Evaluate the following integrals -
Question: Evaluate the following integrals - $\int x \sqrt{x^{2}+x} d x$ Solution: Let $I=\int x \sqrt{x^{2}+x} d x$ Let us assume $x=\lambda \frac{d}{d x}\left(x^{2}+x\right)+\mu$ $\Rightarrow x=\lambda\left[\frac{d}{d x}\left(x^{2}\right)+\frac{d}{d x}(x)\right]+\mu$ We know $\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{\mathrm{n}}\right)=\mathrm{nx}^{\mathrm{n}-1}$ $\Rightarrow x=\lambda\left(2 x^{2-1}+1\right)+\mu$ $\Rightarrow x=\lambda(2 x+1)+\mu$ $\Rightarrow x=2 \lambda x+\lambda+\mu$...
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Question: Mark () against the correct answer in the following: Let $f(x)=e^{\sqrt{x^{2}-1}} \cdot \log (x-1) .$ Then, dom $(f)=?$ A. $(-\infty, 1]$ B. $[-1, \infty)$ C. $(1, \infty)$ D. $(-\infty,-1] \cup(1, \infty)$ Solution: $\mathrm{f}(\mathrm{x})=\mathrm{e}^{\sqrt{\mathrm{x}^{2}-1}} \log (\mathrm{x}-1)$ $x-10$ $\Rightarrow x1$ And $\Rightarrow \mathrm{x}^{2}-1 \geq 0$ $\Rightarrow \mathrm{x}^{2} \geq 1$ $\Rightarrow-1 \leq \mathrm{x} \geq 1$ Taking the intersection we get $\operatorname{Dom}...
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Question: Mark () against the correct answer in the following: Let $f(x) \sqrt{\frac{x-1}{x+4}}$. Then, dom (f) - ? A. $[1,4)$ B. $[1,4]$ C. $(-\infty, 4]$ D. $(-\infty, 1] \cup(4, \infty)$ Solution: $f(x)=\sqrt{\frac{x-1}{x-4}}$ $\sqrt{\frac{x-1}{x-4}} \geq 0$ $\Rightarrow x-1 \geq 0$ $\Rightarrow x \geq 1$ And $x \neq 4$ $x4$ and $x \leq 1$ $\Rightarrow \operatorname{dom}(f)=(-\infty, 1] \cup(4, \infty)$...
Read More →Addition of water to alkynes occurs in acidic medium
Question: Addition of water to alkynes occurs in acidic medium and the presence of Hg2+ ions as a catalyst. Which of the following products will be formed on addition of water to but-1-one under these conditions. Solution: Option (ii) is the answer....
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Question: Mark () against the correct answer in the following: Let $f(x)=\sqrt{9-x^{2}}$. Then, dom $(f)=?$ A. $[-3,3]$ B. $[-\infty,-3]$ C. $[3, \infty)$ D. $(-\infty,-3] \cup(4, \infty)$ Solution: $F(x)=\sqrt{9-x^{2}}$ $\sqrt{9-\mathrm{x}^{2}}$ should be $\geq 0$ $\Rightarrow 9-x^{2} \geq 0$ $\Rightarrow x^{2} \leq 9$ $\Rightarrow-3 \leq x \leq 3$ $\therefore \operatorname{dom}(f)=[-3,3]$...
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Question: Mark () against the correct answer in the following: If $f=\{(1,2),(3,5),(4,1)\}$ and $g=\{(2,3),(5,1),(1,3)\}$ then $(g \circ f)=?$ A. $\{(3,1),(1,3),(3,4)\}$ B. $\{(1,3),(3,1),(4,3)\}$ C. $\{(3,4),(4,3),(1,3)\}$ D. $\{(2,5),(5,2),(1,5)\}$ Solution: $g=\{(2,3),(5,1),(1,3)\}$ $(g \circ f)=\{(\operatorname{dom}(f), 3),(\operatorname{dom}(f), 1),(\operatorname{dom}(f), 3)\}$ $\Rightarrow(g \circ f)=\{(1,3),(3,1),(4,3)\}$...
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Question: Mark () against the correct answer in the following: If $f(x)=x^{2}, g(x)=\tan x$ and $h(x)=\log x$ then $\{h \circ(g \circ f)\}\left(\sqrt{\frac{\pi}{4}}\right)=?$ A. 0 B. 1 C. $\frac{1}{\mathrm{x}}$ D. $\frac{1}{2} \log \frac{\pi}{4}$ Solution: $f(x)=x^{2}, g(x)=\tan x$ and $h(x)=\log x$ $\Rightarrow \mathrm{g}(\mathrm{f}(\mathrm{x}))=\tan (\mathrm{f}(\mathrm{x}))=\tan \left(\mathrm{x}^{2}\right)$ $\Rightarrow \mathrm{h}(\mathrm{g}(\mathrm{f}(\mathrm{x})))=\log (\mathrm{g}(\mathrm{f}...
Read More →Evaluate the following integrals -
Question: Evaluate the following integrals - $\int(2 x-4) \sqrt{x^{2}-4 x+3} d x$ Solution: Let $I=\int(2 x-5) \sqrt{x^{2}-4 x+3} d x$ Let us assume $2 \mathrm{x}-5=\lambda \frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{2}-4 \mathrm{x}+3\right)+\mu$ $\Rightarrow 2 x-5=\lambda\left[\frac{d}{d x}\left(x^{2}\right)-\frac{d}{d x}(4 x)+\frac{d}{d x}(3)\right]+\mu$ $\Rightarrow 2 x-5=\lambda\left[\frac{d}{d x}\left(x^{2}\right)-4 \frac{d}{d x}(x)+\frac{d}{d x}(3)\right]+\mu$ We know $\frac{\mathrm{d}...
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Question: Mark () against the correct answer in the following: If $f(x)=8 x^{3}$ and $g(x)=x^{1 / 3}$ then $(g \circ f)(x)=?$ A. $\mathrm{X}$ B. $2 x$ C. $\frac{\mathrm{x}}{2}$ D. $3 x^{2}$ Solution: $f(x)=8 x^{3}$ $g(x)=x^{1 / 3}$ $\Rightarrow(g \circ f)(x)=(f(x))^{\frac{1}{3}}=\left(8 x^{3}\right)^{\frac{1}{3}}=2 x$...
Read More →Arrange water, ethanol and phenol in increasing
Question: Arrange water, ethanol and phenol in increasing order of acidity and give a reason for your answer. Solution: Increasing order of acidity is: Ethanol water phenol Phenol is more acidic because it forms phenoxide ion after deprotonation and gets stabilized by resonance. Ethanol is less acidic because the ethoxide ion is stabilized by the positive inductive effect. The electron releasing group in ethanol increases the density on oxygen and deprotonation gets difficult to occur. Water is ...
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Question: Mark () against the correct answer in the following: If $f(x)=x^{2}-3 x+2$ then (f of) $(x)=?$ A. $x^{4}$ B. $x^{4}-6 x^{3}$ C. $x^{4}-6 x^{3}+10 x^{2}$ D. None of these Solution: $f(x)=x^{2}-3 x+2$ $\Rightarrow f(x)=x^{2}-2 x-x+2=x(x-2)-1(x-2)$ $\Rightarrow f(x)=(x-2)(x-1)$ $\Rightarrow f(x)=(x-2)(x-1)$ $\Rightarrow f(f(x))=(f(x)-2)(f(x)-1)$ $\Rightarrow f(f(x))=((x-2)(x-1)-2)((x-2)(x-1)-1)$ $\Rightarrow f(f(x))=\left(x^{2}-3 x+2-2\right)\left(x^{2}-3 x+2-1\right)$ $\Rightarrow f(f(x)...
Read More →The carbon-oxygen bond in phenol
Question: The carbon-oxygen bond in phenol is slightly stronger than that in methanol. Why? Solution: (i) Due to the resonance, it develops a partial double bond character in carbon-oxygen bond. Thus, the decrease in the size of the carbon-oxygen bond in phenol. (ii) In phenol, Oxygen is directly attached to a sp2hybridized carbon atom, whereas in methanol, Oxygen is directly attached to an sp3 hybridized carbon atom. Thus, the bond formed between oxygen and sp2 hybridized carbon atom is slightl...
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Question: Mark () against the correct answer in the following: If $f(x)=\sqrt[3]{3-x^{3}}$ then ( $f$ of) $(x)=?$ A $\mathrm{X}^{1 / 3}$ B. $x$ C. $\left(1-\mathrm{x}^{1 / 3}\right)$ D. None of these Solution: $f(x)=\sqrt[3]{3-x^{3}}$...
Read More →Explain why alcohols and ethers of comparable
Question: Explain why alcohols and ethers of comparable molecular mass have different boiling points? Solution: Alcohols and ethers of comparable molecular mass have different boiling points because of the presence of intermolecular hydrogen bonding in alcohols....
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Question: Mark () against the correct answer in the following: If $f(x)=\sqrt[3]{3-x^{3}}$ then ( $f$ of) $(x)=?$ A $\mathrm{X}^{1 / 3}$ B. $x$ C. $\left(1-\mathrm{x}^{1 / 3}\right)$ D. None of these Solution: $f(x)=\sqrt[3]{3-x^{3}}$...
Read More →Explain why p-nitrophenol is more
Question: Explain why p-nitrophenol is more acidic than phenol. Solution: Para-nitrophenol is more acidic than phenol due to the presence of an electron-withdrawing group, -NO2 group, which enhances the acidic strength of the compound by stabilizing the phenoxide ion....
Read More →Explain why low molecular mass
Question: Explain why low molecular mass alcohols are soluble in water. Solution: This is due to the presence of intermolecular hydrogen bonding due to the presence of OH group between alcohol molecules. With an increase in the alkyl group of alcohol or case of high molecular mass alcohols, it suppresses the effect of polar nature of OH group of alcohol. Thus, the solubility of alcohol decreases with increases in molecular size....
Read More →Why is the C—O—H bond angle in alcohols slightly
Question: Why is the COH bond angle in alcohols slightly less than the tetrahedral angle whereas the COC bond angle in ether is slightly greater? Solution: The COH bond angle in alcohols is slightly less than the tetrahedral angle due to the repulsion between the unshared pair of electrons or the lone pair of electrons on the oxygen atom....
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Question: Mark () against the correct answer in the following: If $f(x)=\frac{1}{(1-x)}$ then (f of of) $(x)=?$ A. $\frac{1}{(1-3 x)}$ B. $\frac{x}{(1+3 x)}$ C. $\mathrm{X}$ D. None of these Solution: $f(x)=\frac{1}{1-x}$ $\Rightarrow(f \circ f \circ f)(x)=f(f(f(x)))$ $\Rightarrow \mathrm{f}(\mathrm{f}(\mathrm{x}))=\frac{1}{1-\mathrm{f}(\mathrm{x})}=\frac{1}{1-\frac{1}{1-\mathrm{x}}}=\frac{1-\mathrm{x}}{1-\mathrm{x}-1}=\frac{\mathrm{x}-1}{\mathrm{x}}=1-\frac{1}{\mathrm{x}}$ $\Rightarrow \mathrm{...
Read More →Ethers can be prepared by Williamson synthesis in
Question: Ethers can be prepared by Williamson synthesis in which an alkyl halide is reacted with sodium alkoxide. Di-tert-butyl ether cant be prepared by this method. Explain. Solution: Ethers can be prepared by Williamson synthesis in which an alkyl halide is reacted with sodium alkoxide. Di-tert-butyl ether cant be prepared by this method because in this case, elimination is more favoured over substitution....
Read More →Evaluate the following integrals -
Question: Evaluate the following integrals - $\int(2 x+3) \sqrt{x^{2}+4 x+3} d x$ Solution: Let $I=\int(2 x+3) \sqrt{x^{2}+4 x+3} d x$ Let us assume $2 x+3=\lambda \frac{d}{d x}\left(x^{2}+4 x+3\right)+\mu$ $\Rightarrow 2 \mathrm{x}+3=\lambda\left[\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{2}\right)+\frac{\mathrm{d}}{\mathrm{dx}}(4 \mathrm{x})+\frac{\mathrm{d}}{\mathrm{dx}}(3)\right]+\mu$ $\Rightarrow 2 \mathrm{x}+3=\lambda\left[\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{2}\right)+4 \f...
Read More →The dipole moment of phenol is smaller
Question: The dipole moment of phenol is smaller than that of methanol. Why? Solution: The dipole moment of phenol is smaller than that of methanol due to the electron-withdrawing effect of the phenyl ring. Due to the resonance, the polarity of the C-O bond in phenol decreases....
Read More →In Kolbe’s reaction, instead of phenol,
Question: In Kolbes reaction, instead of phenol, phenoxide ion is treated with carbon dioxide. Why? Solution: In Kolbes reaction, instead of phenol, phenoxide ion is treated with carbon dioxide (a weak electrophile) because phenoxide ion is more reactive towards electrophilic aromatic substitution....
Read More →Nitration is an example of aromatic electrophilic substitution
Question: Nitration is an example of aromatic electrophilic substitution and its rate depends upon the group already present in the benzene ring. Out of benzene and phenol, which one is more easily nitrated and why? Solution: Phenol is more nitrated because of the presence of the hydroxyl group in phenol. Due to the resonance effect caused by OH group, the ortho- and para-positions in the benzene ring becomes electron-rich and therefore, activates it towards electrophilic substitution reaction. ...
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Question: Mark () against the correct answer in the following: If $f\left(x+\frac{1}{x}\right)=x^{2}+\frac{1}{x^{2}}$ then $f(x)=?$ A. $x^{2}$ B. $\left(x^{2}-1\right)$ C. $\left(x^{2}-2\right)$ D. None of these Solution: $f\left(x+\frac{1}{x}\right)=x^{2}+\frac{1}{x^{2}}=\left(x+\frac{1}{x}\right)^{2}-2$ $\Rightarrow \mathrm{f}(\mathrm{x})=\mathrm{x}^{2}-2$...
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