A nucleus A with a finite de-broglie wavelength
Question: A nucleus $A$, with a finite de-broglie wavelength $\lambda_{A}$, undergoes spontaneous fission into two nuclei $\mathrm{B}$ and $\mathrm{C}$ of equal mass. B flies in the same direction as that of A, while C flies in the opposite direction with a velocity equal to half of that of $\mathrm{B}$. the de-Broglie wavelengths $\lambda_{\mathrm{B}}$ and $\lambda_{\mathrm{C}}$ of $\mathrm{B}$ and $\mathrm{C}$ are respectively :(1) $\lambda_{\mathrm{A}}, 2 \lambda_{\mathrm{A}}$(2) $2 \lambda_{...
Read More →The electron gain enthalpy (in kj / mol) of fluorine,
Question: The electron gain enthalpy (in $\mathrm{kj} / \mathrm{mol}$ ) of fluorine, chlorine, bromine and iodine, respectively, are:$-296,-325,-333$ and $-349$$-349,-333,-325$ and $-296$$-333,-349,-325$ and $-296$$-333,-325,-349$ and $-296$Correct Option: , 3 Solution: Chlorine has highest electron gain enthalpy (most negative) among the given elements, the electron gain enthalpy decreases down the group i.e., moves to least negative....
Read More →If for a>0, the feet of perpendiculars
Question: If for $\mathrm{a}0$, the feet of perpendiculars from the points $\mathrm{A}(\mathrm{a},-2 \mathrm{a}, 3)$ and $\mathrm{B}(0,4,5)$ on the plane $l_{\mathrm{X}}+\mathrm{my}+\mathrm{nz}=0$ are points $\mathrm{C}(0,-\mathrm{a},-1)$ and $\mathrm{D}$ respectively, then the length of line segment $C D$ is equal to :(1) $\sqrt{31}$(2) $\sqrt{41}$(3) $\sqrt{55}$(4) $\sqrt{66}$Correct Option: , 4 Solution: C lies on plane $\Rightarrow-m a-n=0 \Rightarrow \frac{m}{n}=-\frac{1}{a} \ldots$ (1) $\o...
Read More →The ratio of the mass densities of nuclei
Question: The ratio of the mass densities of nuclei of ${ }^{40} \mathrm{Ca}$ and ${ }^{16} \mathrm{O}$ is close to :(1) 1(2) $0.1$(3) 5(4) 2Correct Option: 1 Solution: (1) Nuclear density is independent of atomic number....
Read More →Reaction of an inorganic sulphite X with
Question: Reaction of an inorganic sulphite $\mathrm{X}$ with dilute $\mathrm{H}_{2} \mathrm{SO}_{4}$ generates compound Y. Reaction of $Y$ with $\mathrm{NaOH}$ gives $X$. Further, the reaction of $\mathrm{X}$ with $\mathrm{Y}$ and water affords compound Z. Y and Z, respectively, are:$\mathrm{SO}_{2}$ and $\mathrm{Na}_{2} \mathrm{SO}_{3}$$\mathrm{SO}_{3}$ and $\mathrm{NaHSO}_{3}$$\mathrm{SO}_{2}$ and $\mathrm{NaHSO}_{3}$$\mathrm{S}$ and $\mathrm{Na}_{2} \mathrm{SO}_{3}$Correct Option: , 3 Soluti...
Read More →The activity of a radioactive sample falls from
Question: The activity of a radioactive sample falls from $700 \mathrm{~s}^{-1}$ to $500 \mathrm{~s}^{-1}$ in 30 minutes. Its half life is close to:(1) $72 \mathrm{~min}$(2) $62 \mathrm{~min}$(3) $66 \mathrm{~min}$(4) $52 \mathrm{~min}$Correct Option: , 2 Solution: (2) We know that Activity, $A=A_{0} e^{-\lambda t}$ $A=A_{0} e^{-t \operatorname{In} 2 / T_{1 / 2}}\left(\because \lambda=\frac{\operatorname{In}_{2}}{T_{1 / 2}}\right)$ $\Rightarrow 500=700 e^{-t \ln 2 / T_{1 / 2}}$ $\Rightarrow \ope...
Read More →The reaction of NO with
Question: The reaction of $\mathrm{NO}$ with $\mathrm{N}_{2} \mathrm{O}_{4}$ at $250 \mathrm{~K}$ gives :$\mathrm{N}_{2} \mathrm{O}$$\mathrm{NO}_{2}$$\mathrm{N}_{2} \mathrm{O}_{3}$$\mathrm{N}_{2} \mathrm{O}_{5}$Correct Option: , 3 Solution:...
Read More →Given the masses of various atomic particles Mp=1.0072
Question: Given the masses of various atomic particles $m_{p}=1.0072$ $\mathrm{u}, \mathrm{m}_{\mathrm{n}}=1.0087 \mathrm{u}, \mathrm{m}_{\mathrm{e}}=0.000548 \mathrm{u}, \mathrm{m}_{v}^{-}=0, \mathrm{~m}_{\mathrm{d}}=2.0141 \mathrm{u}$, where $\mathrm{p} \equiv$ proton, $\mathrm{n} \equiv$ neutron, $\mathrm{e} \equiv$ electron, $\bar{v} \equiv$ antineutrino and $\mathrm{d} \equiv$ deuteron. Which of the following process is allowed by momentum and energy conservation?(1) $n+n \rightarrow$ deute...
Read More →The number of Cl=O bonds in perchloric acid is,
Question: The number of $\mathrm{Cl}=\mathrm{O}$ bonds in perchloric acid is, "________________" Solution: (3) The structure of perchloric acid is The number $\mathrm{Cl}=\mathrm{O}$ bond in $\mathrm{HClO}_{4}$ is 3 ....
Read More →You are given that mass
Question: You are given that mass of ${ }_{3}^{7} \mathrm{Li}=7.0160 \mathrm{u}$ Mass of ${ }_{2}^{4} \mathrm{He}=4.0026 \mathrm{u}$ and Mass of ${ }_{1}^{1} \mathrm{H}=1.0079 \mathrm{u}$ When $20 \mathrm{~g}$ of ${ }_{3}^{7} \mathrm{Li}$ is converted into ${ }_{2}^{4} \mathrm{He}$ by proton capture, the energy liberated, (in $\mathrm{kWh}$ ), is : $\left[\right.$ Mass of nucleon $\left.=1 \mathrm{GeV} / \mathrm{c}^{2}\right]$ (1) $4.5 \times 10^{5}$(2) $8 \times 10^{6}$(3) $6.82 \times 10^{5}$(...
Read More →Prove the following
Question: If $x=\sum_{n=0}^{\infty}(-1)^{n} \tan ^{2 \mathrm{n}} \theta$ and $y=\sum_{n=0}^{\infty} \cos ^{2 n} \theta$, for $0\theta\frac{\pi}{4}$ then :(1) $x(1+y)=1$(2) $y(1-x)=1$(3) $y(1+x)=1$(4) $x(1-y)=1$Correct Option: , 2 Solution: $y=1+\cos ^{2} \theta+\cos ^{4} \theta+\ldots . .$ $\Rightarrow y=\frac{1}{1-\cos ^{2} \theta} \Rightarrow \frac{1}{y}=\sin ^{2} \theta$ $x=1-\tan ^{2} \theta+\tan ^{4} \theta+\ldots \ldots$ $x=\frac{1}{1-\left(-\tan ^{2} \theta\right)}=\frac{1}{\sec ^{2} \the...
Read More →The presence of soluble fluoride ion upto
Question: The presence of soluble fluoride ion upto $1 \mathrm{ppm}$ concentration in drinking water, is :harmful for teethharmful to skinharmful to bonessafe for teethCorrect Option: , 4 Solution: Fluoride $\left(\mathrm{F}^{-}\right)$ion conc. upto $1 \mathrm{ppm}$ makes the enamel of teeth much harder by converting But above 2 ppm cause brown motting of teeth....
Read More →The value of
Question: The value of $\cos ^{3}\left(\frac{\pi}{8}\right) \cdot \cos \left(\frac{3 \pi}{8}\right)+\sin ^{3}\left(\frac{\pi}{8}\right) \cdot \sin \left(\frac{3 \pi}{8}\right)$ is(1) $\frac{1}{\sqrt{2}}$(2) $\frac{1}{2 \sqrt{2}}$(3) $\frac{1}{2}$(4) $\frac{1}{4}$Correct Option: , 2 Solution: $\cos ^{3} \frac{\pi}{8}\left[4 \cos ^{3} \frac{\pi}{8}-3 \cos \frac{\pi}{8}\right]$ $+\sin ^{3} \frac{\pi}{8}\left[3 \sin \frac{\pi}{8}-4 \sin ^{3} \frac{\pi}{8}\right]$ $=4 \cos ^{6} \frac{\pi}{8}-4 \sin ^...
Read More →The correct statement with respect to dinitrogen is:
Question: The correct statement with respect to dinitrogen is:$\mathrm{N}_{2}$ is paramagnetic in nature.it can combine with dioxygen at $25^{\circ} \mathrm{C}$.liquid dinitrogen is not used in cryosurgery.it can be used as an inert diluent for reactive chemicals.Correct Option: , 4 Solution: (a) $\mathrm{N}_{2}$ is diamagnetic in nature. (c) Liquid $\mathrm{N}_{2}$ is used in cryosurgery. (d) Because of its inertness, it is used where an inert atmosphere is required....
Read More →Prove the following
Question: If $\frac{\sqrt{2} \sin \alpha}{\sqrt{1+\cos 2 \alpha}}=\frac{1}{7}$ and $\sqrt{\frac{1-\cos 2 \beta}{2}}=\frac{1}{\sqrt{10}}$ $\alpha, \beta \in\left(0, \frac{\pi}{2}\right)$, then $\tan (\alpha+2 \beta)$ is equal to Solution: $\frac{\sqrt{2} \sin \alpha}{\sqrt{2} \cos \alpha}=\frac{1}{7}$ and $\sqrt{\frac{1-\cos ^{2} \beta}{2}}=\frac{1}{10}$ $\Rightarrow \quad \frac{\sqrt{2} \sin \beta}{\sqrt{2}}=\frac{1}{\sqrt{10}}$ $\therefore \quad \tan \alpha=\frac{1}{7}$ and $\sin \beta=\frac{1}...
Read More →A radioactive nucleus decays by two different processes.
Question: A radioactive nucleus decays by two different processes. The half life for the first process is $10 \mathrm{~s}$ and that for the second is $100 \mathrm{~s}$. The effective half life of the nucleus is close to :(1) $9 \mathrm{sec}$.(2) $6 \mathrm{sec}$.(3) $55 \mathrm{sec}$(4) $12 \mathrm{sec} .$Correct Option: 1 Solution: (1) Let $\lambda_{1}$ and $\lambda_{2}$ be the decay constants of two process. $N$ be the number of nuclei left undecayed after two process. From the law of radioact...
Read More →Reaction of ammonia with excess
Question: Reaction of ammonia with excess $\mathrm{Cl}_{2}$ gives :$\mathrm{NH}_{4} \mathrm{Cl}$ and $\mathrm{N}_{2}$$\mathrm{NH}_{4} \mathrm{Cl}$ and $\mathrm{HCl}$$\mathrm{NCl}_{3}$ and $\mathrm{NH}_{4} \mathrm{Cl}$$\mathrm{NCl}_{3}$ and $\mathrm{HCl}$Correct Option: , 4 Solution:...
Read More →Acitvities of three radioactive substances A, B and C are represented by the curves A, B and C, in the figure.
Question: Acitvities of three radioactive substances A, B and C are represented by the curves $A, B$ and $C$, in the figure. Then their half-lives $\mathrm{T}_{\frac{1}{2}}(\mathrm{~A}): \mathrm{T}_{\frac{1}{2}}(\mathrm{~B}): \mathrm{T}_{\frac{1}{2}}(\mathrm{C})$ are in the ratio: (1) $2: 1: 1$(2) $3: 2: 1$(3) $2: 1: 3$(4) $4: 3: 1$Correct Option: , 3 Solution: (3) Since, $R=R_{0} e^{-\lambda t}$ $\ln R=\ln R_{0}+(-\lambda \ln t)$ $\lambda=\frac{\ln 2}{t_{1 / 2}}=$ Slope $\lambda_{A}=\frac{6}{10...
Read More →Prove the following
Question: If $L=\sin ^{2}\left(\frac{\pi}{16}\right)-\sin ^{2}\left(\frac{\pi}{8}\right)$ and $M=\cos ^{2}\left(\frac{\pi}{16}\right)-\sin ^{2}\left(\frac{\pi}{8}\right)$, then :(1) $L=-\frac{1}{2 \sqrt{2}}+\frac{1}{2} \cos \frac{\pi}{8}$(2) $L=\frac{1}{4 \sqrt{2}}-\frac{1}{4} \cos \frac{\pi}{8}$(3) $M=\frac{1}{4 \sqrt{2}}+\frac{1}{4} \cos \frac{\pi}{8}$(4) $M=\frac{1}{2 \sqrt{2}}+\frac{1}{2} \cos \frac{\pi}{8}$Correct Option: , 4 Solution: $\mathrm{L}+\mathrm{M}=1-2 \sin ^{2} \frac{\pi}{8}=\cos...
Read More →The reaction in which the hybridisation of the underlined atom is affected is :
Question: The reaction in which the hybridisation of the underlined atom is affected is :Correct Option: , 4 Solution:...
Read More →If the equation
Question: If the equation $\cos ^{4} \theta+\sin ^{4} \theta+\lambda=0$ has real solutions for $\theta$, then $\lambda$ lies in the interval :(1) $\left(-\frac{5}{4},-1\right)$(2) $\left[-1,-\frac{1}{2}\right]$(3) $\left(-\frac{1}{2},-\frac{1}{4}\right]$(4) $\left[-\frac{3}{2},-\frac{5}{4}\right]$Correct Option: , 2 Solution: $\sin ^{4} \theta+\cos ^{4} \theta=-\lambda$ $\Rightarrow\left(\sin ^{2} \theta+\cos ^{2} \theta\right)^{2}-2 \sin ^{2} \theta \cdot \cos ^{2} \theta=-\lambda$ $\Rightarrow...
Read More →Prove the following
Question: If $\$ 0$(1) $\frac{1+\sqrt{3}}{2}$(2) $\frac{1-\sqrt{3}}{2}$(3) $\frac{\sqrt{3}}{2}$(4) $\frac{1}{2}$Correct Option: 1 Solution: $2 \cos \left(\frac{x+y}{2}\right) \cos \left(\frac{x-y}{2}\right)-\left[2 \cos ^{2}\left(\frac{x+y}{2}\right)-1\right]=\frac{3}{2}$ $2 \cos \left(\frac{x+y}{2}\right)\left[\cos \left(\frac{x-y}{2}\right)-\cos \left(\frac{x+y}{2}\right)\right]=\frac{1}{2}$ $2 \cos \left(\frac{x+y}{2}\right)\left[2 \sin \left(\frac{x}{2}\right) \cdot \sin \left(\frac{y}{2}\ri...
Read More →On heating, lead (II) nitrate gives a brown gas (A).
Question: On heating, lead (II) nitrate gives a brown gas (A). The gas (A) on cooling changes to a colourless solid/liquid (B). (B) on heating with NO changes to a blue solid (C). The oxidation number of nitrogen in solid (C) is :$+5$$+2$$+3$$+4$Correct Option: , 3 Solution:...
Read More →Find the Binding energy per neucleon
Question: Find the Binding energy per neucleon for ${ }_{50}^{120} \mathrm{Sn}$. Mass of proton $m_{p}=1.00783 \mathrm{U}$, mass of neutron $m_{n}=1.00867 \mathrm{U}$ and mass of tin nucleus $m_{\text {Sn }}=119.902199 \mathrm{U}$. $($ take $1 \mathrm{U}=931 \mathrm{MeV})$ (1) $7.5 \mathrm{MeV}$(2) $9.0 \mathrm{MeV}$(3) $8.0 \mathrm{MeV}$(4) $8.5 \mathrm{MeV}$Correct Option: , 4 Solution: (4) Mass defect, $\Delta m=\left(50 m_{p}+70 m_{n}\right)-\left(m_{s n}\right)$ $=(50 \times 1.00783+70 \tim...
Read More →Prove the following
Question: If $15 \sin ^{4} \alpha+10 \cos ^{4} \alpha=6$, for some $\alpha \in \mathrm{R}$, then the value of $27 \sec ^{6} \alpha+8 \operatorname{cosec}^{6} \alpha$ is equal to:(1) 350(2) 500(3) 400(4) 250Correct Option: , 4 Solution: $15 \sin ^{4} \alpha+10 \cos ^{4} \alpha=6$ $15 \sin ^{4} \alpha+10 \cos ^{4} \alpha=6\left(\sin ^{2} \alpha+\cos ^{2} \alpha\right)^{2}$ $\left(3 \sin ^{2} \alpha-2 \cos ^{2} \alpha\right)^{2}=0$ $\tan ^{2} \alpha=\frac{2}{3} \cdot \cot ^{2} \alpha=\frac{3}{2}$ $...
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