Solve this
Question: $\sqrt{2}$ is (a) a rational number(b) an irrational number(c) a terminating decimal(d) a nonterminating repeating decimal Solution: Let $\sqrt{2}$ is a rational number. $\therefore \sqrt{2}=\frac{p}{q}$, where $p$ and $q$ are some integers and $\operatorname{HCF}(p, q)=1$ ........(1) $\Rightarrow \sqrt{2} q=p$ $\Rightarrow(\sqrt{2} q)^{2}=p^{2}$ $\Rightarrow 2 q^{2}=p^{2}$ ⇒p2is divisible by 2⇒pis divisible by 2 .... (2)Letp= 2m, wheremis some integer. $\therefore \sqrt{2} q=p$ $\Righ...
Read More →a and b are two positive integers such that the least prime factor of a is 3 and the least prime factor of b is 5
Question: aandbare two positive integers such that the least prime factor ofais 3 and the least prime factor ofbis 5. Then, the least prime factor of (a+b) is(a) 2(b) 3(c) 5(d) 8 Solution: (a) 2Since 5 + 3 = 8, the least prime factor ofa+bhas to be 2, unlessa+bis a prime number greater than 2.Ifa+bis a prime number greater than 2, thena+bmust be an odd number. So, eitheraorbmust be an even number. Ifais even, then the least prime factor ofais 2, which is not 3 or 5. So, neitheranorbcanbe an even...
Read More →Let y=y(x) be the solution curve of the differential equation,
Question: Let $y=y(x)$ be the solution curve of the differential equation, $\left(y^{2}-x\right) \frac{d y}{d x}=1$, satisfying $y(0)=1$. This curve intersects the $x$-axis at a point whose abscissa is:(1) $2-e$(2) $-e$(3) 2(4) $2+e$Correct Option: 1 Solution: The given differential equation is $\frac{d x}{d y}+x=y^{2}$\ Comparing with $\frac{d x}{d y}+P x=Q$, where $P=1, Q=y^{2}$ Now, I.F. $=e^{\int 1 \cdot d y}=e^{y}$ $x \cdot e^{y}=\int\left(y^{2}\right) e^{y} \cdot d y=y^{2} \cdot e^{y}-\int...
Read More →A battery of 3.0 V
Question: A battery of $3.0 \mathrm{~V}$ is connected to a resistor dissipating $0.5$ W of power. If the termial voltage of the battery is $2.5 \mathrm{~V}$, the power dissipated within the internal resistance is:(1) $0.50 \mathrm{~W}$(2) $0.072 \mathrm{~W}$(3) $0.10 \mathrm{~W}$(4) $0.125 \mathrm{~W}$Correct Option: , 3 Solution: (3) When resistor is connected power dissipated, $P_{R}=0.5 \mathrm{~W}$ Emf of battery, $E=3 \mathrm{~V}$ Terminal voltage, $V=2.5 \mathrm{~V}$ $P_{R}=i^{2} R=0.5 \ma...
Read More →The number 1.732 is
Question: The number 1.732 is(a) an irrational number(b) a rational number(c) an integer(d) a whole number Solution: Clearly, 1.732 is a terminating decimal.Hence, a rational number.Hence, the correct answer is option (b)....
Read More →The decimal expansion of the number
Question: The decimal expansion of the number $\frac{14753}{1250}$ will terminate after (a) one decimal place(b) two decimal place(c) three decimal place(d) four decimal place Solution: (d) four decimal places $\frac{14753}{1250}=\frac{14753}{5^{4} \times 2}=\frac{14753 \times 2^{3}}{5^{4} \times 2^{4}}=\frac{118024}{10000}=11.8024$ So, the decimal expansion of the number will terminate after four decimal places....
Read More →The decimal expansion of the rational number
Question: The decimal expansion of the rational number $\frac{37}{2^{2} \times 5}$ will terminate after (a) one decimal place(b) two decimal places(c) three decimal places(d) four decimal places Solution: (b) two decimal places $\frac{37}{2^{2} \times 5}=\frac{37 \times 5}{2^{2} \times 5^{2}}=\frac{185}{100}=1.85$ So, the decimal expansion of the rational number will terminate after two decimal places....
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, $e^{y}\left(\frac{d y}{d x}-1\right)=e^{x}$ such that $y(0)=0$, then $y(1)$ is equal to:(1) $1+\log _{e} 2$(2) $2+\log _{e} 2$(3) $2 e$(4) $\log _{e} 2$Correct Option: 1 Solution: Let $e^{y}=t$ $e^{y} \frac{d y}{d x}=\frac{d t}{d x}$ $\therefore \quad \frac{d t}{d x}-t=e^{x}$ $\left[\because e^{y} \frac{d y}{d x}-e^{y}=e^{x}\right]$ I.F. $=e^{\int-1 . d x}=e^{-x}$ $t\left(e^{-x}\right)=\int e^{x} \cdot e^{-x} d x \Rightarrow e^{...
Read More →Which of the following rational numbers is expressible as a terminating decimal?
Question: Which of the following rational numbers is expressible as a terminating decimal? (a) $\frac{124}{165}$ (b) $\frac{131}{30}$ (c) $\frac{2027}{625}$ (d) $\frac{1625}{462}$ Solution: (c) $\frac{2027}{625}$ $\frac{124}{165}=\frac{124}{5 \times 33}$; we know 5 and 33 are not the factors of 124 . It is in its simplest form and it cannot be expressed as the product of $\left(2^{m} \times 5^{n}\right)$ for some non-negative integers $m, n$ So, it cannot be expressed as a terminating decimal. $...
Read More →Two resistors
Question: Two resistors $400 \Omega$ and $800 \Omega$ are connected in series across a $6 \mathrm{~V}$ battery. The potential difference measured by a voltmeter of $10 \mathrm{k} \Omega$ across $400 \Omega$ resistor is close to:(1) $2 \mathrm{~V}$(2) $1.8 \mathrm{~V}$(3) $2.05 \mathrm{~V}$(4) $1.95 \mathrm{~V}$Correct Option: , 4 Solution: (4) The voltmeter of resistance $10 \mathrm{k} \Omega$ is parallel to the resistance of $400 \Omega$. So, their equivalent resistance is $\frac{1}{R^{\prime}}...
Read More →Let Xk
Question: Let $x^{k}+y^{k}=a^{k},(a, k0)$ and $\frac{d y}{d x}+\left(\frac{y}{x}\right)^{\frac{1}{3}}=0$, then $k$ is:(1) $\frac{3}{2}$(2) $\frac{4}{3}$(3) $\frac{2}{3}$(4) $\frac{1}{3}$Correct Option: , 3 Solution: $k \cdot x^{k-1}+k \cdot y^{k-1} \frac{d y}{d x}=0$ $\Rightarrow \frac{d y}{d x}=-\left(\frac{x}{y}\right)^{k-1}$ $\Rightarrow \frac{d y}{d x}+\left(\frac{x}{y}\right)^{k-1}=0$ $\Rightarrow \quad k-1=-\frac{1}{3}$ $\Rightarrow \quad k=1-\frac{1}{3}=\frac{2}{3}$...
Read More →3.24636363...is
Question: 3.24636363...is(a) an integer(b) a rational number(c) an irrational number(d) none of these Solution: (b) a rational numberIt is a rational number because it is a repeating decimal....
Read More →Model a torch battery of length l to be made up of a thin cylindrical bar
Question: Model a torch battery of length $l$ to be made up of a thin cylindrical bar of radius ' $a$ ' and a concentric thin cylindrical shell of radius ' $b$ ' filled in between with an electrolyte of resistivity $\rho$ (see figure). If the battery is connected to a resistance of value $R$, the maximum Joule heating in $R$ will take place for: (1) $R=\frac{\rho}{2 \pi l}\left(\frac{b}{a}\right)$(2) $R=\frac{\rho}{2 \pi l} \ln \left(\frac{b}{a}\right)$(3) $R=\frac{\rho}{\pi l} \ln \left(\frac{b...
Read More →2.13113111311113...is
Question: 2.13113111311113...is(a) an integer(b) a rational number(c) an irrational number(d) none of these Solution: (c) an irrational numberIt is an irrational number because it is a non-terminating and non-repeating decimal....
Read More →if
Question: If $y=\left(\frac{2}{\pi} x-1\right) \operatorname{cosec} x$ is the solution of the differential equation, $\frac{\mathrm{d} y}{\mathrm{~d} x}+\mathrm{p}(x) y=\frac{2}{\pi} \operatorname{cosec} x, 0x\frac{\pi}{2}$, then the function $\mathrm{p}(x)$ is equal to:(1) $\cot x$(2) cose(3) $\sec x$(4) $\tan x$Correct Option: 1 Solution: $\because y=\left(\frac{2}{\pi} x-1\right) \operatorname{cosec} x$ $\frac{d y}{d x}=\frac{2}{\pi} \operatorname{cosec} x-\left(\frac{2}{\pi} x-1\right) \oper...
Read More →Solve this
Question: 2. $\overline{35}$ is (a) an integer(b) a rational number(c) an irrational number(d) none of these Solution: (b) a rational number $2 . \overline{35}$ is a rational number because it is a repeating decimal....
Read More →π is
Question: is(a) an integer(b) a rational number(c) an irrational number(d) none of these Solution: (c) an irrational number $\pi$ is an irrational number because it is a non-repeating and non-terminating decimal....
Read More →Which of the following is an irrational number?
Question: Which of the following is an irrational number? (a) $\frac{22}{7}$ (b) $3.1416$ (c) $3 . \overline{1416}$ (d) 3.141141114... Solution: (d) 3.141141114...3.141141114 is an irrational number because it is a non-repeating and non-terminating decimal....
Read More →The general solution of the differential equation
Question: The general solution of the differential equation $\sqrt{1+x^{2}+y^{2}+x^{2} y^{2}}+x y \frac{d y}{d x}=0$ is : (where $\mathrm{C}$ is a constant of integration) (1) $\sqrt{1+y^{2}}+\sqrt{1+x^{2}}=\frac{1}{2} \log _{e}\left(\frac{\sqrt{1+x^{2}}+1}{\sqrt{1+x^{2}}-1}\right)+C$(2) $\sqrt{1+y^{2}}-\sqrt{1+x^{2}}=\frac{1}{2} \log _{e}\left(\frac{\sqrt{1+x^{2}}+1}{\sqrt{1+x^{2}}-1}\right)+C$(3) $\sqrt{1+y^{2}}+\sqrt{1+x^{2}}=\frac{1}{2} \log _{e}\left(\frac{\sqrt{1+x^{2}}-1}{\sqrt{1+x^{2}}+1...
Read More →An ideal cell of emf 10 V is connected in circuit shown in figure.
Question: An ideal cell of emf $10 \mathrm{~V}$ is connected in circuit shown in figure. Each resistance is $2 \Omega$. The potential difference (in V) across the capacitor when it is fully charged is _______ Solution: As capacitor is fully charged no current will flow through it. We have the current distribution as shown in the figure. Equivalent resistance, $\mathrm{R}_{\mathrm{eq}}=\left(\frac{4 \times 2}{4+2}\right)+2$ Net current, $i=\frac{10}{\frac{4}{3}+2}=\frac{10 \times 3}{10}=3$ Amp $i...
Read More →A number when divided by 143 leaves 31 as remainder.
Question: A number when divided by 143 leaves 31 as remainder. What will be the remainder when the same number is divided by 13?(a) 0(b) 1(c) 3(d) 5 Solution: (d) 5We know,Dividend = DivisorQuotient + Remainder.It is given that:Divisor = 143Remainder = 13So, the given number is in the form of143x+31, wherexis the quotient.143x+ 31 = 13 (11x) + (1322)+ 5 = 13 (11x+ 2) + 5Thus, the remainder will be 5 when the same number is divided by 13....
Read More →Euclid's division lemma sates that for any positive integers a and b,
Question: Euclid's division lemma sates that for any positive integersaandb, there exist unique integersqandrsuch thata=bq+r, wherermust satisfy(a) 1 rb(b) 0 rb(c) 0rb(d) 0 rb Solution: (c) 0rbEuclid's division lemma states that for any positive integersaandb,there exist unique integersqandrsuch thata=bq+r,wherer must satisfy 0rb...
Read More →The simplest form of
Question: The simplest form of $\frac{1095}{1168}$ is (a) $\frac{17}{26}$ (b) $\frac{25}{26}$ (c) $\frac{13}{16}$ (d) $\frac{15}{16}$ Solution: (d) $\frac{15}{16}$ $\frac{1095}{1168}=\frac{1095 \div 73}{1168 \div 73}=\frac{15}{16}$ Hence, HCF of 1095 and 1168 is 73....
Read More →Let y=y(x) be the solution of the differential
Question: Let $y=y(x)$ be the solution of the differential equation $\cos x \frac{d y}{d x}+2 y \sin x=\sin 2 x, x \in\left(0, \frac{\pi}{2}\right)$. If $y(\pi / 3)=0$, then $y(\pi / 4)$ is equal to :(1) $2-\sqrt{2}$(2) $2+\sqrt{2}$(3) $\sqrt{2}-2$(4) $\frac{1}{\sqrt{2}}-1$Correct Option: , 3 Solution: $\frac{d y}{d x}+2 y \tan x=2 \sin x$ I.F. $=e^{\int 2 \tan x d x}=\sec ^{2} x$ The solution of the differential equation is $y \times$ I.F. $=\int$ I.F $\times 2 \sin x d x+C$ $\Rightarrow y \cdo...
Read More →What is the largest number that divides 245 and 1029, leaving remainder 5 in each case?
Question: What is the largest number that divides 245 and 1029, leaving remainder 5 in each case?(a) 15(b) 16(c) 9(d) 5 Solution: (b) 16We know that the required number divides 240 (245 5) and 1024 (1029 5). Required number = HCF (240, 1024) $240=2 \times 2 \times 2 \times 2 \times 3 \times 5$ $1024=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$ $\therefore \mathrm{HCF}=2 \times 2 \times 2 \times 2=16$...
Read More →