If y=y(x) is the solution of the differential
Question: If $y=y(x)$ is the solution of the differential equation $\frac{5+\mathrm{e}^{x}}{2+y} \cdot \frac{\mathrm{d} y}{\mathrm{~d} x}+\mathrm{e}^{x}=0$ satisfying $y(0)=1$, then a value of $y\left(\log _{\mathrm{e}} 13\right)$ is :(1) 1$(2)-1$(3) 0(4) 2Correct Option: , 2 Solution: $\frac{5+e^{x}}{2+y} \cdot \frac{d y}{d x}=-e^{x}$ $\int \frac{d y}{2+y}=-\int \frac{e^{x}}{5+e^{x}} d x$ $\Rightarrow \log _{e}|2+y| \cdot \log _{e}\left|5+e^{x}\right|=\log _{e} C$ $\Rightarrow\left|(2+y)\left(5...
Read More →What is the largest number that divides 70 and 125,
Question: What is the largest number that divides 70 and 125, leaving remainders 5 and 8 respectively?(a) 13(b) 9(c) 3(d) 585 Solution: (a) 13We know the required number divides 65 (70 5) and 117 (125 8). Required number =HCF (65, 117)we know, $65=13 \times 5$ $117=13 \times 3 \times 3$ $\therefore \mathrm{HCF}=13$...
Read More →The solution of the differential equation
Question: The solution of the differential equation $\frac{d y}{d x}-\frac{y+3 x}{\log _{e}(y+3 x)}+3=0$ is : (where $C$ is a constant of integration.)(1) $x-\frac{1}{2}\left(\log _{e}(y+3 x)\right)^{2}=C$(2) $x-\log _{e}(y+3 x)=C$(3) $y+3 x-\frac{1}{2}\left(\log _{e} x\right)^{2}=C$(4) $x-2 \log _{e}(y+3 x)=C$Correct Option: 1 Solution: Let $y+3 x=t$ $\Rightarrow \frac{d y}{d x}+3=\frac{d t}{d x}$ Putting these value in given differential equation $\frac{d t}{d x}=\frac{t}{\log _{e} t}$ $\Right...
Read More →What is the largest number that divides each one of 1152 and 1664 exactly?
Question: What is the largest number that divides each one of 1152 and 1664 exactly?(a) 32(b) 64(c) 128(d) 256 Solution: (c) 128Largest number that divides each one of 1152 and 1664 =HCF (1152, 1664)We know, $1152=2^{7} \times 3^{2}$ $1164=2^{7} \times 13$ $\therefore \mathrm{HCF}=2^{7}=128$...
Read More →The product of two numbers is 1600 and their HCF is 5. The LCM of the numbers is
Question: The product of two numbers is 1600 and their HCF is 5. The LCM of the numbers is(a) 8000(b) 1600(c) 320(d) 1605 Solution: (c) 320Let the two numbers bexandy.It is given that: $x \times y=1600$ HCF = 5 We know, $\mathrm{HCF} \times \mathrm{LCM}=x \times y$ $\Rightarrow \quad 5 \times \mathrm{LCM}=1600$ $\Rightarrow \quad \therefore \angle C M=\frac{1600}{5}=320$...
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, $x y^{\prime}-y=x^{2}(x \cos x+\sin x), x0$. If $y(\pi)=\pi$, then $y^{\prime \prime}\left(\frac{\pi}{2}\right)+y\left(\frac{\pi}{2}\right)$ is equal to:(1) $2+\frac{\pi}{2}$(2) $1+\frac{\pi}{2}+\frac{\pi^{2}}{4}$(3) $2+\frac{\pi}{2}+\frac{\pi^{2}}{4}$(4) $1+\frac{\pi}{2}$Correct Option: 1 Solution: $\frac{d y}{d x}-\frac{y}{x}=x(x \cos x+\sin x)$ I.F. $=e^{-\int \frac{1}{x} d x}=\frac{1}{x}$ $\therefore \int d\left(\frac{y}{x}...
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 numbers is 54, what is the other number?(a) 36(b) 45(c) 9(d) 81 Solution: (d) 81Let the two numbers bexandy.It is given that:x =54HCF = 27LCM = 162We know, $x \times y=\mathrm{HCF} \times \mathrm{LCM}$ $\Rightarrow 54 \times y=27 \times 162$ $\Rightarrow 54 y=4374$ $\Rightarrow \quad \therefore y=\frac{4374}{54}=81$...
Read More →Consider four conducting materials copper,
Question: Consider four conducting materials copper, tungsten, mercury and aluminium with resistivity $\rho_{C}, \rho_{T}, \rho_{\mathrm{M}}$ and $\rho_{A}$ respectively. Then :(1) $\rho_{C}\rho_{A}\rho_{T}$(2) $\rho_{M}\rho_{A}\rho_{C}$(3) $\rho_{A}\rho_{T}\rho_{C}$(4) $\rho_{A}\rho_{M}\rho_{C}$Correct Option: , 2 Solution: (2) $\rho_{M}=98 \times 10^{-8}$ $\rho_{A}=2.65 \times 10^{-8}$ $\rho_{C}=1.724 \times 10^{-8}$ $\rho_{T}=5.65 \times 10^{-8}$ $\therefore \rho_{M}\rho_{T}\rho_{A}\rho_{C}$...
Read More →LCM of
Question: LCM of $\left(2^{3} \times 3 \times 5\right)$ and $\left(2^{4} \times 5 \times 7\right)$ is (a) 40(b) 560(c) 1680(d) 1120 Solution: (c) 1680 LCM $\left(2^{3} \times 3 \times 5,2^{4} \times 5 \times 7\right)$ LCM = Product of greatest power of each prime factor involved in the numbers $=2^{4} \times 3 \times 5 \times 7$ $=16 \times 3 \times 5 \times 7$ $=1680$...
Read More →Five equal resistances are connected in a network as shown in figure.
Question: Five equal resistances are connected in a network as shown in figure. The net resistance between the points $A$ and $B$ is : (1) $\frac{3 R}{2}$(2) $\frac{\mathrm{R}}{2}$(3) $\mathrm{R}$(4) $2 \mathrm{R}$Correct Option: , 3 Solution: $(3)$ It is balanced wheat stone bridge So, we know that $\mathrm{R}_{1} \mathrm{R}_{\mathrm{A}}=\mathrm{R}_{2} \mathrm{R}_{3}$ $\frac{\mathrm{R}_{1}}{\mathrm{R}_{2}}=\frac{\mathrm{R}_{3}}{\mathrm{R}_{4}}$ $\operatorname{Req}=\frac{2 R \times 2 R}{2 R+2 R}...
Read More →HCF of (23 × 32 × 5), (22 × 33 ×52)
Question: HCF of $\left(2^{3} \times 3^{2} \times 5\right),\left(2^{2} \times 3^{3} \times 5^{2}\right)$ and $\left(2^{4} \times^{3} \times 5^{3} \times 7\right)$ is (a) 30(b) 48(c) 60(d) 105 Solution: (c) 60 HCF $=\left(2^{3} \times 3^{2} \times 5,2^{2} \times 3^{3} \times 5^{2}, 2^{4} \times 3 \times 5^{3} \times 7\right)$ HCF = Product of smallest power of each common prime factor in the numbers $=2^{2} \times 3 \times 5$ $=60$...
Read More →If
Question: If $x^{3} d y+x y d x=x^{2} d y+2 y d x ; y(2)=e$ and $x1$, then $y(4)$ is equal to :(1) $\frac{3}{2}+\sqrt{e}$(2) $\frac{3}{2} \sqrt{e}$(3) $\frac{1}{2}+\sqrt{e}$(4) $\frac{\sqrt{e}}{2}$ Correct Option: , 2 Solution: $x^{3} d y+x y d y x=2 y d x+x^{2} d y$ $\Rightarrow\left(x^{3}-x^{2}\right) d y=(2-x) y d x$ $\Rightarrow \frac{d y}{y}=\frac{2-x}{x^{2}(x-1)} d x$ $\Rightarrow \int \frac{d y}{y}=\int \frac{2-x}{x^{2}(x-1)} d x$.....(i) Let $\frac{2-x}{x^{2}(x-1)}=\frac{A}{x}+\frac{B}{x...
Read More →If a = (22 × 33 × 54) and b
Question: If $a=\left(2^{2} \times 3^{3} \times 5^{4}\right)$ and $b=\left(2^{3} \times 3^{2} \times 5\right)$, then $\operatorname{HCF}(a, b)=$ ? (a) 90(b) 180(c) 360(d) 540 Solution: (b) 180It is given that: $a=\left(2^{2} \times 3^{3} \times 5^{4}\right)$ and $b=\left(2^{3} \times 3^{2} \times 5\right)$ HCF (a,b) = Product of smallest power of each common prime factor in the numbers $=2^{2} \times 3^{2} \times 5$ = 180...
Read More →Which of the following is a pair of co-primes?
Question: Which of the following is a pair of co-primes?(a) (14, 35)(b) (18, 25)(c) (31, 93)(d) (32, 62) Solution: The numbers that do not share any common factor other than 1 are called co-primes.Clearly in option (b),factors of 18 are: 1, 2, 3, 6, 9 and 18factors of 25 are: 1, 5, 25The two numbers do not share any common factor other than 1.They are co-primes to each other....
Read More →Explain why
Question: Explain why $3 . \overline{1416}$ is a rational number. Solution: Since, $3 . \overline{1416}$ is a non-terminating repeating decimal. Hence, is a rational number....
Read More →The solution curve of the differential equation,
Question: The solution curve of the differential equation, $\left(1+e^{-x}\right)\left(1+y^{2}\right) \frac{d y}{d x}=y^{2}$, which passes through the point $(0,1)$, is :(1) $y^{2}+1=y\left(\log _{e}\left(\frac{1+e^{-x}}{2}\right)+2\right)$(2) $y^{2}+1=y\left(\log _{e}\left(\frac{1+e^{x}}{2}\right)+2\right)$(3) $y^{2}=1+y \log _{e}\left(\frac{1+e^{x}}{2}\right)$(4) $y^{2}=1+y \log _{e}\left(\frac{1+e^{-x}}{2}\right)$Correct Option: , 3 Solution: $\int\left(\frac{y^{2}+1}{y^{2}}\right) d y=\int \...
Read More →Write a rational number between
Question: Write a rational number between $\sqrt{3}$ and 2 . Solution: Since, $\sqrt{3}=1.732 \ldots$ So, we may take $1.8$ as the required rational number between $\sqrt{3}$ and 2 . Thus, the required rational number is 1.8...
Read More →A cylindrical wire of radius 0.5 mm
Question: A cylindrical wire of radius $0.5 \mathrm{~mm}$ and conductivity $5 \times 10^{7} \mathrm{~S} / \mathrm{m}$ is subjected to an electric field of $10 \mathrm{mV} / \mathrm{m}$. The expected value of current in the wire will be $\mathrm{x}^{3} \pi \mathrm{mA}$. The value of $x$ is Solution: (5) We know that $\mathbf{J}=\sigma \mathrm{E}$ $\Rightarrow \mathrm{J}=5 \times 10^{7} \times 10 \times 10^{-3}$ $\Rightarrow \mathrm{J}=50 \times 10^{4} \mathrm{~A} / \mathrm{m}^{2}$ Currentflowing ...
Read More →If a curve y=f(x), passing through the point
Question: If a curve $y=f(x)$, passing through the point $(1,2)$, is the solution of the differential equation, $2 x^{2} d y=\left(2 x y+y^{2}\right) d x$, then $f\left(\frac{1}{2}\right)$ is equal to :(1) $\frac{1}{1+\log _{e} 2}$(2) $\frac{1}{1-\log _{e} 2}$(3) $1+\log _{e} 2$(4) $\frac{-1}{1+\log _{e} 2}$Correct Option: 1 Solution: $\frac{d y}{d x}=\frac{2 x y+y^{2}}{2 x^{2}}$ It is homogeneous differential equation. $\therefore \quad$ Put $y=v x$ $\Rightarrow v+x \frac{d v}{d x}=v+\frac{v^{2...
Read More →Show that
Question: Show that $\frac{\sqrt{2}}{3}$ is irrational. Solution: Let $\frac{\sqrt{2}}{3}$ is a rational number. $\therefore \frac{\sqrt{2}}{3}=\frac{p}{q}$, where $p$ and $q$ are some integers and $\operatorname{HCF}(p, q)=1$ .......(1) $\Rightarrow \sqrt{2} q=3 p$ $\Rightarrow(\sqrt{2} q)^{2}=(3 p)^{2}$ $\Rightarrow 2 q^{2}=9 p^{2}$ ⇒p2is divisible by 2⇒p is divisible by 2 ....(2) Letp= 2m, wheremis some integer. $\therefore \sqrt{2} q=3 p$ $\Rightarrow \sqrt{2} q=3(2 m)$ $\Rightarrow(\sqrt{2}...
Read More →A current through a wire depends on time
Question: A current through a wire depends on time as $\mathrm{i}=\alpha_{0} \mathrm{t}+\beta \mathrm{t}^{2}$ where $\alpha_{0}=20 \mathrm{~A} / \mathrm{s}$ and $\beta=8 \mathrm{As}^{-2}$. Find the charge crossed through a section of the wire in $15 \mathrm{~s}$.(1) $2100 \mathrm{C}$(2) $260 \mathrm{C}$(3) $2250 \mathrm{C}$(4) $11250 \mathrm{C}$Correct Option: , 4 Solution: (4) given : $i=\alpha_{0} t+\beta t^{2}$ $\alpha=20 \mathrm{~A} / \mathrm{s}$ and $\beta=8 \mathrm{As}^{-2}$ $\mathrm{t}=15...
Read More →Explain why 0.15015001500015 ... is an irrational number.
Question: Explain why 0.15015001500015 ... is an irrational number. Solution: Irrational numbers are non-terminating non-recurring decimals.Thus, 0.15015001500015 ... is an irrational 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, $\frac{2+\sin x}{y+1} \cdot \frac{d y}{d x}=-\cos x, y0, y(0)=1$. If $y(\pi)=a$ and $\frac{d y}{d x}$ at $x=\pi$ is $b$, then the ordered pair $(a, b)$ is equal to :(1) $\left(2, \frac{3}{2}\right)$(2) $(1,-1)$(3) $(1,1)$(4) $(2,1)$Correct Option: , 3 Solution: The given differential equation is $\frac{2+\sin x}{y+1} \frac{d y}{d x}=-\cos x, y0$ $\Rightarrow \frac{d y}{y+1}=-\frac{\cos x}{2+\sin x} d x$ Integrate both sides, $\...
Read More →Express
Question: Express $0 . \overline{23}$ as a rational number in simplest form. Solution: Let $x$ be $0 . \overline{23}$ $x=0 . \overline{23}$ ...........(1) Multiplying both sides by 100, we get $100 x=23 . \overline{23}$ ..........(2) Subtracting (1) from (2), we get $100 x-x=23 . \overline{23}-0 . \overline{23}$ $\Rightarrow 99 x=23$ $\Rightarrow x=\frac{23}{99}$ Thus, simplest form of $0 . \overline{23}$ as a rational number is $\frac{23}{99}$....
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...
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