Sucrose hydrolyses in acid solution into glucose and fructose following
Question: Sucrose hydrolyses in acid solution into glucose and fructose following first order rate law with a half-life of $3.33 \mathrm{~h}$ at $25^{\circ} \mathrm{C}$. After $9 \mathrm{~h}$, the fraction of sucrose remaining is $\mathrm{f}$. The value of $\log _{10}\left(\frac{1}{\mathrm{f}}\right)$ is____________ $\mathrm{mat} \times 10^{-2}$ (Rounded off to the nearest integer) [Assume: $\ln 10=2.303, \ln 2=0.693$ ] Solution: (81)...
Read More →Find the largest four-digits number which when divided by 4, 7 and 13 leaves a remainder of 3 in each case.
Question: Find the largest four-digits number which when divided by 4, 7 and 13 leaves a remainder of 3 in each case. Solution: Largest 4 digit number is 9999To find the largest 4 digit number divisible by 4, 7 and 13, we find the LCM of 4, 7 and 13 first. $\operatorname{LCM}(4,7,13)=4 \times 7 \times 13=364$ Now, to we divide 9999 by 364 and subtract the remainder from 9999 to get the number completely divisible by 4, 7 and 13. $9999-171=9828$ Because the number leaves the remainder 3, so we ad...
Read More →The value of
Question: The value of $\int_{0}^{2 \pi}[\sin 2 x(1+\cos 3 x)] d x$, where [t] denotes the greatest integer function, is:(1) $\pi$(2) $-\pi$(3) $-2 \pi$(4) $2 \pi$Correct Option: , 2 Solution: $I=\int_{0}^{2 \pi}[\sin 2 x(1+\cos 3 x)] d x$.....(1) $\because \int_{0}^{a} f(x)=\int_{0}^{a} f(a-x) d x$ $\therefore I=\int_{0}^{2 \pi}[-\sin 2 x(1+\cos 3 x)] d x$....(2) From (1) + (2), we get; $2 I=\int_{0}^{2 \pi}(-1) d x \Rightarrow 2 I=-(x)_{0}^{2 \pi} \Rightarrow I=-\pi$...
Read More →Find the greatest number of four digits which is exactly divisible by 15, 24 and 36.
Question: Find the greatest number of four digits which is exactly divisible by 15, 24 and 36. Solution: Prime factorization:15 = 3 5 $24=2^{3} \times 3$ $36=2^{2} \times 3^{2}$ LCM = product of greatest power of each prime factor involved in the numbers $=2^{3} \times 3^{2} \times 5=360$ Now, the greatest four digit number is 9999.On dividing 9999 by 360 we get 279 as remainder.Thus, 9999 279 = 9720 is exactly divisible by 360.Hence, the greatest number of four digits which is exactly divisible...
Read More →A reaction has a half life of 1 min.
Question: A reaction has a half life of $1 \mathrm{~min}$. The time required for $99.9 \%$ completion of the reaction is____________ min. (Round off to the Nearest integer) $[$ Use $: \ln 2=0.69, \ln 10=2.3]$ Solution: (10) $\frac{\mathrm{t}_{99.9 \%}}{\mathrm{t}_{50 \%}}=\frac{\frac{1}{\mathrm{~K}} \ln \frac{100}{0.1}}{\frac{1}{\mathrm{~K}} \ln 2}$ $=\frac{\ln 1000}{\ln 2} \times \mathrm{t}_{50 \%}$ $=\frac{3 \ln 10}{\ln 2} \times 1$ $=\frac{3 \times 2.3}{0.69}=10$...
Read More →The value of the integral
Question: The value of the integral $\int_{0}^{1} x \cot ^{-1}\left(1-x^{2}+x^{4}\right) d x$ is:(1) $\frac{\pi}{2}-\frac{1}{2} \log _{e} 2$(2) $\frac{\pi}{4}-\log _{e} 2$(3) $\frac{\pi}{2}-\log _{e} 2$(4) $\frac{\pi}{4}-\frac{1}{2} \log _{e} 2$Correct Option: , 4 Solution: $\int_{0}^{1} x \cot ^{-1}\left(1-x^{2}+x^{4}\right) d x=\int_{0}^{1} x \tan ^{-1}\left(\frac{1}{1+x^{4}-x^{2}}\right)$ $=\int_{0}^{1} x \tan ^{-1}\left(\frac{x^{2}-\left(x^{2}-1\right)}{1+x^{2}\left(x^{2}-1\right)}\right) d ...
Read More →Find the smallest number which when increased by 17 is exactly divisible by both 468 and 520.
Question: Find the smallest number which when increased by 17 is exactly divisible by both 468 and 520. Solution: The smallest number which when increased by 17 is exactly divisible by both 468 and 520 is obtained by subtracting 17 from the LCM of 468 and 520.Prime factorization of 468 and 520 is: $468=2^{2} \times 3^{2} \times 13$ $520=2^{3} \times 5 \times 13$ $\mathrm{LCM}=$ product of greatest power of each prime factor involved in the numbers $=2^{3} \times 3^{2} \times 5 \times 13=4680$ Th...
Read More →The value of
Question: The value of $\int_{0}^{\pi / 2} \frac{\sin ^{3} x}{\sin x+\cos x} d x$ is:(1) $\frac{\pi-2}{8}$(2) $\frac{\pi-1}{4}$(3) $\frac{\pi-2}{4}$(4) $\frac{\pi-1}{2}$Correct Option: , 2 Solution: Let $I=\int_{0}^{\pi / 2} \frac{\sin ^{3} x d x}{\sin x+\cos x}$.....(1) Use the property $\int_{0}^{a} f(x) d x=\int_{0}^{a} f(a-x) d x$ $\Rightarrow I=\int_{0}^{\pi / 2} \frac{\cos ^{3} x d x}{\sin x+\cos x}$ .....(2) Adding equation (1) and (2), we get $2 I=\int_{0}^{\pi / 2}\left(1-\frac{1}{2} \s...
Read More →in a line of sight ratio communication,
Question: in a line of sight ratio communication, a distance of about $50 \mathrm{~km}$ is kept between the transmitting and receiving antennas. If the height of the receiving antenna is $70 \mathrm{~m}$, then the minimum height of the transmitting antenna should be : (Radius of the Earth $=6.4 \times 10^{6} \mathrm{~m}$ ).(1) $20 \mathrm{~m}$(2) $51 \mathrm{~m}$(3) $32 \mathrm{~m}$(4) $40 \mathrm{~m}$Correct Option: , 3 Solution: (3) $\mathrm{LOS}=\sqrt{2 h_{T} R}+\sqrt{2 h_{R} R}$ or $50 \time...
Read More →Solve the following
Question: $2 \mathrm{NO}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{~g}) \rightleftharpoons 2 \mathrm{NOCl}(\mathrm{s})$ This reaction was studied at $-10^{\circ} \mathrm{C}$ and the following data was obtained $[\mathrm{NO}]_{0}$ and $\left[\mathrm{Cl}_{2}\right]_{0}$ are the initial concentrations and $\mathrm{r}_{0}$ is the initial reaction rate. The overall order of the reaction is _______________ (Round off to the Nearest Integer). Solution: (3) $\mathrm{r}=\mathrm{k}[\mathrm{NO}]^{\mathrm{m}}\lef...
Read More →The wavelength of the carrier waves in a modern optical fiber communication network is close to :
Question: The wavelength of the carrier waves in a modern optical fiber communication network is close to :(1) $2400 \mathrm{~nm}$(2) $1500 \mathrm{~nm}$(3) $600 \mathrm{~nm}$(4) $900 \mathrm{~nm}$Correct Option: , 2 Solution: (2) Carrier waves of wavelength $1500 \mathrm{~nm}$ is used in moderr optical fiber communication....
Read More →Let f(x)=
Question: Let $f(x)=\int_{0}^{x} g(t) d t$, where $g$ is a non-zero even function. If $f(x+5)=g(x)$, then $\int_{0}^{x} f(t) d t$ equals : (1) $\int_{x+5}^{5} g(t) d t$(2) $\int_{5}^{x+5} g(t) d t$(3) $2 \int_{5}^{x+5} g(t) d t$(4) $5 \int_{x+5}^{5} g(t) d t$Correct Option: 1, Solution: $f(x)=\int_{0}^{x} g(g) d t$............(i) $\because g$ is a non-zero even function. $\therefore g(-x)=g(x)$,...........(ii) Given, $f(x+5)=g(x)$...........(iii) From (i) $f^{\prime}(x)=g(x)$ Let, $I=\int_{0}^{x...
Read More →An amplitude modulated wave is represented by the expression
Question: An amplitude modulated wave is represented by the expression $v_{m}=5(1+0.6 \cos 6280 t) \sin \left(211 \times 10^{4} t\right)$ volts The minimum and maximum amplitudes of the amplitude modulated wave are, respectively :(1) $\frac{3}{2} \mathrm{~V}, 5 \mathrm{~V}$(2) $\frac{5}{2} \mathrm{~V}, 8 \mathrm{~V}$(3) $5 \mathrm{~V}, 8 \mathrm{~V}$(4) $3 \mathrm{~V}, 5 \mathrm{~V}$Correct Option: , 2 Solution: (2) From the given expression, $V_{m}=5(1+0.6 \cos 6280 t) \sin \left(211 \times 10^...
Read More →Solve the following
Question: The reaction $2 \mathrm{~A}+\mathrm{B}_{2} \rightarrow 2 \mathrm{AB}$ is an elementary reaction. For a certain quantity of reactants, if the volume of the reaction vessel is reduced by a factor of 3 , the rate of the reaction increases by a factor of _______________(Round off to the Nearest Integer). Solution: (27) Reaction : $2 \mathrm{~A}+\mathrm{B}_{2} \longrightarrow 2 \mathrm{AB}$As the reaction is elementary, the rate of reaction is $\mathrm{r}=\mathrm{K} \cdot[\mathrm{A}]^{2}\le...
Read More →Find the smallest number which when divided by 28 and 32 leaves remainders 8 and 12 respectively.
Question: Find the smallest number which when divided by 28 and 32 leaves remainders 8 and 12 respectively. Solution: Let the required number bex.Using Euclid's lemma,x= 28p+ 8 andx= 32q+ 12, wherepandqare the quotients⇒28p+ 8 =32q+ 12⇒28p =32q+ 4⇒ 7p =8q+ 1 ..... (1)Herep= 8n 1 andq= 7n 1 satisfies (1), wherenis a natural numberOn puttingn= 1, we getp= 8 1 = 7 andq= 7 1 = 6Thus,x= 28p+ 8= 28 7 + 8= 204Hence, the smallest number which when divided by 28 and 32 leaves remainders 8 and 12 is 204....
Read More →For a certain first order reaction 32 % of the reactant is left after 570s.
Question: For a certain first order reaction $32 \%$ of the reactant is left after $570 \mathrm{~s}$. The rate constant of this reaction is $\times 10^{-3} \mathrm{~s}^{-1}$. (Round off to the Nearest Integer). $\left[\right.$ Given : $\left.\log _{10} 2=0.301, \ln 10=2.303\right]$ Solution: (2) For $1^{\text {st }}$ order reaction, $\mathrm{K}=\frac{2.303}{\mathrm{t}} \cdot \log \frac{\left[\mathrm{A}_{0}\right]}{\left[\mathrm{A}_{\mathrm{t}}\right]}=\frac{2.303}{570 \mathrm{sec}} \cdot \log \l...
Read More →Find the least number which when divided by 35, 56 and 91 leaves the same remainder
Question: Find the least number which when divided by 35, 56 and 91 leaves the same remainder 7 in each case. Solution: Least number which can be divided by 35, 56 and 91 is LCM of 35, 56 and 91. Prime factorization of35, 56 and 91 is: 35 = 5 7 $56=2^{3} \times 7$ 91 = 7 13 LCM = product of greatest power of each prime factor involved in the numbers $=2^{3} \times 5 \times 7 \times 13=3640$ Least number which can be divided by 35, 56 and 91 is 3640.Least numberwhich when divided by 35, 56 and 91...
Read More →If f(x)=
Question: If $f(x)=\frac{2-x \cos x}{2+x \cos x}$ and $g(x)=\log _{e} x,(x0)$ then the value of the integral $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} g(f(x)) d x$ is : (1) $\log _{\mathrm{e}} 3$(2) $\log _{\mathrm{e}} \mathrm{e}$(3) $\log _{\mathrm{e}} 2$(4) $\log _{\mathrm{e}} 1$Correct Option: , 4 Solution: $g(f(x))=\log \left(\frac{2-x \cos x}{2+x \cos x}\right), x0$ Let $I=\int_{-\pi / 4}^{\pi / 4} \log \left(\frac{2-x \cos x}{2+x \cos x}\right) d x$ .......(1) Use the property $\int_{a}^{b} f...
Read More →If the highest frequency modulating a carrier
Question: If thehighest frequency modulating a carrier is $5 \mathrm{kHz}_{4}$ then the number of $\mathrm{AM}$ broadcast stations accommodated in a $90 \mathrm{kHz}$ bandwidth are Solution: No. of station $=\frac{\text { Band width }}{2 \times \text { Highest Band width }}$ $\Rightarrow \frac{90}{2 \times 5}$ $\Rightarrow 9$...
Read More →A and B decompose via first order kinetics with half-lives
Question: A and B decompose via first order kinetics with half-lives $54.0 \mathrm{~min}$ and $18.0 \mathrm{~min}$ respectively. Starting from an equimolar non reactive mixture of $\mathrm{A}$ and $\mathrm{B}$, the time taken for the concentration of $A$ to become 16 times that of $B$ is_____________ min. (Round off to the Nearest Integer). Solution: (108)...
Read More →If f(x)=
Question: If $f(x)=\frac{2-x \cos x}{2+x \cos x}$ and $g(x)=\log _{e} x,(x0)$ then the value of the integral $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} g(f(x)) d x$ is : (1) $\log _{\mathrm{e}} 3$(2) $\log _{\mathrm{e}} \mathrm{e}$(3) $\log _{\mathrm{e}} 2$(4) $\log _{\mathrm{e}} 1$Correct Option: , 4 Solution: $g(f(x))=\log \left(\frac{2-x \cos x}{2+x \cos x}\right), x0$...
Read More →Find the largest number which divides 320 and 457 leaving remainders 5 and 7 respectively.
Question: Find the largest number which divides 320 and 457 leaving remainders 5 and 7 respectively. Solution: We know that the required number divides 315 (320 5) and 450 (457 7). Required number =HCF (315, 450)On applying Euclid's lemma, we get: Therefore, theHCFof 315 and 450 is 45.Hence, the required number is 45....
Read More →Draw the output Y in the given combination of gates.
Question: Draw the output $Y$ in the given combination of gates. (1) (2) (3) (4)Correct Option: Solution: (1) Find output expression $y=A \cdot B$ Inputs...
Read More →The value of
Question: The value of $\int_{0}^{2 \pi} \frac{x \sin ^{8} x}{\sin ^{8} x+\cos ^{8} x} d x$ is equal to:(1) $2 \pi$(2) $2 \pi^{2}$(3) $\pi^{2}$(4) $4 \pi$Correct Option: , 3 Solution: $\int_{0}^{2 \pi} \frac{x \sin ^{8} x}{\sin ^{8} x+\cos ^{8} x} d x$ $=\int_{0}^{\pi}\left[\frac{x \sin ^{8} x}{\sin ^{8} x+\cos ^{8} x}+\frac{(2 \pi-x) \sin ^{8} x}{\sin ^{8} x+\cos ^{8} x}\right] d x$ $\left[\because \int_{0}^{2 a} f(x) d x=\int_{0}^{a} f(x) d x+\int_{0}^{a} f(2 a-x) d x\right]$ $=\int_{0}^{\pi} ...
Read More →Find the largest number which divides 438 and 606,
Question: Find the largest number which divides 438 and 606, leaving remainder 6 in each case. Solution: Largest number which divides 438 and 606, leaving remainder 6 is actually the largest number which divides 438 6 = 432 and 606 6 = 600, leaving remainder 0.Therefore, HCF of 432 and 600 gives the largest number.Now, prime factors of 432 and 600 are: $432=2^{4} \times 3^{3}$ $600=2^{3} \times 3 \times 5^{2}$ HCF = product of smallest power of each common prime factor in the numbers $=2^{3} \ti...
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