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Question: A cell $E_{1}$ of emf $6 \mathrm{~V}$ and internal resistance $2 \Omega$ is connected with another cell $\mathrm{E}_{2}$ of emf $4 \mathrm{~V}$ and internal resistance $8 \Omega$ (as shown in the figure). The potential difference across points $\mathrm{X}$ and $\mathrm{Y}$ is -(1) $3.6 \mathrm{~V}$(2) $10.0 \mathrm{~V}$(3) $5.6 \mathrm{~V}$(4) $2.0 \mathrm{~V}$Correct Option: , 2 Solution: emf of $E_{1}=6 \mathrm{v}$ $r_{1}=2 \Omega$ $\operatorname{emf}$ of $E_{2}=4 \Omega$ $r_{2}=8 \O...
Read More →solve this
Question: A cell $E_{1}$ of emf $6 \mathrm{~V}$ and internal resistance $2 \Omega$ is connected with another cell $\mathrm{E}_{2}$ of emf $4 \mathrm{~V}$ and internal resistance $8 \Omega$ (as shown in the figure). The potential difference across points $\mathrm{X}$ and $\mathrm{Y}$ is -(1) $3.6 \mathrm{~V}$(2) $10.0 \mathrm{~V}$(3) $5.6 \mathrm{~V}$(4) $2.0 \mathrm{~V}$Correct Option: , 2 Solution: emf of $E_{1}=6 \mathrm{v}$ $r_{1}=2 \Omega$ $\operatorname{emf}$ of $E_{2}=4 \Omega$ $r_{2}=8 \O...
Read More →Express
Question: Express $0 . \overline{4}$ as a rational number in simplest form. Solution: Let $x$ be $0 . \overline{4}$ $x=0 . \overline{4}$ Multiplying both sides by 10, we get $10 x=4 . \overline{4}$ Subtracting (1) from (2), we get $10 x-x=4 . \overline{4}-0 . \overline{4}$ $\Rightarrow 9 x=4$ $\Rightarrow x=\frac{4}{9}$ Thus, simplest form of $0 . \overline{4}$ as a rational number is $\frac{4}{9}$....
Read More →Let slope of the tangent line to a curve at any
Question: Let slope of the tangent line to a curve at any point $\mathrm{P}(\mathrm{x}, \mathrm{y})$ be given by $\frac{x y^{2}+y}{x}$, If the curve intersects the line $x+2 y=4$ at $x=-2$, then the value of $y$, for which the point $(3, y)$ lies on the curve, is :(1) $-\frac{18}{11}$(2) $-\frac{18}{19}$(3) $-\frac{4}{3}$(4) $\frac{18}{35}$Correct Option: , 2 Solution: $\frac{d y}{d x}=\frac{x y^{2}+y}{x}$ $\Rightarrow \frac{x d y-y d x}{y^{2}}=x d x$ $\Rightarrow-d\left(\frac{x}{y}\right)=d\lef...
Read More →The LCM of two numbers is 1200.
Question: The LCM of two numbers is 1200. Show that the HCF of these numbers cannot be 500. Why? Solution: If the LCM of two numbers is 1200 then, it is not possibleto have their HCF equals to 500.Since, HCF must be a factor of LCM, but 500 is not a factor of 1200....
Read More →If a and b are relatively prime, what is their LCM?
Question: Ifaandbare relatively prime, what is their LCM? Solution: If two numbers are relatively prime then their greatest common factor will be 1. HCF(a,b) = 1Using the formula, Product of two numbers = HCF LCMwe conclude that,ab= 1 LCM LCM =abThus, LCM(a,b) isab....
Read More →Two wires of same length and thickness having specific resistances
Question: Two wires of same length and thickness having specific resistances $6 \Omega \mathrm{cm}$ and $3 \Omega \mathrm{cm}$ respectively are connected in parallel. The effective resistivity is $\rho \Omega \mathrm{cm}$. The value of $\rho$ to the nearest integer, is Solution: (4) $\because$ in parallel $\mathrm{R}_{\mathrm{net}}=\frac{\mathrm{R}_{1} \mathrm{R}_{2}}{\mathrm{R}_{1}+\mathrm{R}_{2}}$ $\frac{\rho \ell}{2 \mathrm{~A}}=\frac{\rho_{1} \frac{\ell}{\mathrm{A}} \times \rho_{2} \frac{\el...
Read More →Give an example of two irrationals whose product is rational.
Question: Give an example of two irrationals whose product is rational. Solution: Let the two irrationals be $4 \sqrt{5}$ and $3 \sqrt{5}$. $(4 \sqrt{5}) \times(3 \sqrt{5})=60$ Thus, product (i.e., 60) is a rational number....
Read More →Give an example of two irrationals whose sum is rational.
Question: Give an example of two irrationals whose sum is rational. Solution: Let the two irrationals be $4-\sqrt{5}$ and $4+\sqrt{5}$. $(4-\sqrt{5})+(4+\sqrt{5})=8$ Thus, sum (i.e., 8) is a rational number....
Read More →Is it possible to have two numbers whose HCF is 25 and LCM is 520?
Question: Is it possible to have two numbers whose HCF is 25 and LCM is 520? Solution: No, it is not possibleto have two numbers whose HCF is 25 and LCM is 520.Since, HCF must be a factor of LCM, but 25 is not a factor of 520....
Read More →The difference between degree and order
Question: The difference between degree and order of differential equation that represents the family of curves given by $y^{2}=a\left(x+\frac{\sqrt{a}}{2}\right), a0$ is Solution: $y^{2}=a\left(x+\frac{\sqrt{a}}{2}\right)$ $2 y y^{\prime}=a$ $y^{2}=2 y y^{\prime}\left(x+\frac{\sqrt{2 y y^{\prime}}}{2}\right)$ $y=2 y^{\prime}\left(x+\frac{\sqrt{y y^{\prime}}}{\sqrt{2}}\right)$ $y-2 x y^{\prime}=\sqrt{2} y^{\prime} \sqrt{y y^{\prime}}$ $\left(y-2 x \frac{d y}{d x}\right)^{2}=2 y\left(\frac{d y}{d...
Read More →The voltage across
Question: The voltage across the $10 \Omega$ resistor in the given circuit is $\mathrm{x}$ volt. The value of ' $x$ ' to the nearest integer is Solution: (70) $\mathrm{R}_{\mathrm{eq}_{1}}=\frac{50 \times 20}{70}=\frac{100}{7}$ $\mathrm{R}_{\mathrm{eq}}=\frac{170}{7}$ $\mathrm{v}_{1}=\left[\frac{\frac{17}{170}}{7}\right] \times 10=70 \mathrm{v}$ Ans. $=70.00$...
Read More →Show that there is no value of n for which
Question: Show that there is no value of $n$ for which $\left(2^{n} \times 5^{n}\right)$ ends in 5 . Solution: We can write: $\left(2^{n} \times 5^{n}\right)=(2 \times 5)^{n}$ $=10^{n}$ For any value ofn, we get 0 in the end. Thus, there is no value of $n$ for which $\left(2^{n} \times 5^{n}\right)$ ends in 5 ....
Read More →Write the decimal expansion of
Question: Write the decimal expansion of $\frac{73}{\left(2^{4} \times 5^{3}\right)}$. Solution: Decimal expansion: $\frac{73}{\left(2^{4} \times 5^{3}\right)}=\frac{73 \times 5}{2^{4} \times 5^{4}}$ $=\frac{365}{(2 \times 5)^{4}}$ $=\frac{365}{(10)^{4}}$ $=\frac{365}{10000}$ $=0.0365$ Thus, the decimal expansion of $\frac{73}{\left(2^{4} \times 5^{3}\right)}$ is $0.0365$....
Read More →If y=y(x) is the solution of the equation
Question: If $y=y(x)$ is the solution of the equation $e^{\sin y} \cos y$ $\frac{d y}{d x}+e^{\sin y} \cos x=\cos x, y(0)=0$; then $1+y\left(\frac{\pi}{6}\right)+\frac{\sqrt{3}}{2} y\left(\frac{\pi}{3}\right)+\frac{1}{\sqrt{2}} y\left(\frac{\pi}{4}\right)$ is equal to Solution: $e^{\sin y} \cos y \frac{d y}{d x}+e^{\sin y} \cos x=\cos x$ Put $e^{\sin y}=t$ $e^{\sin y} \times \cos y \frac{d y}{d x}=\frac{d t}{d x}$ $\Rightarrow \frac{d t}{d x}+t \cos x=\cos x$ I. $F .=e^{\int \cos x d x}=e^{\sin ...
Read More →Simplify
Question: Simplify: $\frac{2 \sqrt{45}+3 \sqrt{20}}{2 \sqrt{5}}$ Solution: $\frac{2 \sqrt{45}+3 \sqrt{20}}{2 \sqrt{5}}=\frac{2 \sqrt{3 \times 3 \times 5}+3 \sqrt{2 \times 2 \times 5}}{2 \sqrt{5}}$ $=\frac{2 \times 3 \sqrt{5}+3 \times 2 \sqrt{5}}{2 \sqrt{5}}$ $=\frac{6 \sqrt{5}+6 \sqrt{5}}{2 \sqrt{5}}$ $=\frac{12 \sqrt{5}}{2 \sqrt{5}}$ $=6$ Hence, simplified form of $\frac{2 \sqrt{45}+3 \sqrt{20}}{2 \sqrt{5}}$ is 6 ....
Read More →In the experiment of Ohm's law,
Question: In the experiment of Ohm's law, a potential difference of $5.0 \mathrm{~V}$ is applied across the end of a conductor of length $10.0 \mathrm{~cm}$ and diameter of $5.00 \mathrm{~mm}$. The measured current in the conductor is $2.00 \quad$ A. The maximum permissible percentage error in the resistivity of the conductor is :-(1) $3.9$(2) $8.4$(3) $7.5$(4) 3Correct Option: 1 Solution: (1) $\mathrm{R}=\frac{\rho \ell}{\mathrm{A}}=\frac{\mathrm{V}}{\mathrm{I}}$ $\rho=\frac{\mathrm{AV}}{\mathr...
Read More →If the rational number
Question: If the rational number $\frac{a}{b}$ has a terminating decimal expansion, what is the condition to be satisfied by $b$ ? Solution: Letxbe a rational number whose decimal expansion terminates. Then, we can express $x$ in the form $\frac{a}{b}$, where $a$ and $b$ are coprime, and prime factorization of $b$ is of the form $\left(2^{m} \times 5^{m}\right)$, where $m$ and $n$ are non negative integers....
Read More →If a and b are relatively prime then what is their HCF?
Question: Ifaandbare relatively prime then what is their HCF? Solution: If two numbers are relatively prime then their greatest common factor will be 1.Thus, HCF(a,b) = 1....
Read More →Seawater at a frequency
Question: Seawater at a frequency $\mathrm{f}=9 \times 10^{2} \mathrm{~Hz}$, has permittivity $\varepsilon=80 \varepsilon_{0}$ and resistivity $\rho=0.25 \Omega \mathrm{m}$. Imagine a parallel plate capacitor is immersed in seawater and is driven by an alternating voltage source $V(t)=V_{0} \sin (2 \pi f t)$. Then the conduction current density becomes $10^{x}$ times the displacement current density after time $\mathrm{t}=\frac{1}{800} \mathrm{~s}$. The value of $\mathrm{x}$ is (Given : $\frac{1...
Read More →What is a composite number?
Question: What is a composite number? Solution: A composite number is a positive integer which is not prime (i.e., which has factors other than 1 and itself)....
Read More →If the product of two numbers is 1050 and their HCF is 25,
Question: If the product of two numbers is 1050 and their HCF is 25, find their LCM. Solution: HCF of two numbers = 25Product of two numbers = 1050Let their LCM bex.Using the formula, Product of two numbers = HCF LCMwe conclude that, $1050=25 \times x$ $x=\frac{1050}{25}$ = 42 Hence, their LCM is 42....
Read More →If a and b are two prime numbers then find LCM(a, b).
Question: Ifaandbare two prime numbers then find LCM(a,b). Solution: Prime factorization:a=ab=bLCM = product of greatest power of each prime factor involved in the numbers =abThus,LCM(a,b) =ab....
Read More →If a and b are two prime numbers then find HCF(a, b).
Question: Ifaandbare two prime numbers then find HCF(a,b). Solution: Prime factorization:a=ab=bHCF = product of smallest power of each common prime factor in the numbers = 1Thus,HCF(a,b) = 1...
Read More →Express 360 as product of its prime factors.
Question: Express 360 as product of its prime factors. Solution: Prime factorization: $360=2^{3} \times 3^{2} \times 5$...
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