For the reaction
Question: For the reaction $2 \mathrm{~A}+\mathrm{B} \rightarrow \mathrm{C}$, the values of initial rate at different reactant concentrations are given in the table below. The rate law for the reaction is : Rate $=k[\mathrm{~A}][\mathrm{B}]^{2}$Rate $=k[\mathrm{~A}]^{2}[\mathrm{~B}]$Rate $=k[\mathrm{~A}][\mathrm{B}]$Rate $=k[\mathrm{~A}]^{2}[\mathrm{~B}]$Correct Option: 1 Solution: $2 \mathrm{~A}+\mathrm{B} \longrightarrow \mathrm{C}$ Rate $=k[\mathrm{~A}]^{x}[\mathrm{~B}\rangle^{y}$ Exp-1, $0.0...
Read More →Let f be a twice differentiable function defined on
Question: Let $f$ be a twice differentiable function defined on $\mathrm{R}$ such that $f(0)=1, f^{\prime}(0)=2$ and $f^{\prime}(x) \neq 0$ for all $\mathrm{X} \in \mathrm{R}$. If $\left|\begin{array}{ll}f(x) f^{\prime}(x) \\ f^{\prime}(x) f^{\prime \prime}(x)\end{array}\right|=0$, for all $\mathrm{x} \in \mathrm{R}$ then the value of $f(1)$ lies in the interval:(1) $(9,12)$(2) $(6,9)$(3) $(3,6)$(4) $(0,3)$Correct Option: , 2 Solution: Given $f(x) f^{\prime \prime}(X)-\left(f^{\prime}(x)\right)^...
Read More →The population P=P(t) at time
Question: The population $P=P(t)$ at time ' $t^{\prime}$ of a certain species follows the differential equation $\frac{d P}{d t}=0.5 \mathrm{P}-450$. If $\mathrm{P}(0)=850$, then the time at which population becomes zero is :(1) $\frac{1}{2} \log _{e} 18$(2) $2 \log _{\mathrm{e}} 18$(3) $\log _{\mathrm{e}} 9$(4) $\log _{\mathrm{e}} 18$Correct Option: , 2 Solution: $\frac{d p}{d t}=\frac{p-900}{2}$ $\int_{850}^{0} \frac{d p}{p-900}=\int_{0}^{t} \frac{d t}{2}$ $\ell n \mid P-900 \|_{850}^{0}=\frac...
Read More →A sample of milk splits after
Question: A sample of milk splits after $60 \mathrm{~min}$. at $300 \mathrm{~K}$ and after $40 \mathrm{~min}$. at $400 \mathrm{~K}$ when the population of lactobacillus acidophilus in it doubles. The activation energy (in $\mathrm{kJ} / \mathrm{mol}$ ) for this process is closest to _______________. (Given, $\mathrm{R}=8.3 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}, \ln \left(\frac{2}{3}\right)=0.4, \mathrm{e}^{-3}=4.0$ ) Solution: (3.98) For a first order reaction, $k t=\ln \frac{[\mathrm{...
Read More →Define
Question: Define (i) rational numbers (ii) irrational numbers (iii) real numbers. Solution: Rational numbers: The numbers of the form $\frac{p}{q}$ where $p, q$ are integers and $q \neq 0$ are called rational numbers. Example: $\frac{2}{3}$ Irrational numbers: The numbers which when expressed in decimal form are expressible as non-terminating and non-repeating decimals are called irrational numbers. Example: $\sqrt{2}$ Real numbers: The numbers which are positive or negative, whole numbers or de...
Read More →Let y=y(x) be the solution of the differential equation
Question: Let $y=y(x)$ be the solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}}=(\mathrm{y}+1)\left((\mathrm{y}+1) \mathrm{e}^{\mathrm{x}^{2} / 2}-\mathrm{x}\right), 0\mathrm{x}2.1$, with $\mathrm{y}(2)=0$. Then the value of $\frac{\mathrm{dy}}{\mathrm{dx}}$ at $\mathrm{x}=1$ is equal to:(1) $\frac{-\mathrm{e}^{3 / 2}}{\left(\mathrm{e}^{2}+1\right)^{2}}$(2) $-\frac{2 \mathrm{e}^{2}}{\left(1+\mathrm{e}^{2}\right)^{2}}$(3) $\frac{\mathrm{e}^{5 / 2}}{\left(1+\mathrm{e}^{2}\righ...
Read More →For following reactions:
Question: For following reactions: it was found that the $E_{a}$ is decrease by $30 \mathrm{~kJ} / \mathrm{mol}$ in the presence of catalyst. If the rate remains unchanged, the activation energy for catalysed reaction is (Assume pre exponential factor is same):$75 \mathrm{~kJ} / \mathrm{mol}$$105 \mathrm{~kJ} / \mathrm{mol}$$135 \mathrm{~kJ} / \mathrm{mol}$$198 \mathrm{~kJ} / \mathrm{mol}$Correct Option: 1 Solution: Given: $k_{1}=k_{2}$ $\mathrm{Ae}^{-\frac{\mathrm{E}_{a_{1}}}{\mathrm{RT}_{1}}}=...
Read More →State whether the given statement is true of false:
Question: State whether the given statement is true of false:(i) The sum of two rationals is always rational.(ii) The product of two rationals is always rational.(iii) The sum of two irrationals is an irrational.(iv) The product of two irrationals is an irrational.(v) The sum of a rational and and irrational is irrational.(vi) The product of a rational and an irrational is irrational. Solution: (i) True(ii) True(iii) False Counter example: $2+\sqrt{3}$ and $2-\sqrt{3}$ are two irrational numbers...
Read More →Solve this
Question: A $100 \mathrm{~V}$ carrier wave is made to vary between $160 \mathrm{~V}$ and $40 \mathrm{~V}$ by a modulating signal. What is the modulation index?(1) $0.3$(2) $0.5$(3) $0.6$(4) $0.4$Correct Option: 3, Solution: (3) Maximum amplitude $=\mathrm{E}_{m}+\mathrm{E}_{c}=160$ $E_{m}+100=160$ $E_{m}^{\prime}=160-100=60$ Modulation index, $\mu=\frac{E_{m}}{E_{c}}=\frac{60}{100}$ $\mu=0.6$...
Read More →(i) Give an example of two irrationals whose sum is rational.
Question: (i) Give an example of two irrationals whose sum is rational.(ii) Give an examples of two irrationals whose product is rational. Solution: (i) Let $(2+\sqrt{3)},(2-\sqrt{3})$ be two irrationals. $\therefore(2+\sqrt{3})+(2-\sqrt{3})=4=$ rational number (ii) Let $2 \sqrt{3}, 3 \sqrt{3}$ be two irrationals. $\therefore 2 \sqrt{3} \times 3 \sqrt{3}=18=$ rational number...
Read More →The differential equation satisfied by the system of parabolas
Question: The differential equation satisfied by the system of parabolas $y^{2}=4 a(x+a)$ is:(1) $y\left(\frac{d y}{d x}\right)_{2}^{2}-2 x\left(\frac{d y}{d x}\right)-y=0$(2) $y\left(\frac{d y}{d x}\right)_{2}^{2}-2 x\left(\frac{d y}{d x}\right)+y=0$(3) $y\left(\frac{d y}{d x}\right)^{2}+2 x\left(\frac{d y}{d x}\right)-y=0$(4) $y\left(\frac{d y}{d x}\right)+2 x\left(\frac{d y}{d x}\right)-y=0$Correct Option: , 3 Solution: $y^{2}=4 a x+4 a^{2}$ differentiate with respect to $x$ $\Rightarrow 2 y ...
Read More →Prove that
Question: Prove that $\frac{2}{\sqrt{7}}$ is irrational. Solution: $\frac{2}{\sqrt{7}}=\frac{2}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}}=\frac{2}{7} \sqrt{7}$ Let $\frac{2}{7} \sqrt{7}$ is a rational number. $\therefore \frac{2}{7} \sqrt{7}=\frac{p}{q}$, where $p$ and $q$ are some integers and $\operatorname{HCF}(p, q)=1$ $\Rightarrow 2 \sqrt{7} q=7 p$ $\Rightarrow(2 \sqrt{7} q)^{2}=(7 p)^{2}$ $\Rightarrow 7\left(4 q^{2}\right)=49 p^{2}$ $\Rightarrow 4 q^{2}=7 p^{2}$ $\Rightarrow q^{2}$ is div...
Read More →Consider the following plots of rate constant versus
Question: Consider the following plots of rate constant versus $\frac{1}{T}$ for four different reactions. Which of the following orders is correct for the activation energies of these reactions? $E_{b}E_{a}E_{d}E_{c}$$E_{a}E_{c}E_{d}E_{b}$$E_{c}E_{a}E_{d}E_{b}$$E_{b}E_{d}E_{c}E_{a}$Correct Option: , 3 Solution: Arrhenius equation, $k=\mathrm{A} e^{-\mathrm{Ea} / \mathrm{RT}}$ $\log k=\log \mathrm{A}-\frac{\mathrm{E}_{a}}{2.303 \mathrm{RT}}$ slope $=-\frac{E_{a}}{2.303 R}$ $\therefore$ More nega...
Read More →If the curve y=y(x) is the solution
Question: If the curve $y=y(x)$ is the solution of the differential equation $2\left(x^{2}+x^{5 / 4}\right) d y-y\left(x+x^{1 / 4}\right) d x=2 x^{9 / 4} d x, x0$ which passes through the point $\left(1,1-\frac{4}{3} \log _{e} 2\right)$, then the value of $y(16)$ is equal to:(1) $4\left(\frac{31}{3}+\frac{8}{3} \log _{\mathrm{e}} 3\right)$(2) $\left(\frac{31}{3}+\frac{8}{3} \log _{\mathrm{e}} 3\right)$(3) $4\left(\frac{31}{3}-\frac{8}{3} \log _{\mathrm{e}} 3\right)$(4) $\left(\frac{31}{3}-\frac{...
Read More →To double the covering range of a TV transmittion tower, its height should be multiplied by:
Question: To double the covering range of a TV transmittion tower, its height should be multiplied by:(1) $\frac{1}{\sqrt{2}}$(2) 2(3) 4(4) $\sqrt{2}$Correct Option: , 3 Solution: (3) As we know, Range $=\sqrt{2 \mathrm{hR}}$ therefore to double the range height ' $h$ ' should be 4 times....
Read More →Prove that
Question: Prove that $\frac{1}{\sqrt{3}}$ is irrational. Solution: Let $\frac{1}{\sqrt{3}}$ be rational. $\therefore \frac{1}{\sqrt{3}}=\frac{a}{b}$, where $a, b$ are positive integers having no common factor other than 1 $\therefore \sqrt{3}=\frac{b}{a}$ .................(1) Since $a, b$ are non-zero integers, $\frac{b}{a}$ is rational. Thus, equation (1) shows that $\sqrt{3}$ is rational. This contradicts the fact that $\sqrt{3}$ is rational. Hence, $\frac{1}{\sqrt{3}}$ is irrational....
Read More →An amplitude modulated signal is plotted below:
Question: An amplitude modulated signal is plotted below: Which one of the following best describes the above signal?(1) $\left(9+\sin \left(2.5 \pi \times 10^{5} \mathrm{t}\right)\right) \sin \left(2 \pi \times 10^{4} \mathrm{t}\right) \mathrm{V}$(2) $\left(1+9 \sin \left(2 \pi \times 10^{4} \mathrm{t}\right)\right) \sin \left(2.5 \pi \times 10^{5} \mathrm{t}\right) \mathrm{V}$(3) $\left(9+\sin \left(2 \pi \times 10^{4} \mathrm{t}\right)\right) \sin \left(2.5 \pi \times 10^{5} \mathrm{t}\right)...
Read More →The rate of a certain biochemical reaction at physiological temperature (T)
Question: The rate of a certain biochemical reaction at physiological temperature $(T)$ occurs $10^{6}$ times faster with enzyme than without. The change in the activation energy upon adding enzyme is:$-6(2.303) \mathrm{RT}$$-6 R T$$+6(2.303) \mathrm{RT}$+6RTCorrect Option: 1 Solution:...
Read More →An amplitude modulated signal is given by
Question: An amplitude modulated signal is given by $\mathrm{V}(\mathrm{t})=10[1+0.3$ $\left.\cos \left(2.2 \times 10^{4} \mathrm{t}\right)\right] \sin \left(5.5 \times 10^{5} \mathrm{t}\right) .$ Here $\mathrm{t}$ is in seconds. The sideband frequencies (in $\mathrm{kHz}$ ) are, [Given $\pi=22 / 7]$(1) 1785 and 1715(2) $178.5$ and $171.5$(3) $89.25$ and $85.75$(4) $892.5$ and $857.5$Correct Option: , 3 Solution: $v(t)=10\left\lfloor 1+u .5 \cos \left(2.2 \times 10^{\top}\right)\right\rfloor$ $\...
Read More →Prove that
Question: Prove that $5 \sqrt{2}$ is irrational. Solution: Let $5 \sqrt{2}$ is a rational number. $\therefore 5 \sqrt{2}=\frac{p}{q}$, where $p$ and $q$ are some integers and $\operatorname{HCF}(p, q)=1$ ......(1) $\Rightarrow 5 \sqrt{2} q=p$ $\Rightarrow(5 \sqrt{2} q)^{2}=p^{2}$ $\Rightarrow 2\left(25 q^{2}\right)=p^{2}$ $\Rightarrow p^{2}$ is divisible by 2 $\Rightarrow p$ is divisible by 2 .....(2) Letp= 2m, wheremis some integer. $\therefore 5 \sqrt{2} q=2 m$ $\Rightarrow(5 \sqrt{2} q)^{2}=(...
Read More →For the reaction
Question: For the reaction $2 \mathrm{H}_{2}(\mathrm{~g})+2 \mathrm{NO}(\mathrm{g}) \rightarrow \mathrm{N}_{2}(\mathrm{~g})+2 \mathrm{H}_{2} \mathrm{O}(\mathrm{g})$ the observed rate expression is, rate $=\mathrm{k}_{\mathrm{f}}[\mathrm{NO}]^{2}\left[\mathrm{H}_{2}\right]$. The rate expression for the reverse reaction is:$\mathrm{k}_{\mathrm{b}}\left[\mathrm{N}_{2}\right]\left[\mathrm{H}_{2} \mathrm{O}\right]^{2}$$\mathrm{k}_{\mathrm{b}}\left[\mathrm{N}_{2}\right]\left[\mathrm{H}_{2} \mathrm{O}\...
Read More →Prove that
Question: Prove that $(5-2 \sqrt{3})$ is an irrational number. Solution: Let $x=5-2 \sqrt{3}$ be a rational number. $x=5-2 \sqrt{3}$ $\Rightarrow x^{2}=(5-2 \sqrt{3})^{2}$ $\Rightarrow x^{2}=(5)^{2}+(2 \sqrt{3})^{2}-2(5)(2 \sqrt{3})$ $\Rightarrow x^{2}=25+12-20 \sqrt{3}$ $\Rightarrow x^{2}-37=-20 \sqrt{3}$ $\Rightarrow \frac{37-x^{2}}{20}=\sqrt{3}$ Since $x$ is a rational number, $x^{2}$ is also a rational number. $\Rightarrow 37-x^{2}$ is a rational number $\Rightarrow \frac{37-x^{2}}{20}$ is a...
Read More →During the nuclear explosion, one of the products is
Question: During the nuclear explosion, one of the products is ${ }^{90} \mathrm{Sr}$ with half life of $6.93$ years. If $1 \mu \mathrm{g}$ of ${ }^{90} \mathrm{Sr}$ was absorbed in the bones of a newly born baby in place of $\mathrm{Ca}$, how much time, in years, is require to reduce it by $90 \%$ if it is not lost metabolically ______________ . Solution:...
Read More →Let y=y(x) be the solution of the differential equation
Question: Let $y=y(x)$ be the solution of the differential equation $\cos x(3 \sin x+\cos x+3) d y=$ $(1+y \sin x(3 \sin x+\cos x+3)) d x$ $0 \leq x \leq \frac{\pi}{2}, y(0)=0 .$ Then,$y\left(\frac{\pi}{3}\right)$ is equal to: (1) $2 \log _{e}\left(\frac{2 \sqrt{3}+9}{6}\right)$(2) $2 \log _{e}\left(\frac{2 \sqrt{3}+10}{11}\right)$(3) $2 \log _{\mathrm{e}}\left(\frac{\sqrt{3}+7}{2}\right)$(4) $2 \log _{e}\left(\frac{3 \sqrt{3}-8}{4}\right)$Correct Option: 2, Solution: Sol. $\cos x(3 \sin x+\cos ...
Read More →The modulation frequency of an AM radio station
Question: The modulation frequency of an AM radio station is 250 $\mathrm{kHz}$, which is $10 \%$ of the carrier wave. If another AM station approaches you for license what broadcast frequency will you allot?(1) $2750 \mathrm{kHz}$(2) $2900 \mathrm{kHz}$(3) $2250 \mathrm{kHz}$(4) $2000 \mathrm{kHz}$Correct Option: , 4 Solution: (4) According to question, modulation frequency, $250 \mathrm{~Hz}$ is $10 \%$ of carrier wave $\mathrm{f}_{\text {carricr }}=\frac{250}{0.1}=2500 \mathrm{KHZ}$ $\therefo...
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