The radius of an atom is of the order of 1 Å
Question: The radius of an atom is of the order of 1 Å and radius of nucleus is of the order of fermi. How many magnitudes higher is the volume of the atom as compared to the volume of the nucleus? Solution: Radius of atom = 1 Å = 10-10m Radius of nucleus = 1 fermi = 10-15m Volume of atom = 4/3Ra3 Volume of nucleus = 4/3Rn3 Vatom/Vnucleus = 1015 Mass of one mole of carbon atom = 12 g = 1.67 10-27kg...
Read More →Why do we have different units
Question: Why do we have different units for the same physical quantity? Solution: We have different units for the same physical quantity because they differ from place to place....
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 $f(x)=\frac{x(x+1)}{o^{x}}$ defined on $[-1,0]$ is (a) $0.5$ (b) $\frac{1+\sqrt{5}}{2}$ (c) $\frac{1-\sqrt{5}}{2}$ (d) $-0.5$ Solution: (C) $\frac{1-\sqrt{5}}{2}$ Given: $f(x)=\frac{x(x+1)}{e^{x}}$ Differentiating the given function with respect tox, we get $f^{\prime}(x)=\frac{e^{x}(2 x+1)-x(x+1) e^{x}}{\left(e^{x}\right)^{2}}$ $\Rightarrow f^{\prime}(x)=\frac{(2 x+1)-x(x+1)}{e^{x}}$ $\Rightarrow f^{\prime}(x)=\frac{2 x+1-x^{2}-x}{e...
Read More →Which of the following are not a unit
Question: Which of the following are not a unit of time? (a) second (b) parsec (c) year (d) light year Solution: The correct answer is (b) parsec and (d) light year...
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 $f(x)=\frac{x(x+1)}{o^{x}}$ defined on $[-1,0]$ is (a) $0.5$ (b) $\frac{1+\sqrt{5}}{2}$ (c) $\frac{1-\sqrt{5}}{2}$ (d) $-0.5$ Solution: (C) $\frac{1-\sqrt{5}}{2}$ Given: $f(x)=\frac{x(x+1)}{e^{x}}$ Differentiating the given function with respect tox, we get $f^{\prime}(x)=\frac{e^{x}(2 x+1)-x(x+1) e^{x}}{\left(e^{x}\right)^{2}}$ $\Rightarrow f^{\prime}(x)=\frac{(2 x+1)-x(x+1)}{e^{x}}$ $\Rightarrow f^{\prime}(x)=\frac{2 x+1-x^{2}-x}{e...
Read More →Which of the following ratios express pressure?
Question: Which of the following ratios express pressure? (a) Force/area (b) Energy/volume (c) Energy/area (d) Force/volume Solution: The correct answer is (a) force/area and (b) energy/volume...
Read More →If Planck’s constant (h) and speed of light in vacuum (c)
Question: If Plancks constant (h) and speed of light in vacuum (c) are taken as two fundamental quantities, which of the following can, also, be taken to express length, mass, and time in terms of the three chosen fundamental quantities? (a) mass of the electron (me) (b) universal gravitational constant (G) (c) charge of the electron (e) (d) mass of proton (mp) Solution: The correct answer is (a) mass of electron (b) universal gravitational constant and (d) mass of proton...
Read More →When the tangent to the curve y = x log x is parallel
Question: When the tangent to the curvey=xlogxis parallel to the chord joining the points (1, 0) and (e,e), the value ofxis (a) $e^{1 / 1-e}$ (b) $e^{(e-1)(2 e-1)}$ (c) $e^{\frac{2 e-1}{e-1}}$ (d) $\frac{e-1}{e}$ Solution: (a) $e^{1 / 1-e}$ Given: $y=f(x)=x \log x$ Differentiating the given function with respect tox,we get $f^{\prime}(x)=1+\log x$ $\Rightarrow$ Slope of the tangent to the curve $=1+\log x$ Also, Slope of the chord joining the points $(1,0)$ and $(e, e),(m)=\frac{e}{e-1}$ The tan...
Read More →When the tangent to the curve y = x log x is parallel
Question: When the tangent to the curvey=xlogxis parallel to the chord joining the points (1, 0) and (e,e), the value ofxis (a) $e^{1 / 1-e}$ (b) $e^{(e-1)(2 e-1)}$ (c) $e^{\frac{2 e-1}{e-1}}$ (d) $\frac{e-1}{e}$ Solution: (a) $e^{1 / 1-e}$ Given: $y=f(x)=x \log x$ Differentiating the given function with respect tox,we get $f^{\prime}(x)=1+\log x$ $\Rightarrow$ Slope of the tangent to the curve $=1+\log x$ Also, Slope of the chord joining the points $(1,0)$ and $(e, e),(m)=\frac{e}{e-1}$ The tan...
Read More →Photon is quantum of radiation with
Question: Photon is quantum of radiation with energy E = hv where v is frequency and h is Plancks constant. The dimensions of h are the same as that of: (a) linear impulse (b) angular impulse (c) linear momentum (d) angular momentum Solution: The correct option is (b) angular impulse and d) angular momentum...
Read More →If P, Q, R are physical quantities,
Question: If P, Q, R are physical quantities, having different dimensions, which of the following combinations can never be a meaningful quantity? (a) (P Q)/R (b) PQ R (c) PQ/R (d) (PR Q2)/R (e) (R + Q)/P Solution: The correct answer is d) (PR Q2)/R and e) (R + Q)/P...
Read More →On the basis of dimensions,
Question: On the basis of dimensions, decide which of the following relations for the displacement of a particle undergoing simple harmonic motion is not correct: (a) y = a sin 2t/T (b) y = a sin vt (c) y = a/T sin (t/a) (d) y = a2 [sin (2 t/T) cos (2t/T)] Solution: The correct answer is b) y = a sin vt and c) y = a/T sin (t/a)...
Read More →The value of c in Rolle's theorem when
Question: The value ofcin Rolle's theorem when $f(x)=2 x^{3}-5 x^{2}-4 x+3, x \in[1 / 3,3]$ is (a) 2 (b) $-\frac{1}{3}$ (c) $-2$ (d) $\frac{2}{3}$ Solution: (a) 2 Given: $f(x)=2 x^{3}-5 x^{2}-4 x+3$ Differentiating the given function with respect tox, we get $f^{\prime}(x)=6 x^{2}-10 x-4$ $\Rightarrow f^{\prime}(c)=6 c^{2}-10 c-4$ $\therefore f^{\prime}(c)=0$ $\Rightarrow 3 c^{2}-5 c-2=0$ $\Rightarrow 3 c^{2}-6 c+c-2=0$ $\Rightarrow 3 c(c-2)+c-2=0$ $\Rightarrow(3 c+1)(c-2)=0$ $\Rightarrow c=2, \...
Read More →The value of c in Rolle's theorem when
Question: The value ofcin Rolle's theorem when $f(x)=2 x^{3}-5 x^{2}-4 x+3, x \in[1 / 3,3]$ is (a) 2 (b) $-\frac{1}{3}$ (c) $-2$ (d) $\frac{2}{3}$ Solution: (a) 2 Given: $f(x)=2 x^{3}-5 x^{2}-4 x+3$ Differentiating the given function with respect tox, we get $f^{\prime}(x)=6 x^{2}-10 x-4$ $\Rightarrow f^{\prime}(c)=6 c^{2}-10 c-4$ $\therefore f^{\prime}(c)=0$ $\Rightarrow 3 c^{2}-5 c-2=0$ $\Rightarrow 3 c^{2}-6 c+c-2=0$ $\Rightarrow 3 c(c-2)+c-2=0$ $\Rightarrow(3 c+1)(c-2)=0$ $\Rightarrow c=2, \...
Read More →If momentum (P), area (A), and time (T)
Question: If momentum (P), area (A), and time (T) are taken to be fundamental quantities, then energy has the dimensional formula (a) (P1A-1T1) (b) (P2A1T1) (c) (P1A-1/2T1) (d) (P1A1/2T-1) Solution: The correct answer is d) (P1A1/2T-1)...
Read More →Young’s modulus of steel is
Question: Youngs modulus of steel is 1.9 x 1011N/m2. When expressed in CGS units of dyne/cm2, it will be equal to (1 N = 105dyne, 1 m2= 104cm2) . (a) 1.9 xlO10 (b) 1.91012 (c) 1.9 xlO12 (d) 1.9 xlO13 Solution: The correct answer is (c) 1.9 1012...
Read More →Rolle's theorem is applicable in case
Question: Rolle's theorem is applicable in case of $\phi(x)=a^{\sin x}, aa$ in (a) any interval(b) the interval [0, ](c) the interval (0, /2)(d) none of these Solution: (b) the interval $[0, \pi]$ The given function is $\phi(x)=a^{\sin x}$, where $a0$. Differentiating the given function with respect tox,we get $f^{\prime}(x)=\log a\left(\cos x a^{\sin x}\right)$ $\Rightarrow f^{\prime}(c)=\log a\left(\cos c a^{\sin c}\right)$ Let $f^{\prime}(c)=0$ $\Rightarrow \log a\left(\cos c a^{\sin c}\right...
Read More →The mean length of an object is 5 cm.
Question: The mean length of an object is 5 cm. Which of the following measurements is most accurate? (a) 4.9 cm (b) 4.805 cm (c) 5.25 cm (d) 5.4 cm Solution: The correct answer is (a) 5.00 mm...
Read More →Rolle's theorem is applicable in case
Question: Rolle's theorem is applicable in case of $\phi(x)=a^{\sin x}, aa$ in (a) any interval(b) the interval [0, ](c) the interval (0, /2)(d) none of these Solution: (b) the interval $[0, \pi]$ The given function is $\phi(x)=a^{\sin x}$, where $a0$. Differentiating the given function with respect tox,we get $f^{\prime}(x)=\log a\left(\cos x a^{\sin x}\right)$ $\Rightarrow f^{\prime}(c)=\log a\left(\cos c a^{\sin c}\right)$ Let $f^{\prime}(c)=0$ $\Rightarrow \log a\left(\cos c a^{\sin c}\right...
Read More →If x and y are acute angles such that
Question: If x and y are acute angles such that $\cos x=\frac{13}{14}$ and $\cos y=\frac{1}{7}$ prove that $(x-y)=-\frac{\pi}{3}$ Solution: Given $\cos x=\frac{13}{14}$ and $\cos y=\frac{1}{7}$ Now we will calculate value of sinx and siny $\sin x=\sqrt{\left(1-\cos ^{2} x\right)} \Rightarrow \sqrt{\left(1-\left(\frac{13}{14}\right)^{2}\right)}=\sqrt{\left(\frac{196-169}{196}\right)} \Rightarrow \sqrt{\left(\frac{27}{196}\right)}=\frac{3 \sqrt{3}}{14}$ siny $=\sqrt{\left(1-\cos ^{2} y\right)} \Ri...
Read More →Which of the following
Question: Which of the following measurements is most precise? (a) 5.00 mm (b) 5.00 cm (c) 5.00 m (d) 5.00 km Solution: (a) Key concept: Precision is the degree to which several measurements provide answers very close to each other. It is an indicator of the scatter in the data. The lesser the scatter, higher the precision. Let us first check the units. In all the options magnitude is same but units of measurement are different. As here 5.00 mm has the smallest unit. All given measurements are c...
Read More →If from Lagrange's mean value theorem, we have
Question: If from Lagrange's mean value theorem, we have $f^{\prime}\left(x_{1}\right)=\frac{f^{\prime}(b)-f(a)}{b-a}$, then (a) $ax_{1} \leq b$ (b) $a \leq x_{1}b$ (c) $ax_{1}b$ (d) $a \leq x_{1} \leq b$ Solution: (c) $ax_{1}b$ In the Lagrange's mean value theorem, $c \in(a, b)$ such that $f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}$. So, if there is $x_{1}$ such that $f^{\prime}\left(x_{1}\right)=\frac{f(b)-f(a)}{b-a}$, then $x_{1} \in(a, b)$. $\Rightarrow ax_{1}b$...
Read More →If from Lagrange's mean value theorem, we have
Question: If from Lagrange's mean value theorem, we have $f^{\prime}\left(x_{1}\right)=\frac{f^{\prime}(b)-f(a)}{b-a}$, then (a) $ax_{1} \leq b$ (b) $a \leq x_{1}b$ (c) $ax_{1}b$ (d) $a \leq x_{1} \leq b$ Solution: (c) $ax_{1}b$ In the Lagrange's mean value theorem, $c \in(a, b)$ such that $f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}$. So, if there is $x_{1}$ such that $f^{\prime}\left(x_{1}\right)=\frac{f(b)-f(a)}{b-a}$, then $x_{1} \in(a, b)$. $\Rightarrow ax_{1}b$...
Read More →If x and y are acute such that
Question: If x and y are acute such that $\sin x=\frac{1}{\sqrt{5}}$ and $\sin y=\frac{1}{\sqrt{10}}$ prove that $(x+y)=\frac{\pi}{4}$ Solution: Givensin $x=\frac{1}{\sqrt{5}}$ andsiny $=\frac{1}{\sqrt{10}}$ Now we will calculate value of cos x and cosy $\cos x=\sqrt{\left(1-\sin ^{2} x\right)} \Rightarrow \sqrt{\left(1-\left(\frac{1}{\sqrt{5}}\right)^{2}\right)}=\sqrt{\left(\frac{5-1}{5}\right)} \Rightarrow \sqrt{\left(\frac{4}{5}\right)}=\frac{2}{\sqrt{5}}$ $\cos y=\sqrt{\left(1-\sin x^{2}\rig...
Read More →You measure two quantities as A = 1.0 m ± 0.2 m,
Question: You measure two quantities as A = 1.0 m 0.2 m, B = 2.0 m 0.2 m. We should report correct value for AB as (a) 1.4 m 0.4 m (b) 1.41 m 0.15 m (c) 1.4 m + 0.3 m (d) 1.4 m 0.2 m Solution: The correct answer is (d) 1.4 m 0.2 m...
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