A function
Question: A function $f(x)=1+\frac{1}{x}$ is defined on the closed interval $[1,3]$. A point in the interval, where the function satisfies the mean value theorem, is___________ Solution: The function $f(x)=1+\frac{1}{x}$ is defined on the interval $[1,3]$. $f(x)$ is continuous on $[1,3]$ and differentiable on $(1,3)$. So, by mean value theorem there must exist at least one real number $c \in(1,3)$ such that $f^{\prime}(c)=\frac{f(3)-f(1)}{3-1}$ $\Rightarrow-\frac{1}{c^{2}}=\frac{\frac{4}{3}-2}{3...
Read More →The displacement of a particle is given
Question: The displacement of a particle is given by x = (t-2)2where x is in metres and t is seconds. The distance covered by the particle in first 4 seconds is (a) 4 m (b) 8 m (c) 12 m (d) 16 m Solution: The correct answer is (b) 8 m...
Read More →A vehicle travels half the distance
Question: A vehicle travels half the distance L with speed V1and the other half with speed V2, then its average speed is (a) (V1+V2)/2 (b) (2V1+V2)/(V1+V2) (c) (2V1V2)/(V1+V2) (d) L(V1+V2)/V1V2 Solution: The correct answer is c) (2V1V2)/(V1+V2)...
Read More →In one dimensional motion,
Question: In one dimensional motion, instantaneous speed v satisfies 0 v v0 (a )The displacement in time T must always take non-negative values. (b) The displacement x in time T satisfies -v()T x v0 (c) The acceleration is always a non-negative number. (d) The motion has no turning points. Solution: The correct answer is (b) the displacement in time T must always take non-negative values...
Read More →If the solve the problem
Question: If $f(x)=e^{x} \sin x$ in $[0, \pi]$, then $c$ in Rolle's theorem is (a) $\pi / 6$ (b) $\pi / 4$ (c) $\pi / 2$ (d) $3 \pi / 4$ Solution: (d) $3 \pi / 4$ The given function is $f(x)=e^{x} \sin x$. Differentiating the given function with respect tox, we get $f^{\prime}(x)=e^{x} \cos x+\sin x e^{x}$ $\Rightarrow f^{\prime}(c)=e^{c} \cos c+\sin c e^{c}$ Now, $e^{x} \cos x$ is continuous and derivable in $R$. Therefore, it is continuous on $[0, \pi]$ and derivable on $(0, \pi)$. $\therefore...
Read More →Prove that
Question: If $\sin x=\frac{12}{3}$ and $\sin y=\frac{4}{5}$ where $\frac{\pi}{2}x\pi$ and $0y\frac{\pi}{2}$ find the values of (i) $\sin (x+y)$ (ii) $\cos (x+y)$ (iii) $\tan (x-y)$ Solution: Given $\sin x=\frac{12}{13}$ and $\sin y=\frac{4}{5}$, Here we will find values of cosx and cosy $\cos x=\sqrt{\left(1-\sin ^{2} x\right)} \Rightarrow \sqrt{\left(1-\left(\frac{12}{13}\right)^{2}\right)}=\sqrt{\left(\frac{169-144}{169}\right)} \Rightarrow \sqrt{\left(\frac{25}{169}\right)}=\frac{5}{13}$ $\co...
Read More →If the solve the problem
Question: If $f(x)=e^{x} \sin x$ in $[0, \pi]$, then $c$ in Rolle's theorem is (a) $\pi / 6$ (b) $\pi / 4$ (c) $\pi / 2$ (d) $3 \pi / 4$ Solution: (d) $3 \pi / 4$ The given function is $f(x)=e^{x} \sin x$. Differentiating the given function with respect tox, we get $f^{\prime}(x)=e^{x} \cos x+\sin x e^{x}$ $\Rightarrow f^{\prime}(c)=e^{c} \cos c+\sin c e^{c}$ Now, $e^{x} \cos x$ is continuous and derivable in $R$. Therefore, it is continuous on $[0, \pi]$ and derivable on $(0, \pi)$. $\therefore...
Read More →Einstein’s mass-energy relation emerging
Question: Einsteins mass-energy relation emerging out of his famous theory of relativity relates mass (m) to energy (E) as E= mc2, where c is speed of light in vacuum. At the nuclear level, the magnitudes of energy are very small. The energy at nuclear level is usually measured in MeV, where 1 MeV = 1.6 x 10-13J; the masses are measured in unified atomic mass unit (u) where, 1 u = 67 x 10-27kg. (a) Show that the energy equivalent of 1 u is 931.5 MeV. (b) A student writes the relation as 1 u = 93...
Read More →(a) How many astronomical units (AU) make 1 parsec?
Question: (a) How many astronomical units (AU) make 1 parsec? (b) Consider the sun like a star at a distance of 2 parsecs. When it is seen through a telescope with 100 magnification, what should be the angular size of the star? Sun appears to be (1/2) from the earth. Due to atmospheric fluctuations,eyecannot resolve objects smaller than 1 arc minute. (c) Mars has approximately half of the earths diameter. When it is closest to the earth it is at about 1/2 AU from the earth. Calculate at what siz...
Read More →In an experiment to estimate the size
Question: In an experiment to estimate the size of a molecule of oleic acid, 1mL of oleic acid is dissolved in 19mL of alcohol. Then 1mL of this solution is diluted to 20mL by adding alcohol. Now, 1 drop of this diluted solution is placed on water in a shallow trough. The solution spreads over the surface of water forming one molecule thick layer. Now, lycopodium powder is sprinkled evenly over the film we can calculate the thickness of the film which will give us the size of oleic acid molecule...
Read More →An artificial satellite is revolving around a planet
Question: An artificial satellite is revolving around a planet of mass M and radius R, in a circular orbit of radius r. From Keplers third law about the period of a satellite around a common central body, square of the period of revolution T is proportional to the cube of the radius of the orbit r. Show using dimensional analysis, that T = k/R r3/g where k is a dimensionless constant and g is acceleration due to gravity. Solution: From Keplers third law, we know that T2 a3 where T2 is the square...
Read More →If the velocity of light c,
Question: If the velocity of light c, Plancks constant h and gravitational constant G are taken as fundamental quantities then express mass, length and time in terms of dimensions of these quantities. Solution: Principle of homogeneity is used for solving this problem. [h] = [ML2T1] [c] = [LT-1] [G] = [M-1L3T-2] Let m = kcahbGc Solving the above we get, m = kc1/2h1/2G-1/2= kch/G Let L = kcahbGc Solving the above we get, L = kc-3/2h1/2G1/2= khG/c3 Let T = cahbGc Solving the above we get, L = kc-5...
Read More →In the expression P = E l2 m–5 G–2, E, m
Question: In the expression P = E l2m5G2, E, m, l and G denote energy, mass, angular momentum and gravitational constant, respectively. Show that P is a dimensionless quantity. Solution: From the problem, P = E l2m5G2 E is the energy = [ML2T-2] m is the mass = [M] L is the angular momentum = [ML2T1] G is the gravitational constant = [M-1L2T2] Substituting the values we get, [P] = [M0L0T0]...
Read More →A physical quantity X is related
Question: A physical quantity X is related to four measurable quantities a, b, c and d as follows: X = a2b3c5/2d2. The percentage error in the measurement of a, b, c and d are 1%, 2%, 3% and 4%, respectively. What is the percentage error in quantity X? If the value of X calculated on the basis of the above relation is 2.763, to what value should you round off the result. Solution: The given physical quantity, X = a2b3c5/2d2 Percentage error in X = (∆x/x)(100) Percentage error in a = (∆a/a)(100) ...
Read More →The volume of a liquid flowing out per second
Question: The volume of a liquid flowing out per second of a pipe of length l and radius r is written by a student as $v=\frac{\pi}{8} \frac{P r^{4}}{\eta l} \quad$ where $\mathrm{P}$ is the pressure difference between the two ends of the pipe and $\mathrm{n}$ is coefficient of viscosity of the liquid having dimensional formula $\mathrm{ML}^{-1} \mathrm{~T}^{-1}$. Check whether the equation is dimensionally correct. Solution: Dimension of the given physical quantity is [V] = dimension of volume/...
Read More →The value of c in Rolle's theorem for the function
Question: The value of $c$ in Rolle's theorem for the function $f(x)=x^{3}-3 x$ in the interval $[0, \sqrt{3}]$ is Solution: (a) 1 The given function is $f(x)=x^{3}-3 x$. This is a polynomial function, which is continuous and derivable in $R$. Therefore, the function is continuous on $[0, \sqrt{3}]$ and derivable on $(0, \sqrt{3})$. Differentiating the given function with respect tox,we get $f^{\prime}(x)=3 x^{2}-3$ $\Rightarrow f^{\prime}(c)=3 c^{2}-3$ $\therefore f^{\prime}(c)=0$ $\Rightarrow ...
Read More →The value of c in Rolle's theorem for the function
Question: The value of $c$ in Rolle's theorem for the function $f(x)=x^{3}-3 x$ in the interval $[0, \sqrt{3}]$ is Solution: (a) 1 The given function is $f(x)=x^{3}-3 x$. This is a polynomial function, which is continuous and derivable in $R$. Therefore, the function is continuous on $[0, \sqrt{3}]$ and derivable on $(0, \sqrt{3})$. Differentiating the given function with respect tox,we get $f^{\prime}(x)=3 x^{2}-3$ $\Rightarrow f^{\prime}(c)=3 c^{2}-3$ $\therefore f^{\prime}(c)=0$ $\Rightarrow ...
Read More →A new system of units is proposed
Question: A new system of units is proposed in which unit of mass is kg, unit of length m and unit of time s. How much will 5 J measure in this new system? Solution: Let Q be the physical quantity = n1u1 = n2u2 Let M1, L1, T1 and M2,L2,T2 be the units of mass, length, and time for the given two systems. n2 = n1 [U] = [ML2T2] M1 = 1 kg L1 = 1 m T1 = 1 s M2 = kg L2 = m T1 = s Substituting the values we get, n2 = 52/2J...
Read More →If the unit of force is 100 N,
Question: If the unit of force is 100 N, unit of length is 10 m and unit of time is 100 s, what is the unit of mass in this system of units? Solution: Force [F] = 100 N Length [L] = 10 m Time [t] = 100 s [F] = [MLT2] Substituting the values, we get M = 105kg...
Read More →The value of c in Lagrange's mean value theorem
Question: The value of $c$ in Lagrange's mean value theorem for the function $f(x)=x(x-2)$ when $x \in[1,2]$ is (a) 1(b) 1/2(c) 2/3(d) 3/2 Solution: (d) $\frac{3}{2}$ We have $f(x)=x(x-2)$ It can be rewritten as $f(x)=x^{2}-2 x$. We know that a polynomial function is everywhere continuous and differentiable. Since $f(x)$ is a polynomial, it is continuous on $[1,2]$ and differentiable on $(1,2)$. Thus, $f(x)$ satisfies both the conditions of Lagrange's theorem on $[1,2]$. So, there must exist at ...
Read More →The vernier scale of a travelling microscope
Question: The vernier scale of a travelling microscope has 50 divisions which coincide with 49 main scale divisions. If each main scale division is 0.5 mm, calculate the minimum inaccuracy in the measurement of distance. Solution: The minimum inaccuracy in the measurement of distance = (1/50)(0/5)mm = 0.01 mm...
Read More →The value of c in Lagrange's mean value theorem
Question: The value of $c$ in Lagrange's mean value theorem for the function $f(x)=x(x-2)$ when $x \in[1,2]$ is (a) 1(b) 1/2(c) 2/3(d) 3/2 Solution: (d) $\frac{3}{2}$ We have $f(x)=x(x-2)$ It can be rewritten as $f(x)=x^{2}-2 x$. We know that a polynomial function is everywhere continuous and differentiable. Since $f(x)$ is a polynomial, it is continuous on $[1,2]$ and differentiable on $(1,2)$. Thus, $f(x)$ satisfies both the conditions of Lagrange's theorem on $[1,2]$. So, there must exist at ...
Read More →The distance of a galaxy is of the order of 1025 m.
Question: The distance of a galaxy is of the order of 1025 m. Calculate the order of magnitude of time taken by light to reach us from the galaxy. Solution: Distance of the galaxy = 1025m Speed of light = 3 108m/s Time taken, t is t = distance/speed = 3.33 1016s...
Read More →Which of the following time measuring
Question: Which of the following time measuring devices is most precise? (a) A wallclock (b) A stop watch (c) A digital watch (d) An atomic clock Given reason for your answer. Solution: Option (d) is correct because a clock can measure time correctly up to one second. A stop watch can measure time correctly up to a fraction of a second. A digital watch can measure time up to a fraction of second whereas an atomic clock is the most accurate timekeeper and is based on characteristic frequencies of...
Read More →Name the device used for measuring
Question: Name the device used for measuring the mass of atoms and molecules. Solution: Mass spectrograph is the device that is used for measuring the mass of atoms and molecules....
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