In a building there are 15 bulbs of 45 W
Question: In a building there are 15 bulbs of $45 \mathrm{~W}, 15$ bulbs of $100 \mathrm{~W}$, 15 small fans of $10 \mathrm{~W}$ and 2 heaters of $1 \mathrm{~kW}$. The voltage of electric main is $220 \mathrm{~V}$. The minimum fuse capacity (rated value) of the building will be:(1) $10 \mathrm{~A}$(2) $25 \mathrm{~A}$(3) $15 \mathrm{~A}$(4) $20 \mathrm{~A}$Correct Option: , 4 Solution: (4) Net Power, $P$ $=15 \times 45+15 \times 100+15 \times 10+2 \times 1000$ $=15 \times 155+2000 \mathrm{~W}$ P...
Read More →Find the simplest form of
Question: Find the simplest form of $\frac{148}{185}$. Solution: $\frac{148}{185}=\frac{148 \div 37}{185 \div 37}=\frac{4}{5}(\because$ HCF of 148 and 185 is 37$)$ Hence, the simplest form is $\frac{4}{5}$....
Read More →Examine whether
Question: Examine whether $\frac{17}{30}$ is a terminating decimal. Solution: $\frac{17}{30}=\frac{17}{2 \times 3 \times 5}$ We know that 2, 3 and 5 are not the factors of 17. So, $\frac{17}{30}$ is in its simplest form. Also, $30=2 \times 3 \times 5 \neq\left(2^{m} \times 5^{n}\right)$ Hence, $\frac{17}{30}$ is a non-terminating decimal....
Read More →The HCF of two numbers is 27 and their LCM is 162.
Question: The HCF of two numbers is 27 and their LCM is 162. If one of the number is 81, find the other. Solution: Let the two numbers be $x, y$. It is given that:x= 81HCF = 27 and LCM = 162 We know, Product of two numbers $=\mathrm{HCF} \times \mathrm{LCM}$ $\Rightarrow \quad x \times y=27 \times 162$ $\Rightarrow \quad 81 \times y=4374$ $\Rightarrow \quad y=\frac{4374}{81}=54$ Hence, the other numberyis 54....
Read More →Show that any number of the form 4n, n ∈ N
Question: Show that any number of the form 4n,nNcan never end with the digit 0. Solution: If 4nends with 0, then it must have 5 as a factor.But we know the only prime factor of 4nis 2.Also we know from the fundamental theorem of arithmetic that prime factorisation of each number is unique.Hence, 4ncan never end with the digit 0....
Read More →The current
Question: The current $I_{1}$ (in $A$ ) flowing through $1 \Omega$ resistor in the following circuit is: (1) $0.4$(2) $0.5$(3) $0.2$(4) $0.25$Correct Option: , 3 Solution: (3)...
Read More →if cos x
Question: If $\cos x \frac{d y}{d x}-y \sin x=6 x,\left(0x\frac{\pi}{2}\right)$ and $y\left(\frac{\pi}{3}\right)=0$, then $y\left(\frac{\pi}{6}\right)$ is equal to: (1) $\frac{\pi^{2}}{2 \sqrt{3}}$(2) $-\frac{\pi^{2}}{2}$(3) $-\frac{\pi^{2}}{2 \sqrt{3}}$(4) $-\frac{\pi^{2}}{4 \sqrt{3}}$Correct Option: , 3 Solution: $\cos x d y-(\sin x) y d x=6 x d x$ $\Rightarrow \int d(y \cos x)=\int 6 x d x \Rightarrow y \cos x=3 x^{2}+C \ldots(1)$ Given, $y\left(\frac{\pi}{3}\right)=0$ Putting $x=\frac{\pi}{3...
Read More →Prove that
Question: $0 . \overline{68}+0 . \overline{73}=?$ (a) $1 . \overline{41}$ (b) $1 . \overline{42}$ (c) $0 . \overline{141}$ (d) None of these Solution: (b) $1 . \overline{42}$...
Read More →On dividing a positive integer n by 9, we get 7 as remainder. What will be the remainder if
Question: On dividing a positive integernby 9, we get 7 as remainder. What will be the remainder if (3n 1) is divided by 9?(a) 1(b) 2(c) 3(d) 4 Solution: (b) 2Letqbe the quotient.It is given that:remainder = 7On applying Euclid's algorithm, i.e. dividingnby 9, we haven =9q+ 7⇒ 3n=27q+21⇒3n1 = 27q+20 $\Rightarrow 3 n-1=9 \times 3 q+9 \times 2+2$ $\Rightarrow 3 n-1=9 \times(3 q+2)+2$ So, when (3n 1) is divided by 9, we get the remainder 2....
Read More →In the figure shown, the current in the $
Question: In the figure shown, the current in the $10 \mathrm{~V}$ battery is close to(1) $0.71$ A from positive to negative terminal(2) $0.42 \mathrm{~A}$ from positive to negative terminal(3) $0.21$ A from positive to negative terminal(4) $0.36 \mathrm{~A}$ from negative to positive terminalCorrect Option: , 3 Solution: Using Kirchoff's loop law in loop $A B C D$ $-5 i_{2}-10\left(i_{1}+i_{2}\right)-2 i_{2}+20=0$ $\Rightarrow-10 i_{1}-17 i_{2}+20=0$ ..(1) Using Kirchoff's loop law in loop $B E...
Read More →Ine solution of the differential equation
Question: Ine solution of the differential equation $x \frac{d y}{d x}+2 y=x^{2}$ $(x \neq 0)$ with $y(1)=1$, is:(1) $\mathrm{y}=\frac{4}{5} x^{3}+\frac{1}{5 x^{2}}$(2) $\mathrm{y}=\frac{x^{3}}{5}+\frac{1}{5 x^{2}}$(3) $\mathrm{y}=\frac{x^{2}}{4}+\frac{3}{4 x^{2}}$(4) $\mathrm{y}=\frac{3}{4} x^{2}+\frac{1}{4 x^{2}}$Correct Option: , 3 Solution: $\frac{d y}{d x}+\frac{2}{x} y=x$ and $y(1)=1$ (given) Since, the above differential equation is the linear differential equation, then $I . F=e^{\int \f...
Read More →Which of the following has terminating decimal expansion?
Question: Which of the following has terminating decimal expansion? (a) $\frac{32}{91}$ (b) $\frac{19}{80}$ (c) $\frac{23}{45}$ (d) $\frac{25}{42}$ Solution: (b) $\frac{19}{80}$ $\frac{19}{80}=\frac{19}{2^{4} \times 5}$ We know 2 and 5 are not factors of 19, so it is in its simplest form. And $\left(2^{4} \times 5\right)=\left(2^{m} \times 5^{n}\right)$ Hence, $\frac{19}{80}$ is a terminating decimal....
Read More →The decimal representation of
Question: The decimal representation of $\frac{71}{150}$ is (a) a terminating decimal(b) a non-terminating, repeating decimal(b) a non-terminating and non-repeating decimal(d) none of these Solution: (b) a non-terminating, repeating decimal $\frac{71}{150}=\frac{71}{2 \times 3 \times 5^{2}}$ We know that 2, 3 or 5 are not factors of 71.So, it is in its simplest form. And, $\left(2 \times 3 \times 5^{2}\right) \neq\left(2^{m} \times 5^{n}\right)$ $\therefore \frac{71}{150}=0.47 \overline{3}$ Henc...
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, $\left(x^{2}+1\right)^{2} \frac{d y}{d x}+2 x\left(x^{2}+1\right) y=1$ such that $y(0)=0$. If $\sqrt{a} y(1)=\frac{\pi}{32}$, then the value of ' $a$ ' is : (1) $\frac{1}{4}$(2) $\frac{1}{2}$(3) 1(4) $\frac{1}{16}$Correct Option: , 4 Solution: $\left(1+x^{2}\right)^{2} \frac{d y}{d x}+2 x\left(1+x^{2}\right) y=1$ $\Rightarrow \frac{d y}{d x}+\left(\frac{2 x}{1+x^{2}}\right) y=\frac{1}{\left(1+x^{2}\right)^{2}}$ Since, the above...
Read More →What is the least number that divisible by all the natural numbers from 1 to 10 (both inclusive)?
Question: What is the least number that divisible by all the natural numbers from 1 to 10 (both inclusive)?(a) 100(b) 1260(c) 2520(d) 5040 Solution: (c) 2520We have to find the least number that is divisible by all numbers from 1 to 10. $\therefore$ LCM $(1$ to 10$)=2^{3} \times 3^{2} \times 5 \times 7=2520$ Thus, 2520 is the least number that is divisible by every element and is equal to the least common multiple....
Read More →Prove that
Question: $(2+\sqrt{2})$ is (a) an integer(b) a rational number(c) an irrational number(d) none of these Solution: (c) an irrational number $2+\sqrt{2}$ is an irrational number. if it is rational, then the difference of two rational is rational $\therefore(2+\sqrt{2})-2=\sqrt{2}=$ irrational...
Read More →if
Question: If $\frac{d y}{d x}=\frac{x y}{x^{2}+y^{2}} ; y(1)=1 ;$ then a value of $x$ satisfying $y(x)=e$ is:(1) $\frac{1}{2} \sqrt{3} e$(2) $\frac{e}{\sqrt{2}}$(3) $\sqrt{2} e$(4) $\sqrt{3} e$Correct Option: , 4 Solution: The given differential equation, $\frac{d y}{d x}=\frac{x y}{x^{2}+y^{2}}$ Put $y=v x \Rightarrow \frac{d y}{d x}=v+x \frac{d v}{d x}$ Then, $v+x \frac{d v}{d x}=\frac{v x^{2}}{x^{2}+v^{2} x^{2}}=\frac{v}{1+v^{2}}$ $\Rightarrow \quad \frac{1+v^{2}}{v^{3}} d v=-\frac{1}{x} d x$...
Read More →In the circuit, given in the figure currents in different branches and value of one resistor are shown.
Question: In the circuit, given in the figure currents in different branches and value of one resistor are shown. Then potential at point $B$ with respect to the point $A$ is : (1) $+2 \mathrm{~V}$(2) $-2 \mathrm{~V}$(3) $-1 \mathrm{~V}$(4) $+1 V$Correct Option: , 4 Solution: Let us assume the potential at $A=V_{A}=0$ Using Kirchoff's junction rule at $C$, we get $i_{1}+i_{3}=i_{2}$ $1 \mathrm{~A}+i_{3}=2 \mathrm{~A} \Rightarrow i_{3}=2 \mathrm{~A}$ Now using Kirchoff's loop law along $A C D B$ ...
Read More →If for x >0, y=y(x) is the solution
Question: If for $x \geq 0, y=y(x)$ is the solution of the differential equation, $(x+1) d y=\left((x+1)^{2}+y-3\right) d x, y(2)=0$ then $y(3)$ is equal to__________. Solution: $(x+1) d y \quad((x+1)+(y \quad 3)) d x \quad 0$ $\Rightarrow \frac{d y}{d x}=(1+x)+\left(\frac{y-3}{1+x}\right)$ $\frac{d y}{d x}-\frac{1}{(1+x)} y=(1+x)-\frac{3}{(1+x)}$ I.F. $=e^{-\int \frac{1}{(1+x)} d x}=\frac{1}{(1+x)}$ $\therefore \quad \frac{d}{d x}\left(\frac{y}{1+x}\right)=1-\frac{3}{(1+x)^{2}}$ $\frac{y}{1+x}=...
Read More →An electrical power line, having a total resistance
Question: An electrical power line, having a total resistance of $2 \Omega$, delivers $1 \mathrm{~kW}$ at $220 \mathrm{~V}$. The efficiency of the transmission line is approximately:(1) $72 \%$(2) $91 \%$(3) $85 \%$(4) $96 \%$Correct Option: 4, Solution: (2) Given : Power, $P=1 \mathrm{~kW}=1000 \mathrm{~W}$ $R=2 \Omega, V=220 \mathrm{~V}$ Current, $I=\frac{P}{V}=\frac{1000}{220}$ $P_{\text {loss }}=I^{2} R=\left(\frac{1000}{220}\right)^{2} \times 2$ $\therefore$ Efficiency $=\frac{1000}{1000+P_...
Read More →The differential equation of the family of curves,
Question: The differential equation of the family of curves, $x^{2}=4 b(y+b), b \in R$, is:(1) $x\left(y^{\prime}\right)^{2}=x+2 y y^{\prime}$(2) $x\left(y^{\prime}\right)^{2}=2 y y^{\prime}-x$(3) $x y^{\prime \prime}=y^{\prime}$(4) $x\left(y^{\prime}\right)^{2}=x-2 y y^{\prime}$Correct Option: 1, Solution: Since, $x^{2}=4 b(y+b)$ $x^{2}=4 b y+4 b^{2}$ $2 x=4 b y^{\prime}$ $\Rightarrow \quad b=\frac{x}{2 y^{\prime}}$ So, differential equation is $x^{2}=\frac{2 x}{y^{\prime}} \cdot y+\left(\frac{...
Read More →Solve this
Question: Four resistances $40 \Omega, 60 \Omega, 90 \Omega$ and $110 \Omega$ make the arms of a quadrilateral $A B C D$. Across $A C$ is a battery of emf $40 \mathrm{~V}$ and internal resistance negligible. The potential difference across $B D$ in $\mathrm{V}$ is ___________ Solution: Current through $A B, i_{1}=\frac{40}{40+60}=0.4$ Current through $A D, i_{2}=\frac{40}{90+110}=\frac{1}{5}$ Using KVL in BAD loop $V_{B}+i_{1}(40)-i_{2}(90)=V_{D}$ $\Rightarrow V_{B}-V_{D}=\frac{1}{5}(90)-\frac{4...
Read More →Let y=y(x) be a solution of the differential equation,
Question: Let $y=y(x)$ be a solution of the differential equation, $\sqrt{1-x^{2}} \frac{d y}{d x}+\sqrt{1-y^{2}}=0,|x|1 .$ If $y\left(\frac{1}{2}\right)=\frac{\sqrt{3}}{2}$, then $y\left(\frac{-1}{\sqrt{2}}\right)$ is equal to: (1) $\frac{\sqrt{3}}{2}$(2) $-\frac{1}{\sqrt{2}}$(3) $\frac{1}{\sqrt{2}}$(4) $-\frac{\sqrt{3}}{2}$Correct Option: , 3 Solution: The given differential eqn. is $\frac{d y}{\sqrt{1-y^{2}}}+\frac{d x}{\sqrt{1-x^{2}}}=0 \Rightarrow \sin ^{-1} y+\sin ^{-1} x=c$ At $x=\frac{1}...
Read More →Solve this
Question: $\frac{1}{\sqrt{2}}$ is (a) a fraction(b) a rational number(c) an irrational number(d) none of these Solution: (c) an irrational number $\frac{1}{\sqrt{2}}$ is an irrational number....
Read More →The value of current 1 flowing from A to C in the circuit diagram is :
Question: The value of current $i_{1}$ flowing from $A$ to $C$ in the circuit diagram is : (1) $2 \mathrm{~A}$(2) $4 \mathrm{~A}$(3) $1 \mathrm{~A}$(4) $5 \mathrm{~A}$Correct Option: 3, Solution: (3) The equivalent circuit can be drawn as Voltage across $A C=8 \mathrm{~V}$ Resistance $R_{A C}=4+4=8 \Omega$ $i_{1}=\frac{V}{R_{A C}}=\frac{8}{4+4}=1 \mathrm{Amp}$...
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