Solve this
Question: Let $f: N \rightarrow R: f(x)=4 x^{2}+12 x+15 .$ Show that $f: N \rightarrow$ range $(f)$ is invertible. Find $f^{-1} .$ Solution: To Show: that $\mathrm{f}$ is invertible To Find: Inverse of $f$ [NOTE: Any functions is invertible if and only if it is bijective functions (i.e. one-one and onto)] one-one function: A function $\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}$ is said to be a one-one function or injective mapping if different elements of A have different images in B. Thus fo...
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Question: Let $f: N \rightarrow R: f(x)=4 x^{2}+12 x+15 .$ Show that $f: N \rightarrow$ range $(f)$ is invertible. Find $f^{-1} .$ Solution: To Show: that $\mathrm{f}$ is invertible To Find: Inverse of $f$ [NOTE: Any functions is invertible if and only if it is bijective functions (i.e. one-one and onto)] one-one function: A function $\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}$ is said to be a one-one function or injective mapping if different elements of A have different images in B. Thus fo...
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Question: Let $R_{+}$be the set of all positive real numbers. show that the function $f: R_{+} \rightarrow[-5, \infty]: f(x)=\left(9 x^{2}+6 x-5\right)$ is invertible. Find $\mathrm{f}^{-1}$. Solution: To Show: that $\mathrm{f}$ is invertible To Find: Inverse of $f$ [NOTE: Any functions is invertible if and only if it is bijective functions (i.e. one-one and onto)] one-one function: A function $f: A \rightarrow B$ is said to be a one-one function or injective mapping if different elements of $A$...
Read More →Evaluate the following integrals -
Question: Evaluate the following integrals - $\int(x+1) \sqrt{2 x^{2}+3} d x$ Solution: Let $\mathrm{I}=\int(\mathrm{x}+1) \sqrt{2 \mathrm{x}^{2}+3} \mathrm{dx}$ Let us assume $x+1=\lambda \frac{d}{d x}\left(2 x^{2}+3\right)+\mu$ $\Rightarrow \mathrm{x}+1=\lambda\left[\frac{\mathrm{d}}{\mathrm{dx}}\left(2 \mathrm{x}^{2}\right)+\frac{\mathrm{d}}{\mathrm{dx}}(1)\right]+\mu$ $\Rightarrow \mathrm{x}+1=\lambda\left[2 \frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{2}\right)+\frac{\mathrm{d}}{\mathrm{...
Read More →Show that the function
Question: Show that the function $f$ on $A=R-\left\{\frac{-4}{3}\right\}$ into itself, defined by $f(x)=\frac{4 x}{(3 x+4)}$ is one-one and onto. Hence, find $f^{-1}$. Solution: To Show: that $\mathrm{f}$ is one-one and onto To Find: Inverse of $f$ [NOTE: Any functions is invertible if and only if it is bijective functions (i.e. one-one and onto)] one-one function: A function $f: A \rightarrow B$ is said to be a one-one function or injective mapping if different elements of A have different imag...
Read More →Alkyl fluorides are synthesised by heating
Question: Alkyl fluorides are synthesised by heating an alkyl chloride/bromide in presence of ____________ or ____________. (i) Ca F2 (ii) CoF2 (iii) Hg2F2 (iv) NaF Solution: Option (ii) and (iii) are the answers....
Read More →Alkyl halides are prepared from alcohols
Question: Alkyl halides are prepared from alcohols by treating with (i) HCl + ZnCl2 (ii) Red P + Br2 (iii) H2SO4+ KI (iv) All the above Solution: Option (i)HCl + ZnCl2 and (ii)Red P + Br2 are the answers...
Read More →Which of the following compounds can be
Question: Which of the following compounds can be classified as aryl halides? (i)p-ClC6H4CH2CH(CH3)2 (ii) p-CH3CHCl(C6H4)CH2CH3 (iii) o-BrH2C-C6H4CH(CH3)CH2CH3 (iv) C6H5-Cl Solution: Option (i)p-ClC6H4CH2CH(CH3)2 and (iv) C6H5-Clare the answers....
Read More →Show that the function
Question: Show that the function $f$ on $A=R-\left\{\frac{2}{3}\right\}$, defined as $f(x)=\frac{4 x+3}{6 x-4}$ is one-one and onto. Hence, find $f^{-1}$. Solution: To Show: that $\mathrm{f}$ is one-one and onto To Find: Inverse of $f$ [NOTE: Any functions is invertible if and only if it is bijective functions (i.e. one-one and onto)] one-one function: A function $f: A \rightarrow B$ is said to be a one-one function or injective mapping if different elements of A have different images in B. Thus...
Read More →Which of the following are secondary bromides?
Question: Which of the following are secondary bromides? (i) (CH3)2 CHBr (ii) (CH3)3C CH2Br (iii) CH3CH(Br)CH2CH3 (iv) (CH3)2CBrCH2CH3 Solution: Option (i)(CH3)2 CHBr and (iii)CH3CH(Br)CH2CH3are the answers....
Read More →Which of the following compounds
Question: Which of the following compounds are gem-dihalides? (i) Ethylidene chloride (ii) Ethylene dichloride (iii) Methylene chloride (iv) Benzyl chloride Solution: Option (i)Ethylidene chloride and (iii)Methylene chlorideare the answers....
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Question: If $f(x)=\frac{(4 x+3)}{(6 x-4)}, x \neq \frac{2}{3}$, show that (f o f) $(x)=x$ for all $x \neq \frac{2}{3}$ Hence, find $\mathrm{f}^{-1}$. Solution: To Show: that $f$ o $f(x)=x$ Finding (f o f) (x)...
Read More →Ethylene chloride and ethylidene chloride
Question: Ethylene chloride and ethylidene chloride are isomers. Identify the correct statements. (i) Both the compounds form the same product on treatment with alcoholic KOH. (ii) Both the compounds form the same product on treatment with aq.NaOH. (iii) Both the compounds form the same product on reduction. (iv) Both the compounds are optically active. Solution: Option (i) and (iii) are the answers....
Read More →Haloalkanes contain halogen atom (s) attached
Question: Haloalkanes contain halogen atom (s) attached to the sp3 hybridised carbon atom of an alkyl group. Identify haloalkane from the following compounds. (i) 2-Bromopentane (ii) Vinyl chloride (chloroethene) (iii) 2-chloroacetophenone (iv) Trichloromethane Solution: Option (i) and (iv) are the answers....
Read More →Which of the following statements are correct
Question: Which of the following statements are correct about the kinetics of this reaction? (i) The rate of reaction depends on the concentration of only (b). (ii) The rate of reaction depends on the concentration of both (a) and (b). (iii) Molecularity of reaction is one. (iv) Molecularity of reaction is two. Solution: Option (i) and (iii) are the answers....
Read More →Which of the following statements are correct
Question: Which of the following statements are correct about the mechanism of this reaction? (i) A carbocation will be formed as an intermediate in the reaction. (ii) OHwill attach the substrate (b) from one side and Cl- will leave it simultaneously from the other side. (iii) An unstable intermediate will be formed in which OH and Cl will be attached by weak bonds. (iv) The reaction proceeds through an SN1 mechanism. Solution: Option (i) and (iv) are the answers....
Read More →Which of the following statements are correct
Question: Which of the following statements are correct about the reaction intermediate? (i) Intermediate (c) is unstable because in this carbon is attached to 5 atoms. (ii) Intermediate (c) is unstable because carbon atom is sp2 hybridised. (iii) Intermediate (c) is stable because carbon atom is sp2 hybridised. (iv) Intermediate (c) is less stable than the reactant (b). Solution: Option (i) and (iv) are the answers....
Read More →Which of the following statements
Question: Which of the following statements are correct about this reaction? (i) The given reaction follows the SN2 mechanism. (ii) (b) and (d) have the opposite configuration. (iii) (b) and (d) have the same configuration. (iv) The given reaction follows the SN1 mechanism. Solution: Option (i) and (ii) are the answers....
Read More →Which of the statements are correct about t
Question: Which of the statements are correct about the above reaction? (i) (a) and (e) both are nucleophiles. (ii) In (c) carbon atom is sp3 hybridised. (iii) In (c) carbon atom is sp2 hybridised. (iv) (a) and (e) both are electrophiles. Solution: Option (i) and (iii) are the answers....
Read More →Which is the correct increasing order of boiling points
Question: Which is the correct increasing order of boiling points of the following compounds? 1-Bromoethane, 1-Bromopropane, 1-Bromobutane, Bromobenzene (i) Bromobenzene 1-Bromobutane 1-Bromopropane 1-Bromoethane (ii) Bromobenzene 1-Bromoethane 1-Bromopropane 1-Bromobutane (iii) 1-Bromopropane 1-Bromobutane 1-Bromoethane Bromobenzene (iv) 1-Bromoethane 1-Bromopropane 1-Bromobutane Bromobenzene Solution: Option (iv)1-Bromoethane 1-Bromopropane 1-Bromobutane Bromobenzeneis the answer...
Read More →Evaluate the following integrals -
Question: Evaluate the following integrals - $\int(x+1) \sqrt{x^{2}-x+1} d x$ Solution: Let $I=\int(x+1) \sqrt{x^{2}-x+1} d x$ Let us assume $x+1=\lambda \frac{d}{d x}\left(x^{2}-x+1\right)+\mu$ $\Rightarrow x+1=\lambda\left[\frac{d}{d x}\left(x^{2}\right)-\frac{d}{d x}(x)+\frac{d}{d x}(1)\right]+\mu$ We know $\frac{\mathrm{d}}{\mathrm{dx}}\left(\mathrm{x}^{\mathrm{n}}\right)=\mathrm{n} \mathrm{x}^{\mathrm{n}-1}$ and derivative of a constant is 0 . $\Rightarrow x+1=\lambda\left(2 x^{2-1}-1+0\rig...
Read More →Which is the correct increasing order of boiling points
Question: Which is the correct increasing order of boiling points of the following compounds? 1-Iodobutane, 1-Bromobutane, 1-Chlorobutane, Butane (i) Butane 1-Chlorobutane 1-Bromobutane 1-Iodobutane (ii) 1-Iodobutane 1-Bromobutane 1-Chlorobutane Butane (iii) Butane 1-Iodobutane 1-Bromobutane 1-Chlorobutane (iv) Butane 1-Chlorobutane 1-Iodobutane 1-Bromobutane Solution: Option (i)Butane 1-Chlorobutane 1-Bromobutane 1-Iodobutaneis the answer....
Read More →Prove the following
Question: (i) (a) (b) (c) (ii) (b) (a) (c) (iii) (c) (b) (a) (iv) (a) (c) (b) Solution: Option (iii) is the answer....
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Question: (i) (c) (b) (a) (ii) (b) (c) (a) (iii) (a) (c) (b) (iv) (a) (b) (c) Solution: Option (iv)(a) (b) (c) is the answer....
Read More →Solve the following
Question: (i) (a) (b) (c) (ii) (a) (c) (b) (iii) (c) (b) (a) (iv) (b) (c) (a) Solution: Option (iv) is the answer....
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