The number of subshells associated with n=4 and m=-2 quantum numbers is :
Question: The number of subshells associated with $n=4$ and $m=-2$ quantum numbers is : 82164Correct Option: , 2 Solution: For $n=4$ possible values of $l=0,1,2,3$; only $l=2$ and $l=3$ can have $m=-2 .$ So possible subshells are $2 .$...
Read More →If $lpha$ and $eta$ be the coefficients of
Question: If $\alpha$ and $\beta$ be the coefficients of $x^{4}$ and $x^{2}$ respectively in the expansion of $\left(x+\sqrt{x^{2}-1}\right)^{6}+\left(x-\sqrt{x^{2}-1}\right)^{6}$, then: (1) $\alpha+\beta=60$(2) $\alpha+\beta=-30$(3) $\alpha-\beta=60$(4) $\alpha-\beta=-132$Correct Option: , 4 Solution: Using Binomial expansion $(x+a)^{n}+(x-a)^{n}=2\left(T_{1}+T_{3}+T_{5}+T_{7} \cdots\right)$ $\therefore\left(x+\sqrt{x^{2}-1}\right)^{6}+\left(x-\sqrt{x^{2}-1}\right)^{6}=2\left(T_{1}+T_{3}+T_{5}+...
Read More →If $lpha$ and $eta$ be the coefficients of
Question: If $\alpha$ and $\beta$ be the coefficients of $x^{4}$ and $x^{2}$ respectively in the expansion of $\left(x+\sqrt{x^{2}-1}\right)^{6}+\left(x-\sqrt{x^{2}-1}\right)^{6}$, then: (1) $\alpha+\beta=60$(2) $\alpha+\beta=-30$(3) $\alpha-\beta=60$(4) $\alpha-\beta=-132$Correct Option: , 4 Solution: Using Binomial expansion...
Read More →The figure that is not a direct manifestation of the quantum nature of atoms is :
Question: The figure that is not a direct manifestation of the quantum nature of atoms is :Correct Option: , 4 Solution: (a), (b) and (c) are according to quantum theory but (d) is statement of kinetic theory of gases....
Read More →In a Frank-Hertz experiment,
Question: In a Frank-Hertz experiment, an electron of energy $5.6 \mathrm{eV}$ passes through mercury vapour and emerges with an energy $0.7 \mathrm{eV}$. The minimum wavelength of photons emitted by mercury atoms is close to :(1) $1700 \mathrm{~nm}$(2) $2020 \mathrm{~nm}$(3) $220 \mathrm{~nm}$(4) $250 \mathrm{~nm}$Correct Option: , 4 Solution: (4) Using, wavelength, $\lambda=\frac{12375}{\Delta \mathrm{E}}$ or, $\lambda=\frac{12375}{4.9}=250 \mathrm{~nm}$...
Read More →In a Frank-Hertz experiment,
Question: In a Frank-Hertz experiment, an electron of energy $5.6 \mathrm{eV}$ passes through mercury vapour and emerges with an energy $0.7 \mathrm{eV}$. The minimum wavelength of photons emitted by mercury atoms is close to :(1) $1700 \mathrm{~nm}$(2) $2020 \mathrm{~nm}$(3) $220 \mathrm{~nm}$(4) $250 \mathrm{~nm}$Correct Option: , 4 Solution: (4) Using, wavelength, $\lambda=\frac{12375}{\Delta \mathrm{E}}$...
Read More →The coefficient of $x^{7}$ in the expression
Question: The coefficient of $x^{7}$ in the expression $(1+x)^{10}+x(1+x)^{9}$ $+x^{2}(1+x)^{8}+\ldots+x^{10}$ is:(1) 210(2) 330(3) 120(4) 420Correct Option: , 2 Solution: The given series is in G.P. then $S_{n}=\frac{a\left(1-r^{n}\right)}{1-r}$ $\frac{(1+x)^{10}\left[1-\left(\frac{x}{1+x}\right)^{11}\right]}{\left(1-\frac{x}{1+x}\right)}$ $\Rightarrow \frac{(1+x)^{10}\left[(1+x)^{11}-x^{11}\right]}{(1+x)^{11} \times \frac{1}{(1+x)}}=(1+x)^{11}-x^{11}$ $\therefore \quad$ Coefficient of $x^{7}$ ...
Read More →If the volume and the surface area of a sphere are numerically the same, the nits radius is
Question: If the volume and the surface area of a sphere are numerically the same, the nits radius is(a) 1 unit(b) 2 units(c) 3 units(d) 4 units Solution: (c) 3 unitsWe have: $\frac{4}{3} \pi r^{3}=4 \pi r^{2}$ $\Rightarrow \frac{1}{3} r=1$ $\Rightarrow r=3$ units...
Read More →A cone, a hemisphere and a cylinder stand on equal bases and have the same height.
Question: A cone, a hemisphere and a cylinder stand on equal bases and have the same height. The ratio of their volumes is(a) 1 : 2 : 3(b) 2 : 1 : 3(c) 2 : 3 : 1(d) 3 : 2 : 1 Solution: (a) 1 : 2 : 3The cone, hemisphere and the cylinder stand on equal bases and have the same height.We know that radius and height of a hemisphere are the same.Hence, the height of the cone and the cylinder will be the common radius.i.e.,r = hRatio of the volumes of the cone, hemisphere and the cylinder:. $\frac{\fra...
Read More →If the sum of the coefficients of all
Question: If the sum of the coefficients of all even powers of $x$ in the product $\left(1+x+x^{2}+\ldots+x^{2 n}\right)\left(1-x+x^{2}-x^{3}+\ldots+x^{2 n}\right)$ is 61 , then $n$ is equal to___________. Solution: Let $\left(1-x+x^{2} \ldots . . x^{2 n}\right)\left(1+x+x^{2} \ldots . . x^{2 n}\right)$ $=a_{0}+a_{1} x+a_{2} x^{2}+\ldots . .$ put $x=1$ $1(2 n+1)=a_{0}+a_{1}+a_{2}+\ldots . . a_{2 n}$.......(1) put $x=-1$ $(2 n+1) \times 1=a_{0}-a_{1}+a_{2}+\ldots \ldots a_{2 n} \quad \ldots$ (2) ...
Read More →A ball weighing 10 g is moving with a velocity of 90 ms
Question: A ball weighing $10 \mathrm{~g}$ is moving with a velocity of $90 \mathrm{~ms}^{-1}$. If the uncertainty in its velocity is $5 \%$, then the uncertainty in its position is $\times 10^{-33} \mathrm{~m}$. (Rounded off to the nearest integer) [Given :$h=6.63 \times 10^{-34} \mathrm{Js}$ ] Solution: (1) $m=10 g=10^{-2} \mathrm{Kg}$ $v=90 \mathrm{~m} / \mathrm{sec}$ $\Delta \mathrm{v}=\mathrm{v} \times 5 \%=90 \times \frac{5}{100}=4.5 \mathrm{~m} / \mathrm{sec}$ $\mathrm{m} \cdot \Delta \ma...
Read More →A cone and a hemisphere have equal bases and equal volumes.
Question: A cone and a hemisphere have equal bases and equal volumes. The ratio of their heights is(a) 1 : 2(b) 2 : 1(c) 4 : 1 (d) $\sqrt{2}: 1$ Solution: (b) 2 : 1 Let the radii of the cone and the hemisphere berand their respective heights behandH. Then $\frac{1}{3} \pi \times r^{2} \times h=\frac{2}{3} \pi \times r^{2} \times H$ $\Rightarrow \frac{h}{H}=\frac{2}{1}$...
Read More →In a Frank-Hertz experiment, an electron of energy 5.6 eV
Question: In a Frank-Hertz experiment, an electron of energy $5.6 \mathrm{eV}$ passes through mercury vapour and emerges with an energy $0.7 \mathrm{eV}$. The minimum wavelength of photons emitted by mercury atoms is close to :(1) $1700 \mathrm{~nm}$(2) $2020 \mathrm{~nm}$(3) $220 \mathrm{~nm}$(4) $250 \mathrm{~nm}$Correct Option: , 4 Solution: (4) Using, wavelength, $\lambda=\frac{12375}{\Delta \mathrm{E}}$ or, $\lambda=\frac{12375}{4.9}=250 \mathrm{~nm}$...
Read More →A hemispherical bowl of radius 9 cm contains a liquid. This liquid is to be filled into cylindrical small bottles of diameter 3 cm and height 4 cm.
Question: A hemispherical bowl of radius 9 cm contains a liquid. This liquid is to be filled into cylindrical small bottles of diameter 3 cm and height 4 cm. How many bottles will be needed to empty thee bowl?(a) 27(b) 35(c) 54(d) 63 Solution: (c) 54 Number of bottles $=\frac{\text { volume of bowl }}{\text { volume of } 1 \text { bottle }}$ $=\frac{\frac{2}{3} \times \pi \times 9^{3}}{\pi \times 1.5^{2} \times 4}$ $=\frac{2 \times 729}{2.25 \times 12}=54$...
Read More →The greatest positive integer k,
Question: The greatest positive integer $k$, for which $49^{k}+1$ is a factor of the sum $49^{125}+49^{124}+\ldots+49^{2}+49+1$, is:(1) 32(2) 63(3) 60(4) 65Correct Option: , 2 Solution: $\frac{(49)^{126}-1}{48}=\frac{\left((49)^{63}+1\right)\left(49^{63}-1\right)}{48}\left[\because S_{n}=\frac{a\left(r^{n}-1\right)}{r-1}\right]$ $\therefore \mathrm{K}=63$...
Read More →When a certain photosensitive surface is illuminated with monochromatic light of frequency v,
Question: When a certain photosensistive surface is illuminated with monochromatic light of frequency $v$, the stopping potential for the photo current is $-\mathrm{V}_{0} / 2$. When the surface is illuminated by monochromatic light of frequency $v / 2$, the stoppoing potential is $-\mathrm{V}_{0}$. The threshold frequency for photoelectric emission is :(1) $\frac{5 v}{3}$(2) $\frac{4}{3} v$(3) $2 v$(4) None of the aboveCorrect Option: , 4 Solution: (4)...
Read More →The orbital having two radial as well as two angular nodes is
Question: The orbital having two radial as well as two angular nodes is$5 \mathrm{~d}$$4 \mathrm{f}$3p$4 \mathrm{~d}$Correct Option: 1 Solution: $A \cdot N \cdot=$ $\mathrm{R} \cdot \mathrm{N}=\mathrm{n}-\ell-1$...
Read More →If the constant term in the binomial expansion of
Question: If the constant term in the binomial expansion of $\left(\sqrt{x} \frac{k}{x^{2}}\right)^{10}$ is 405, then $|k|$ equals:(1) 9(2) 1(3) 3(4) 2Correct Option: , 3 Solution: General term $=T_{r+1}={ }^{10} C_{r}(\sqrt{x})^{10-r} \cdot\left(-\frac{k}{x^{2}}\right)^{\eta}$ $={ }^{10} C_{r}(-k)^{r} \cdot x^{\frac{10-r}{2}-2 r}={ }^{10} C_{r}(-k)^{r} \cdot x^{\frac{10-5 r}{2}}$ Since, it is constant term, then $\frac{10-5 r}{2}=0 \Rightarrow r=2$ $\therefore{ }^{10} C_{2}(-k)^{2}=405$ $\Right...
Read More →The volumes of the two spheres are in the ratio 64 : 27 and the sum of their radii is 7 cm.
Question: The volumes of the two spheres are in the ratio 64 : 27 and the sum of their radii is 7 cm. The difference of their total surface areas is(a) 38 cm2(b) 58 cm2(c) 78 cm2(d) 88 cm2 Solution: (d) 88 cm2Suppose that the radii of the spheres arercm and (7 r)cm. Then we have: $\frac{\frac{4}{3} \pi(7-r)^{3}}{\frac{4}{3} \pi r^{3}}=\frac{64}{27}$ $\Rightarrow \frac{(7-\tau)}{r}=\frac{4}{3}$ $\Rightarrow 21-3 r=4 r$ $\Rightarrow 21=7 r$ $\Rightarrow r=3 \mathrm{~cm}$ Now, the radii of the two ...
Read More →Electromagnetic radiation of wavelength
Question: Electromagnetic radiation of wavelength $663 \mathrm{~nm}$ is just sufficient to ionize the atom of metal $\mathrm{A}$. The ionization energy of metal $\mathrm{A}$ in $\mathrm{kJmol}^{-1}$ is _____________. (Rounded off to the nearest integer) $\left[\mathrm{h}=6.63 \times 10^{-34} \mathrm{Js}, \mathrm{c}=3.00 \times 10^{8} \mathrm{~ms}^{-1}, \mathrm{~N}_{\mathrm{A}}=6.02 \times 10^{23} \mathrm{~mol}^{-1}\right]$ Solution: (180) Whergy req. to ionize an atom of metal ' $A^{\prime}=\fra...
Read More →If {p} denotes the fractional part of the number p,
Question: If $\{p\}$ denotes the fractional part of the number $p$, then $\left\{\frac{3^{200}}{8}\right\}$, is equal to :(1) $\frac{5}{8}$(2) $\frac{7}{8}$(3) $\frac{3}{8}$(4) $\frac{1}{8}$Correct Option: , 4 Solution: $\frac{3^{200}}{8}=\frac{1}{8}\left(9^{100}\right)$ $=\frac{1}{8}(1+8)^{100}=\frac{1}{8}\left[1+n \cdot 8+\frac{n(n+1)}{2} \cdot 8^{2}+\ldots .\right]$ $=\frac{1}{8}+$ Integer $\therefore\left\{\frac{3^{200}}{8}\right\}=\left\{\frac{1}{8}+\right.$ integer $\}=\frac{1}{8}$...
Read More →In a photoelectric experiment,
Question: In a photoelectric experiment, the wavelength of the light incident on a metal is changed from $300 \mathrm{~nm}$ to $400 \mathrm{~nm}$. The decrease in the stopping potential is close to: $\left(\frac{\mathrm{hc}}{\mathrm{e}}=1240 \mathrm{~nm}-\mathrm{V}\right)$(1) $0.5 \mathrm{~V}$(2) $1.5 \mathrm{~V}$(3) $1.0 \mathrm{~V}$(4) $2.0 \mathrm{~V}$Correct Option: 3, Solution: (3) Let $\phi=$ work function of the metal, $\frac{\mathrm{hc}}{\lambda_{1}}=\phi+\mathrm{eV}_{1}$ ...(1) $\frac{\...
Read More →If {p} denotes the fractional part of the number p,
Question: If $\{p\}$ denotes the fractional part of the number $p$, then $\left\{\frac{3^{200}}{8}\right\}$, is equal to :(1) $\frac{5}{8}$(2) $\frac{7}{8}$(3) $\frac{3}{8}$(4) $\frac{1}{8}$Correct Option: , 4 Solution: $\frac{3^{200}}{8}=\frac{1}{8}\left(9^{100}\right)$...
Read More →The radius of a hemispherical balloon increases from 6 cm to 12 cm as air is being pumped into it.
Question: The radius of a hemispherical balloon increases from 6 cm to 12 cm as air is being pumped into it. The ratio of the surface areas of the balloons in two cases is(a) 1 : 4(b) 1 : 3(c) 2 : 3(d) 1 : 2 Solution: (a) 1 : 4 Ratio of the surface areas of balloon $=\frac{2 \pi \times 6^{2}}{2 \pi \times 12^{2}}=\frac{36}{144}=\frac{1}{4}$...
Read More →A spherical ball of radius 3 cm is melted and recast into three spherical balls.
Question: A spherical ball of radius 3 cm is melted and recast into three spherical balls. The radii of two of these balls are 1.5 cm and 2 cm. The radius of the third ball is(a) 1 cm(b) 1.5 cm(c) 2.5 cm(d) 0.5 cm Solution: (c) 2.5 cmLetrcm be the radius of the third ball.Volume of the original ball = volume of the three balls $\frac{4}{3} \pi \times 3^{3}=\frac{4}{3} \pi \times 1.5^{3}+\frac{4}{3} \pi \times 2^{3}+\frac{4}{3} \pi r^{3}$ $\Rightarrow 27=3.375+8+\mathrm{r}^{3}$ $\Rightarrow r^{3}...
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