If X/Y is a rational number,
Question: If $\frac{x}{y}$ is a rational number, then $\mathrm{y}$ is always a whole number. Solution: False. If $x / y$ is a rational numbers, then $y$ is not equal to 0 . But 0 is a whole number....
Read More →Prove the following
Question: $(1 / 5) \times[(2 / 7)+(3 / 8)]=[(1 / 5) \times(2 / 7)]+$______. Solution: $(1 / 5) \times[(2 / 7)+(3 / 8)]=[(1 / 5) \times(2 / 7)]+[(1 / 5) \times(3 / 8)]$ $\because$ From the rule of distributive law of multiplication $[a \times(b+c)=(a \times b)+(a \times c)]$...
Read More →Express each of the following in the form (a + ib):
Question: Express each of the following in the form (a + ib): $\frac{(1-i)^{3}}{\left(1-i^{3}\right)}$ Solution: Given: $\frac{(1-i)^{3}}{\left(1-i^{3}\right)}$ The above equation can be re-written as $=\frac{(1)^{3}-(i)^{3}-3(1)^{2}(i)+3(1)(i)^{2}}{\left(1-i \times i^{2}\right)}$ $\left[\because(a-b)^{3}=a^{3}-b^{3}-3 a^{2} b+3 a b^{2}\right]$ $=\frac{1-i^{3}-3 i+3 i^{2}}{[1-i(-1)]}\left[\because i^{2}=-1\right]$ $=\frac{1-i \times i^{2}-3 i+3(-1)}{(1+i)}$ $=\frac{1-i(-1)-3 i-3}{1+i}$ $=\frac{-...
Read More →The rational number
Question: The rational number $10.11$ in the form $\frac{p}{q}$ is - Solution: The rational number $10.11$ in the from $\mathrm{p} / \mathrm{q}$ is $\underline{1011 / 100}$....
Read More →Express each of the following in the form (a + ib)
Question: Express each of the following in the form (a + ib): $\frac{(2+3 i)^{2}}{(2-i)}$ Solution: Given: $\frac{(2+3 i)^{2}}{(2-i)}$ Now, we rationalize the above equation by multiply and divide by the conjugate of (2 i) $=\frac{(2+3 i)^{2}}{(2-i)} \times \frac{(2+i)}{(2+i)}$ $=\frac{(2+3 i)^{2}(2+i)}{(2-i)(2+i)}$ $=\frac{\left(4+9 i^{2}+12 i\right)(2+i)}{(2)^{2}-(i)^{2}}$ $\left[\because(a+b)(a-b)=\left(a^{2}-b^{2}\right)\right]$ $=\frac{[4+9(-1)+12 i](2+i)}{4-i^{2}}\left[\because \cdot i^{2}...
Read More →The multiplicative inverse
Question: The multiplicative inverse of $\frac{4}{3}$ is________. Solution: The multiplicative inverse of $4 / 3$ is $3 / 4$....
Read More →Fill in the blanks.
Question: Fill in the blanks. (i) A tap can fill a tank in 6 hours. The part of the tank filled in 1 hour is ......... (ii) A and B working together can finish a piece of work in 6 hours while A alone can do it in 9 hours. B alone can do it in ......... hours. (iii) A can do a work in 16 hours and B alone can do it in 24 hours. If A, B and C working together can finish it in 8 hours, then C alone can finish it in ......... hours. (iv) If A's one day's work is $\frac{3}{20}$, then A can finish th...
Read More →Express each of the following in the form (a + ib)
Question: Express each of the following in the form (a + ib) $\frac{(3-2 i)(2+3 i)}{(1+2 i)(2-i)}$ Solution: Given: $\frac{(3-2 i)(2+3 i)}{(1+2 i)(2-i)}$ Firstly, we solve the given equation $=\frac{3(2)+3(3 i)-2 i(2)+(-2 i)(3 i)}{(1)(2)+1(-i)+2 i(2)+2 i(-i)}$ $=\frac{6+9 i-4 i-6 i^{2}}{2-i+4 i-2 i^{2}}$ $=\frac{6+5 i-6(-1)}{2+3 i-2(-1)}$ $=\frac{6+6+5 i}{2+3 i+2}$ $=\frac{12+5 i}{4+3 i}$ Now, we rationalize the above by multiplying and divide by the conjugate of 4 + 3i $=\frac{12+5 i}{4+3 i} \t...
Read More →Mark (✓) against the correct answer:
Question: Mark (✓) against the correct answer: A works twice as fast as B. If both of them can together finish a peice of work in 12 hours, then B alone can do it in (a) 24 hours (b) 27 hours (c) 36 hours (d) 18 hours Solution: (c) 36 hours Suppose B take s x hour s to complete the work. $\therefore$ B's 1 hour work $=\frac{1}{\mathrm{x}}$ A works twice as fast as B. $\therefore$ A's 1 hour work $=\frac{2}{x}$ $\frac{1}{12}=\frac{1}{\mathrm{x}}+\frac{2}{\mathrm{x}}=\frac{3}{\mathrm{x}}$ $\Righta...
Read More →The reciprocal of
Question: The reciprocal of $\frac{-5}{7}$ is_______. Solution: The reciprocal of $-5 / 7$ is $\underline{-7 / 5}$....
Read More →Mark (✓) against the correct answer:
Question: Mark (✓) against the correct answer: A pump can fill a cistern in 2 hours. Due to a leak in the tank it takes $2 \frac{1}{3}$ hours to fill it. The leak can empty the full tank in (a) 7 hours (b) 14 hours (c) 8 hours (d) 3 hours Solution: (b) 14 hours A pump fills a tank in 2 hours. Part of tank filled by the pump in one hour $=\frac{1}{2}$ Let $\mathrm{x}$ hours be the time required for water to leak from the cistern. Part of tank drained by the leak in one hour $=-\frac{1}{x}$ Time r...
Read More →Rational numbers can be added
Question: Rational numbers can be added or multiplied in any-. Solution: order Rational numbers can be added or multiplied in any order and this concept is known as commutative property....
Read More →Solve this
Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$, when $\mathrm{x}=\mathrm{e}^{\theta}\left(\theta+\frac{1}{\theta}\right)$ and $\mathrm{y}=\mathrm{e}^{-\theta}\left(\theta-\frac{1}{\theta}\right)$ Solution: $\operatorname{as} x=e^{\theta}\left(\theta+\frac{1}{\theta}\right)$ Differentiating it with respect to $\theta$ using the product rule, $\frac{\mathrm{dx}}{\mathrm{d} \theta}=\mathrm{e}^{\theta} \frac{\mathrm{d}}{\mathrm{d} \theta}\left(\theta+\frac{1}{\theta}\right)+\left(\theta+\frac{1}{\...
Read More →The negative of a negative rational
Question: The negative of a negative rational number is always a-rationalnumber. Solution: positive Let x be a positive rational number. Then, x be a negative rational number. Now, negative of a negative rational number = (- x)= x =positive rational number....
Read More →Mark (✓) against the correct answer:
Question: Mark (✓) against the correct answer: A can do a piece of work in 14 days and B is 40% more efficient than A. In how many days can B finish it? (a) 10 days (b) $7 \frac{1}{2}$ days (c) $5 \frac{1}{4}$ days (d) $5 \frac{3}{5}$ days Solution: (a) 10 days A's 1 day work $=\frac{1}{14}$ B is $40 \%$ more efficient than A. $\therefore$ B's 1 day work $=\frac{140}{100} \times \frac{1}{14}=\frac{1}{10}$ B takes 10 days to complete the work....
Read More →Mark (✓) against the correct answer:
Question: Mark (✓) against the correct answer: A can finish a piece of work in 12 hours while B can finish it in 15 hours. How long will both take to finish it, working together? (a) 9 hours (b) $6 \frac{2}{3}$ hours (c) $6 \frac{3}{4}$ hours (d) $8 \frac{1}{3}$ hours Solution: (b) $6 \frac{2}{3}$ hours A's 1 hour work $=\frac{1}{12}$ B's 1 hour work $=\frac{1}{15}$ $(\mathrm{~A}+\mathrm{B})$ 's 1 hour work $=\frac{1}{12}+\frac{1}{15}=\frac{9}{60}=\frac{3}{20}$ Time taken by $\mathrm{A}$ and $\m...
Read More →Mark (✓) against the correct answer:
Question: Mark (✓) against the correct answer: The rates of working of two tapes A and B are in the ratio 2 : 3. The ratio of the time taken by A and B respectively to fill a given cistern is (a) 2 : 3 (b) 3 : 2 (c) 4 : 9 (d) 9 : 4 Solution: (b) 3:2 Rates at which taps $\mathrm{A}$ and $\mathrm{B}$ work $=2: 3$ The ratio of time taken by taps $\mathrm{A}$ and $\mathrm{B}$ to fill the cistern $=\frac{1}{\text { Rates at which taps } \mathrm{A} \text { and } \mathrm{B} \text { work }}$ $=\frac{1}{...
Read More →Solve this
Question: Find $\frac{d y}{d x}$, when $x=a(\cos \theta+\theta \sin \theta)$ and $y=a(\sin \theta-\cos \theta)$ Solution: the given equation are $x=a(\cos \theta+\theta \sin \theta)$ Then $\frac{\mathrm{dx}}{\mathrm{d} \theta}=\mathrm{a}\left[\frac{\mathrm{d}}{\mathrm{d} \theta} \cos \theta+\frac{\mathrm{d}}{\mathrm{d} \theta}(\theta \sin \theta)\right]$ $=a\left[-\sin \theta+\frac{\theta d}{d \theta}(\sin \theta)+\sin \theta \frac{d}{d \theta}(\theta)\right]$ $=a[-\sin \theta+\theta \cos \theta...
Read More →A pipe can fill a cistern in 9 hours.
Question: A pipe can fill a cistern in 9 hours. Due to a leak in its bottom, the cistern fills up in 10 hours. If the cistern is full, in how much time will it be emptied by the leak? Solution: Time taken by the pipe to fill the cistern $=9$ hours Part of the cistern filled in one hour $=\frac{1}{9}$ Suppose the leak empties the full cistern in x hours. Part of the cistern emptied in one hour $=-\frac{1}{x}$ (negative sign implies a leak) Time taken by the cistern to fill completely due to the l...
Read More →There are rational numbers between
Question: There are rational numbers between any two rational numbers. Solution: Infinite There are infinite rational numbers between any two rational numbers....
Read More →2 men or 3 women can do a piece of work in 16 days.
Question: 2 men or 3 women can do a piece of work in 16 days. In how many days can 4 men and 6 women do the same work? Solution: Work of 2 men $=$ Work of 3 women $\Rightarrow$ Work of 1 man $=\frac{3}{2}$ women Three women can do a piece of work in 16 days. As 4 men and 6 women $=\left(4 \times \frac{3}{2}\right)$ women $+6$ women $=6$ women $+6$ women $=12$ women Also, 3 women can do the work in 16 days. So, work done by 3 women in one day $=\frac{1}{16}$ $\therefore$ Work done by 1 woman in o...
Read More →Find the values
Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$, when $\mathrm{x}=\frac{3 \text { at }}{1+\mathrm{t}^{2}}$ and $\mathrm{y}=\frac{3 \mathrm{at}^{2}}{1+\mathrm{t}^{2}}$ Solution: $\mathrm{aS} \mathrm{X}=\frac{3 \mathrm{at}}{1+\mathrm{t}^{2}}$ Differentiating it with respect to $t$ using quotient rule, $\frac{d x}{d t}=\left[\frac{\left(\left(1+t^{2}\right) \frac{d(3 a t)}{d t}-3 a t \frac{d\left(1+t^{2}\right)}{d t}\right)}{\left(1+t^{2}\right)^{2}}\right]$ $=\left[\frac{\left(1+t^{2}\right)(3 a)...
Read More →Find the values
Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$, when $\mathrm{x}=\frac{3 \text { at }}{1+\mathrm{t}^{2}}$ and $\mathrm{y}=\frac{3 \mathrm{at}^{2}}{1+\mathrm{t}^{2}}$ Solution: $\mathrm{aS} \mathrm{X}=\frac{3 \mathrm{at}}{1+\mathrm{t}^{2}}$ Differentiating it with respect to $t$ using quotient rule, $\frac{d x}{d t}=\left[\frac{\left(\left(1+t^{2}\right) \frac{d(3 a t)}{d t}-3 a t \frac{d\left(1+t^{2}\right)}{d t}\right)}{\left(1+t^{2}\right)^{2}}\right]$ $=\left[\frac{\left(1+t^{2}\right)(3 a)...
Read More →Tap A can fill a cistern in 8 hours and tap B can empty it in 12 hours.
Question: Tap A can fill a cistern in 8 hours and tap B can empty it in 12 hours. How long will it take to fill the cistern if both of them are opened together? Solution: Tap A can fill a cistern in 8 hours. Part of cistern filled by Tap A in 1 hour $=\frac{1}{8}$ Tap B empties the cistern in 12 hours. Part of cistern emptied by Tap B in 1 hour $=-\frac{1}{12}$ (negative sign shows that tap B drains the tank) Part of cistern filled in one hour when both taps are opened together $=\frac{1}{8}-\fr...
Read More →-5/7 is______ than -3.
Question: $-5 / 7$ is _________than $-3$ Solution: $-5 / 7$ is more than $-3 .$...
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