Evaluate the following integrals:
Question: Evaluate the following integrals: $\int(x+1) e^{x} \log \left(x e^{x}\right) d x$ Solution: Let $I=\int(x+1) e^{x} \log \left(x e^{x}\right) d x$ $\mathrm{Xe}^{\mathrm{x}}=\mathrm{t}$ $\left(1 \times e^{x}+x e^{x}\right) d x=d t$ $(x+1) e^{x} d x=d t$ $I=\int \log t d t$ $=\int 1 \times \log t d t$ Using integration by parts, $=\log \mathrm{t} \int \mathrm{dt}-\int \frac{\mathrm{d}}{\mathrm{dt}} \log \mathrm{t} \int \mathrm{dt}$ $=\mathrm{t} \log \mathrm{t}-\int \frac{1}{\mathrm{t}} \m...
Read More →Prove that
Question: Let $f: R \rightarrow R$ be defined by $f(x)=\left\{\begin{array}{ccc}2 x+3, \text { when } x-2 \\ x^{2}-2, \text { when } -2 \leq x \leq 3 \\ 3 x-1, \text { when } x3\end{array}\right.$ Find (i) $f(2)$ (ii) $f(4)$ (iii) $f(-1)$ (iv) $f(-3)$. Solution: i) $f(2)$ Since $f(x)=x^{2}-2$, when $x=2$ $\therefore f(2)=(2)^{2}-2=4-2=2$ $\therefore f(2)=2$ ii)f(4) Since $f(x)=3 x-1$, when $x=4$ $\therefore f(4)=(3 \times 4)-1=12-1=11$ $\therefore f(4)=11$ iii)f( - 1) Since $f(x)=x^{2}-2$, when ...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int x\left(\frac{\sec 2 x-1}{\sec 2 x+1}\right) d x$ Solution: Let $\mathrm{I}=\int \mathrm{x}\left(\frac{\sec 2 \mathrm{x}-1}{\sec 2 \mathrm{x}+1}\right) \mathrm{dx}$ it can be written $\mathrm{n}$ terms of $\cos \mathrm{x}$ $=\int x\left(\frac{1-\cos 2 x}{1+\cos 2 x}\right) d x$ $=\int x\left(\frac{\sec ^{2} x}{\cos ^{2} x}\right) d x$ $=\int x \tan ^{2} x d x$ $=\int x\left(\sec ^{2} x-1\right) d x$ $=\int x \sec ^{2} x-\int x d x$ Using integrati...
Read More →Electrolytic refining is used to purify
Question: Electrolytic refining is used to purify which of the following metals? (i) Cu and Zn (ii) Ge and Si (iii) Zr and Ti (iv) Zn and Hg Solution: Option (i)Cu and Zn is the answer....
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int x \tan ^{2} x d x$ Solution: Let $I=\int x \tan ^{2} x d x$ $=\int x\left(\sec ^{2} x-1\right) d x$ $=\int x \sec ^{2} x d x-\int x d x$ Using integration by parts, $=x \int \sec ^{2} x d x-\int \frac{d}{d x} x \int \sec ^{2} x d x-\frac{x^{2}}{2}$ We know that, $\int \sec ^{2} x d x=\tan x$ $=x \tan x-\int \tan x d x-\frac{x^{2}}{2}$ $=x \tan x-\log |\sec x|-\frac{x^{2}}{2}+c$...
Read More →Give an example of a function which is
Question: Give an example of a function which is (i) one - one but not onto (ii) one - one and onto (iii) neither one - one nor onto (iv) onto but not one - one. Solution: (i) one - one but not onto $f(x)=6 x$ For One - One $f\left(x_{1}\right)=6 x_{1}$ $f\left(x_{2}\right)=6 x_{2}$ put $f\left(x_{1}\right)=f\left(x_{2}\right)$ we get $6 x_{1}=6 x_{2}$ Hence, if $f\left(x_{1}\right)=f\left(x_{2}\right), x_{1}=x_{2}$ Function $f$ is one - one For Onto $f(x)=6 x$ let $f(x)=y$, such that $y \in N$ ...
Read More →In the metallurgy of aluminium
Question: In the metallurgy of aluminium ________________. (i) Al3+ is oxidised to Al (s). (ii) graphite anode is oxidised to carbon monoxide and carbon dioxide. (iii) the oxidation state of oxygen changes in the reaction at the anode. (iv) the oxidation state of oxygen changes in the overall reaction involved in the process Solution: Option (ii)graphite anode is oxidised to carbon monoxide and carbon dioxideis the answer....
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \sin ^{-1} \sqrt{x} d x$ Solution: Let $I=\sin ^{-1} \sqrt{x} d x$ $\sqrt{\mathrm{X}}=\mathrm{t} ; \mathrm{x}=\mathrm{t}^{2}$ $\mathrm{d} \mathrm{x}=2 \mathrm{tdt}$ $=\sin ^{-1} \mathrm{t} 2 \mathrm{t} \mathrm{dt}$ Using integration by parts, $=\sin ^{-1} t \int 2 t d t-\int \frac{d}{d t} \sin ^{-1} t \int 2 t d t$ We know that, $\frac{\mathrm{d}}{\mathrm{dt}} \sin ^{-1} \mathrm{t}=\frac{\mathrm{t}}{\sqrt{1-\mathrm{t}^{2}}}$ $=\mathrm{t}^{2} \sin...
Read More →Brine is electrolysed by using inert electrodes.
Question: Brine is electrolysed by using inert electrodes. The reaction at anode is ________. (i) Cl(aq.) 1/2Cl2 (g) + e ; Ecell = 1.36V (ii) 2H2O(l) O2(g0 + 4H+ + 4e- ; Ecell = 1.23V (iii) Na+(aq) + e- Na(s) ; Ecell = 2.71V (iv) H+(aq) + e- 1/2H2(g) ; Ecell = 0.00V Solution: Option (i)Cl(aq.) 1/2Cl2 (g) + e ; Ecell = 1.36Vis the answer....
Read More →In the extraction of copper from
Question: In the extraction of copper from its sulphide ore, the metal is formed by the reduction of Cu2O with (i) FeS (ii) CO (iii) Cu2S (iv) SO2 Solution: Option (iii)Cu2Sis the answer....
Read More →Zone refining is based on
Question: Zone refining is based on the principle that ___________. (i) impurities of low boiling metals can be separated by distillation. (ii) impurities are more soluble in molten metal than in solid metal. (iii) different components of a mixture are differently adsorbed on an absorbent. (iv) vapours of the volatile compound can be decomposed in pure metal. Solution: Option (ii) impurities are more soluble in molten metal than in solid metal is the answer....
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \sec ^{-1} \sqrt{x} d x$ Solution: Let $\mathrm{I}=\int \sec ^{-1} \sqrt{\mathrm{x}} \mathrm{dx}$ $\sqrt{x}=t ; x=t^{2}$ $\mathrm{d} \mathrm{x}=2 \mathrm{tdt}$ $I=\int 2 t s e c^{-1} t d t$ Using integration by parts, $=2\left[\sec ^{-1} \mathrm{t} \int \mathrm{tdt}-\int \frac{\mathrm{d}}{\mathrm{dt}} \mathrm{sec}^{-1} \mathrm{t} \int \mathrm{tdt}\right]$ We know that, $\frac{\mathrm{d}}{\mathrm{dt}} \sec ^{-1} \mathrm{t}=\frac{1}{\mathrm{t} \sqr...
Read More →Define each of the following:
Question: Define each of the following: (i) injective function (ii) surjective function (iii) bijective function (iv) many - one function (v) into function Give an example of each type of functions. Solution: 1)injective function Definition: $A$ function $f: A \rightarrow B$ is said to be a one - one function or injective mapping if different elements of $A$ have different $f$ images in $B$. A function $f$ is injective if and only if whenever $f(x)=f(y), x=y$. Example: $f(x)=x+9$ from the set of...
Read More →A number of elements are available in earth’s
Question: A number of elements are available in earths crust but most abundant elements are ____________. (i) Al and Fe (ii) Al and Cu (iii) Fe and Cu (iv) Cu and Ag Solution: Option (i) is the answer....
Read More →Which of the following reactions is an example
Question: Which of the following reactions is an example of autoreduction? (i) Fe3O4 + 4CO 3Fe + 4CO2 (ii) Cu2O + C 2Cu + CO (iii) Cu2+ (aq) + Fe (s) Cu (s) + Fe2+ (aq) (iv) Cu2O + 1/2Cu2S 3Cu + 1/2SO2 Solution: Option (iv) is the answer....
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \operatorname{cosec}^{3} x d x$ Solution: Let $I=\int \operatorname{cosec}^{3} x d x$ $=\int \operatorname{cosec} x \times \operatorname{cosec}^{2} x d x$ Using integration by parts, $=\operatorname{cosec} x \int \operatorname{cosec}^{2} x d x-\int \frac{d}{d x} \operatorname{cosec} x \int \operatorname{cosec}^{2} x d x$ We know that, $\int \operatorname{cosec}^{2} x d x=-\cot x$ and $\frac{d}{d x} \operatorname{cosec} x=\operatorname{cosec} x \c...
Read More →When copper ore is mixed with silica,
Question: When copper ore is mixed with silica, in a reverberatory furnace copper matte is produced. The copper matte contains ____________. (i) sulphides of copper (II) and iron (II) (ii) sulphides of copper (II) and iron (III) (iii) sulphides of copper (I) and iron (II) (iv) sulphides of copper (I) and iron (III) Solution: Option (iii) is the answer....
Read More →In the extraction of chlorine by electrolysis of brine ____________.
Question: In the extraction of chlorine by electrolysis of brine ____________. (i) oxidation of Cl ion to chlorine gas occurs. (ii) reduction of Cl ion to chlorine gas occurs. (iii) For the overall reaction, ∆Gᶱ has a negative value. (iv) a displacement reaction takes place. Solution: Option (iii) is the answer....
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\log x}{(x+1)^{2}} d x$ Solution: We know that integration by parts is given by: $\int \mathrm{UV}=\mathrm{U} \int \mathrm{V} \mathrm{dv}-\int \frac{\mathrm{d}}{\mathrm{dx}} \mathrm{U} \int \mathrm{V} \mathrm{dv}$ Choosing $\log x$ as first function and $\frac{1}{(x+1)^{2}}$ as second function we get, $\int \frac{\log x}{(x+1)^{2}} d x=\log x \int\left(\frac{1}{(x+1)^{2}}\right) d x-\int\left(\frac{d}{d x}(\log x) \int \frac{1}{(x+1)^{2}} d...
Read More →Define a function. What do you mean by the domain and range of a function?
Question: Define a function. What do you mean by the domain and range of a function? Give examples. Solution: Definition: A relation $R$ from a set $A$ to a set $B$ is called a function if each element of $A$ has a unique image in B. It is denoted by the symbol $f: A \rightarrow B$ which reads ' $f$ ' is a function from $A$ to $B$ ' $f$ ' maps $A$ to $B$. Let $\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}$, then the set $\mathrm{A}$ is known as the domain of $\mathrm{f} \$ the set $\mathrm{B}$ i...
Read More →What is the role of activated charcoal
Question: What is the role of activated charcoal in a gas mask used in coal mines? Solution: Activated charcoal acts as a good adsorbent as it is porous. In coal mines, the activated charcoal in a gas mask provides fresh air for inhaling....
Read More →Why does the white precipitate of silver
Question: Why does the white precipitate of silver halide become coloured in the presence of dye eosin? Solution: The surface of silver halide acts as a good adsorbent. It can adsorb the pigments of eosin dye, which is coloured. Thus silver halide appears coloured....
Read More →What happens when dialysis is prolonged?
Question: What happens when dialysis is prolonged? Solution: If the dialysis continues for a long time, the traces of electrolyte present in blood also get completely removed and blood coagulation occurs....
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{x \cos ^{-1} x}{\sqrt{1-x^{2}}} d x$ Solution: Let $I=\int \frac{x \cos ^{-1} x}{\sqrt{1-x^{2}}} d x$ Let $\mathrm{t}=\cos ^{-1} \mathrm{x}$ $\mathrm{dt}=\frac{1}{\sqrt{1-\mathrm{x}^{2}}} \mathrm{dx}$ Also, $\cos t=x$ Thus, $I=-\int t \cos t d t$ Now let us solve this by 'by parts' method Using integration by parts, $I=-t\left(\int \cos t d t-\int \frac{d}{d t} t \int \cos t d t\right)$ Let $\mathrm{U}=\mathrm{t} ; \mathrm{d} \mathrm{u}=\ma...
Read More →Why do physisorption and chemisorption
Question: Why do physisorption and chemisorption behave differently with rising in temperature? Solution: In physisorption, with an increase in temperature, this bond between adsorbent and adsorbate weaker and the amount of adsorbate decrease whereas in chemisorption an amount of activation energy required for the formation of a bond between adsorbent and adsorbate which is achieved by increasing the temperature....
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