The decomposition of formic acid on gold surface follows first order kinetics.
Question: The decomposition of formic acid on gold surface follows first order kinetics. If the rate constant at $300 \mathrm{~K}$ is $1.0 \times 10^{-3} \mathrm{~s}^{-1}$ and the activation energy $\mathrm{E}_{\mathrm{a}}=11.488 \mathrm{~kJ} \mathrm{~mol}^{-1}$, the rate constant at $200 \mathrm{~K}$ is___________ $\times 10^{-5} \mathrm{~s}^{-1}$. (Round of to the Nearest Integer). ( Given : $R=8.314 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$ ) Solution: (10) $\mathrm{K}_{300}=10^{-4} \m...
Read More →If for all real triplets
Question: If for all real triplets $(a, b, c), f(x)=a+b x+c x^{2} ;$ then $\int_{0}^{1} f(x) d x$ is equal to:(1) $2\left\{3 f(1)+2 f\left(\frac{1}{2}\right)\right\}$(2) $\frac{1}{2}\left\{f(1)+3 f\left(\frac{1}{2}\right)\right\}$(3) $\frac{1}{3}\left\{f(0)+f\left(\frac{1}{2}\right)\right\}$(4) $\frac{1}{6}\left\{f(0)+f(1)+4 f\left(\frac{1}{2}\right)\right\}$Correct Option: , 4 Solution: $\int_{0}^{1}\left(a+b x+c x^{2}\right) d x=a x+\frac{b x^{2}}{2}+\left.\frac{c x^{3}}{3}\right|_{0} ^{1}=a+\...
Read More →Find the simplest form of:
Question: Find the simplest form of: (i) $\frac{69}{92}$ (ii) $\frac{473}{645}$ (iii) $\frac{1095}{1168}$ (iv) $\frac{368}{496}$ Solution: (i) Prime factorisation of 69 and 92 is:69 = 3 23 $92=2^{2} \times 23$ Therefore, $\frac{69}{92}=\frac{3 \times 23}{2^{2} \times 23}=\frac{3}{2^{2}}=\frac{3}{4}$ Thus, simplest form of $\frac{69}{92}$ is $\frac{3}{4}$. (ii) Prime factorisation of 473 and 645 is: 473 = 11 43645 = 3 5 43 Therefore, $\frac{473}{645}=\frac{11 \times 43}{3 \times 5 \times 43}=\fra...
Read More →The maximum and minimum amplitude of an amplitude modulated
Question: The maximum and minimum amplitude of an amplitude modulated wave is $16 \mathrm{~V}$ and $8 \mathrm{~V}$ respectively. The modulation index for this amplitude modulated wave is $\mathrm{x} \times 10^{-2}$. The value of $\mathrm{x}$ is Solution: (33) $\mathrm{A}_{\mathrm{m}}=\frac{\mathrm{A}_{\max }-\mathrm{A}_{\min }}{2}$ $\mathrm{A}_{\mathrm{C}}=\frac{\mathrm{A}_{\max }+\mathrm{A}_{\min }}{2} \quad\left[\begin{array}{c}\mathrm{A}_{\max }=16 \mathrm{~V} \\ \mathrm{~A}_{\min }=8 \mathrm...
Read More →The maximum and minimum amplitude of an amplitude modulated
Question: The maximum and minimum amplitude of an amplitude modulated wave is $16 \mathrm{~V}$ and $8 \mathrm{~V}$ respectively. The modulation index for this amplitude modulated wave is $\mathrm{x} \times 10^{-2}$. The value of $\mathrm{x}$ is Solution: (33) $\mathrm{A}_{\mathrm{m}}=\frac{\mathrm{A}_{\max }-\mathrm{A}_{\min }}{2}$ $\mathrm{A}_{\mathrm{C}}=\frac{\mathrm{A}_{\max }+\mathrm{A}_{\min }}{2} \quad\left[\begin{array}{c}\mathrm{A}_{\max }=16 \mathrm{~V} \\ \mathrm{~A}_{\min }=8 \mathrm...
Read More →The truth table for the following logic circuit is:
Question: The truth table for the following logic circuit is: (1) (2) (3) (4) Correct Option: , 4 Solution: (4) If $A=B=0$ then output $y=1$ If $A=B=1$ then output $y=1$...
Read More →Can two numbers have 15 as their HCF and 175 as their LCM? Give reason.
Question: Can two numbers have 15 as their HCF and 175 as their LCM? Give reason. Solution: No,Since, LCM is always a multiple of HCF.But 175 is not a multiple of 15.Hence, two numbers cannot have 15 as their HCF and 175 as their LCM....
Read More →The HCF of two numbers is 18 and their product is 12960.
Question: The HCF of two numbers is 18 and their product is 12960. Find their LCM. Solution: HCF of two numbers = 18Product of two numbers = 12960Let their LCM bex.Using the formula, Product of two numbers = HCF LCMwe conclude that12960 = 18 x $x=\frac{12960}{18}$ = 720 Hence, their LCM is 720....
Read More →if I =
Question: If $I=\int_{1}^{2} \frac{d x}{\sqrt{2 x^{3}-9 x^{2}+12 x+4}}$, then:(1) $\frac{1}{8}I^{2}\frac{1}{4}$(2) $\frac{1}{9}I^{2}\frac{1}{8}$(3) $\frac{1}{16}I^{2}\frac{1}{9}$(4) $\frac{1}{6}I^{2}\frac{1}{2}$Correct Option: , 2 Solution: $f(x)=\frac{1}{\sqrt{2 x^{3}-9 x^{2}+12 x+4}}$ $f^{\prime}(x)=\frac{-1}{2}\left(\frac{\left(6 x^{2}-18 x+12\right)}{\left(2 x^{3}-9 x^{2}-12 x+4\right)^{3 / 2}}\right)$ $=\frac{-6(x-1)(x-2)}{2\left(2 x^{3}-9 x^{2}+12 x+4\right)^{3 / 2}}$ $f(1)=\frac{1}{3}$ an...
Read More →if a message signal of frequency
Question: if a message signal of frequency ' $f_{\mathrm{m}}$ ' is amplitude modulated with a carrier signal of frequency $\mathrm{I}_{c}^{\prime}$ and radiated through an antenna, the wavelength of the corresponding signal in air is:(1) $\frac{c}{f_{0}+f_{m}}$(2) $\frac{c}{f_{e}-f_{m}}$(3) $\frac{c}{f_{\mathrm{m}}}$(4) $\frac{\mathrm{C}}{f_{C}}$Correct Option: , 4 Solution: (4) Given frequency of massage signa $]=f_{m}$ frequency of carrier signal $=f_{c}$ the wavelength of the corresponding si...
Read More →The LCM of two numbers is 9 times their HCF.
Question: The LCM of two numbers is 9 times their HCF. The sum of LCM and HCF is 500. Find their HCF. Solution: Let the HCF of two numbers bex.Then, LCM = 9xAccording to the question, $\mathrm{LCM}+\mathrm{HCF}=500$ $\Rightarrow 9 x+x=500$ $\Rightarrow 10 x=500$ $\Rightarrow x=50$ Hence, the HCF of two numbers is 50....
Read More →The value of
Question: The value of $\alpha$ for which $4 \alpha \int_{-1}^{2} e^{-\alpha|x|} d x=5$, is :(1) $\log _{e} 2$(2) $\log _{e}\left(\frac{3}{2}\right)$(3) $\log _{e} \sqrt{2}$(4) $\log _{e}\left(\frac{4}{3}\right)$Correct Option: 1 Solution: $4 \alpha\left\{\int_{-1}^{0} e^{\alpha x} d x+\int_{0}^{2} e^{-\alpha x} d x\right\}=5$ $\Rightarrow 4 \alpha\left\{\left.\frac{e^{\alpha x}}{\alpha}\right|_{-1} ^{0}+\left.\frac{e^{-\alpha x}}{-\alpha}\right|_{0} ^{2}\right\}=5$ $\Rightarrow 4 \alpha\left\{\...
Read More →Two solids dissociate as follows
Question: Two solids dissociate as follows The total pressure when both the solids dissociate simultaneously is:$\sqrt{x+y}$ atm$2(\sqrt{x+y})$ atm$(x+y)$ atm$x^{2}+y^{2}$ atmCorrect Option: , 2 Solution:...
Read More →The HCF of two number a and b is 5 and their LCM is 200. Find the product ab.
Question: The HCF of two numberaandbis 5 and their LCM is 200. Find the productab. Solution: Let the two numbers beaandb.Product of two numbers = HCF LCM⇒ab= 5 200⇒ab= 1000Hence,the productabis 1000....
Read More →If f(a+b+1-x)=f(x),
Question: If $f(a+b+1-x)=f(x)$, for all $x$, where $a$ and $b$ are fixed positive real numbers, then $\frac{1}{a+b} \int_{a}^{b} x(f(x)+f(x+1)) d x$ is equal to:(1) $\int_{a+1}^{b+1} f(x) d x$(2) $\int_{a-1}^{b-1} f(x) d x$(3) $\int_{a-1}^{b-1} f(x+1) d x$(4) $\int_{a+1}^{b+1} f(x+1) d x$Correct Option: , 3 Solution: $I=\frac{1}{(a+b)} \int_{a}^{b} x[f(x)+f(x+1)] d x$ $x \rightarrow a+b-x$ $I=\frac{1}{(a+b)} \int_{a}^{b}(a+b-x)[f(a+b-x)+f(a+b+1-x)] d x$ $I=\frac{1}{(a+b)} \int_{a}^{b}(a+b-x)[f(x...
Read More →Given below are two statements :
Question: Given below are two statements : Statement I : A speech signal of $2 \mathrm{kHz}$ is used to modulate a carrier signal of $1 \mathrm{MHz}$. The bandwidth requirement for the signal is $4 \mathrm{kHz}$. Statement II : The side band frequencies are $1002 \mathrm{kHz}$ and $998 \mathrm{kHz}$. In the light of the above statements, choose the correct answer from the options given below :(1) Both statement I and statement II are false(2) Statement I is false but statement II is true(3) Stat...
Read More →Given below are two statements :
Question: Given below are two statements : Statement I : A speech signal of $2 \mathrm{kHz}$ is used to modulate a carrier signal of $1 \mathrm{MHz}$. The bandwidth requirement for the signal is $4 \mathrm{kHz}$. Statement II : The side band frequencies are $1002 \mathrm{kHz}$ and $998 \mathrm{kHz}$. In the light of the above statements, choose the correct answer from the options given below :(1) Both statement I and statement II are false(2) Statement I is false but statement II is true(3) Stat...
Read More →Two positive integers a and b can be written as a = x3y2 and b = xy3,
Question: Two positive integers a and $b$ can be written as $a=x^{3} y^{2}$ and $b=x y^{3}$, where $x$ and $y$ are prime numbers. Find $\operatorname{HCF}(a, b)$ and $L C M(a, b)$. Solution: It is given that, $a=x^{3} y^{2}$ and $b=x y^{3}$, where $x$ and $y$ are prime numbers. $\mathrm{LCM}(a, b)=\mathrm{LCM}\left(x^{3} y^{2}, x y^{3}\right)$ $=$ The highest of indices of $x$ and $y$ $=x^{3} y^{3}$ $\operatorname{HCF}(a, b)=\operatorname{HCF}\left(x^{3} y^{2}, x y^{3}\right)$ $=$ The lowest of ...
Read More →A signal of 0.1 kW is transmitted in a cable.
Question: A signal of $0.1 \mathrm{~kW}$ is transmitted in a cable. The attenuation of cable is $-5 \mathrm{~dB}$ per $\mathrm{km}$ and cable length is $20 \mathrm{~km}$. the power received at receiver is $10^{-\mathrm{x}} \mathrm{W}$. The value of $\mathrm{X}$ is $\left[\right.$ Gain in $\left.d B=10 \log _{10}\left(\frac{P_{0}}{P_{i}}\right)\right]$ Solution: Power of signal transmitted : $P_{i}=0.1 \mathrm{KW}=100 \mathrm{w}$ Rate of attenuation $=-5 \mathrm{~dB} / \mathrm{Km}$ Total length o...
Read More →Find HCF and LCM of 404 and 96 and verify that HCF × LCM = product
Question: Find HCF and LCM of 404 and 96 and verify that HCF LCM = product of two given numbers. Solution: Prime factorisation:404 = 2 2 10196 = 2 2 2 2 2 3HCF (404, 96) = 2 2= 4LCM (404, 96) = 2 2 2 2 2 3 101= 9696Now, LCM HCF = 9696 4= 38784Product of 404 and 96 = 404 96= 38784Hence, HCF LCM = product of two given numbers....
Read More →The value of
Question: The value of $\int_{-\pi / 2}^{\pi / 2} \frac{1}{1+e^{\sin x}} d x$ is:(1) $\frac{\pi}{4}$(2) $\pi$(3) $\frac{\pi}{2}$(4) $\frac{3 \pi}{2}$Correct Option: , 3 Solution: $I=\int_{-\pi / 2}^{\pi / 2} \frac{1}{1+e^{\sin x}} d x$ $=\int_{-\pi / 2}^{0} \frac{1}{1+e^{\sin x}} d x+\int_{0}^{\pi / 2} \frac{1}{1+e^{\sin x}} d x$ $=\int_{0}^{\pi / 2}\left(\frac{1}{1+e^{\sin x}}+\frac{1}{1+e^{-\sin x}}\right) d x$ $=\int_{0}^{\pi / 2} \frac{1+e^{\sin x}}{1+e^{\sin x}} d x=\frac{\pi}{2}$...
Read More →Solve the following
Question: $\Lambda_{\mathrm{m}}^{\circ}$ for $\mathrm{NaCl}, \mathrm{HCl}$ and $\mathrm{NaA}$ are $126.4,425.9$ and $100.5 \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}$, respectively. If the conductivity of $0.001$ M HA is $5 \times 10^{-5} \mathrm{~S} \mathrm{~cm}^{-1}$, degree of dissociation of HA is :$0.50$0.250.1250.75Correct Option: Solution:...
Read More →A TV transmission tower antenna is at a height
Question: Correct Option: , 2 Solution: $C=\overline{A+B}$ $C=\bar{A} \cdot B$...
Read More →Let {x} and [x] denote the fractional part
Question: Let $\{x\}$ and $[x]$ denote the fractional part of $x$ and the greatest integer $\leq x$ respectively of a real number $x$. If $\int_{0}^{n}\{x\} d x, \int_{0}^{n}[x] d x$ and $10\left(n^{2}-n\right),(n \in \mathbf{N}, n1)$ are three consecutive terms of a G.P., then $n$ is equal to______. Solution: $\int_{0}^{n}\{x\} d x=n \int_{0}^{1} x \cdot d x=\frac{n}{2}$ $\Rightarrow \int_{0}^{n}[x] d x=\int_{0}^{n}(x-\{x\}) d x=\frac{n^{2}}{2}-\frac{n}{2}$ According to the questions, $\frac{n}...
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