Tick (✓) the correct answer:
Question: Tick $(\checkmark)$ the correct answer: A man can do a piece of work in 5 days. He and his son working together can finish it in 3 days. In how many days can the son do it alone? (a) $6 \frac{1}{2}$ days (b) 7 days (c) $7 \frac{1}{2}$ days (d) 8 days Solution: (c) $7 \frac{1}{2}$ daysA man can do a work in 5 days. The man's 1 day work $=\frac{1}{5}$ The man and the son can do the work in 3 days. The man and his son's 1 day work $=\frac{1}{3}$ Let the son's 1 day work be $\frac{1}{x}$. ...
Read More →Simplify each of the following and express it in the form (a + ib) :
Question: Simplify each of the following and express it in the form (a + ib) : $(4-3 i)^{-1}$ Solution: Given: $(4-3 i)^{-1}$ We can re- write the above equation as $=\frac{1}{4-3 i}$ Now, rationalizing $=\frac{1}{4-3 i} \times \frac{4+3 i}{4+3 i}$ $=\frac{4+3 i}{(4-3 i)(4+3 i)} \ldots$ (i) Now, we know that, $(a+b)(a-b)=\left(a^{2}-b^{2}\right)$ So, eq. (i) become $=\frac{4+3 i}{(4)^{2}-(3 i)^{2}}$ $=\frac{4+3 i}{16-9 i^{2}}$ $=\frac{4+3 i}{16-9(-1)}\left[\because i^{2}=-1\right]$ $=\frac{4+3 i...
Read More →Which of the following is an example
Question: Which of the following is an example of distributive property of multiplication over addition for rational numbers. (a) $-(1 / 4) \times\{(2 / 3)+(-4 / 7)\}=[-(1 / 4) \times(2 / 3)]+[(-1 / 4) \times(-4 / 7)]$ (b) $-(1 / 4) \times\{(2 / 3)+(-4 / 7)\}=[(1 / 4) \times(2 / 3)]-(-4 / 7)$ (c) $-(1 / 4) \times\{(2 / 3)+(-4 / 7)\}=(2 / 3)+(-1 / 4) \times(-4 / 7)$ (d) $-(1 / 4) \times\{(2 / 3)+(-4 / 7)\}=\{(2 / 3)+(-4 / 7)\}-(1 / 4)$ Solution: (a) $-(1 / 4) \times\{(2 / 3)+(-4 / 7)\}=[-(1 / 4) ...
Read More →Tick (✓) the correct answer:
Question: Tick (✓) the correct answer: A alone can do a piece of work in 10 days and B alone can do it in 15 days. In how many days will A and B together do the same work? (a) 5 days (b) 6 days (c) 8 days (d) 9 days Solution: (b) 6 days A can do a work in 10 days. A's 1 day work $=\frac{1}{10}$ B can do a work in 15 days. B's 1 day work $=\frac{1}{15}$ $(\mathrm{A}+\mathrm{B})$ 's 1 day work $=\frac{1}{10}+\frac{1}{15}=\frac{5}{30}=\frac{1}{6}$ $\mathrm{A}$ and $\mathrm{B}$ together will take 6 ...
Read More →The reciprocal of
Question: The reciprocal of $(-3 / 8) \times(-7 / 13)$ is (a) $104 / 21$ (b) $-104 / 21$ (c) $21 / 104$ (d) $-21 / 104$ Solution: (a) $104 / 21$ $=(-3 \times-7) /(8 \times 13)$ $=(21 / 104)$ Reciprocal of $21 / 104$ is $104 / 21$...
Read More →Pipe A can fill a cistern in 6 hours and pipe B can fill it in 8 hours.
Question: Pipe A can fill a cistern in 6 hours and pipe B can fill it in 8 hours. Both the pipes are opened and after two hours, pipe A is closed. How much time will B take to fill the remaining part of the tank? Solution: Pipe A can fill a cistern in 6 hours. Pipe B can fill a cistern in 8 hours. Part of the cistern filled by pipe A in one hour $=\frac{1}{6}$ Part of the cistern filled by pipe B in one hour $=\frac{1}{8}$ Part of the cistern filled by pipes A and B in one hour $=\frac{1}{6}+\fr...
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: A pipe can fill a cistern in 9 hours. Part of the cistern filled by the pipe in one hour $=\frac{1}{9}$ Let the leak empty the cistern in $\mathrm{x}$ hours. Part of the cistern emptied by the leak in one hour $=-\frac{1}{x}$ (The leak drains out the water) Considering the leak, the tank is filled in 10 h...
Read More →Simplify each of the following and express it in the form (a + ib)
Question: Simplify each of the following and express it in the form (a + ib) $\left(-2-\frac{1}{3} \mathrm{i}\right)^{3}$ Solution: Given: $\left(-2-\frac{1}{3} i\right)^{3}$ We know that, $(-a-b)^{3}=-a^{3}-3 a^{2} b-3 a b^{2}-b^{3} \ldots$ (i) So, on replacing a by 2 and b by 1/3i in eq. (i), we get $-(2)^{3}-3(2)^{2}\left(\frac{1}{3} i\right)-3(2)\left(\frac{1}{3} i\right)^{2}-\left(\frac{1}{3} i\right)^{3}$ $=-8-4 i-6\left(\frac{1}{9} i^{2}\right)-\left(\frac{1}{27} i^{3}\right)$ $=-8-4 i-\f...
Read More →If y is the reciprocal of rational number x,
Question: If y is the reciprocal of rational number x, then the reciprocal of y will be (a)x (b) y (c) $\frac{x}{y}$ (d) $\frac{y}{x}$ Solution: (a) If $y$ be the reciprocal of rational number $x$, i.e. $y=\frac{1}{x}$ or $x=\frac{1}{y}$. Hence, the reciprocal of y will be x....
Read More →A cistern has two inlets A and B which can fill it in 12 minutes and 15 minutes respectively.
Question: A cistern has two inlets A and B which can fill it in 12 minutes and 15 minutes respectively. An outlet C can empty the full cistern in 10 minutes. If all the three pipes are opened together in the empty tank, how much time will they take to fill the tank completely? Solution: Inlet A can fill the cistern in 12 minutes. Inlet B can fill the cistern in 15 minutes. Outlet $\mathrm{C}$ empties the filled cistern in 10 minutes. Part of the cistern filled by inlet A in one minute $=\frac{1}...
Read More →Simplify each of the following and express it in the form (a + ib)
Question: Simplify each of the following and express it in the form (a + ib) : $(-3+5 i)^{3}$ Solution: Given: $(-3+5 i)^{3}$ We know that, $(-a+b)^{3}=-a^{3}+3 a^{2} b-3 a b^{2}+b^{3} \ldots$ (i) So, on replacing a by 3 and b by 5i in eq. (i), we get $-(3)^{3}+3(3)^{2}(5 i)-3(3)(5 i)^{2}+(5 i)^{3}$ $=-27+3(9)(5 i)-3(3)\left(25 i^{2}\right)+125 i^{3}$ $=-27+135 i-225 i^{2}+125 i^{3}$ $=-27+135 i-225 \times(-1)+125 i \times i^{2}$ $=-27+135 i+225-125 i\left[\because i^{2}=-1\right]$ $=198+10 i$...
Read More →Three taps A, B and C can fill an overhead tank in 6 hours, 8 hours and 12 hours respectively.
Question: Three taps A, B and C can fill an overhead tank in 6 hours, 8 hours and 12 hours respectively. How long would the three taps take to fill the empty tank, if all of them are opened together? Solution: Time taken by tap A to fill the tank $=6$ hours Time taken by tap B to fill the tank $=8$ hours Time taken by tap C to fill the tank $=12$ hours A fills $\frac{1}{6}$ of the tank in one hour. B fills $\frac{1}{8}$ of the tank in one hour. C fills $\frac{1}{12}$ of the tank in one hour. Par...
Read More →The reciprocal of any rational number
Question: The reciprocal of any rational number $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$ is (a) $\frac{p}{q}$ (b)1 (c)0 (d) $\frac{q}{p}$ Solution: (d) The reciprocal of any rational number $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$ is $\frac{q}{p}$...
Read More →Pipe A can fill an empty tank in 5 hours while pipe B can empty the full tank in 6 hours.
Question: Pipe A can fill an empty tank in 5 hours while pipe B can empty the full tank in 6 hours. If both are opened at the same time in the empty tank, how much time will they take to fill it up completely? Solution: Pipe A can fill a tank in 5 hours. Pipe B can empty a full tank in 6 hours. Pipe A fills $\frac{1}{5}$ of the tank in one hour. Pipe B empties $\frac{1}{6}$ of the tank in one hour. Part of the tank filled in one hour using both pipes A and B $=\frac{1}{5}-\frac{1}{6}=\frac{6-5}{...
Read More →Solve this
Question: If $\mathrm{y}=(\tan \mathrm{x})^{(\tan \mathrm{x})^{(\tan x)} \infty}$, prove that $\frac{\mathrm{dy}}{\mathrm{dx}}=2$ at $\mathrm{x}=\frac{\pi}{4}$ Solution: Here, $y=(\tan x)^{(\tan x)^{(\tan x)}-^{\infty}}$ $y=(\tan x)^{y}$ By taking log on both sides, $\log y=\log (\tan x)^{y}$ $\log y=y(\log \tan x)$ Differentiating both sides with respect to $x$ using the product rule and chain rule, $\frac{1}{y} \frac{d y}{d x}=y \frac{d(\log \tan x)}{d x}+\log \tan x \frac{d y}{d x}$ $\frac{1}...
Read More →Simplify each of the following and express it in the form (a + ib) :
Question: Simplify each of the following and express it in the form (a + ib) : $(5-2 i)^{2}$ Solution: Given: $(5-2 i)^{2}$ We know that $(a-b)^{2}=a^{2}+b^{2}-2 a b \ldots(i)$ So, on replacing a by 5 and b by 2i in eq. (i), we get $(5)^{2}+(2 i)^{2}-2(5)(2 i)$ $=25+4 \mathrm{i}^{2}-20 \mathrm{i}$ $=25-4-20 \mathrm{i}\left[\because \mathrm{i}^{2}=-1\right]$ $=21-20 \mathrm{i}$...
Read More →Pipes A and B can fill an empty tank in 10 hours and 15 hours respectively.
Question: Pipes A and B can fill an empty tank in 10 hours and 15 hours respectively. If both are opened together in the empty tank, how much time will they take to fill it completely? Solution: A can fill a tank in 10 hours. B can fill a tank in 15 hours. Pipe A fills $\frac{1}{10}$ of the tank in one hour. Pipe B fills $\frac{1}{15}$ of the tank in one hour. Part of tank filled by pipes A and B together $=\frac{1}{10}+\frac{1}{15}=\frac{3+2}{30}=\frac{5}{30}=\frac{1}{6}$ Thus, pipes A and B re...
Read More →The reciprocal of 0 is
Question: The reciprocal of 0 is (a) 1 (b) -1 (c) 0 (d) Not defined Solution: (d) The reciprocal of 0 is not defined....
Read More →A and B can do a piece of work in 12 days, B and C in 15 days, and C and A in 20 days.
Question: A and B can do a piece of work in 12 days, B and C in 15 days, and C and A in 20 days. How much time will A alone take to finish the job? Solution: $(\mathrm{A}+\mathrm{B})$ can complete the work in 12 days. $(\mathrm{B}+\mathrm{C})$ can complete the work in 15 days. $(\mathrm{C}+\mathrm{A})$ can complete the work in 20 days. $(\mathrm{A}+\mathrm{B})$ 's 1 day work $=\frac{1}{12}$ $(\mathrm{~B}+\mathrm{C})$ 's 1 day work $=\frac{1}{15}$ $(\mathrm{C}+\mathrm{A})$ 's 1 day work $=\frac{1...
Read More →Simplify each of the following and express it in the form (a + ib) :
Question: Simplify each of the following and express it in the form (a + ib) : $(2+\sqrt{-3})^{2}$ Solution: Given: $(2-\sqrt{-3})^{2}$ We know that $(a-b)^{2}=a^{2}+b^{2}-2 a b \ldots(i)$ So, on replacing a by 2 and b by $\sqrt{-3}$ in eq. (i), we get $(2)^{2}+(\sqrt{-3})^{2}-2(2)(\sqrt{-3})$ $=4+(-3)-4 \sqrt{-3}$ $=4-3-4 \sqrt{-3}$ $=1-4 \sqrt{3} \mathrm{i}^{2}\left[\because \mathrm{i}^{2}=-1\right]$ $=1-4 \mathrm{i} \sqrt{3}$...
Read More →The reciprocal of -1 is
Question: The reciprocal of -1 is (a) 1 (b) -1 (c) 0 (d) Not defined Solution: (b) The reciprocal of -1 is the number itself....
Read More →A and B can do a piece of work in 18 days; B and C can do it in 24 days while C and A can finish it in 36 days.
Question: A and B can do a piece of work in 18 days; B and C can do it in 24 days while C and A can finish it in 36 days. In how many days can A, B, C finish it, if they all work together? Solution: Time needed by $\mathrm{A}$ and $\mathrm{B}$ to finish the work $=18$ days Time needed by $\mathrm{B}$ and $\mathrm{C}$ to finish the work $=24$ days Time needed by $\mathrm{C}$ and $\mathrm{A}$ to finish the work $=36$ days Work done by $\mathrm{A}$ and $\mathrm{B}$ in one day $=\frac{1}{18}$ Work d...
Read More →The reciprocal of 1 is ;
Question: The reciprocal of 1 is ; (a) 1 (b) -1 (c) 0 (d) Not defined Solution: (a) The reciprocal of 1 is the number itself....
Read More →A, B and C can do a piece of work in 15, 12 and 20 days respectively.
Question: A, B and C can do a piece of work in 15, 12 and 20 days respectively. They started the work together, but C left after 2 days. In how many days will the remaining work be completed by A and B? Solution: Time taken by $\mathrm{A}=15$ days Time taken by $\mathrm{B}=12$ days Time taken by $\mathrm{C}=20$ days Work $d$ by $\mathrm{A}$ in one day $=\frac{1}{15}$ Work done by $\mathrm{B}$ in one day $=\frac{1}{12}$ Work done by $\mathrm{C}$ in one day $=\frac{1}{20}$ Work done in one day by ...
Read More →Simplify each of the following and express it in the form a + ib
Question: Simplify each of the following and express it in the form a + ib $(-2+\sqrt{-3})(-3+2 \sqrt{-3})$ Solution: Given: $(-2+\sqrt{-3})(-3+2 \sqrt{-3})$ We re write the above equation $(-2+\sqrt{(-1) \times 3})(-3+2 \sqrt{(-1) \times 3})$ $=\left(-2+\sqrt{3 i^{2}}\right)\left(-3+2 \sqrt{3 i^{2}}\right)$ $\left[\because, i^{2}=-1\right]$ $=(-2+i \sqrt{3})(-3+2 i \sqrt{3})$ Now, open the brackets $=-2 \times(-3)+(-2) \times 2 \mathrm{i} \sqrt{3}+\mathrm{i} \sqrt{3} \times(-3)+\mathrm{i} \sqrt...
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