Evaluate the integral:
Question: Evaluate the integral: $\int x^{2} \sqrt{a^{6}-x^{6}} d x$ Solution: Key points to solve the problem: - Such problems require the use of method of substitution along with method of integration by parts. By method of integration by parts if we have $\int \mathrm{f}(\mathrm{x}) \mathrm{g}(\mathrm{x}) \mathrm{d} \mathrm{x}=\mathrm{f}(\mathrm{x}) \int \mathrm{g}(\mathrm{x}) \mathrm{dx}-\int \mathrm{f}^{\prime}(\mathrm{x})\left(\int \mathrm{g}(\mathrm{x}) \mathrm{dx}\right) \mathrm{dx}$ - T...
Read More →Which of the following structures is enantiomeric
Question: Which of the following structures is enantiomeric with the molecule (A) given below : Solution: Option (i) is the answer....
Read More →In which of the following molecules carbon
Question: In which of the following molecules carbon atom marked with an asterisk (*) is asymmetric? (i) (a), (b), (c), (d) (ii) (a), (b), (c) (iii) (b), (c), (d) (iv) (a), (c), (d) Solution: Option (ii)(a), (b), (c)is the answer....
Read More →Find the value
Question: Let $A=\{1,2,3), B=\{4,5,6,7)$ and let $f=\{(1,4),(2,5),(3,6)\}$ be a function from $A$ to $B .$ State whether $f$ is one-one. Solution: To state: Whether f is one-one Given: $f=\{(1,4),(2,5),(3,6)\}$ Here the function is defined from $A \rightarrow B$ For a function to be one-one if the images of distinct elements of $A$ under $f$ are distinct i.e. 1,2 and 3 must have a distinct image. From $f=\{(1,4),(2,5),(3,6)\}$ we can see that 1,2 and 3 have distinct image. Therefore $\mathrm{f}$...
Read More →Find the value
Question: Let $A=\{1,2,3), B=\{4,5,6,7)$ and let $f=\{(1,4),(2,5),(3,6)\}$ be a function from $A$ to $B .$ State whether $f$ is one-one. Solution: To state: Whether f is one-one Given: $f=\{(1,4),(2,5),(3,6)\}$ Here the function is defined from $A \rightarrow B$ For a function to be one-one if the images of distinct elements of $A$ under $f$ are distinct i.e. 1,2 and 3 must have a distinct image. From $f=\{(1,4),(2,5),(3,6)\}$ we can see that 1,2 and 3 have distinct image. Therefore $\mathrm{f}$...
Read More →Arrange the following compounds
Question: Arrange the following compounds in increasing order of their boiling points. (i) (b) (a) (c) (ii) (a) (b) (c) (iii) (c) (a) (b) (iv) (c) (b) (a) Solution: Option (iii) (c) (a) (b)is the answer....
Read More →Arrange the following compounds
Question: Arrange the following compounds in the increasing order of their densities. (i) (a) (b) (c) (d) (ii) (a) (c) (d) (b) (iii) (d) (c) (b) (a) (iv) (b) (d) (c) (a) Solution: Option (i)(a) (b) (c) (d)is the answer....
Read More →Solve this
Question: Let $f: R \rightarrow R: f(x)=10 x+7$. Find the function $g: R \rightarrow R: g \circ f=f \circ g=I_{g}$ Solution: To find: the function $g: R \rightarrow R: g \circ f=f \circ g=I_{g}$ Formula used: (i) $g$ o $f=g(f(x))$ (ii) $f \circ g=f(g(x))$ Given: $f: R \rightarrow R: f(x)=10 x+7$ We have, $f(x)=10 x+7$ Let $f(x)=y$ $\Rightarrow y=10 x+7$ $\Rightarrow y-7=10 x$ $\Rightarrow x=\frac{y-7}{10}$ Let $g(y)=\frac{y-7}{10}$ where $g: R \rightarrow R$ $g \circ f=g(f(x))=g(10 x+7)=\frac{(1...
Read More →Which reagent will you use for
Question: Which reagent will you use for the following reaction? CH3CH2CH2CH3 CH3CH2CH2CH2Cl + CH3CH2CHClCH3 (i) Cl2/UV light (ii) NaCl + H2SO4 (iii) Cl2 gas in dark (iv) Cl2 gas in the presence of iron in dark Solution: Option (i)Cl2/UV light is the answer....
Read More →Which of the following is the halogen
Question: Which of the following is the halogen exchange reaction? Solution: Option (i) is the answer....
Read More →Toluene reacts with a halogen in the presence
Question: Toluene reacts with a halogen in the presence of iron (III) chloride giving ortho and para halo compounds. The reaction is (i) Electrophilic elimination reaction (ii) Electrophilic substitution reaction (iii) Free radical addition reaction (iv) Nucleophilic substitution reaction Solution: Option (ii) Electrophilic substitution reaction is the answer....
Read More →Prove that
Question: Let $f(x)=8 x^{3}$ and $g(x)=x^{1 / 3}$. Find $g$ of and $f \circ g$. Solution: To find: $g$ o $f$ and $f \circ g$ Formula used: (i) $f \circ g=f(g(x))$ (ii) $g \circ f=g(f(x))$ Given: (i) $f(x)=8 x^{3}$ (ii) $g(x)=x^{1 / 3}$ We have, $g \circ f=g(f(x))=g\left(8 x^{3}\right)$ $g \circ f=\left(8 x^{3}\right)^{\frac{1}{3}}=2 x$ $f \circ g=f(g(x))=f\left(x^{1 / 3}\right)$ $f \circ g=8\left(x^{\frac{1}{3}}\right)^{3}=8 x$ Ans) $g \circ f=2 x$ and $f \circ g=8 x$...
Read More →Evaluate the integral:
Question: Evaluate the integral: $\int x \sqrt{x^{4}+1} d x$ Solution: Key points to solve the problem: - Such problems require the use of method of substitution along with method of integration by parts. By method of integration by parts if we have $\int \mathrm{f}(\mathrm{x}) \mathrm{g}(\mathrm{x}) \mathrm{dx}=\mathrm{f}(\mathrm{x}) \int \mathrm{g}(\mathrm{x}) \mathrm{dx}-\int \mathrm{f}^{\prime}(\mathrm{x})\left(\int \mathrm{g}(\mathrm{x}) \mathrm{dx}\right) \mathrm{dx}$ - To solve the integr...
Read More →Identify the compound Y in
Question: Identify the compound Y in the following reaction. Solution: Option (i) is the answer....
Read More →Which of the following alcohols will yield the corresponding
Question: Which of the following alcohols will yield the corresponding alkyl chloride on reaction with concentrated HCl at room temperature? Solution: Option (iv) is the answer....
Read More →Solve this
Question: Let $A=\{1,2,3,4\}$ and $f=\{(1,4),(2,1)(3,3),(4,2)\} .$ Write down (f of). Solution: To find: $f$ o $f$ Formula used: $f$ o $f=f(f(x))$ Given: (i) $f=\{(1,4),(2,1)(3,3),(4,2)\}$ We have, $f \circ f(1)=f(f(1))=f(4)=2$ $f \circ f(2)=f(f(2))=f(1)=4$ $f$ of $(3)=f(f(3))=f(3)=3$ $f \circ f(4)=f(f(4))=f(2)=1$ Ans) $f$ of $f=\{(1,2),(2,4),(3,3),(4,1)\}$...
Read More →The order of reactivity of following
Question: The order of reactivity of following alcohols with halogen acids is ___________. (i) (A) (B) (C) (ii) (C) (B) (A) (iii) (B) (A) (C) (iv) (A) (C) (B) Solution: Option (ii)(C) (B) (A) is the answer....
Read More →Prove that
Question: Let $f=\{(1,2),(3,5),(4,1)\}$ and $g=\{(1,3),(2,3),(5,1))$. Write down $g$ o $f$. Solution: To find: $g$ of Formula used: $g$ o $f=g(f(x))$ Given: (i) $f=\{(1,2),(3,5),(4,1)\}$ (ii) $g=\{(1,3),(2,3),(5,1)\}$ We have, $g \circ f(1)=g(f(1))=g(2)=3$ $g \circ f(3)=g(f(3))=g(5)=1$ $g \circ f(4)=g(f(4))=g(1)=3$ Ans) $\mathrm{g}$ o $\mathrm{f}=\{(1,3),(3,1),(4,3)\}$...
Read More →Solve this
Question: Let $f: R \rightarrow R: f(x)=3 x+2$, find $f\{f(x)\}$ Solution: To find: $f\{f(x)\}$ Formula used: (i) $f$ o $f=f(f(x))$ Given: (i) $f: R \rightarrow R: f(x)=3 x+2$ We have, $f\{f(x)\}=f(f(x))=f(3 x+2)$ $f \circ f=3(3 x+2)+2$ $=9 x+6+2$ $=9 x+8$ Ans) $f\{f(x)\}=9 x+8$...
Read More →Evaluate the integral:
Question: Evaluate the integral: $\int \sqrt{3-2 x-2 x^{2}} d x$ Solution: Key points to solve the problem: - Such problems require the use of method of substitution along with method of integration by parts. By method of integration by parts if we have $\int \mathrm{f}(\mathrm{x}) \mathrm{g}(\mathrm{x}) \mathrm{d} \mathrm{x}=\mathrm{f}(\mathrm{x}) \int \mathrm{g}(\mathrm{x}) \mathrm{dx}-\int \mathrm{f}^{\prime}(\mathrm{x})\left(\int \mathrm{g}(\mathrm{x}) \mathrm{d} \mathrm{x}\right) \mathrm{dx...
Read More →Solve this
Question: Let $f: R \rightarrow R: f(x)=3 x+2$, find $f\{f(x)\}$ Solution: To find: $f\{f(x)\}$ Formula used: (i) $f$ o $f=f(f(x))$ Given: (i) $f: R \rightarrow R: f(x)=3 x+2$ We have, $f\{f(x)\}=f(f(x))=f(3 x+2)$ $f \circ f=3(3 x+2)+2$ $=9 x+6+2$ $=9 x+8$ Ans) $f\{f(x)\}=9 x+8$...
Read More →Prove that
Question: Let $f: R \rightarrow R: f(x)=\left(3-x^{3}\right)^{1 / 3}$. Find $f$ o $f$. Solution: To find: $f$ of Formula used: (i) $f$ o $f=f(f(x))$ Given: (i) $f: R \rightarrow R: f(x)=\left(3-x^{3}\right)^{1 / 3}$ We have, $f \circ f=f(f(x))=f\left(\left(3-x^{3}\right)^{1 / 3}\right)$ $f \circ f=\left[3-\left\{\left(3-x^{3}\right)^{1 / 3}\right\}^{3}\right]^{1 / 3}$ $=\left[3-\left(3-x^{3}\right)\right]^{1 / 3}$ $=\left[3-3+x^{3}\right]^{1 / 3}$ $=\left[x^{3}\right]^{1 / 3}$ $=x$ Ans) $f$ o $f...
Read More →Evaluate the integral:
Question: Evaluate the integral: $\int \sqrt{2 x^{2}+3 x+4} d x$ Solution: Key points to solve the problem: - Such problems require the use of method of substitution along with method of integration by parts. By method of integration by parts if we have $\int \mathrm{f}(\mathrm{x}) \mathrm{g}(\mathrm{x}) \mathrm{dx}=\mathrm{f}(\mathrm{x}) \int \mathrm{g}(\mathrm{x}) \mathrm{dx}-\int \mathrm{f}^{\prime}(\mathrm{x})\left(\int \mathrm{g}(\mathrm{x}) \mathrm{dx}\right) \mathrm{dx}$ - To solve the in...
Read More →Prove that
Question: Let $f: R \rightarrow R$ and $g: R \rightarrow R$ defined by $f(x)=x^{2}$ and $g(x)=(x+1)$. Show that $g \circ f \neq f \circ g$. Solution: To prove: $\mathrm{g}$ of $\mathrm{f} \neq \mathrm{fog}$ Formula used: (i) $f \circ g=f(g(x))$ (ii) $g \circ f=g(f(x))$ Given: (i) $f: R \rightarrow R: f(x)=x^{2}$ (ii) $g: R \rightarrow R: g(x)=(x+1)$ We have, $f \circ g=f(g(x))=f(x+7)$ $f \circ g=(x+7)^{2}=x^{2}+14 x+49$ $g \circ f=g(f(x))=g\left(x^{2}\right)$ $g \circ f=\left(x^{2}+1\right)=x^{2...
Read More →Prove that
Question: Let $f(x)=x+7$ and $g(x)=x-7, x \in$ R. Find $(f \circ g)(7)$ Solution: To find: (f o g) (7) Formula used: $f \circ g=f(g(x))$ Given: (i) $f(x)=x+7$ (ii) $g(x)=x-7$ We have, $f \circ g=f(g(x))=f(x-7)=[(x-7)+7]$ $\Rightarrow X$ (f o g) $(x)=x$ $(f \circ g)(7)=7$ Ans). (f o g) (7) =7...
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