The curve amongst the family of curves represented
Question: The curve amongst the family of curves represented by the differential equation, $\left(x^{2}-y^{2}\right) d x+2 x y d y=0$ which passes through $(1,1)$, is:(1) a circle with centre on the $x$-axis.(2) an ellipse with major axis along the $y$-axis.(3) a circle with centre on the $y$-axis.(4) a hyperbola with transverse axis along the $x$-axis.Correct Option: 1 Solution: $y^{2} d x-2 x y d y=x^{2} d x$ $2 x y d y-y^{2} d x=-x^{2} d x$ $d\left(x y^{2}\right)=-x^{2} d x$ $\frac{x d\left(y...
Read More →The series combination of two batteries,
Question: The series combination of two batteries, both of the same emf $10 \mathrm{~V}$, but different internal resistance of $20 \Omega$ and $5 \Omega$, is connected to the parallel combination of two resistors $30 \Omega$ and $\mathrm{R} \Omega$. The voltage difference across the battery of internal resistance $20 \Omega$ is zero, the value of $R$ (in $\Omega$ ) is _______ Solution: The resistance of $30 \Omega$ is in parallel with $R$. Their effective resistance $\frac{1}{R^{\prime}}=\frac{1...
Read More →Show that
Question: Show that $(4+3 \sqrt{2})$ is irrational. Solution: Let (4 + 32) be a rational number. Then both $(4+3 \sqrt{2})$ and 4 are rational. $\Rightarrow(4+3 \sqrt{2}-4)=3 \sqrt{2}=$ rational $[\because$ Difference of two rational numbers is rational] $\Rightarrow 3 \sqrt{2}$ is rational. $\Rightarrow \frac{1}{3}(3 \sqrt{2})$ is rational. $\quad[\because$ Product of two rational numbers is rational] $\Rightarrow \sqrt{2}$ is rational. This contradicts the fact that $\sqrt{2}$ is irrational (w...
Read More →if
Question: If $\frac{\mathrm{d} y}{\mathrm{~d} x}+\frac{3}{\cos ^{2} x} y=\frac{1}{\cos ^{2} x}, x \in\left(\frac{-\pi}{3}, \frac{\pi}{3}\right)$, and $y\left(\frac{\pi}{4}\right)=\frac{4}{3}$, then $y\left(-\frac{\pi}{4}\right)$ equals: (1) $\frac{1}{3}+e^{6}$(2) $\frac{1}{3}$(3) $-\frac{4}{3}$(4) $\frac{1}{3}+\mathrm{e}^{3}$Correct Option: 1 Solution: Given, $\frac{d y}{d x}+\frac{3}{\cos ^{2} x} y=\frac{1}{\cos ^{2} x}$ $\frac{d y}{d x}=\sec ^{2} x(1-3 y)$ $\Rightarrow \int \frac{d y}{(1-3 y)}...
Read More →Show that one and only one out of n, (n + 2) and (n + 4) is divisible by 3,
Question: Show that one and only one out ofn, (n+ 2) and (n+ 4) is divisible by 3, wherenis any positive integer. Solution: Let qbe quotient andrbe the remainder.On applying Euclid's algorithm, i.e. dividingnby 3, we haven =3q+r 0r 3⇒ n =3q+ r r =0, 1 or 2⇒n =3qor n= (3q+1) orn =(3q+2)Case 1: Ifn =3q, thennis divisible by 3.Case 2: Ifn =(3q+1), then (n +2) = 3q+3 = 3(3q+1), which is clearly divisible by 3. In this case, (n +2) is divisible by 3.Case 3: Ifn =(3q+2), then (n +4) = 3q+6 = 3(q +2),...
Read More →Show that every positive odd integer is of the form
Question: Show that every positive odd integer is of the form (4q+ 1) or (4q+ 3) for some integerq. Solution: Letabe the given positive odd integer.On dividingaby 4,letqbe the quotient andrthe remainder.Therefore,by Euclid's algorithm we havea =4q+r 0r 4⇒ a =4q+ r r = 0,1,2,3⇒ a =4q,a =4q+1, a =4q+2, a =4q+ 3But, 4qand 4q+2 = 2 (2q+1) = evenThus, whenais odd, it is of the form (4q+ 1) or (4q+3) for some integerq....
Read More →Four resistances
Question: Four resistances of $15 \Omega, 12 \Omega^{2}, 4 \Omega$ and $10 \Omega$ respectively in cyclic order to form Wheatstone's network. The resistance that is to be connected in parallel with the resistance of $10 \Omega$ to balance the network is_____$\Omega$ Solution: As per Wheatstone bridge balance condition $\frac{P}{Q}=\frac{S}{R}$ Let resistance $R^{\prime}$ is connected in parallel with resistance S of $10 \Omega$ $\therefore \frac{15}{12}=\frac{10 R^{\prime}}{\frac{10+R^{\prime}}{...
Read More →satisfies the differential equation,
Question: Let $f:[0,1] \rightarrow \mathrm{R}$ be such that $f(x y)=f(x) f(y)$, for all $x, y \in[0,1]$, and $f(0) \neq 0$. If $y=y(x)$ satisfies the differential equation, $\frac{d y}{d x}=f(x)$ with $y(0)=1$, then $y\left(\frac{1}{4}\right)+y\left(\frac{3}{4}\right)$ is equal to:(1) 3(2) 4(3) 2(4) 5Correct Option: 1 Solution: $f(x y)=f(x) \cdot f(y)$....(1) Put $x=y=0$ in (1) to get $f(0)=1$ Put $x=y=1$ in (1) to get $f(1)=0$ or $f(1)=1$ $f(1)=0$ is rejected else $y=1$ in (1) gives $f(x)=0$ $\...
Read More →Prove that
Question: Prove that $\sqrt{3}$ is an irrational number. Solution: Let $\sqrt{3}$ be rational and its simplest form be $\frac{a}{b}$. Then, $a, b$ are integers with no common factors other than 1 and $b \neq 0$. Now $\sqrt{3}=\frac{a}{b} \Rightarrow 3=\frac{a^{2}}{b^{2}}$ [on squaring both sides] $\Rightarrow 3 b^{2}=a^{2}$ ... (1) $\Rightarrow 3$ divides $a^{2} \quad$ [since 3 divides $3 b^{2}$ ] $\Rightarrow 3$ divides $a \quad$ [since 3 is prime, 3 divides $a^{2} \Rightarrow 3$ divides $a$ ] ...
Read More →The length of a potentiometer wire
Question: The length of a potentiometer wire is $1200 \mathrm{~cm}$ and it carries a current of $60 \mathrm{~mA}$. For a cell of emf $5 \mathrm{~V}$ and internal resistance of $20 \Omega$, the null point on it is found to be at $1000 \mathrm{~cm}$. The resistance of whole wire is:(1) $80 \Omega$(2) $120 \Omega$(3) $60 \Omega$(4) $100 \Omega$Correct Option: , 4 Solution: Let $R$ be the resistance of the whole wire Potential gradient for the potentiometer wire $' A B^{\prime}=-\frac{d V}{d \ell}=\...
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, $x \frac{\mathrm{d} y}{\mathrm{~d} x}+2 y=x^{2}$ satisfying $y(1)=1$, then $y\left(\frac{1}{2}\right)$ is equal to:(1) $\frac{7}{64}$(2) $\frac{1}{4}$(3) $\frac{49}{16}$(4) $\frac{13}{16}$Correct Option: , 3 Solution: Since, $x \frac{d y}{d x}+2 y=x^{2}$ $\Rightarrow \quad \frac{d y}{d x}+\frac{2}{x} y=x$ I.F. $\quad=e^{\int \frac{2}{x} d x}=e^{2 \ln x}=e^{\ln x^{2}}=x^{2}$. Solution of differential equation is: $y \cdot x^{2}=\...
Read More →Find the largest number which divides 546 and 764,
Question: Find the largest number which divides 546 and 764, leaving remainders 6 and 8 respectively. Solution: We know the required number divides 540 (546 6) and 756 (764 8), respectively.Required largest number =HCF (540, 756)Prime factorisation: $540=2 \times 2 \times 3 \times 3 \times 3 \times 5=2^{2} \times 3^{2} \times 5$ $756=2 \times 2 \times 3 \times 3 \times 3 \times 7=2^{2} \times 3^{3} \times 7$ $\therefore \mathrm{HCF}=2^{2} \times 3^{3}=108$ Hence, the largest number is 108 ....
Read More →The balancing length for a cell is
Question: The balancing length for a cell is $560 \mathrm{~cm}$ in a potentiometer experiment. When an external resistance of $10 \Omega$ is connected in parallel to the cell, the balancing length changes by $60 \mathrm{~cm}$. If the internal resistance of the cell is $\frac{N}{10} \Omega$, where $\mathrm{N}$ is an integer then value of $\mathrm{N}$ is _______ Solution: (12) We know that $E \propto \ell$ where $l$ is the balancing length $\therefore \mathrm{E}=k(560)$ ....(1) When the balancing ...
Read More →Find the HCF and LCM of
Question: Find the HCF and LCM of $\frac{8}{9}, \frac{10}{27}$ and $\frac{16}{81}$. Solution: HCF of fractions $=\frac{\text { HCF of Numerators }}{\text { LCM of Denominators }}$ $\mathrm{LCM}$ of fractions $=\frac{\text { LCM of Numerators }}{\text { HCF of Denominators }}$ Prime factorisation of the numbers given in the numerators are as follows: $8=2 \times 2 \times 2$ $10=2 \times 5$ $16=2 \times 2 \times 2 \times 2$ HCF of Numerators $=2$ $L C M$ of Numerators $=2^{4} \times 5=80$ Prime fa...
Read More →The general solution of the differential equation
Question: The general solution of the differential equation $\left(y^{2}-x^{3}\right) \mathrm{dx}-x y d y=0(x \uparrow 0)$ is : (where $c$ is a constant of integration)(1) $y^{2}-2 x^{2}+c x^{3}=0$(2) $y^{2}+2 x^{3}+c x^{2}=0$(3) $y^{2}+2 x^{2}+c x^{3}=0$(4) $y^{2}-2 x^{3}+c x^{2}=0$Correct Option: , 2 Solution: Given differential equation can be written as, $y^{2} d x-x y d y=x^{3} d x$ $\Rightarrow \frac{(y d x-x d y) y}{x^{2}}=x d x \Rightarrow-y d\left(\frac{y}{x}\right)=x d x$ $\Rightarrow-...
Read More →Consider the differential equation,
Question: Consider the differential equation, $y^{2} d x+\left(x-\frac{1}{y}\right) d y=0$. If value of $y$ is 1 when $x=1$, then the value of $x$ for which $y=2$, is :(1) $\frac{5}{2}+\frac{1}{\sqrt{e}}$(2) $\frac{3}{2}-\frac{1}{\sqrt{e}}$(3) $\frac{1}{2}+\frac{1}{\sqrt{e}}$(4) $\frac{3}{2}-\sqrt{e}$.Correct Option: , 2 Solution: Consider the differential equation, $y^{2} d x+\left(x-\frac{1}{y}\right) d y=0$ $\Rightarrow \frac{d x}{d y}+\left(\frac{1}{y^{2}}\right) x=\frac{1}{y^{3}}$ $\mathrm{...
Read More →Find the HCF of 1008 and 1080 by prime factorization method.
Question: Find the HCF of 1008 and 1080 by prime factorization method. Solution: Prime factorisation: $1008=2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 7=2^{4} \times 3^{2} \times 7$ $1080=2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 5=2^{3} \times 3^{3} \times 5$ HCF = Product of smallest power of each common prime factor in the number $=2^{3} \times 3^{2}=72$...
Read More →Give prime factorisation of 4620.
Question: Give prime factorisation of 4620. Solution: Prime factorisation: $4620=2 \times 2 \times 3 \times 5 \times 7 \times 11=2^{2} \times 3 \times 5 \times 7 \times 11$...
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 $(2+\sqrt{2})$ and $(2-\sqrt{2})$ be two irrational numbers. Sum $=(2+\sqrt{2})+(2-\sqrt{2})=2+\sqrt{2}+2-\sqrt{2}=4$, which is a rational number....
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, $\frac{d y}{d x}+y \tan x=2 x+x^{2} \tan x, x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, such that $\mathrm{y}(0)=1$ Then :(1) $y\left(\frac{\pi}{4}\right)+y\left(-\frac{\pi}{4}\right)=\frac{\pi^{2}}{2}+2$(2) $y,\left(\frac{\pi}{4}\right)+y \cdot\left(\frac{\pi}{4}\right)=-\sqrt{2}$(3) $y\left(\frac{\pi}{4}\right)-y\left(-\frac{\pi}{4}\right)=\sqrt{2}$(4) $y^{\prime}\left(\frac{\pi}{4}\right)-y^{\prime}\left(-\frac{\pi}{4}...
Read More →In the figure, potential difference between A and B is
Question: In the figure, potential difference between $A$ and $B$ is: (1) $10 \mathrm{~V}$(2) $5 \mathrm{~V}$(3) $15 \mathrm{~V}$(4) zeroCorrect Option: 1 Solution: (1) The given circuit has two $10 k \Omega$ resistances in parallel, so we can reduce this parallel combination to a single equivalent resistance of $5 \mathrm{k} \Omega$. Diode is in forward bias. So it will behave like a conducting wire. $V_{A}-V_{B}=\frac{30}{5+10} \times 5=10 \mathrm{~V}$...
Read More →Find the HCF and LCM of 12, 15, 18, 27.
Question: Find the HCF and LCM of 12, 15, 18, 27. Solution: Prime factorisation: $12=2 \times 2 \times 3=2^{2} \times 3$ $15=3 \times 5$ $18=2 \times 3 \times 3=2 \times 3^{2}$ $27=3 \times 3 \times 3=3^{3}$ Now,HCF = Product of smallest power of each common prime factor in the number= 3LCM = Product of greatest power of each prime factor involved in the number $=2^{2} \times 3^{3} \times 5=540$...
Read More →Prove that
Question: Prove that $(2+\sqrt{3})$ is irrational. Solution: Let $(2+\sqrt{3})$ be rational. Then, both $(2+\sqrt{3})$ and 2 are rational. $\therefore\{(2+\sqrt{3})-2\}$ is rational [ $\because$ Difference of two rational is rational] $\Rightarrow \sqrt{3}$ is rational. This contradicts the fact that $\sqrt{3}$ is irrational. The contradiction arises by assuming $(2+\sqrt{3})$ is rational. Hence, $(2+\sqrt{3})$ is irrational....
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 $\frac{\mathrm{dy}}{\mathrm{dx}}=(\tan x-y) \sec ^{2} x, \mathrm{x} \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, such that $\mathrm{y}(0)=0$ then $y\left(-\frac{\pi}{4}\right)$ is equal to :(1) $\mathrm{e}-2$ (2) $\frac{1}{2}-e$(3) $2+\frac{1}{e}$(4) $\frac{1}{e}-2$Correct Option: 1 Solution: $\frac{d y}{d x}+y \sec ^{2} x=\sec ^{2} \times \tan x$ Given equation is linear differential equation. IF $=e^{\int \sec ^{2} x d x}=e^...
Read More →Which of the following numbers are irrational?
Question: Which of the following numbers are irrational? (a) $\sqrt{2}$ (b) $\sqrt[3]{6}$ (c) $3.142857$ (d) $2 . \overline{3}$ (e) $\pi$ (f) $\frac{22}{7}$ (g) $0.232332333 \ldots$ (h) $5.27 \overline{41}$ Solution: (a) $\sqrt{2}$ is irrational ( $\because$ if $p$ is prime, then $\sqrt{p}$ is irrational). (b) $\sqrt[3]{6}=\sqrt[3]{2} \times \sqrt[3]{3}$ is irrational. (c) $3.142857$ is rational because it is a terminating decimal. (d) $2 . \overline{3}$ is rational because it is a non-terminati...
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