The salary of an officer is increased by 25%.
Question: The salary of an officer is increased by 25%. By what per cent should the new salary be decreased to restore the original salary? Solution: Let the original salary be Rs 100 Then, after increment of 25% the salary becomes $=100\left(1+\frac{25}{100}\right)=100\left(\frac{125}{100}\right)=R s 125$ To restore the original salary, let the new salary be decreased by x%. Thus, we get $125\left(1-\frac{x}{100}\right)=100$ $\Rightarrow\left(1-\frac{x}{100}\right)=\frac{100}{125}=\frac{4}{5}$ ...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\tan ^{-1}\left(\frac{a+b x}{b-a x}\right)$ Solution: $y=\tan ^{-1}\left(\frac{a+b x}{b-a x}\right)$ Dividing numerator and denominator by $b$ $y=\tan ^{-1}\left(\frac{\frac{a}{b}+x}{1-\frac{a}{b} x}\right)$ Using, $\tan ^{-1} x+\tan ^{-1} y=\tan ^{-1}\left(\frac{x+y}{1-x y}\right)$ $y=\tan ^{-1} \frac{a}{b}+\tan ^{-1} x$ Differentiating w.r.t $x$ we get $\frac{d y}{d x}=\frac{d}{d x}\left(\tan ^{-1} \frac{a}{b}+\tan ^{-1} x\...
Read More →Solve this
Question: Let $f=\{(0,-5),(1,-2),(3,4),(4,7)\}$ be a linear function from $Z$ into $Z$. Write an expression for f. Solution: Given that: $f=\{(0,-5),(1,-2),(3,4),(4,7)\}$ be a function from $Z$ to $Z$ defined by linear function. We know that, linear functions are of the form y = mx + b Let f(x) = ax + b, for some integers a, b Here, $(0,-5) \in f$ $\Rightarrow \mathrm{f}(0)=-5$ $\Rightarrow \mathrm{a}(0)+\mathrm{b}=-5$ $\Rightarrow \mathrm{b}=-5 \ldots(\mathrm{i})$ Similarly, $(1,-2) \in f$ $\Ri...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\tan ^{-1}\left(\frac{a+b \tan x}{b-a \tan x}\right)$ Solution: $y=\tan ^{-1}\left(\frac{a+b \tan x}{b-a \tan x}\right)$ Dividing numerator and denominator by $b$ $y=\tan ^{-1}\left(\frac{\frac{\mathrm{a}}{\mathrm{b}}+\tan \mathrm{x}}{1-\frac{\mathrm{a}}{\mathrm{b}} \tan \mathrm{x}}\right)$ $y=\tan ^{-1}\left(\frac{\tan \left(\tan ^{-1} \frac{\mathrm{a}}{\mathrm{b}}\right)+\tan \mathrm{x}}{1-\tan \left(\tan ^{-1} \frac{\mathr...
Read More →Find the percentage of pure gold in 22-carat gold,
Question: Find the percentage of pure gold in 22-carat gold, if 24-carat gold is 100% pure. Solution: 22 carat gold contains 22 parts pure gold out of 24 parts. Also, 24 carat gold is given to be $100 \%$ pure. $\therefore$ Percentage of pure gold in 22 carat gold $=\left(\frac{22}{24} \times 100\right) \%$ $=91 \frac{2}{3} \%$ Hence, 22 carat gold contains $91 \frac{2}{3} \%$ of pure gold....
Read More →Of the three metals X, Y and Z, X reacts with cold water,
Question: Of the three metals X, Y and Z, X reacts with cold water, Y with hot water and Z with steam only. Identify X, Y and Z and also arrange them in order of increasing reactivity. Solution: The answer is based on the relative positions of the metals in the reactivity series. The reactivity with water decreases down the series. Metal X is Na or K Metal Y is Mg or Ca Metal Z is Al or Fe Order of increasing reactivity : Z Y X....
Read More →Divide Rs 7000 among A, B and C such that A gets 50%
Question: Divide Rs 7000 among A, B and C such that A gets 50% of what B gets and B gets 50% of what C gets. Solution: Let Rs $x$ be the amount of money recieved by C. Then, amount of money B gets $=(50 \%$ of Rs $x)$ Amount of money A gets $=(50 \%$ of B $)$ $=(25 \%$ of Rs $x)$ Now, $x+(50 \%$ of Rs $x)+(25 \%$ of Rs $x)=$ Rs 7000 $\Rightarrow x+\left(x \times \frac{50}{100}\right)+\left(x \times \frac{25}{100}\right)=\mathrm{Rs} 7000$ $\Rightarrow x+\frac{50 x}{100}+\frac{25 x}{100}=\mathrm{R...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\tan ^{-1}\left(\frac{\sqrt{x}+\sqrt{a}}{1-\sqrt{x a}}\right)$ Solution: $y=\tan ^{-1}\left(\frac{\sqrt{x}+\sqrt{a}}{1-\sqrt{x a}}\right)$ Using, $\tan ^{-1} x+\tan ^{-1} y=\tan ^{-1}\left(\frac{x+y}{1-x y}\right)$ $y=\tan ^{-1} \sqrt{x}+\tan ^{-1} \sqrt{a}$ Differentiating w.r.t $x$ we get $\frac{d y}{d x}=\frac{d}{d x}\left(\tan ^{-1} \sqrt{x}+\tan ^{-1} \sqrt{a}\right)$ $\frac{d y}{d x}=\frac{1}{1+(\sqrt{x})^{2}} \frac{d}{...
Read More →(i) Given below are the steps lor extraction of copper from its ore.
Question: (i) Given below are the steps lor extraction of copper from its ore. Write the reaction involved. (a) Roasting of copper (1) sulphide (b) Reduction of copper (1) oxide with copper (1) sulphide. (c) Electrolytic refining (ii) Draw a neat and well labelled diagram for electrolytic refining of copper. Solution: This reaction in which one of the reactants (Cu2S) carries the reduction of the product (Cu2O) is known as auto-reduction. (c) Reactions taking place in electrorefensing are : (ii)...
Read More →Let g = {(1, 2), (2, 5), (3, 8), (4, 10), (5, 12), (6, 12)}
Question: Let g = {(1, 2), (2, 5), (3, 8), (4, 10), (5, 12), (6, 12)}. Is g a function? If yes, its domain range. If no, give reason. Solution: Given: g = {(1, 2), (2, 5), (3, 8), (4, 10), (5, 12), (6, 12)} We know that A function f from set A to set B is a correspondence (rule) which associates elements of set A to elements of set B such that: (i) all elements of set $A$ are associated with an element in set $B$. (ii) an element of set $A$ is associated with a unique element in set $B$ Here, we...
Read More →Explain the following
Question: Explain the following (a) Reactivity of Al decreases if it is dipped in cone. HNO3 (b) Carbon cannot reduce the oxides of Na or Mg. (c) NaCl is not a conductor of electricity in solid state whereas it does conduct electricity in aqueous solution as in molten state (d) Iron articles are galvanised. (e) Metals like Na, K, Ca and Mg are never found in their free state in nature. Solution: (a) When Al metal is dipped in cone. HNO3for sometime, it is oxidised initially to aluminium oxide (A...
Read More →Gunpowder contains 75% nitre and 10% sulphur.
Question: Gunpowder contains 75% nitre and 10% sulphur. Find the amount of gunpowder which carries 9 kg nitre. What amount of gunpowder would contain 2.5 kg sulphur? Solution: Let $x$ be the amount of gunpowder. Amount of nitre $=75 \%$ Let $x \mathrm{~kg}$ be the amount of gunpowder containing $9 \mathrm{~kg}$ of nitre. i.e., $(75 \%$ of $\mathrm{x})=9 \mathrm{~kg}$ $\Rightarrow\left(x \times \frac{75}{100}\right)=9$ $\Rightarrow \frac{75 x}{100}=9$ $\Rightarrow x=\left(9 \times \frac{100}{75}\...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\tan ^{-1}\left(\frac{a+x}{1-a x}\right)$ Solution: $y=\tan ^{-1}\left(\frac{a+x}{1-a x}\right)$ Using, $\tan ^{-1} x+\tan ^{-1} y=\tan ^{-1}\left(\frac{x+y}{1-x y}\right)$ $y=\tan ^{-1} x+\tan ^{-1} a$ Differentiating w.r.t $\mathrm{x}$ we get $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\tan ^{-1} \mathrm{x}+\tan ^{-1} \mathrm{a}\right)$ $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{1+\mathrm{x}^{2}}+0$ ...
Read More →Give the steps involved in the extraction
Question: Give the steps involved in the extraction of metals of low and medium reactivity from their respective sulphide ores. Solution: Extraction of Metals present low in the Activity series Silver (Ag), gold (Au) and platinum (Pt) generally occur in the free or native state. This means that they can be isolated rather easily. Metals like copper (Cu) and mercury (Hg) are comparitively more reactive and occur in combined states. For example, the ore of mercury is cinnabar (HgS) while that of c...
Read More →Balanced diet should contain 12% of proteins, 25% of fats and 63% of carbohydrates.
Question: Balanced diet should contain 12% of proteins, 25% of fats and 63% of carbohydrates. If a child needs 2600 calories in his food daily, find in calories the amount of each of these in his daily food intake. Solution: Amount of protein $=12 \%$ of 2600 $=\left(2600 \times \frac{12}{100}\right)$ $=312 \mathrm{cal}$ Amount of fat $=25 \%$ of 2600 $=\left(2600 \times \frac{25}{100}\right)$ $=650 \mathrm{cal}$ Amount of carbohydrate $=63 \%$ of 2600 $=\left(2600 \times \frac{63}{100}\right)$ ...
Read More →Let A = {2, 3, 5, 7} and B = {3, 5, 9, 13, 15}.
Question: Let $A=\{2,3,5,7\}$ and $B=\{3,5,9,13,15\} .$ Let $f=\{(x, y): x \in A, y \in B$ and $y=$2 x-1\}$ Write f in roster form. Show that f is a function from A to B. Find the domain and range of f. Solution: Given: A = {2, 3, 5, 7} and B = {3, 5, 9, 13, 15} $f=\{(x, y): x \in A, y \in B$ and $y=2 x-1\}$ For $x=2$ $y=2 x-1$ $y=2(2)-1$ $y=3 \in B$ For $x=3$ $y=2 x-1$ $y=2(3)-1$ $y=5 \in B$ For $x=5$ $y=2 x-1$ $y=2(5)-1$ $y=9 \in B$ For $x=7$ $y=2 x-1$ $y=2(7)-1$ $\mathrm{y}=13 \in \mathrm{B}$...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\sin ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)+\sec ^{-1}\left(\frac{1+x^{2}}{1-x^{2}}\right), x \in R$ Solution: $y=\sin ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)+\sec ^{-1}\left(\frac{1+x^{2}}{1-x^{2}}\right)$ Using, $\sec ^{-1} x=\frac{1}{\cos ^{-1} x}$ $y=\sin ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)+\cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right)$ Using, $\cos ^{-1} x+\sin ^{-1} x=\frac{\pi}{2}$ $y=\frac{\pi}{2}$ Differe...
Read More →An alloy contains 40% copper, 32% nickel and rest zinc.
Question: An alloy contains 40% copper, 32% nickel and rest zinc. Find the mass of zinc in one kg of the alloy. Solution: Mass of the alloy $=1 \mathrm{~kg}$ Percentage of copper $=40 \%$ Percentage of nickel $=32 \%$ Percentage of zinc $=\{100-(40+32)\} \%$ = 28% $\therefore$ Mass of zinc in $1 \mathrm{~kg}$ of alloy $=\left(\frac{28}{100} \times 1\right) \mathrm{kg}$ $=0.28 \mathrm{~kg}=0.28 \times 1000 \mathrm{~g}=280 \mathrm{~g}$...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\cos ^{-1}\left(\frac{1-x^{2 n}}{1+x^{2 n}}\right), 0x\infty$ Solution: $y=\cos ^{-1}\left\{\frac{1-x^{2 n}}{1+x^{2 n}}\right\}$ Let $x^{n}=\tan \theta$ Now $y=\cos ^{-1}\left\{\frac{1-\tan ^{2} \theta}{1+\tan ^{2} \theta}\right\}$ Using $\frac{1-\tan ^{2} \theta}{1+\tan ^{2} \theta}=\cos 2 \theta$ $y=\cos ^{-1}\{\cos 2 \theta\}$ Considering the limits, $0x\infty$ $0x^{n}\infty$ $0\theta\frac{\pi}{2}$ Now, $y=\cos ^{-1}(\cos ...
Read More →A non-metal A which is the largest constituent of air,
Question: A non-metal A which is the largest constituent of air, when heated with H2in 1 : 3 ratio in the presence of catalyst (Fe) gives a gas B. On heating with O2it gives an oxide C. If this oxide is passed into water in the presence of air, it gives an acid D which acts as a strong oxidising agent. (a) Identify A, B, C and D. (b) To which group of periodic table does this non-metal belong ? Solution: (a) The available information suggests that the non-metal A is nitrogen (N) and its molecula...
Read More →The value of a machine depreciates every year by 20%.
Question: The value of a machine depreciates every year by 20%. If the present value of the machine be Rs 160000, what was its value last year? Solution: Let Rs $x$ be the value of the machine last year. Then, present value $=80 \%$ of $\operatorname{Rs} x$ $=\operatorname{Rs}\left(x \times \frac{80}{100}\right)$ $=\operatorname{Rs} \frac{4 x}{5}$ Now, $\frac{4 \mathrm{x}}{5}=160000$ $\Rightarrow \mathrm{x}=\left(160000 \times \frac{5}{4}\right)$ $\Rightarrow \mathrm{x}=40000 \times 5=200000$ He...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $\sin ^{-1}\left(\frac{1}{\sqrt{1+x^{2}}}\right)$ Solution: $y=\sin ^{-1}\left\{\frac{1}{\sqrt{1+x^{2}}}\right\}$ Let $x=\cot \theta$ Now $y=\sin ^{-1}\left\{\frac{1}{\sqrt{1+\cot ^{2} \theta}}\right\}$ Using, $1+\cot ^{2} \theta=\operatorname{cosec}^{2} \theta$ Now $y=\sin ^{-1}\left\{\frac{1}{\sqrt{\operatorname{cosec}^{2} \theta}}\right\}$ $y=\sin ^{-1}\left\{\frac{1}{\operatorname{cosec} \theta}\right\}$ $y=\sin ^{-1}(\sin...
Read More →Solve this
Question: Let $A=\{0,1,2\}$ and $B=\{3,5,7,9\} .$ Let $f=\{(x, y): x \in A, y \in B$ and $y=2 x+3\} .$ Write f as a set of ordered pairs. Show that f is function from A to B. Find dom (f) and range (f). Solution: Given: $A=\{0,1,2\}$ and $B=\{3,5,7,9\}$ $f=\{(x, y): x \in A, y \in B$ and $y=2 x+3\}$ For $x=0$ $y=2 x+3$ $y=2(0)+3$ $y=3 \in B$ For $x=1$ $y=2 x+3$ $y=2(1)+3$ $y=5 \in B$ For $x=2$ $y=2 x+3$ $y=2(2)+3$ $y=7 \in B$ $\therefore f=\{(0,3),(1,5),(2,7)\}$ (0, 5), (0, 7), (0, 9), (1, 3), (...
Read More →A solution of CuSO4 was kept in an iron pot.
Question: A solution of CuSO4was kept in an iron pot. After few days, the iron pot was found to have a number of holes in it. Explain the reason in terms of reactivity. Write the equation of the reaction involved. Solution: Iron (Fe) is placed above copper (Cu) in the reactivity series. This means that a chemical reaction had occurred between iron (iron pot) and aqueous CuSO4solution. $\mathrm{Fe}(s)+\mathrm{CuSO}_{4}(a q) \longrightarrow \mathrm{FeSO}_{4}(a q)+\mathrm{Cu}(s)$ Since iron was con...
Read More →The population of a town increases by 8% annually.
Question: The population of a town increases by 8% annually. If the present population is 54000, what was it a year ago? Solution: Let $x$ be the population of the town a year ago. Then, present population $=108 \%$ of $x$ $=\left(x \times \frac{108}{100}\right)$ $=\frac{27 x}{25}$ Now, $\frac{27 x}{25}=54000$ $\Rightarrow x=\left(54000 \times \frac{25}{27}\right)$ $\Rightarrow x=50000$ Hence, the population of the town a year ago was 50000 ....
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