In how many days can both finish the work, working together?
Question: A can do $\frac{2}{3}$ of a certain work in 16 days and $B$ can do $\frac{1}{4}$ of the same work in 3 days. In how many days can both finish the work, working together? Solution: A can do $\frac{2}{3}$ work in 16 days So, work done by A in one day $=\frac{2}{48}=\frac{1}{24}$ B can do $\frac{1}{4}$ work in 3 days So, work done by B in one day $=\frac{1}{12}$ Work done jointly by $\mathrm{A}$ and $\mathrm{B}$ in one day $=\frac{1}{24}+\frac{1}{12}=\frac{1+2}{24}=\frac{3}{24}=\frac{1}{8...
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Question: If $y=(\sin x)^{(\sin x)^{(\sin x)^{-\infty}}}$, prove that $\frac{d y}{d x}=\frac{y^{2} \cot x}{(1-y \log \sin x)}$ Solution: Here, $y=(\sin x)^{(\sin x)^{(\sin x)^{-\infty}}}$ $y=(\sin x)^{y}$ By taking log on both sides, $\log y=\log (\sin x)^{y}$ $\log y=y(\log \sin x)$ Differentiating both sides with respect to $x$ by using product rule, $\frac{1}{y} \frac{d y}{d x}=y \frac{d(\log \sin x)}{d x}+\log \sin x \frac{d y}{d x}$ $\frac{1}{y} \frac{d y}{d x}=\frac{y}{\sin x} \frac{d(\sin...
Read More →If x is any rational number,
Question: If x is any rational number, then x + 0 is equal to (a)x (b) 0 (c)-x (d) Not defined Solution: (a) If x is any rational number, then x + 0 = x [0 is the additive identity]...
Read More →A can do a piece of work in 14 days while B can do it in 21 days.
Question: A can do a piece of work in 14 days while B can do it in 21 days. They began together and worked at it for 6 days. Then, A fell ill and B had to complete the remaining work alone. In how many days was the work completed? Solution: Time taken by $\mathrm{A}$ to complete the work $=14$ days Work done by $\mathrm{A}$ in one day $=\frac{1}{14}$ Time taken by $\mathrm{B}$ to complete the work $=21$ days Work done by $\mathrm{B}$ in one day $=\frac{1}{21}$ Work done jointly by $\mathrm{A}$ a...
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 $(3+4 i)(2-3 i)$ Solution: Given: $(3+4 i)(2-3 i)$ Firstly, we open the brackets $3 \times 2+3 \times(-3 i)+4 i \times 2-4 i \times 3 i$ $=6-9 i+8 i-12 i^{2}$ $=6-i-12(-1)\left[\because, i^{2}=-1\right]$ $=6-i+12$ $=18-i$...
Read More →A and B can finish a piece of work in 16 days and 12 days respectively.
Question: A and B can finish a piece of work in 16 days and 12 days respectively. A started the work and worked at it for 2 days. He was then joined by B. Find the total time taken to finish the work. Solution: Time taken by $\mathrm{A}$ to complete the work $=16$ days Work done per day by $\mathrm{A}=\frac{1}{16}$ Time taken by $\mathrm{B}$ to complete the work $=12$ days Work done per day by $\mathrm{B}=\frac{1}{12}$ Work done per day by $\mathrm{A}$ and $\mathrm{B}=\frac{1}{12}+\frac{1}{16}=\...
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Question: If $\mathrm{y}=\sqrt{\tan \mathrm{x}+\sqrt{\tan \mathrm{x}+\sqrt{\tan \mathrm{x}+\ldots \operatorname{tos} \infty}}}$, prove that $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\sec ^{2} \mathrm{x}}{2 \mathrm{y}-1}$ Solution: $y=\sqrt{\tan x+\sqrt{\tan x+\sqrt{\tan x+\cdots \text { to } \infty}}}$ $y=\sqrt{\tan x+y}$ On squaring both sides, $y^{2}=\tan x+y$ Differentiating both sides with respect to $x$, $2 y \frac{d y}{d x}=\sec ^{2} x+\frac{d y}{d x}$ $\frac{d y}{d x}(2 y-1)=\sec ^{2} x$ $\f...
Read More →The multiplicative inverse of
Question: The multiplicative inverse of $-1 \frac{1}{7}$ is (a) $8 / 7$ (b) $-8 / 7$ (c) $7 / 8$ (d) $7 /-8$ Solution: (d) $7 /-8$ $=-1 \frac{1}{7}=-8 / 7$ $=7 /-8$ $[\because$ reciprocal $]$...
Read More →vA, B and C working together can finish a piece of work in 8 hours.
Question: A, B and C working together can finish a piece of work in 8 hours. A alone can do it in 20 hours and B alone can do it in 24 hours. In how many hours will C alone do the same work? Solution: A can complete the work in $20 \mathrm{~h}$. Work done per hour by A $=\frac{1}{20}$ B can complete the work in $24 \mathrm{~h} .$ Work done per hour by B $=\frac{1}{24}$ It takes $8 \mathrm{~h}$ to complete the work if A, B and C work together. Work done together per hour by A,B and C $=\frac{1}{8...
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 $(5+\sqrt{-3})(5-\sqrt{-3})$ Solution: Given: $(5+\sqrt{-3})(5-\sqrt{-3})$ We re write the above equation $(5+\sqrt{(-1) \times 3})(5-\sqrt{(-1) \times 3})$ $=\left(5+\sqrt{3 i^{2}}\right)\left(5-\sqrt{3 i^{2}}\right)\left(\because, i^{2}=-1\right]$ $=(5+i \sqrt{3})(5-i \sqrt{3})$ Now, we know that, $(a+b)(a-b)=\left(a^{2}-b^{2}\right)$ Here, $a=5$ and $b=i \sqrt{3}$ $=(5)^{2}-(i \sqrt{3})^{2}$ $=25-\left(3 i^{2}\right)$ ...
Read More →A can do a piece of work in 24 hours while B alone can do it in 16 hours.
Question: A can do a piece of work in 24 hours while B alone can do it in 16 hours. If A, B and C working together can finish it in 8 hours, in how many hours can C alone finish the work? Solution: Time taken by A to complete the piece of work $=24 \mathrm{~h}$ Work done per hour by $\mathrm{A}=\frac{1}{24}$ Time taken by $\mathrm{B}$ to complete the work $=16 \mathrm{~h}$ Work done per hour by $\mathrm{B}=\frac{1}{16}$ Total time taken when $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ work togethe...
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Question: If $y=\sqrt{\log x+\sqrt{\log x+\sqrt{\log x+\ldots \text { to } \infty}}}$, prove that $(2 y-1) \frac{d y}{d x}=\frac{1}{x}$ Solution: $y=\sqrt{\log x+\sqrt{\log x+\sqrt{\log x+\cdots \text { to } \infty}}}$ $y=\sqrt{\log x+y}$ On squaring both sides, $y^{2}=\log x+y$ Differentiating both sides with respect to $x$, $2 y \frac{d y}{d x}=\frac{1}{x}+\frac{d y}{d x}$ $\frac{d y}{d x}(2 y-1)=\frac{1}{x}$ $\frac{d y}{d x}=\frac{1}{x(2 y-1)}$ Hence proved....
Read More →– (-x) is same as
Question: (-x) is same as (a) $-x$ (b) $x$ (c) $1 / x$ (d) $-1 / x$ Solution: (b) -(-x) = x Negative of negative rational number is equal to positive rational number....
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 : $(1-i)^{2}(1+i)-(3-4 i)^{2}$ Solution: Given: $(1-i)^{2}(1+i)-(3-4 i)^{2}$ $=\left(1+i^{2}-2 i\right)(1+i)-\left(9+16 i^{2}-24 i\right)$ $\left[\because(a-b)^{2}=a^{2}+b^{2}-2 a b\right]$ $=(1-1-2 i)(1+i)-(9-16-24 i)\left[\because i^{2}=-1\right]$ $=(-2 i)(1+i)-(-7-24 i)$ Now, we open the brackets $-2 i \times 1-2 i \times i+7+24 i$ $=-2 i-2 i^{2}+7+24 i$ $=-2(-1)+7+22 i\left[\because, i^{2}=-1\right]$ $=2+7+22 i$ $=9+2...
Read More →A, B and C can do a piece of work in 10 days, 12 days and 15 days respectively.
Question: A, B and C can do a piece of work in 10 days, 12 days and 15 days respectively. How long will they take to finish it if they work together? Solution: Time taken by A to complete the work $=10$ days Time taken by B to complete the work $=12$ days Time taken by C to complete the work $=15$ days Work done per day by A $=\frac{1}{10}$ Work done per day by B $=\frac{1}{12}$ Work done per day by C $=\frac{1}{15}$ Total work done per day $=\frac{1}{10}+\frac{1}{12}+\frac{1}{15}=\frac{6+5+4}{6...
Read More →To get the product 1 , we should multiply
Question: To get the product 1 , we should multiply $(8 / 21)$ by (a) $8 / 21$ (b) $-8 / 21$ (c) $21 / 8$ (d) $-21 / 8$ Solution: (c) 21/8 Because, $=(8 / 21) \times(21 / 8)$ $=(8 \times 21) /(21 \times 8)$ $=168 / 168$ $=1$...
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Question: If $\mathrm{y}=\sqrt{\cos \mathrm{x}+\sqrt{\cos \mathrm{x}+\sqrt{\cos \mathrm{x}+\ldots \text { to } \infty}}}$, prove that $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\sin \mathrm{x}}{1-2 \mathrm{y}}$. Solution: Here, $y=\sqrt{\cos x+\sqrt{\cos x+\sqrt{\cos x+\cdots \text { to } \infty}}}$ $y=\sqrt{\cos x+y}$ On squaring both sides, $y^{2}=\cos x+y$ Differentiating both sides with respect to $x$, $2 y \frac{d y}{d x}=-\sin x+\frac{d y}{d x}$ $\frac{d y}{d x}(2 y-1)=-\sin x$ $\frac{d y}{d x...
Read More →Two motor mechanics, Raju and Siraj, working together can overhaul a scooter in 6 hours.
Question: Two motor mechanics, Raju and Siraj, working together can overhaul a scooter in 6 hours. Raju alone can do the job in 15 hours. In how many hours can Siraj alone do it? Solution: Time taken by Raju $=15 \mathrm{~h}$ Work done by Raju in $1 \mathrm{~h}=\frac{1}{15}$ Time taken by Raju and Siraj working together $=6 \mathrm{~h}$ Work done by Raju and Siraj in $1 \mathrm{~h}=\frac{1}{6}$ Work done by Siraj in $1 \mathrm{~h}=($ work done by Raju and Siraj $)-($ work done by Raju $)$ $=\fra...
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 : $(8-4 i)-(-3+5 i)$ Solution: Given: $(8-4 i)-(-3+5 i)$ Firstly, we open the brackets $8-4 i+3-5 i$ $=11-9 i$...
Read More →A and B, working together can finish a piece of work in 6 days, while A alone can do it in 9 days.
Question: A and B, working together can finish a piece of work in 6 days, while A alone can do it in 9 days. How much time will B alone take to finish it? Solution: Time taken by $\mathrm{A}$ and $\mathrm{B}$ to finish a piece of work $=6$ days Work done per day by $\mathrm{A}$ and $\mathrm{B}=\frac{1}{6}$ Time taken by $\mathrm{A}$ alone $=9$ days Work done per day by A alone $=\frac{1}{9}$ Work done per day by $\mathrm{B}=($ work done by $\mathrm{A}$ and $\mathrm{B})-($ work done by $\mathrm{A...
Read More →If x + 0 = 0 + x = x, which is rational number,
Question: If x + 0 = 0 + x = x, which is rational number, then 0 is called (a) identity for addition of rational numbers (b) additive inverse of x (c) multiplicative inverse of x (d) reciprocal of x Solution: (a) We know that, the sum of any rational number and zero (0) is the rational number itself. Now, x + 0 = 0+ x= x, which is a rational number, then 0 is called identity for addition of rational numbers....
Read More →Ravi can do a piece of work in 15 hours while Raman can do it in 12 hours.
Question: Ravi can do a piece of work in 15 hours while Raman can do it in 12 hours. How long will both take to do it, working together? Solution: Time taken by Ravi $=15 \mathrm{~h}$ Time taken by Raman $=12 \mathrm{~h}$ Work done per hour by Ravi $=\frac{1}{15}$ Work done per hour by Raman $=\frac{1}{12}$ Work done per hour by Ravi and Raman together $=\frac{1}{15}+\frac{1}{12}=\frac{9}{60}=\frac{3}{20}$ $\therefore$ Time taken by Ravi and Raman together to finish the work $=\frac{20}{3} \math...
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 : $(-5+6 i)-(-2+i)$ Solution: Given: $(-5+6 i)-(-2+i)$ Firstly, we open the brackets $-5+6 i+2-i$ $=-3+5 i$...
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Question: If $\mathrm{y}=\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+\sqrt{\mathrm{x}+\ldots \text { to } \infty}}}$, prove that $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{2 \mathrm{y}-1}$. Solution: Here, $y=\sqrt{x+\sqrt{x+\sqrt{x+\cdots} \text { to } \infty}}$ $y=\sqrt{x+y}$ On squaring both sides, $y^{2}=x+y$ Differentiating both sides with respect to $x$, $2 y \frac{d y}{d x}=1+\frac{d y}{d x}$ $\frac{d y}{d x}(2 y-1)=1$ $\frac{d y}{d x}=\frac{1}{2 y-1}$ Hence proved....
Read More →Multiplicative inverse of a negative rational number is
Question: Multiplicative inverse of a negative rational number is (a) a positive rational number (b) a negative rational number (c) 0 (d) 1 Solution: (b) We know that, the product of two rational numbers is 1, taken they are multiplication inverse of each other, e.g. Suppose, p is negative rational number, i.e. $\frac{1}{p}$ is the multiplicative inverse of-p, then, $-p \times \frac{1}{-p}=1$ Hence, multiplicative inverse of a negative rational number is a negative rational number....
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