A and B can do a piece of work in 15 days; B and C in 12 days; C and A in 20 days.
Question: A and B can do a piece of work in 15 days; B and C in 12 days; C and A in 20 days. How many days will be taken by A, B and C working together to finish the work? Solution: $(\mathrm{A}+\mathrm{B})$ can do a work in 15 days. $\therefore(\mathrm{A}+\mathrm{B})$ 's 1 day work $=\frac{1}{15}$ $(\mathrm{B}+\mathrm{C})$ can do a work in 12 days. $\therefore(\mathrm{B}+\mathrm{C})$ 's 1 day work $=\frac{1}{12}$ $(\mathrm{C}+\mathrm{A})$ can do a work in 20 days. $\therefore(\mathrm{C}+\mathrm...
Read More →Express each of the following in the form (a + ib):
Question: Express each of the following in the form (a + ib): $\frac{(3-4 i)}{(4-2 i)(1+i)}$ Solution: Given: $\frac{3-4 i}{(4-2 i)(1+i)}$ Solving the denominator, we get $\frac{3-4 i}{(4-2 i)(1+i)}=\frac{3-4 i}{4(1)+4(i)-2 i(1)-2 i(i)}$ $=\frac{3-4 i}{4+4 i-2 i-2 i^{2}}$ $=\frac{3-4 i}{4+2 i-2(-1)}$ $=\frac{3-4 i}{6+2 i}$ Now, we rationalize the above by multiplying and divide by the conjugate of 6 + 2i $=\frac{3-4 i}{6+2 i} \times \frac{6-2 i}{6-2 i}$ $=\frac{(3-4 i)(6-2 i)}{(6+2 i)(6-2 i)}$ N...
Read More →A can do a piece of work in 10 days while B alone can do it in 15 days.
Question: A can do a piece of work in 10 days while B alone can do it in 15 days. In how many days can both finish the same work? Solution: A can do a piece of work in 10 days. A's 1 day work $=\frac{1}{10}$ B can do a piece of work in 15 days. B's 1 day work $=\frac{1}{15}$ $(\mathrm{A}+\mathrm{B})$ 's 1 day work $=\frac{1}{10}+\frac{1}{15}=\frac{3+2}{30}=\frac{5}{30}=\frac{1}{6}$ $\mathrm{A}$ and $\mathrm{B}$ working together can complete the work in 6 days....
Read More →Tick (✓) the correct answer:
Question: Tick (✓) the correct answer: Two pipes can fill a tank in 10 hours and 12 hours respectively, while a third pipe empties the full tank in 20 hours. If all the three pipes operate simultaneously, in how much time will the tank be full? (a) 7 hrs 15 min (b) 7 hrs 30 min (c) 7 hrs 45 min (d) 8 hrs Solution: (b) 7 hours 30 minutes Part of the tank filled by the first pipe in one hour $=\frac{1}{10}$ Part of the tank filled by the second pipe in one hour $=\frac{1}{12}$ Part of the tank fil...
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Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$, when $x=\frac{e^{t}+e^{-t}}{2}$ and $y=\frac{e^{t}-e^{-t}}{2}$ Solution: $\operatorname{as} x=\frac{e^{\theta}+e^{\theta}}{2}$ Differentiating it with respect to $t$ $\frac{\mathrm{dx}}{\mathrm{dt}}=\frac{1}{2}\left[\frac{\mathrm{d}\left(\mathrm{e}^{\mathrm{t}}\right)}{\mathrm{dt}}+\frac{\mathrm{d}\left(\mathrm{e}^{-\mathrm{t}}\right)}{\mathrm{dt}}\right]$ $=\frac{1}{2}\left[\mathrm{e}^{\mathrm{t}}+\mathrm{e}^{-\mathrm{t}} \frac{\mathrm{d}(-\math...
Read More →For rational numbers
Question: For rational numbers $(a / b),(c / d)$ and $(e / f)$ we have $(a / b) \times((c / d)+(e / f))=$ ____+____. Solution: For rational numbers $(a / b),(c / d)$ and $(e / f)$ we have $(a / b) \times((c / d)+(e / f))=((a / b) \times(c / d))$ $+((a / b) \times(e / f))$...
Read More →Tick (✓) the correct answer:
Question: Tick (✓) the correct answer: A pump can fill a tank in 2 hours. Due to a leak in the tank it takes $2 \frac{1}{3}$ hours to fill the tank. The leak can empty the full tank in (a) $2 \frac{1}{3}$ hours (b) 7 hours (c) 8 hours (d) 14 hours Solution: (d) 14 hours A pump can fill a tank in 2 hours. Part of the tank filled by the pump in one hour $=\frac{1}{2}$ Suppose the leak empties a full tank in x hours. Part of the tank emptied by the leak in one hour $=-\frac{1}{x}$ Part of tank fill...
Read More →The negative of 1 is—————-.
Question: The negative of 1 is-. Solution: -1 The negative of 1 is -1...
Read More →Express each of the following in the form (a + ib)
Question: Express each of the following in the form (a + ib): $\frac{(-2+5 i)}{(3-5 i)}$ Solution: Given: $\frac{-2+5 i}{3-5 i}$ Now, rationalizing $=\frac{-2+5 i}{3-5 i} \times \frac{3+5 i}{3+5 i}$ $=\frac{(-2+5 i)(3+5 i)}{(3-5 i)(3+5 i)} \ldots(\mathrm{i})$ Now, we know that, $(a+b)(a-b)=\left(a^{2}-b^{2}\right)$ So, eq. (i) become $=\frac{(-2+5 i)(3+5 i)}{(3)^{2}-(5 i)^{2}}$ $=\frac{-2(3)+(-2)(5 i)+5 i(3)+5 i(5 i)}{9-25 i^{2}}$ $=\frac{-6-10 i+15 i+25 i^{2}}{9-25(-1)}\left[\because i^{2}=-1\r...
Read More →Prove the following
Question: $(213 \times 657)^{-1}=213^{-1} \times$________. Solution: Let us assume the missing number be $x$ Then, $=1 /(213 \times 657)=(1 / 213) \times(x)$ $X=213 /(213 / 657)$ $X=1 / 657$ $X=657^{-1}$ So, $(213 \times 657)^{-1}=213^{-1} \times \underline{657^{-1}}$...
Read More →Tick (✓) the correct answer:
Question: Tick (✓) the correct answer: A tap can fill a cistern in 8 hours and another tap can empty the full cistern in 16 hours. If both the taps are open, the time taken to fill the cistern is (a) $5 \frac{1}{3}$ hours (b) 10 hours (c) 16 hours (d) 20 hours Solution: (c) 16 hours A tap can fill a cistern in 8 hours. Part of cistern filled in one hour $=\frac{1}{8}$ A tap can empty the cistern in 16 hours. Part of cistern emptied in one hour $=-\frac{1}{16}$ (negative sign shows that the ciste...
Read More →Solve this
Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$, when $x=a(1-\cos \theta)$ and $y=a(\theta+\sin \theta)$ at $\theta=\frac{\pi}{2}$ Solution: as $x=a(1-\cos \theta)$ $\frac{\mathrm{dx}}{\mathrm{d} \theta}=\frac{\mathrm{d}[\mathrm{a}(1-\cos \theta)]}{\mathrm{d} \theta}=\mathrm{a}(\sin \theta)$ And $y=a(\theta+\sin \theta)$ $\frac{d y}{d \theta}=\frac{d(\theta+\sin \theta)}{d \theta}=a(1+\cos \theta)$ $\therefore \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\frac{\mathrm{dy}}{\mathrm{d} \theta}}{\frac{\m...
Read More →The reciprocal of
Question: The reciprocal of $(2 / 5) \times(-4 / 9)$ is__________. Solution: $=(2 \times-4) /(5 \times 9)$ $=-8 / 45$ Reciprocal $=-45 / 8$ Hence, the reciprocal of $(2 / 5) \times(-4 / 5)$ is $\underline{-45 / 8}$....
Read More →Tick (✓) the correct answer:
Question: Tick (✓) the correct answer: Two pipes can fill a tank in 20 minutes and 30 minutes respectively. If both the pipes are opened simultaneously, then hte tank will be filled in (a) 10 minutes (b) 12 minutes (c) 15 minutes (d) 25 minutes Solution: (b) 12 minutes First pipe can fill a tank in 20 minutes. Second pipe can fill the tank in 30 minutes. Part of tank filled by the first pipe in one minute $=\frac{1}{20}$ Part of tank filled by the second pipe in one minute $\frac{1}{30}$ Part of...
Read More →Express each of the following in the form (a + ib):
Question: Express each of the following in the form (a + ib): $\frac{(5+\sqrt{2} i)}{(1-\sqrt{2} i)}$ Solution: Given: $\frac{5+\sqrt{2} i}{1-\sqrt{2} i}$ Now, rationalizing $=\frac{5+\sqrt{2} i}{1-\sqrt{2} i} \times \frac{1+\sqrt{2} i}{1+\sqrt{2} i}$ $=\frac{(5+\sqrt{2} i)(1+\sqrt{2} i)}{(1-\sqrt{2} i)(1+\sqrt{2} i)} \ldots$ (i) Now, we know that, $(a+b)(a-b)=\left(a^{2}-b^{2}\right)$ So, eq. (i) become $=\frac{(5+\sqrt{2} i)(1+\sqrt{2} i)}{(1)^{2}-(\sqrt{2} i)^{2}}$ $=\frac{5(1)+5(\sqrt{2} i)+...
Read More →If y is the reciprocal of x,
Question: If $y$ is the reciprocal of $x$, then the reciprocal of $y^{2}$ in terms of $x$ will be_________. Solution: If $y$ be the reciprocal of $x$, then the reciprocal of $y^{2}$ in terms of $x$ will be $\underline{x}^{2}$. From the question, $(1 / x)=y$ Then, Reciprocal of $y^{2}=1 / y^{2}$ Substitute $(1 / x)$ in the place of $y$, $=1 /(1 / x)^{2}$ $=x^{2} / 1$ $=x^{2}$...
Read More →Tick (✓) the correct answer:
Question: Tick (✓) the correct answer: A can do a piece of work in 25 days, which B alone can do in 20 days. A started the work and was joined by B after 10 days. The work lasted for (a) $12 \frac{1}{2}$ days (b) 15 days (c) $16 \frac{2}{3}$ days (d) 14 days Solution: (c) $16 \frac{2}{3}$ days A can do the piece of work in 25 days. Work done by A in 1 day $=\frac{1}{25}$ B can do the same work in 20 days. Work done by B in 1 day $=\frac{1}{20}$ A alone completes $\frac{10}{25}, \mathrm{i}, \math...
Read More →Solve this
Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$, when $x=b \sin ^{2} \theta$ and $y=a \cos ^{2} \theta$ Solution: as $x=b \sin ^{2} \theta$ Then $\frac{\mathrm{dx}}{\mathrm{d} \theta}=\frac{\mathrm{d}\left(\mathrm{b} \sin ^{2} \theta\right)}{\mathrm{d} \theta}=2 \mathrm{~b} \sin \theta \cos \theta$ And $y=a \cos ^{2} \theta$ $\frac{\mathrm{dy}}{\mathrm{d} \theta}=\mathrm{d}\left(\operatorname{acos}^{2} \theta\right)=-2 \operatorname{acos} \theta \sin \theta$ $\therefore \frac{\mathrm{dy}}{\math...
Read More →Express each of the following in the form (a + ib):
Question: Express each of the following in the form (a + ib): $\frac{(3+4 i)}{(4+5 i)}$ Solution: Given: $\frac{3+4 i}{4+5 i}$ Now, rationalizing $=\frac{3+4 i}{4+5 i} \times \frac{4-5 i}{4-5 i}$ $=\frac{(3+4 i)(4-5 i)}{(4+5 i)(4-5 i)}$ Now, we know that $(a+b)(a-b)=\left(a^{2}-b^{2}\right)$ So, eq. (i) become $=\frac{(3+4 i)(4-5 i)}{(4)^{2}-(5 i)^{2}}$ $=\frac{3(4)+3(-5 i)+4 i(4)+4 i(-5 i)}{16-25 i^{2}}$ $=\frac{12-15 i+16 i-20 i^{2}}{16-25(-1) \quad\left[\because i^{2}=-1\right]}$ $=\frac{12+i...
Read More →Tick (✓) the correct answer:
Question: Tick (✓) the correct answer: A can do a piece of work in 20 days which B alone can do in 12 days. B worked at it for 9 days. A can finish the remaining work in (a) 3 days (b) 5 days (c) 7 days (d) 11 days Solution: (b) 5 days A can complete the work in 20 days. Work done by A in 1 day $=\frac{1}{20}$ B can complete the work in 12 days. Work done by B in 1 day $=\frac{1}{12}$ In 9 days, B completes $\frac{9}{12}$, i.e., $\frac{3}{4}$ of the work and leaves $1-\frac{3}{4}$, i. e., $\frac...
Read More →Solve this
Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$, when $x=a e^{\theta}(\sin \theta-\cos \theta), y=a e^{\theta}(\sin \theta+\cos \theta)$ Solution: $\operatorname{as} x=a e^{\theta}(\sin \theta-\cos \theta)$ Differentiating it with respect to $\theta$ $\frac{d x}{d \theta}=a\left[e^{\theta} \frac{d(\sin \theta-\cos \theta)}{d \theta}+(\sin \theta-\cos \theta) \frac{d\left(e^{\theta}\right)}{d \theta}\right]$ $=a\left[e^{\theta}(\cos \theta+\sin \theta)+(\sin \theta-\cos \theta) e^{\theta}\right]...
Read More →The numbers ————–and————–are their
Question: The numbers andare their own reciprocal. Solution: The numbers $\underline{1}$ and $\underline{-1}$ are their own reciprocal. Reciprocal of $1=1 / 1=1$ Reciprocal of $-1=1 /-1=-1$...
Read More →Zero has_______reciprocal.
Question: Zero has_______reciprocal. Solution: Zero has no reciprocal. The reciprocal of $0=1 / 0$ $=$ Undefined...
Read More →Express each of the following in the form (a + ib)
Question: Express each of the following in the form (a + ib): $\frac{1}{(4+3 \mathrm{i})}$ Solution: Given: $\frac{1}{4+3 i}$ Now, rationalizing $=\frac{1}{4+3 i} \times \frac{4-3 i}{4-3 i}$ $=\frac{4-3 i}{(4+3 i)(4-3 i)} \ldots(\mathrm{i})$ Now, we know that, $(a+b)(a-b)=\left(a^{2}-b^{2}\right)$ So, eq. (i) become $=\frac{4-3 i}{(4)^{2}-(3 i)^{2}}$ $=\frac{4-3 i}{16-9 i^{2}}$ $=\frac{4-3 i}{16-9(-1)}\left[\because i^{2}=-1\right]$ $=\frac{4-3 i}{16+9}$ $=\frac{4-3 i}{25}$ $=\frac{4}{25}-\frac{...
Read More →The reciprocal of a negative
Question: The reciprocal of a negative rational number is_______. Solution: The reciprocal of a positive rational number is positive rational number. Let us take positive rational number $2 / 3$ The reciprocal of this positive rational number is $3 / 2$ (positive rational number)...
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