produces 5 beats per second, where sounded with another unmarked tuning fork B. If B is loaded with wax the number of beats is again 5 per second. What is the frequency of the tuning fork B when not loaded? Solution: When the tuning fork B is loaded with the wax, the frequency of the tuning fork becomes less than its original frequency. When the tuning fork is marked 512 Hz, lets assume that the tuning fork B is marked to 517 Hz because it produces 5 bea...
Read More →A sonometer wire is vibrating in resonance
Question: A sonometer wire is vibrating in resonance with a tuning fork. Keeping the tension applied same, the length of the wire is doubled. Under what conditions would the tuning fork still be is resonance with the wire? Solution: The length of the wire used in a sonometer is twice when it vibrates. Therefore, if the tuning fork resonates at L, the sonometer resonates at 2L. The following equation is used to express the frequency of the sonometer $f=\frac{n}{2 L} \sqrt{\frac{T}{\mu}}=\frac{n v...
Read More →Which of the following statements are
Question: Which of the following statements are true for a stationary wave? (a) every particle has a fixed amplitude which is different from the amplitude of its nearest particle (b) all the particles cross their mean position at the same time (c) all the particles are oscillating with the same amplitude (d) there is no net transfer of energy across any plane (e) there are some particles which are always at rest Solution: The correct answers are (a) every particle has a fixed amplitude which is ...
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