Show that the function
Question: Show that the function $f: R \rightarrow R: f(x)=\left\{\begin{array}{l}1, \text { if } x \text { is rational } \\ -1, \text { if } x \text { is irrational }\end{array}\right.$ is many - one into. Find (i) $f\left(\frac{1}{2}\right)$ (ii) $f(\sqrt{2})$ (iii) $f(\pi)$ (iv) $f(2+\sqrt{3})$. Solution: (i) $\mathrm{f}\left(\frac{1}{2}\right)$ Here, $x=1 / 2$, which is rational $\therefore f(1 / 2)=1$ (ii) $\mathrm{f}(\sqrt{2})$ Here, $x=\sqrt{2}$, which is irrational $\therefore f(\sqrt{2}...
Read More →In a cyclotrimetaphosphoric acid molecule,
Question: In a cyclotrimetaphosphoric acid molecule, how many single and double bonds are present? (i) 3 double bonds; 9 single bonds (ii) 6 double bonds; 6 single bonds (iii) 3 double bonds; 12 single bonds (iv) Zero double bonds; 12 single bonds Solution: Option (iii)3 double bonds; 12 single bonds is the answer....
Read More →In qualitative analysis when H2S is passed
Question: In qualitative analysis when H2S is passed through an aqueous solution of salt acidified with dil. HCl, a black precipitate is obtained. On boiling the precipitate with dil. HNO3, it forms a solution of blue colour. Addition of excess of aqueous solution of ammonia to this solution gives _________. (i) a deep blue precipitate of Cu (OH)2 (ii) a deep blue solution of [Cu (NH3)4]2+ (iii) a deep blue solution of Cu(NO3)2 (iv) a deep blue solution of Cu(OH)2.Cu(NO3)2 Solution: Option (ii)a...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int x \sin x \cos 2 x d x$ Solution: Let $I=\int x \sin x \cos 2 x d x=\frac{1}{2} \int x \times 2 \sin x \cos 2 x d x$ Using integration by parts, $=\frac{1}{2} \int x(\sin (x+2 x)-\sin (2 x-x)) d x$ $=\frac{1}{2} \int x(\sin 3 x-\sin x) d x$ Using integration by parts, $=\frac{1}{2}\left(x \int(\sin 3 x-\sin x) d x-\int \frac{d}{d x} x \int(\sin 3 x-\sin x) d x\right) d x$ $=\frac{1}{2}\left[x\left(\frac{-\cos 3 x}{3}+\cos x\right)-\int-\left(\frac...
Read More →On addition of conc. H2SO4 to a chloride salt,
Question: On addition of conc. H2SO4 to a chloride salt, colourless fumes are evolved but in case of an iodide salt, violet fumes come out. This is because (i) H2SO4 reduces HI to I2 (ii) HI is of violet colour (iii) HI gets oxidised to I2 (iv) HI changes to HIO3 Solution: Option (iii)HI gets oxidised to I2is the answer....
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int x^{3} \tan ^{-1} x d x$ Solution: Let $I=\int x^{3} \tan ^{-1} x d x$ Using integration by parts, We know that, $\frac{d}{d x} \tan ^{-1} x=\frac{1}{2\left(1+x^{2}\right)}$ $=\tan ^{-1} \mathrm{x} \int \mathrm{x}^{3} \mathrm{dx}-\int\left(\frac{1}{1+\mathrm{x}^{2}}\right) \int \mathrm{x}^{3} \mathrm{dx}$ $=\tan ^{-1} x \frac{x^{4}}{4}-\frac{1}{4} \int \frac{x^{4}}{1+x^{2}} d x$ $\frac{1}{4} \int \frac{\mathrm{x}^{4}}{1+\mathrm{x}^{2}} \mathrm{dx}...
Read More →Find the domain and range of the real function, defined
Question: Find the domain and range of the real function, defined by $f(x)=\frac{x^{2}}{\left(1+x^{2}\right)}$ Show that f is many - one. Solution: For domain $\left(1+x^{2}\right) \neq 0$ $\Rightarrow x^{2} \neq-1$ $\Rightarrow \mathrm{dom}(\mathrm{f})=\mathrm{R}$ For the range of $x$ : $\Rightarrow y=\frac{x^{2}+1-1}{x^{2}+1}=1-\frac{1}{x^{2}+1}$ $y_{\min }=0($ when $x=0)$ $y_{\max }=1(w h e n x=\infty)$ $\therefore$ range of $f(x)=[0,1)$ For many one the lines cut the curve in 2 equal valued ...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \tan ^{-1}(\sqrt{x}) d x$ Solution: Let $I=\int \tan ^{-1}(\sqrt{x}) d x$ $x=t^{2}$ $\mathrm{d} x=2 \mathrm{tdt}$ $I=\int 2 t \tan ^{-1} t d t$ Using integration by parts, $=2\left(\tan ^{-1} \mathrm{t} \int \mathrm{tdt}-\int \frac{\mathrm{d}}{\mathrm{dt}} \tan ^{-1} \mathrm{t} \int \mathrm{t} \mathrm{dt}\right)$ We know that, $\frac{d}{d t} \tan ^{-1} t=\frac{1}{2\left(1+t^{2}\right)}$ $=2\left[\frac{\mathrm{t}^{2}}{2} \tan ^{-1} \mathrm{t}-\int...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\left(x \tan ^{-1} x\right)}{\left(1+x^{2}\right)^{3 / 2}} d x$ Solution: Let $I=\int \frac{x \tan ^{-1} x}{\left(1+x^{2}\right)^{\frac{3}{2}}} d x$ $\tan ^{-1} \mathrm{x}=\mathrm{t}$ $\frac{1}{1+x^{2}} d x=d t$ $I=\int \frac{t \tan t}{\sqrt{1+\tan ^{2} t}} d t$ We know that, $\sqrt{1+\tan ^{2} t}=\sec t$ $=\int \frac{\mathrm{t} \tan t}{\sec t} \mathrm{dt}$ $=\int \mathrm{t} \frac{\sin t}{\cos t} \cos t \mathrm{dt}$ $=\int \mathrm{t} \sin \...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int\left(e^{\log x}+\sin x\right) \cos x d x$ Solution: Let $\mathrm{I}=\int\left(\mathrm{e}^{\log \mathrm{x}}+\sin \mathrm{x}\right) \cos \mathrm{x} \mathrm{dx}$ $=\int(x+\sin x) \cos x d x$ $=\int x \cos x d x+\int \sin x \cos x d x$ Using integration by parts, $=x \int \cos x d x-\int \frac{d}{d x} x \int \cos x d x+\frac{1}{2} \int \sin 2 x d x$ $=x \times \sin x-\int \sin x d x+\frac{1}{2}\left(\frac{-\cos 2 x}{2}\right)+c$ $=x \sin x+\cos x-\fr...
Read More →Assertion: Hydrometallurgy involves dissolving
Question: Assertion: Hydrometallurgy involves dissolving the ore in a suitable reagent followed by precipitation by a more electropositive metal. Reason: Copper is extracted by hydrometallurgy. (i) Both assertion and reason are true and the reason is the correct explanation of assertion. (ii) Both assertion and reason are true but the reason is not the correct explanation of assertion. (iii) The assertion is true but the reason is false. (iv) The assertion is false but the reason is true. (v) As...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int x^{2} \tan ^{-1} x d x$ Solution: Let $\mathrm{I}=\int \mathrm{x}^{2} \tan ^{-1} \mathrm{x} \mathrm{dx}$ Using integration by parts, Taking inverse function as first function and algebraic function as second function, $=\tan ^{-1} x \int x^{2} d x-\int\left(\frac{1}{1+x^{2}}\right) \int x^{2} d x$ $=\tan ^{-1} x \frac{x^{3}}{3}-\frac{1}{3} \int \frac{x^{3}}{1+x^{2}} d x$ $=\tan ^{-1} x \frac{x^{3}}{3}-\frac{1}{3} \int x-\frac{x}{1+x^{2}} d x$ $=\...
Read More →Assertion: Zone refining method is very usefu
Question: Assertion: Zone refining method is very useful for producing semiconductors. Reason: Semiconductors are of high purity. (i) Both assertion and reason are true and the reason is the correct explanation of assertion. (ii) Both assertion and reason are true but the reason is not the correct explanation of assertion. (iii) The assertion is true but the reason is false. (iv) The assertion is false but the reason is true. (v) Assertion and reason both are wrong. Solution: Option (ii) is corr...
Read More →Assertion: Sulphide ores are concentrated
Question: Assertion: Sulphide ores are concentrated by Froth Flotation method. Reason: Cresols stabilise the froth in the Froth Flotation Method. (i) Both assertion and reason are true and the reason is the correct explanation of assertion. (ii) Both assertion and reason are true but the reason is not the correct explanation of assertion. (iii) The assertion is true but the reason is false. (iv) The assertion is false but the reason is true. (v) Assertion and reason both are wrong. Solution: Opt...
Read More →Assertion: Zirconium can be purified by Van Arkel method.
Question: Assertion: Zirconium can be purified by Van Arkel method. Reason: ZrI4 is volatile and decomposes at 1800K. (i) Both assertion and reason are true and the reason is the correct explanation of assertion. (ii) Both assertion and reason are true but the reason is not the correct explanation of assertion. (iii) The assertion is true but the reason is false. (iv) The assertion is false but the reason is true. (v) Assertion and reason both are wrong. Solution: Option (i) is correct....
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int(x+1) \log x d x$ Solution: Let $I=\int(x+1) \log x d x$ Using integration by parts, $=\log x \int(x+1) d x-\int \frac{d}{d x} \log x \int(x+1) d x$ We know that, $\frac{\mathrm{d}}{\mathrm{dx}} \log \mathrm{x}=\frac{1}{\mathrm{x}}$ $=\log x\left(\frac{x^{2}}{2}+x\right)-\int \frac{1}{x}\left(\frac{x^{2}}{2}+x\right) d x$ $=\left(\frac{x^{2}}{2}+x\right) \log x-\int \frac{x}{2} d x-\int d x$ $=\left(\frac{x^{2}}{2}+x\right) \log x-\frac{x^{2}}{4}-...
Read More →Assertion: Nickel can be purified
Question: Assertion: Nickel can be purified by the Mond process. Reason: Ni (CO)4 is a volatile compound which decomposes at 460K to give pure Ni. (i) Both assertion and reason are true and the reason is the correct explanation of assertion. (ii) Both assertion and reason are true but the reason is not the correct explanation of assertion. (iii) The assertion is true but the reason is false. (iv) The assertion is false but the reason is true. (v) Assertion and reason both are wrong. Solution: Op...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right) d x$ Solution: Let $I=\int \tan ^{-1}\left(\frac{2 x}{1-x^{2}}\right) d x$ $\mathrm{x}=\tan \theta \Rightarrow \mathrm{dx}=\sec ^{2} \theta \mathrm{d} \theta$ $I=\int \tan ^{-1}\left(\frac{2 \tan \theta}{1-2 \tan \theta^{2}}\right) \sec ^{2} \theta d \theta$ We know that, $\frac{2 \tan \theta}{1-2 \tan \theta^{2}}=\tan 2 \theta$ $=\int \tan ^{-1}(\tan 2 \theta) \sec ^{2} \theta d \theta$ $\int 2 \theta \...
Read More →Which of the following relations are functions? Give reasons
Question: Which of the following relations are functions? Give reasons. In case of a function, find its domain and range (i) $f=\{(-1,2),(1,8),(2,11),(3,14)\}$ (ii) $g=\{(1,1),(1,-1),(4,2),(9,3),(16,4)\}$ (iii) $h=\{(a, b),(b, c),(c, b),(d, c)\}$ Solution: For a relation to be a function each element of $1^{\text {st }}$ set should have different image in the second set(Range) i) (i) $f=\{(-1,2),(1,8),(2,11),(3,14)\}$ Here, each of the first set element has different image in second set. $\there...
Read More →Match the items of Column I with
Question: Match the items of Column I with items of Column II and assign the correct code : Code : (i) A (2) B (3) C (4) D (1) (ii) A (1) B (2) C (3) D (5) (iii) A (5) B (4) C (3) D (2) (iv) A (4) B (5) C (3) D (2) Solution: Option (i)A (2) B (3) C (4) D (1) is the answer....
Read More →Match the items of Column I with
Question: Match the items of Column I with the items of Column II and assign the correct code : Code : (i) A (3) B (4) C (2) D (1) (ii) A (5) B (4) C (3) D (2) (iii) A (2) B (3) C (4) D (5) (iv) A (1) B (2) C (3) D (4) Solution: Option (i)A (3) B (4) C (2) D (1) is the answer....
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right) d x$ Solution: Let $I=\int \cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right) d x$ $\mathrm{d} \mathrm{x}=\sec ^{2} \mathrm{t} \mathrm{dt}$ $d x=\sec ^{2} t d t$ $I=\int \cos ^{-1}\left(\frac{1-\tan ^{2} t}{1+\tan ^{2} t}\right) \sec ^{2} t d t$ We know that $\frac{1-\tan ^{2} t}{1+\tan ^{2} t}=\cos 2 t$ $=\int \cos ^{-1}(\cos 2 t) \sec ^{2} t d t$ $=\int 2 t \sec ^{2} t d t$ Using integration by parts, ...
Read More →Match items of Column I with the items
Question: Match items of Column I with the items of Column II and assign the correct code : Code : (i) A (4) B (2) C (3) D (1) (ii) A (2) B (3) C (1) D (5) (iii) A (1) B (2) C (3) D (4) (iv) A (3) B (4) C (5) D (1) Solution: Option (i)A (4) B (2) C (3) D (1)is the answer....
Read More →Find the domain and range of the function
Question: Find the domain and range of the function $F: R \rightarrow R: f(x)=x^{2}+1$ Solution: Since the function $f(x)$ can accept any values as per the given domain $R$, therefore, the domain of the function $f(x)=x^{2}+1$ is $R$. The minimum value of $f(x)=1$ $\Rightarrow$ Range of $f(x)=[-1, \infty]$ i.e range $(f)=\{y \in R: y \geq 1\}$ Ans: dom $(f)=R$ and range $(f)=\{y \in R: y \geq 1\}$...
Read More →Match the items of Column I with
Question: Match the items of Column I with the items of Column II and assign the correct code : Code : (i) A (1) B (2) C (4) D (5) (ii) A (4) B (3) C (1) D (2) (iii) A (3) B (4) C (2) D (1) (iv) A (5) B (4) C (3) D (2) Solution: Option (ii) is the answer....
Read More →