The rate of a reaction decreased by
Question: The rate of a reaction decreased by $3.555$ times when the temperature was changed from $40^{\circ} \mathrm{C}$ to $30^{\circ} \mathrm{C}$. The activation energy (in $\mathrm{kJ} \mathrm{mol}^{-1}$ ) of the reaction is_____________ .Take; $\mathrm{R}=$ $8.314 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1} \operatorname{In} 3.555=1.268$ Solution: (100) The Arrhenices equation is $k=A e^{\frac{E_{\mathrm{a}}}{R T}}$ Assuming $A$ and $E_{a}$ to be independent of temperature $\ln \frac{k_...
Read More →Express each of the following as a fraction in simplest form:
Question: Express each of the following as a fraction in simplest form: (i) $0 . \overline{8}$ (ii) $2 . \overline{4}$ (iii) $0 . \overline{24}$ (iv) $0.1 \overline{2}$ (v) $2.2 \overline{4}$ (vi) $0 . \overline{365}$ Solution: (i) Let $x=0 . \overline{8}$ x =0.888 ...(1)10x= 8.888 ...(2)On subtracting equation (1) from (2), we get $9 x=8 \Rightarrow x=\frac{8}{9}$ $\therefore 0 . \overline{8}=\frac{8}{9}$ (ii) Let $x=2 . \overline{4}$ x =2.444 ...(1)10x=24.444 ...(2)On subtracting equation (1) ...
Read More →A TV transmission tower has a height
Question: A TV transmission tower has a height of $140 \mathrm{~m}$ and the height of the receiving antenna is $40 \mathrm{~m}$. What is the maximum distance upto which signals can be broadcasted from this tower in LOS (Line of Sight) mode? (Given : radius of earth $=6.4 \times 10^{6} \mathrm{~m}$ ).(1) $65 \mathrm{~km}(2)$(2) $48 \mathrm{~km}$(3) $80 \mathrm{~km}$(4) $40 \mathrm{~km}$Correct Option: 1 Solution: (1) Maximum distance upto which signal can be broadcasted $d_{\max }=\sqrt{2 \mathrm...
Read More →Consider the following reactions
Question: Consider the following reactions $\mathrm{A} \rightarrow \mathrm{P} 1 ; \mathrm{B} \rightarrow \mathrm{P} 2 ; \mathrm{C} \rightarrow \mathrm{P} 3 ; \mathrm{D} \rightarrow \mathrm{P} 4$ The order of the above reactions are (i), (ii), (iii), and (iv), respectively. The following graph is obtained when $\log$ [rate] vs. $\log$ [conc.] are plotted : Among the following, the correct sequence for the order of the reactions is:(iv) $$ (i) $$ (ii) $$ (iii)(i) $$ (ii) $$ (iii) $$ (iv)(iii) $$ (...
Read More →Which of the following is true for
Question: Which of the following is true for $y(x)$ that satisfies the differential equation $\frac{d y}{d x}=x y-1+x-y ; y(0)=0$(1) $y(1)=e^{-\frac{1}{2}}-1$(2) $y(1)=e^{\frac{1}{2}}-e^{-\frac{1}{2}}$(3) $y(1)=1$(4) $y(1)=e^{\frac{1}{2}}-1$Correct Option: 1 Solution: $\frac{d y}{d x}=(1+y)(x-1)$ $\frac{d y}{(y+1)}=(x-1) d x$ Integrate $\ln (y+1)=\frac{x^{2}}{2}-x+c$ $(0,0) \Rightarrow \mathrm{c}=0 \Rightarrow \mathrm{y}=\mathrm{e}^{\left(\frac{x^{2}}{2}-\mathrm{x}\right)}-1$...
Read More →In a communication system operating at wavelength
Question: In a communication system operating at wavelength 800 $\mathrm{nm}$, only one percent of source frequency is available as signal bandwidth. The number of channels accomodated for transmitting TV signals of band width $6 \mathrm{MHz}$ are (Take velocity of light $\left.\mathrm{c}=3 \times 10^{8} \mathrm{~m} / \mathrm{s}, \mathrm{h}=6.6 \times 10^{-34} \mathrm{~J}-\mathrm{s}\right)$(1) $3.75 \times 10^{6}$(2) $3.86 \times 10^{6}$(3) $6.25 \times 10^{5}$(4) $4.87 \times 10^{5}$Correct Opt...
Read More →If y=y(x) is the solution of the differential equation
Question: If $y=y(x)$ is the solution of the differential equation $\frac{d y}{d x}+(\tan x) y=\sin x, 0 \leq x \leq \frac{\pi}{3}$, with $\mathrm{y}(0)=0$, then $\mathrm{y}\left(\frac{\pi}{4}\right)$ equal to :(1) $\frac{1}{4} \log _{\mathrm{e}} 2$(2) $\left(\frac{1}{2 \sqrt{2}}\right) \log _{\mathrm{e}} 2$(3) $\log _{\mathrm{e}} 2$(4) $\frac{1}{2} \log _{\mathrm{e}} 2$Correct Option: , 2 Solution: $\frac{d y}{d x}+(\tan x) y=\sin x ; 0 \leq x \leq \frac{\pi}{3}$ I. $F .=e^{\int \tan x d x}=e^{...
Read More →in an amplitude modulator circuit
Question: in an amplitude modulator circuit, the carrier wave is given by, $\mathrm{C}(t)=4 \sin (20000 \pi t)$ while modulating signal is given by, $m(t)=2 \sin (2000 \pi t)$. The values of modulation index and lower side band frequency are :(1) $0.5$ and $10 \mathrm{kHz}$(2) $0.4$ and $10 \mathrm{kHz}$(3) $0.3$ and $9 \mathrm{kHz}$(4) $0.5$ and $9 \mathrm{kHz}$Correct Option: , 4 Solution: (4) Modulation index, $\mu=\frac{A_{m}}{A_{c}}=\frac{2}{4}=0.5$ Given, $f_{e}=\frac{20000 \pi}{2 \pi}=100...
Read More →Without actual division, show that each of the following rational numbers is a non-terminating repeating decimal:
Question: Without actual division, show that each of the following rational numbers is a non-terminating repeating decimal: (i) $\frac{11}{\left(2^{3} \times 3\right)}$ (ii) $\frac{73}{\left(2^{2} \times 3^{3} \times 5\right)}$ (iii) $\frac{129}{\left(2^{2} \times 5^{7} \times 7^{5}\right)}$ (iv) $\frac{9}{35}$ (v) $\frac{77}{210}$ (vi) $\frac{32}{147}$ (vii) $\frac{29}{343}$ (viii) $\frac{64}{455}$ Solution: (i) $\frac{11}{2^{3} \times 3}$ We know either 2 or 3 is not a factor of 11, so it is i...
Read More →If y=y(x) is the solution of the differential equation,
Question: If $y=y(x)$ is the solution of the differential equation, $\frac{\mathrm{dy}}{\mathrm{dx}}+2 \mathrm{y} \tan \mathrm{x}=\sin \mathrm{x}, \mathrm{y}\left(\frac{\pi}{3}\right)=0$, then the maximum value of the function $\mathrm{y}(\mathrm{x})$ over $\mathbb{R}$ is equal to:(1) 8(2) $\frac{1}{2}$(3) $-\frac{15}{4}$(4) $\frac{1}{8}$Correct Option: , 4 Solution: $\frac{d y}{d x}+2 y \tan x=\sin x$ I.F. $=\mathrm{e}^{\int 2 \tan x d x}=\mathrm{e}^{2 \operatorname{tn} \sec x}$ I.F. $=\sec ^{2...
Read More →Given below in the left column are different modes
Question: Given below in the left column are different modes of communication using the kinds of waves given in the right column. From the options given below, find the most appropriate match between entries in the left and the right column.(1) $A-Q, B-S, C-R, D-P$(2) $0.4$ and $10 \mathrm{kHz}$(3) $0.3$ and $9 \mathrm{kHz}$(4) $0.5$ and $9 \mathrm{kHz}$Correct Option: , 3 Solution: (3) Optical Fibre Communication - Infrared Light Radar - Radio Waves Sonar - Ultrasound Mobile Phones - Microwaves...
Read More →The rate constant (k) of a reaction is measured at different temperatures (T),
Question: The rate constant $(k)$ of a reaction is measured at different temperatures $(T)$, and the data are plotted in the given figure. The activation energy of the reaction in $\mathrm{kJ} \mathrm{mol}^{-1}$ is : ( $R$ is gas constant) $2 / R$$1 / R$$R$$2 R$Correct Option: , 4 Solution: Arrhenius equation: $k=A e^{-E a / R T}$\ $\ln k=\ln A-\left(\frac{E_{a}}{R}\right) \frac{1}{T}$ $\ln k=\ln A-\left(\frac{E_{a}}{R \times 10^{3}}\right) \times \frac{10^{3}}{T}$ Slope of graph $=\frac{-E_{a}}...
Read More →The physical sizes of the transmitter and receiver antenna in a communication system are:
Question: The physical sizes of the transmitter and receiver antenna in a communication system are:(1) independent of both carrier and modulation frequency(2) inversely proportional to carrier frequency(3) inversely proportional to modulation frequency(4) proportional to carrier frequencyCorrect Option: , 2 Solution: (2) Size of antenna,' $l=\frac{\lambda}{4} .$ As $\lambda=\frac{C}{f} \therefore l \propto \frac{1}{f}$...
Read More →A signal
Question: A signal Acos\omegat is transmitted using $v_{0} \sin \omega$ modulated (AM) signal is:(1) $v_{0} \sin \omega_{0} t+\frac{\mathrm{A}}{2} \sin$ $\left(\omega_{0}-\omega\right) t+\frac{\mathrm{A}}{2}\left(\omega_{0}+\omega\right) t$(2) $v_{0} \sin \left[\omega_{0}(1+0.01\right.$ Asin\omegat $t]$(3) $v_{0} \sin \omega_{0} \mathrm{t}+\mathrm{A} \cos \omega \mathrm{t}$(4) $\left(v_{0}+\mathrm{A}\right) \cos \omega t \sin \omega_{0} t$Correct Option: 1 Solution: (1) The equation of amplitude...
Read More →A flask contains a mixture of compounds A and B. Both compounds decompose by first-order kinetics.
Question: A flask contains a mixture of compounds A and B. Both compounds decompose by first-order kinetics. The halflives for $\mathrm{A}$ and $\mathrm{B}$ are $300 \mathrm{~s}$ and $180 \mathrm{~s}$, respectively. If the concentrations of $A$ and $B$ are equal initially, the time required for the concentration of $A$ to be four times that of $\mathrm{B}$ (in $\mathrm{s}$ ) is: (Use $\ln 2=0.693$ )180900300120Correct Option: , 2 Solution:...
Read More →The integral
Question: The integral $\int_{\pi / 6}^{\pi / 4} \frac{\mathrm{d} x}{\sin 2 x\left(\tan ^{5} x+\cot ^{5} x\right)}$ equals :(1) $\frac{1}{20} \tan ^{-1}\left(\frac{1}{9 \sqrt{3}}\right)$(2) $\frac{1}{10}\left(\frac{\pi}{4}-\tan ^{-1}\left(\frac{1}{9 \sqrt{3}}\right)\right)$(3) $\frac{\pi}{40}$(4) $\frac{1}{5}\left(\frac{\pi}{4}-\tan ^{-1}\left(\frac{1}{3 \sqrt{3}}\right)\right)$Correct Option: , 2 Solution: $I=\int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \frac{d x}{\sin 2 x\left(\tan ^{5} x+\cot ^{5} x\...
Read More →Without actual division, show that each of the following rational numbers is a terminating decimal.
Question: Without actual division, show that each of the following rational numbers is a terminating decimal. Express each in decimal form: (i) $\frac{23}{\left(2^{3} \times 5^{2}\right)}$ (ii) $\frac{24}{125}$ (iii) $\frac{171}{800}$ (iv) $\frac{15}{1600}$ (v) $\frac{17}{320}$ (vi) $\frac{19}{3125}$ Solution: (i) $\frac{23}{2^{3} \times 5^{2}}=\frac{23 \times 5}{2^{3} \times 5^{3}}=\frac{115}{1000}=0.115$ We know either 2 or 5 is not a factor of 23, so it is in its simplest form. Moreover, it i...
Read More →The value of the integral
Question: The value of the integral $\int_{-2}^{2} \frac{\sin ^{2} x}{\left[\frac{x}{\pi}\right]+\frac{1}{2}} d x$ (where $[x]$ denotes the greatest integer less than or equal to x) is: (1) 0(2) $\sin 4$(3) 4(4) $4-\sin 4$Correct Option: 1 Solution: $\operatorname{Let} f(x)=\frac{\sin ^{2} x}{\left[\frac{x}{\pi}\right]+\frac{1}{2}}$ So, $f(-x)=\frac{\sin ^{2}(-x)}{\left[\frac{-x}{\pi}\right]+\frac{1}{2}}$ $\because[-x]=-1-[x]$ $\Rightarrow f(-x)=\frac{\sin ^{2} x}{-1-\left[\frac{x}{\pi}\right]+\...
Read More →The value of
Question: The value of $\int_{-\pi / 2}^{\pi / 2} \frac{d x}{[x]+[\sin x]+4}$, where $[\mathrm{t}]$ denotes the greatest integer less than or equal to $t$, is:(1) $\frac{1}{12}(7 \pi+5)$(2) $\frac{1}{12}(7 \pi-5)$(3) $\frac{3}{20}(4 \pi-3)$(4) $\frac{3}{10}(4 \pi-3)$Correct Option: , 3 Solution: $I=\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \frac{d x}{[x]+[\sin x]+4}$ $=\int_{\frac{-\pi}{2}}^{-1} \frac{d x}{-2-1+4}+\int_{-1}^{0} \frac{d x}{-1-1+4}+\int_{0}^{1} \frac{d x}{0+0+4}+\int_{1}^{\frac{\pi}{2...
Read More →If
Question: If $\int_{0}^{x} f(\mathrm{t}) \mathrm{d} \mathrm{t}=x^{2}+\int_{x}^{1} \mathrm{t}^{2} f(\mathrm{t}) \mathrm{dt}$, then $f^{\prime}(1 / 2)$ is: (1) $\frac{24}{25}$(2) $\frac{18}{25}$(3) $\frac{4}{5}$(4) $\frac{6}{25}$Correct Option: 1 Solution: $\int_{0}^{x} f(t) d t=x^{2}+\int_{x}^{1} t^{2} f(t) d t$ $\Rightarrow \quad f(x)=2 x-x^{2} f(x)$ $\Rightarrow \quad f(x)=\frac{2 x}{1+x^{2}}$ $\Rightarrow \quad f^{\prime}(x)=\frac{2\left(1-x^{2}\right)}{\left(1+x^{2}\right)^{2}}$ Then, $f^{\pr...
Read More →The number of molecules with energy greater than the threshold energy for
Question: The number of molecules with energy greater than the threshold energy for a reaction increases five fold by a rise of temperature from $27^{\circ} \mathrm{C}$ to $42^{\circ} \mathrm{C}$. Its energy of activation in $\mathrm{J} / \mathrm{mol}$ is______________. (Take $\ln 5=1.6094$; $\mathrm{R}=8.314 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$ ) Solution: $(84297.48)$ $\because k=A \mathrm{e}^{-E_{a} / R T}$ $\ln \frac{k_{2}}{k_{1}}=\frac{E_{a}}{R}\left(\frac{1}{T_{1}}-\frac{1}{T_{...
Read More →If 75 % of a first order reaction was completed in 90 minutes,
Question: If $75 \%$ of a first order reaction was completed in 90 minutes, $60 \%$ of the same reaction would be completed in approximately (in minutes) ______________ (Take : $\log 2=0.30 ; \log 2.5=0.40$ ) Solution: (60)...
Read More →If
Question: If $\int_{0}^{\pi / 3} \frac{\tan \theta}{\sqrt{2 \mathrm{k} \sec \theta}} \mathrm{d} \theta=1-\frac{1}{\sqrt{2}},(\mathrm{k}0)$ then the value of $\mathrm{k}$ is:(1) 4(2) $\frac{1}{2}$(3) 1(4) 2Correct Option: , 4 Solution: Let, $I=\int_{0}^{\pi / 3} \frac{\tan \theta}{\sqrt{2 k \sec \theta}} d \theta$ $=\frac{1}{\sqrt{2 k}} \int_{0}^{\pi / 3} \frac{\sin \theta}{\sqrt{\cos \theta}} d \theta$ Let $\cos \theta=t^{2}$ $\therefore \quad \sin \theta d \theta=-2 t d t$ Hence, integral becom...
Read More →Six bells commence tolling together and toll at intervals of 2, 4, 6, 8, 10, 12, minutes respectively.
Question: Six bells commence tolling together and toll at intervals of 2, 4, 6, 8, 10, 12, minutes respectively. How many times do they toll together in 30 hours? Solution: Six bells toll together at intervals of 2, 4, 6, 8, 10 and 12 minutes, respectively.Prime factorisation: $2=2$ $4=2 \times 2$ $6=2 \times 3$ $8=2 \times 2 \times 2$ $10=2 \times 5$ $12=2 \times 2 \times 3$ $\therefore L C M(2,4,6,8,10,12)=2^{3} \times 3 \times 5=120$ Hence, after every 120 minutes (i.e. 2 hours), they will to...
Read More →The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively.
Question: The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively. If they all change simultaneously at 8 a.m. then at what time will they again change simultaneously? Solution: We find the LCM of 48, 72 and 108 first to get the time after which they will blink together again. Hence, LCM $=2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3=432$ So, they will blink again at 432 seconds past 8:00 am or, $\frac{432}{60}=7$ min...
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