Two stars of masses m and 2 m at a distance $d$ rotate about their common
Question: Two stars of masses $\mathrm{m}$ and $2 \mathrm{~m}$ at a distance $d$ rotate about their common centre of mass in free space. The period of revolution is -(1) $2 \pi \sqrt{\frac{\mathrm{d}^{3}}{3 \mathrm{Gm}}}$(2) $\frac{1}{2 \pi} \sqrt{\frac{3 G m}{d^{3}}}$(3) $\frac{1}{2 \pi} \sqrt{\frac{\mathrm{d}^{3}}{3 \mathrm{Gm}}}$(4) $2 \pi \sqrt{\frac{3 G m}{d^{3}}}$Correct Option: 1 Solution: (1) $\Rightarrow \frac{\mathrm{G}(\mathrm{m})(2 \mathrm{~m})}{\mathrm{d}^{2}}=\mathrm{m} \omega^{2} ...
Read More →Consider the following reaction
Question: Consider the following reaction $\mathrm{MnO}_{4}^{-}+8 \mathrm{H}^{+}+5 \mathrm{e}^{-} \rightarrow \mathrm{Mn}^{+2}+4 \mathrm{H}_{2} \mathrm{O}, \mathrm{E}^{\circ}=1.51 \mathrm{~V}$ The quantity of electricity required in Faraday to reduce five moles of $\mathrm{MnO}_{4}^{-}$is Solution: (25) $\mathrm{MnO}_{4}^{-}+8 \mathrm{H}^{+}+5 \mathrm{e}^{-} \rightarrow \mathrm{Mn}^{+2}+4 \mathrm{H}_{2} \mathrm{O}$ 1 mole of $\mathrm{MnO}_{4}^{-}$require 5 faraday charge 5 moles of $\mathrm{MnO}...
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Question: If $x=\frac{-1}{2}$ is a root of the quadratic equation $3 x^{2}+2 k x+3=0$, find the values of $k$. Solution: Since, $x=\frac{-1}{2}$ is a root of the quadratic equation $3 x^{2}+2 k x+3=0$, then, it must satisfies the equation. $3\left(-\frac{1}{2}\right)^{2}+2 k\left(-\frac{1}{2}\right)+3=0$ $\Rightarrow 3\left(\frac{1}{4}\right)-k+3=0$ $\Rightarrow \frac{3}{4}-k+3=0$ $\Rightarrow \frac{3-4 k+12}{4}=0$ $\Rightarrow 3-4 k+12=0$ $\Rightarrow 4 k=15$ $\Rightarrow k=\frac{15}{4}$ Hence,...
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Question: Copper reduces $\mathrm{NO}_{3}^{\text {-into }} \mathrm{NO}$ and $\mathrm{NO}_{2}$ depending upon the concentration of $\mathrm{HNO}_{3}$ in solution. (Assuming fixed $\left[\mathrm{Cu}^{2+}\right]$ and $\mathrm{P}_{\mathrm{NO}}=\mathrm{P}_{\mathrm{NO}_{2}}$ ), the $\mathrm{HNO}_{3}$ concentration at which the thermodynamic tendency for reduction of $\mathrm{NO}_{3}^{-}$into $\mathrm{NO}$ and $\mathrm{NO}_{2}$ by copper is same is $10^{x} \mathrm{M}$. The value of $2 \mathrm{x}$ is___...
Read More →Lat A =
Question: Let $\mathrm{A}=\left[\begin{array}{cc}\mathrm{i} -\mathrm{i} \\ -\mathrm{i} \mathrm{i}\end{array}\right], \mathrm{i}=\sqrt{-1}$.Then, the system of linear equations $\mathrm{A}^{8}\left[\begin{array}{l}\mathrm{x} \\ \mathrm{y}\end{array}\right]=\left[\begin{array}{c}8 \\ 64\end{array}\right]$ has :(1) A unique solution(2) Infinitely many solutions(3) No solution(4) Exactly two solutionsCorrect Option: , 3 Solution: $A=\left[\begin{array}{cc}i -i \\ -i i\end{array}\right]$ $A^{2}=\left...
Read More →The magnitude of the change in oxidising power of
Question: The magnitude of the change in oxidising power of the $\mathrm{MnO}_{4}^{-} / \mathrm{Mn}^{2+}$ couple is $\mathrm{x} \times 10^{-4} \mathrm{~V}$, if the $\mathrm{H}^{+}$concentration is decreased from $1 \mathrm{M}$ to $10^{-4} \mathrm{M}$ at $25^{\circ} \mathrm{C}$. (Assume concentration of $\mathrm{MnO}_{4}^{-}$and $\mathrm{Mn}^{2+}$ to be same on change in $\mathrm{H}^{+}$concentration). The value of $\mathrm{x}$ is______________ . (Rounded off to the nearest integer) [Given : $\fr...
Read More →Consider two satellites
Question: Consider two satellites $S_{1}$ and $S_{2}$ with periods of revolution $1 \mathrm{hr}$. and $8 \mathrm{hr}$. respectively revolving around a planet in circular orbits. The ratio of angular velocity of satellite $S_{1}$ to the angular velocity of satellite $S_{2}$ is -(1) $8: 1$(2) $1: 8$(3) $2: 1$(4) $1: 4$Correct Option: 1 Solution: (1) We know that $\omega=\frac{2 \pi}{T}$ given : Ratio of time period $\frac{\mathrm{T}_{1}}{\mathrm{~T}_{2}}=\frac{1}{8}$ $\Rightarrow \omega \propto \f...
Read More →The electrode potential of
Question: The electrode potential of $\mathrm{M}^{2+} / \mathrm{M}$ of $3 \mathrm{~d}$ - series elements shows positive value for:ZnCoFeCuCorrect Option: , 4 Solution:...
Read More →The molar conductivities at infinite dilution of barium chloride,
Question: The molar conductivities at infinite dilution of barium chloride, sulphuric arid and hydrochloric acid are 280,860 and $426 \mathrm{Scm}^{2} \mathrm{~mol}^{-1}$ respectively. The molar conductivity at infinite dilution of barium sulphate is ______________ .$\mathrm{Scm}^{2} \mathrm{~mol}^{-1}$ (Round off to the Nearest Integer). Solution: From Kohlrausch's law $\Lambda_{\mathrm{m}}^{\infty}\left(\mathrm{BaSO}_{4}\right)=\lambda_{\mathrm{m}}^{\infty}\left(\mathrm{Ba}^{2+}\right)+\lambda...
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Question: For the reaction $2 \mathrm{Fe}^{3+}(\mathrm{aq})+2 \mathrm{I}^{-}(\mathrm{aq}) \rightarrow 2 \mathrm{Fe}^{2+}(\mathrm{aq})+\mathrm{I}_{2}(\mathrm{~s})$ the magnitude of the standard molar free energy change, $\Delta_{\mathrm{r}} \mathrm{G}_{\mathrm{m}}^{\circ}=-_{\mathrm{kJ}} \mathrm{J}$ (Round off to the Nearest Integer). Solution: (45)...
Read More →Four identical particles of equal masses
Question: Four identical particles of equal masses $1 \mathrm{~kg}$ made to move along the circumference of a circle of radius $1 \mathrm{~m}$ under the action of their own mutual gravitational attraction. The speed of each particle will be(1) $\frac{\sqrt{(1+2 \sqrt{2}) G}}{2}$(2) $\sqrt{G(1+2 \sqrt{2})}$(3) $\sqrt{\frac{G}{2}(2 \sqrt{2}-1)}$(4) $\sqrt{\frac{G}{2}(1+2 \sqrt{2})}$Correct Option: 1, Solution: $\Rightarrow$ By resolving force $\mathrm{F}_{2}$, we get $\Rightarrow \mathrm{F}_{1}+\m...
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Question: A $\mathrm{KC}$ solution of conductivity $0.14 \mathrm{~S} \mathrm{~m}^{-1}$ shows a resistance of $4.19 \Omega$ in a conductivity cell. If the same cell is filled with an $\mathrm{HCl}$ solution, the resistance drops to $1.03 \Omega$. The conductivity of the $\mathrm{HC} 1$ solution is____________ $\times 10^{-2} \mathrm{~S} \mathrm{~m}^{-1}$. (Round off to the Nearest Integer). Solution: (57) $\kappa=\frac{1}{R} \cdot G^{*}$ For same conductivity cell, $\mathrm{G}^{*}$ is constant an...
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Question: A $5.0 \mathrm{~m}$ moldm $^{-3}$ aqueous solution of $\mathrm{KCl}$ has a conductance of $0.55 \mathrm{mS}$ when measured in a cell constant $1.3 \mathrm{~cm}^{-1}$. The molar conductivity of this solution is________________ . $\mathrm{mSm}^{2} \mathrm{~mol}^{-1}$. (Round off to the Nearest Integer) Solution: (14.3) Given conc ${ }^{\mathrm{n}}$ of $\mathrm{KCl}=\frac{\mathrm{m} \cdot \mathrm{mol}}{\mathrm{L}}$ $\therefore$ Conductance $(\mathrm{G})=0.55 \mathrm{mS}$ Cell constant $\l...
Read More →The angular momentum of a planet of mass M moving around the sun in
Question: The angular momentum of a planet of mass $M$ moving around the sun in an elliptical orbit is $\vec{L}$. The magnitude of the areal velocity of the planet is : (1) $\frac{4 \mathrm{~L}}{\mathrm{M}}$(2) $\frac{\mathrm{L}}{\mathrm{M}}$(3) $\frac{2 L}{M}$(4) $\frac{\mathrm{L}}{2 \mathrm{M}}$Correct Option: , 4 Solution: For small displacement ds of the planet its area can be written as $\mathrm{dA}=\frac{1}{2} \mathrm{rd} \ell$ $=\frac{1}{2} \mathrm{rds} \sin \theta$ A.vel $=\frac{\mathrm{...
Read More →If the roots of the equations
Question: If the roots of the equations $a x^{2}+2 b x+c=0$ and $b x^{2}-2 \sqrt{a c} x+b=0$ are simultaneously real then prove that $b^{2}=a c$. Solution: It is given that the roots of the equation $a x^{2}+2 b x+c=0$ are real. $\therefore D_{1}=(2 b)^{2}-4 \times a \times c \geq 0$ $\Rightarrow 4\left(b^{2}-a c\right) \geq 0$ $\Rightarrow b^{2}-a c \geq 0$ $\ldots \ldots(1)$ Also, the roots of the equation $b x^{2}-2 \sqrt{a c} x+b=0$ are real. $\therefore D_{2}=(-2 \sqrt{a c})^{2}-4 \times b ...
Read More →If the angular velocity of earth's spin is increased such that the bodies at the
Question: If the angular velocity of earth's spin is increased such that the bodies at the equator start floating, the duration of the day would be approximately : (Take : $\mathrm{g}=10 \mathrm{~ms}^{-2}$, the radius of earth, $\mathrm{R}=6400 \times 10^{3} \mathrm{~m}$, Take $\pi=3.14)$(1) 60 minutes(2) does not change(3) 1200 minutes(4) 84 minutesCorrect Option: , 4 Solution: (4) For objects to float $\mathrm{mg}=\mathrm{m} \omega^{2} \mathrm{R}$ $\omega=$ angular velocity of earth. $\mathrm{...
Read More →Which one of the following statements is not a tautology?
Question: Which one of the following statements is not a tautology?(1) $(p \vee q) \rightarrow(p \vee(\sim q))$(2) $(p \wedge q) \rightarrow(\sim p) \vee q$(3) $p \rightarrow(p \vee q)$(4) $(p \wedge q) \rightarrow p$Correct Option: 1 Solution: By truth table :...
Read More →The element with Z=120
Question: The element with $Z=120$ (not yet discovered) will be an/a:Inner-transition metalAlkaline earth metalAlkali metalTransition metalCorrect Option: , 2 Solution: Elements with $Z=120$ will belong to alkaline earth metals. Its electronic configuration may be represented as [Og] $8 s^{2}$....
Read More →If a and b are real and a ≠ b then show that the roots of the equation
Question: If $a$ and $b$ are real and $a \neq b$ then show that the roots of the equation $(a-b) x^{2}+5(a+b) x-2(a-b)=0$ are real and unequal. Solution: The given equation is $(a-b) x^{2}+5(a+b) x-2(a-b)=0$. $\therefore D=[5(a+b)]^{2}-4 \times(a-b) \times[-2(a-b)]$ $=25(a+b)^{2}+8(a-b)^{2}$ Since $a$ and $b$ are real and $a \neq b$, so $(a-b)^{2}0$ and $(a+b)^{2}0$. $\therefore 8(a-b)^{2}0$ ..........(1) (Product of two positive numbers is always positive) Also, $25(a+b)^{2}0$ ..........(2) (Pr...
Read More →The time period of a satellite in a circular orbit of radius
Question: The time period of a satellite in a circular orbit of radius $\mathrm{R}$ is $\mathrm{T}$. The period of another satellite in a circular orbit of radius $9 \mathrm{R}$ is:(1) $9 \mathrm{~T}$(2) $27 \mathrm{~T}$(3) $12 \mathrm{~T}$(4) $3 \mathrm{~T}$Correct Option: , 2 Solution: (2) $\mathrm{T}^{2} \propto \mathrm{R}^{3}$ $\left(\frac{\mathrm{T}^{\prime}}{\mathrm{T}}\right)^{2}=\left(\frac{9 \mathrm{R}}{\mathrm{R}}\right)^{3}$ $\mathrm{~T}^{2}=\mathrm{T}^{2} \times 9^{3}$ $\mathrm{~T}=\...
Read More →The contrapositive of the statement
Question: The contrapositive of the statement "If you are born in India, then you are a citizen of India", is :(1) If you are not a citizen of India, then you are not born in India.(2) If you are a citizen of India, then you are born in India.(3) If you are born in India, then you are not a citizen of India.(4) If you are not born in India, then you are not a citizen of India.Correct Option: 1 Solution: S: "If you are born in India, then you are a citizen of India." Contrapositive of $p \rightar...
Read More →In the above sequence of reactions, A and D, respectively, are:
Question: In the above sequence of reactions, $\mathrm{A}$ and $\mathrm{D}$, respectively, are:$\mathrm{KI}$ and $\mathrm{KMnO}_{4}$$\mathrm{MnO}_{2}$ and $\mathrm{KIO}_{3}$$\mathrm{KIO}_{3}$ and $\mathrm{MnO}_{2}$$\mathrm{KI}$ and $\mathrm{K}_{2} \mathrm{MnO}_{4}$Correct Option: , 2 Solution:...
Read More →In the above sequence of reactions, $mathrm{A}$ and $mathrm{D}$, respectively, are:
Question: In the above sequence of reactions, $\mathrm{A}$ and $\mathrm{D}$, respectively, are:$\mathrm{KI}$ and $\mathrm{KMnO}_{4}$$\mathrm{MnO}_{2}$ and $\mathrm{KIO}_{3}$$\mathrm{KIO}_{3}$ and $\mathrm{MnO}_{2}$$\mathrm{KI}$ and $\mathrm{K}_{2} \mathrm{MnO}_{4}$Correct Option: , 2 Solution:...
Read More →if
Question: If $p \rightarrow(p \wedge \sim q)$ is false, then the truth values of $p$ and $q$ are respectively:(1) $\mathrm{F}, \mathrm{F}$(2) $\mathrm{T}, \mathrm{F}$(3) $\mathrm{T}, \mathrm{T}$(4) $\mathrm{F}, \mathrm{T}$Correct Option: , 3 Solution:...
Read More →A geostationary satellite is orbiting around an arbitary planet
Question: A geostationary satellite is orbiting around an arbitary planet ' $\mathrm{P}$ ' at a height of $11 \mathrm{R}$ above the surface of ' $P$ ', $R$ being the radius of ' $P$ '. The time period of another satellite in hours at a height of $2 \mathrm{R}$ from the surface of ' $\mathrm{P}$ ' is has the time period of 24 hours.(1) $6 \sqrt{2}$(2) $\frac{6}{\sqrt{2}}$(3) 3(4) 5Correct Option: Solution: (3) $\mathrm{T} \propto \mathrm{R}^{3 / 2}$ $\frac{24}{\mathrm{~T}}=\left(\frac{12 \mathrm{...
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