Evaluate the following integrals -
Question: Evaluate the following integrals - $\int(x-2) \sqrt{2 x^{2}-6 x+5} d x$ Solution: Let $I=\int(x-2) \sqrt{2 x^{2}-6 x+5} d x$ Let us assume $x-2=\lambda \frac{d}{d x}\left(2 x^{2}-6 x+5\right)+\mu$ $\Rightarrow \mathrm{x}-2=\lambda\left[\frac{\mathrm{d}}{\mathrm{dx}}\left(2 \mathrm{x}^{2}\right)-\frac{\mathrm{d}}{\mathrm{dx}}(6 \mathrm{x})-\frac{\mathrm{d}}{\mathrm{dx}}(5)\right]+\mu$ $\Rightarrow \mathrm{x}-2=\lambda\left[2 \frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{2}\right)-6 \f...
Read More →Mark (√) against the correct answer in the following:
Question: Mark () against the correct answer in the following: Let A and B be two non - empty sets and let $f:(A \times B) \rightarrow(B \times A): f(a, b)=(b, a) .$ Then, $f$ is A. one - one and into B. one - one and onto C. many - one and into D. many - one and onto Solution: SINCE, f(a, b) = (b, a).There is no same value of y at different values of xfunction is one one $\therefore$ Range $(A \times B) \neq$ Codomain $(B \times A)$ $\Rightarrow$ function is into...
Read More →Write the IUPAC name of the following
Question: Write the IUPAC name of the following compounds. Solution: (A) IUPAC name of the compound is 3-Ethyl-5-methyl hexane-2,4-diol. (B) IUPAC name of the compound is1-Methoxy-3-nitrocyclohexane....
Read More →Mark (√) against the correct answer in the following:
Question: Mark () against the correct answer in the following: Let $\mathrm{f}: \mathrm{N} \rightarrow \mathrm{N}: \mathrm{f}(\mathrm{x})=\left\{\begin{array}{l}\frac{1}{2}(\mathrm{n}+1), \text { when } \mathrm{n} \text { is odd } \\ \frac{\mathrm{n}}{2}, \text { when } \mathrm{n} \text { is even. }\end{array}\right.$ Then, $f$ is A. one - one and into B. one - one and onto C. many - one and into D. many - one and onto Solution: $f(1)=1$ $f(2)=1$ $f(3)=2$ $f(4)=2$ $f(5)=3$ $f(6)=3$ Since at diff...
Read More →What is the structure and IUPAC
Question: What is the structure and IUPAC name of glycerol? Solution: Structure of Glycerol...
Read More →Which of the following are benzylic alcohols?
Question: Which of the following are benzylic alcohols? Solution: Option (ii) and (iii) are the answers....
Read More →Phenol can be distinguished from ethanol
Question: Phenol can be distinguished from ethanol by the reactions with _________. (i) Br2/water (ii) Na (iii) Neutral FeCl3 (iv) All the above Solution: Option (i) and (iii) are the answers....
Read More →Which of the following reagents can be used
Question: Which of the following reagents can be used to oxidise primary alcohols to aldehydes? (i) CrO3 in an anhydrous medium. (ii) KMnO4 in acidic medium. (iii) Pyridinium chlorochromate. (iv) Heat in the presence of Cu at 573K Solution: Option (i), (iii) and (iv) are the answers....
Read More →Mark (√) against the correct answer in the following:
Question: Mark () against the correct answer in the following: Let $A=R-\{3\}$ and $B=R-\{1\} .$ Then $f: A \rightarrow A: f(x)=\frac{(x-2)}{(x-3)}$ is A. one - one and into B. one - one and onto C. many - one and into D. many - one and onto Solution: $\mathrm{f}: \mathrm{A} \rightarrow \mathrm{A}: \mathrm{f}(\mathrm{x})=\frac{(\mathrm{x}-2)}{(\mathrm{x}-3)}$ In this function $x=3$ and $y=1$ are the asymptotes of this curve and these are not included in the functions of the domain and range resp...
Read More →Which of the following reactions
Question: Which of the following reactions will yield phenol? Solution: Option (A), (B) and (C)...
Read More →Which of the following are used to convert
Question: Which of the following are used to convert RCHO into RCH2OH? (i) H2/Pd (ii) LiAlH4 (iii) NaBH4 (iv) Reaction with RMgX followed by hydrolysis Solution: Option (i), (ii) and (iii) are the answers....
Read More →Mark (√) against the correct answer in the following
Question: Mark () against the correct answer in the following $f: C \rightarrow R: f(z)=|z|$ is A. one - one and into B. one - one and onto C. many - one and into D. many - one and onto Solution: Here in this range the lines cut the curve in 2 equal valued points of $y$ therefore the function $f(z)=|z|$ is not one - one $\Rightarrow f(z)=$ many one Range of $f(z)=[0, \infty) \neq R$ (codomain) $\therefore f(z)$ is not onto. $\Rightarrow f(z)=$ into Hence, $f(z)=|z|$ is many one and into...
Read More →Arrange the following compounds in increasing
Question: Arrange the following compounds in increasing order of boiling point. Propan-1-ol, butan-1-ol, butan-2-ol, pentan-1-ol (i) Propan-1-ol, butan-2-ol, butan-1-ol, pentan-1-ol (ii) Propan-1-ol, butan-1-ol, butan-2-ol, pentan-1-ol (iii) Pentan-1-ol, butan-2-ol, butan-1-ol, propan-1-ol (iv) Pentan-1-ol, butan-1-ol, butan-2-ol, propan-1-ol Solution: Option (i)Propan-1-ol, butan-2-ol, butan-1-ol, pentan-1-olis the answer...
Read More →Mark the correct increasing order
Question: Mark the correct increasing order of reactivity of the following compounds with HBr/HCl. (i) a b c (ii) b a c (iii) b c a (iv) c b a Solution: Option (iii) b c a is the answer....
Read More →Evaluate the following integrals -
Question: Evaluate the following integrals - $\int(4 x+1) \sqrt{x^{2}-x-2 x} d x$ Solution: Let $I=\int(4 x+1) \sqrt{x^{2}-x-2} d x$ Let us assume $4 x+1=\lambda \frac{d}{d x}\left(x^{2}-x-2\right)+\mu$ $\Rightarrow 4 x+1=\lambda\left[\frac{d}{d x}\left(x^{2}\right)-\frac{d}{d x}(x)-\frac{d}{d x}(2)\right]+\mu$ We know $\frac{d}{d x}\left(x^{n}\right)=n x^{n-1}$ and derivative of a constant is 0 . $\Rightarrow 4 x+1=\lambda\left(2 x^{2-1}-1-0\right)+\mu$ $\Rightarrow 4 x+1=\lambda(2 x-1)+\mu$ $\...
Read More →Mark the correct order of decreasing
Question: Mark the correct order of decreasing acid strength of the following compounds. (i) e d b a c (ii) b d a c e (iii) d e c b a (iv) e d c b a Solution: Option (ii)b d a c eis the answer....
Read More →Mark (√) against the correct answer in the following:
Question: Mark () against the correct answer in the following: $f: R \rightarrow R: f(x)=\cos x$ is A. one - one and into B. one - one and onto C. many - one and into D. many - one and onto Solution: $f(x)=\cos x$ $y=\cos x$ Here in this range the lines cut the curve in many equal valued points of $y$ therefore the function $f(x)=$ cosx is not one - one. $\Rightarrow f(x)=$ many one Range of $f(x)=[-1,1] \neq R$ (codomain) $\therefore f(x)$ is not onto. $\Rightarrow f(x)=$ into Hence, $f(x)=\cos...
Read More →Which of the following is most acidic?
Question: Which of the following is most acidic? (i) Benzyl alcohol (ii) Cyclohexanol (iii) Phenol (iv) m-Chlorophenol Solution: Option (iv) m-Chlorophenol is the answer....
Read More →Phenol is less acidic than
Question: Phenol is less acidic than ______________. (i) ethanol (ii) o-nitrophenol (iii) o-methyl phenol (iv) o-methoxy phenol Solution: Option (ii)o-nitrophenol is the answer....
Read More →Mark (√) against the correct answer in the following
Question: Mark () against the correct answer in the following $f:\left[\frac{-\pi}{2}, \frac{\pi}{2}\right] \rightarrow[-1,1]: f(x)=\sin x$ is A. one - one and into B. one - one and onto C. many - one and into D. many - one and onto Solution: $f:\left[\frac{-\pi}{2}, \frac{\pi}{2}\right] \rightarrow[-1,1]: f(x)=\sin x$ Here in this range, the function is NOT repeating its value, Therefore it is one - one. Range = Codomain $\therefore$ Function is onto Hence, option B is the correct choice....
Read More →Which of the following compounds will react
Question: Which of the following compounds will react with sodium hydroxide solution in water? (i) C6H5OH (ii) C6H5CH2OH (iii) (CH3)3 COH (iv) C2H5OH Solution: Option (i)C6H5OH is the answer...
Read More →Which of the following species
Question: Which of the following species can act as the strongest base? Solution: Option (ii) is the answer....
Read More →Mark (√) against the correct answer in the following:
Question: Mark () against the correct answer in the following: $f: R^{+} \rightarrow R^{+}: f(x)=e^{x}$ is A. many - one and into B. many - one and onto C. one - one and into D. one - one and onto Solution: $f(x)=e^{x}$ Since the function $f(x)$ is monotonically increasing from the domain $R^{+} \rightarrow R^{+}$ $\therefore f(x)$ is one -one Range of $f(x)=(1, \infty)=R^{+}$(codomain) $\therefore f(x)$ is onto $\therefore f: R^{+} \rightarrow R^{+}: f(x)=e^{x}$ is one - one onto....
Read More →IUPAC name of the compound is
Question: IUPAC name of the compound is ______________. (i) 1-methoxy-1-methylethane (ii) 2-methoxy-2-methylethane (iii) 2-methoxypropane (iv) isopropylmethyl ether Solution: Option (iii)2-methoxypropane is the answer....
Read More →IUPAC name of m-cresol is ___________.
Question: IUPAC name of m-cresol is ___________. (i) 3-methylphenol (ii) 3-chlorophenol (iii) 3-methoxyphenol (iv) benzene-1,3-diol Solution: Option (i)3-methylphenol is the answer....
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