Rajan can do a piece of work in 24 days while Amit can do it in 30 days.
Question: Rajan can do a piece of work in 24 days while Amit can do it in 30 days. In how many days can they complete it, if they work together? Solution: Work done by Rajan in 1 day $=\frac{1}{24}$ Work done by Amit in 1 day $=\frac{1}{30}$ Work done by Amit and Rajan together in 1 day $=\frac{1}{24}+\frac{1}{30}=\frac{54}{720}=\frac{3}{40}$ $\therefore$ They can complete the work in $\frac{40}{3}$ days, i.e., $13 \frac{1}{3}$ days if they work together....
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 : $(3+\sqrt{-16})-(4-\sqrt{-9})$ Solution: Given: $(3+\sqrt{-16})-(4-\sqrt{-9})$ We re - write the above equation $(3+\sqrt{(-1) \times 16})(-1)(4-\sqrt{(-1) \times 9})$ $=\left(3+\sqrt{16 i^{2}}\right)-\left(4-\sqrt{9 i^{2}}\right)\left[\because i^{2}=-1\right]$ $=(3+4 i)-(4-3 i)$ Now, we open the brackets, we get $3+4 i-4+3 i$ $=-1+7 i$...
Read More →The additive inverse of
Question: The additive inverse of $\frac{-7}{19}$ is (a) $-7 / 19$ (b) $7 / 19$ (c) $19 / 7$ (d) $-19 / 7$ Solution: Additive inverse of $(-7 / 19)$ is (b) $(7 / 19)$ The additive inverse of the rational number $-a / b$ is $a / b$ and vice-versa....
Read More →Solve this
Question: If $\mathrm{y}=1+\frac{\alpha}{\left(\frac{1}{\mathrm{x}}-\alpha\right)}+\frac{\beta / \mathrm{x}}{\left(\frac{1}{\mathrm{x}}-\alpha\right)\left(\frac{1}{\mathrm{x}}-\beta\right)}+\frac{\gamma / \mathrm{x}^{2}}{\left(\frac{1}{\mathrm{x}}-\alpha\right)\left(\frac{1}{\mathrm{x}}-\beta\right)\left(\frac{1}{\mathrm{x}}-\gamma\right)}$, find $\frac{\mathrm{dy}}{\mathrm{dx}}$ Solution: Given, $\mathrm{y}=1+\frac{\alpha}{\left(\frac{1}{\mathrm{x}}-\alpha\right)}+\frac{\beta / \mathrm{x}}{\lef...
Read More →One (1) is
Question: One (1) is (a) the identity for addition of rational numbers (b) the identity for subtraction of rational numbers (c) the identity for multiplication of rational numbers (d) the identity for division of rational numbers Solution: (c) One (1) is the identity for multiplication of rational numbers. That means, If a is a rational number. Then, a-1 = 1-a = a Note One (1) is the multiplication identity for integers and whole number also....
Read More →Fill in the blanks.
Question: Fill in the blanks. (i) If 3 persons can do a piece of work in 4 days, then 4 persons can do it in ......... days. (ii) If 5 pipes can fill tank in 144 minutes, then 6 pipes can fill it in ......... minutes. (iii) A car covers a certain distance in 1 hr 30 minutes at 60 km per hour. If it moves at 45 km per hour, it will take ......... hours. (iv) If 8 oranges cost Rs 20.80, the cost of 5 oranges is Rs ......... (v) The weight of 12 sheets of a paper is 50 grams. How many sheets will w...
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 : $2(3+4 i)+i(5-6 i)$ Solution: Given: $2(3+4 i)+i(5-6 i)$ Firstly, we open the brackets $2 \times 3+2 \times 4 i+i \times 5-i \times 6 i$ $=6+8 i+5 i-6 i^{2}$ $=6+13 i-6(-1)\left[\because, i^{2}=-1\right]$ $=6+13 i+6$ $=12+13 i$...
Read More →Zero (0) is
Question: Zero (0) is (a) the identity for addition of rational numbers (b) the identity for subtraction of rational numbers (c) the identity for multiplication of rational numbers (d) the identity for division of rational numbers Solution: (a) Zero (0) is the identity for addition of rational numbers. That means, If a is a rational number. Then, a+0=0+a = a Note Zero (0) is also the additive identity for integers and whole number as well....
Read More →Prove that
Question: Solution: L.H.S $=\sum_{n=1}^{13}\left(i^{n}+i^{n+1}\right)$ $=i^{1}+i^{2}+i^{3}+i^{4}+i^{5}+i^{6}+\ldots . .+i^{13}+i^{14}$ Since $i^{4^{4 n}}=1$ $\Rightarrow i^{4 n+1}=i$ $\Rightarrow i^{4 n+2}=-1$ $\Rightarrow i^{4 n+3}=-1$ $=i-1-i+1+i-1 \ldots \ldots+i-1$ As, all terms will get cancel out consecutively except the first two terms. So that will get remained will be the answer. Activate Windows $=\mathrm{i}-1$ L.H.S = R.H.S Hence proved....
Read More →Which of the following expressions shows
Question: Which of the following expressions shows that rational numbers are associative under multiplication. (a) $[(2 / 3) \times((-6 / 7) \times(3 / 5))]=[((2 / 3) \times(-6 / 7)) \times(3 / 5)]$ (b) $[(2 / 3) \times((-6 / 7) \times(3 / 5))]=[(2 / 3) \times((3 / 5) \times(-6 / 7))]$ (c) $[(2 / 3) \times((-6 / 7) \times(3 / 5))]=[((3 / 5) \times(2 / 3)) \times(-6 / 7)]$ (d) $[((2 / 3) \times(-6 / 7)) \times(3 / 5)]=[((-6 / 7) \times(2 / 3)) \times(3 / 5)]$ Solution: (a) $[(2 / 3) \times((-6 / ...
Read More →Prove the following
Question: $-\frac{3}{8}+\frac{1}{7}=\frac{1}{7}+\left[\frac{-3}{8}\right]$ is an example to show that (a) addition of rational numbers is commutative (b) rational numbers are closed under addition (c) addition of rational numbers is associative (d) rational numbers are distributive under addition Solution: (a) addition of rational numbers is commutative. The arrangement of above rational numbers is in the form of Commutative law of addition $[a+b=b+a]$...
Read More →Which of the following is not true?
Question: Which of the following is not true? (a) rational numbers are closed under addition (b) rational numbers are closed under subtraction (c) rational numbers are closed under multiplication (d) rational numbers are closed under division Solution: (d) Rational numbers are not closed under division. As, 1 and 0 are the rational numbers but $\frac{1}{0}$ is not defined....
Read More →Mark (✓) against the correct answer:
Question: Mark (✓) against the correct answer: xandyvary inversely. Whenx= 15, theny= 6. What will be the value ofywhenx= 9? (a) 10 (b) 15 (c) 54 (d) 135 Solution: (a) 10 Since $x$ and $y \operatorname{var} y$ inversely, $x y=$ constant. Now, $15 \times 6=9 \times y_{1}$ $\Rightarrow \frac{90}{9}=y_{1}$ $\Rightarrow 10=y_{1}$ Value ofy= 10, whenx= 9....
Read More →Solve this
Question: If $y=x^{\tan x}+\sqrt{\frac{x^{2}+1}{2}}$, find $\frac{d y}{d x}$ Solution: Given $y=x^{\tan x}+\sqrt{\frac{x^{2}+1}{2}}$ $y=e^{\tan x \log x}+e^{\frac{1}{2} \log \frac{x^{2}+1}{2}}$ $\frac{d y}{d x}=e^{\tan x \log x} \frac{d}{d x}(\tan x \log x)+e^{\frac{1}{2} \log \frac{x^{2}+1}{2}} \frac{d}{d x}\left(\frac{1}{2} \log \frac{x^{2}+1}{2}\right)$ $\frac{d y}{d x}=x^{\tan x}\left[\frac{\tan x}{x}+\sec ^{2} \log x\right]+\sqrt{\frac{x^{2}+1}{2}}\left(\frac{1}{2} \times \frac{2}{x^{2}+1} ...
Read More →The numerical expression
Question: The numerical expression $(3 / 8)+(-5 / 7)=(-19 / 56)$ shows that (a) rational numbers are closed under addition. (b) rational numbers are not closed under addition. (c) rational numbers are closed under multiplication. (d) addition of rational numbers is not commutative. Solution: (b) We have $\frac{3}{8}+\frac{(-5)}{7}=\frac{-19}{56}$ Show that rational numbers are closed under addition. $\left[\frac{3}{8}\right.$ and $\frac{-5}{7}$ are rational numbers and their addition is $\frac{-...
Read More →Mark (✓) against the correct answer:
Question: Mark (✓) against the correct answer: xandyvary directly. Whenx= 3, theny= 36. What will be the value ofxwheny= 96? (a) 18 (b) 12 (c) 8 (d) 4 Solution: (c) 8 $x$ and $y$ var $y$ directly. Then $x=k y$, where $k$ is the constant of proportionality. $\Rightarrow k=\frac{x}{y}$ Now, $\frac{3}{36}=\frac{x_{1}}{96}$ $\Rightarrow \frac{96 \times 3}{36}=x_{1}$ $\Rightarrow 8=x_{1}$ $\therefore$ Value of $x=8$...
Read More →Mark (✓) against the correct answer:
Question: Mark (✓) against the correct answer: Rashmi types 510 words in half an hour. How many words would she type in 10 minutes? (a) 85 (b) 150 (c) 170 (d) 153 Solution: (c) 170 words Letxbe the number of words typed by Rashmi in 10 minutes. Less time will be taken to type less number of words. So, it is a case of direct variation. Now, $\frac{510}{30}=\frac{x}{10}$ $\Rightarrow x=170$ Rashmi will type 170 words in 10 minutes....
Read More →Mark (✓) against the correct answer:
Question: Mark (✓) against the correct answer: A car is travelling at an average speed of 60 km per hour. How much distance will it cover in 1 hour 12 minutes? (a) 50 km (b) 72 km (c) 63 km (d) 67.2 km Solution: (b) 72 km Letxkm be the distance covered in 1 h 12 min. Now, 1 h 12 min = (60+12) min = 72 min More distance will be covered in more time. So, it is a cas of direct proportion. Now, $\frac{60}{60}=\frac{x}{72}$ $\Rightarrow x=72 \mathrm{~km}$ $\therefore$ The car will cover a distance of...
Read More →A number of the form
Question: A number of the form $\frac{p}{q}$ is said to be a rational number, if (a) p, q are integers (b) $\mathrm{p}, \mathrm{q}$ are integers and $q \neq 0$ (c) $\mathrm{p}, \mathrm{q}$ are integers and $p \neq 0$ (d) $\mathrm{p}, \mathrm{q}$ are integers and $p \neq 0$, also $q \neq 0$ Solution: (b) A number of the form $\frac{p}{q}$ is said to be a rational number, if $p$ and $q$ are integers and...
Read More →Mark (✓) against the correct answer:
Question: Mark (✓) against the correct answer: 35 men can reap a field in 8 days. In how many days can 20 men reap it? (a) 14 days (b) 28 days (c) $87 \frac{1}{2}$ days (d) none of these Solution: (a) 14 days Let 20 men takexdays to reap the field. Clearly, less number of men will take more days. So, it is a case of inverse proportion. Now, $8 \times 35=x \times 20$ $\Rightarrow x=\frac{8 \times 35}{20}$ $\Rightarrow x=14$ 20 men can reap the field in 14 days....
Read More →A number which can be expressed as
Question: A number which can be expressed as $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$ is (a) natural number (b) whole number (c) integer (d) rational number Solution: (d) A number which can be expressed as $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$ is a rational number....
Read More →Mark (✓) against the correct answer:
Question: Mark (✓) against the correct answer: 35 men can reap a field in 8 days. In how many days can 20 men reap it? (a) 14 days (b) 28 days (c) $87 \frac{1}{2}$ days (d) none of these Solution: (a) 14 days Let 20 men takexdays to reap the field. Clearly, less number of men will take more days. So, it is a case of inverse proportion. Now, $8 \times 35=x \times 20$ $\Rightarrow x=\frac{8 \times 35}{20}$ $\Rightarrow x=14$ 20 men can reap the field in 14 days....
Read More →Solve this
Question: If $\mathrm{x}=\mathrm{e}^{\mathrm{x} / \mathrm{y}}$, prove that $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{x}-\mathrm{y}}{\mathrm{x} \log \mathrm{x}}$ Solution: $x=e^{x} / y$ Taking logon both sides, $\log x=\log e^{x} / y$ $\log x=\frac{x}{y} \ldots \ldots$ (i) [ Since $\log e^{a}=a$ ] or, $y=\frac{x}{\log x}$ (ii) Differentiating the given equation with respect to $x$, $\frac{d y}{d x}=\frac{\log x \frac{d}{d x}(x)-x \frac{d}{d x}(\log x)}{(\log x)^{2}}$ $\frac{d y}{d x}=\frac{\...
Read More →Mark (✓) against the correct answer:
Question: Mark (✓) against the correct answer: 14 workers can build a wall in 42 days. One worker can build it in (a) 3 days (b) 147 days (c) 294 days (d) 588 days Solution: (d) 588 days Let one worker takexdays to build the wall. Clearly, one worker will take more days to finish the work. So, it is a case of inverse proportion Now, $14 \times 42=1 \times x$ $\Rightarrow x=14 \times 42$ $\Rightarrow x=588$ One worker can build the wall in 588 days....
Read More →Prove that
Question: Prove that $\mathrm{i}^{53}+\mathrm{i}^{72}+\mathrm{i}^{93}+\mathrm{i}^{102}=2 \mathrm{i}$ Solution: L.H.S $=\mathrm{i}^{53}+\mathrm{i}^{72}+\mathrm{i}^{93}+\mathrm{i}^{102}$ $=i^{4 \times 13+1}+i^{4 \times 18}+i^{4 \times 23+1}+i^{4 \times 25+2}$ Since $i^{4 n}=1$ $\Rightarrow i^{4 n+1}=i$ (where $n$ is any positive integer) $\Rightarrow i^{4 n+2}=-1$ $\Rightarrow i^{4 n+3}=-1$ $=i+1+i+i^{2}$ $=i+1+i-1$ $=2 \mathrm{i}$ L.H.S = R.H.S Hence proved....
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