Prove that
Question: $f: R \rightarrow R: f(x)=\left\{\begin{array}{r}1, \text { if } x \text { is rational } \\ -1, \text { if } x \text { is rational }\end{array}\right.$ Show that $\mathrm{f}$ is many-one and into. Solution: To prove: function is many-one and into Given: $f: R \rightarrow R: f(x)=\left\{\begin{array}{c}1, \text { if } x \text { is rational } \\ -1, \text { if } x \text { is irrational }\end{array}\right.$ We have, $f(x)=1$ when $x$ is rational It means that all rational numbers will hav...
Read More →Evaluate the integral:
Question: Evaluate the integral: 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 $\int f(x) g(x) d x=f(x) \int g(x) d x-\int f^{\prime}(x)\left(\int g(x) d x\right) d x$ - To solve the integrals of the form: $\int \sqrt{a x^{2}+b x+c} d x$ after applying substitution and integration by parts we have direct formulae as described below: $\int \sqrt{a^{2}-x^{2...
Read More →Find the value
Question: Let $f: R \rightarrow R: f(x)=10 x+3$. Find $f^{-1}$. Solution: To find: $\mathrm{f}^{-1}$ Given: $f: R \rightarrow R: f(x)=10 x+3$ We have, $f(x)=10 x+3$ Let $f(x)=y$ such that $y \in R$ $\Rightarrow y=10 x+3$ $\Rightarrow y-3=10 x$ $\Rightarrow x=\frac{y-3}{10}$ $\Rightarrow f^{-1}=\frac{y-3}{10}$ Ans) $f^{-1}(y)=\frac{y-3}{10}$ for all $y \in R$...
Read More →Evaluate the integral:
Question: Evaluate the integral: $\int \sqrt{16 x^{2}+25} d x$ Solution: Key points to solve the problem: - Such problems require the use of the method of substitution along with a method of integration by parts. By the 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 solv...
Read More →Solve this
Question: Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}: \mathrm{f}(\mathrm{x})=\frac{2 \mathrm{x}-7}{4}$ be an invertible function. Find $\mathrm{f}^{-1}$. Solution: To find: $\mathrm{f}^{-1}$ Given: $f: R \rightarrow R: f(x)=\frac{2 x-7}{4}$ We have, $f(x)=\frac{2 x-7}{4}$ Let $f(x)=y$ such that $y \in R$ $\Rightarrow y=\frac{2 x-7}{4}$ $\Rightarrow 4 y=2 x-7$ $\Rightarrow 4 y+7=2 x$ $\Rightarrow x=\frac{4 y+7}{2}$ $\Rightarrow f^{-1}=\frac{4 y+7}{2}$ Ans) $f^{-1}(y)=\frac{4 y+7}{2}$ for ...
Read More →Evaluate the integral:
Question: Evaluate the integral: $\int \sqrt{9-x^{2}} d x$ Solution: Key points to solve the problem: - Such problems require the use of the method of substitution along with a method of integration by parts. By the 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 th...
Read More →Show that the function
Question: Show that the function $f: R \rightarrow R: f(x)=1+x^{2}$ is many-one into. Solution: To prove: function is many-one into Given: $f: R \rightarrow R: f(x)=1+x^{2}$ We have, $f(x)=1+x^{2}$ For, $f\left(x_{1}\right)=f\left(x_{2}\right)$ $\Rightarrow 1+x_{1}^{2}=1+x_{2}^{2}$ $\Rightarrow x_{1}^{2}=x_{2}^{2}$ $\Rightarrow x_{1}^{2}-x_{2}^{2}=0$ $\Rightarrow\left(x_{1}-x_{2}\right)\left(x_{1}+x_{2}\right)=0$ $\Rightarrow x_{1}=x_{2}$ or, $x_{1}=-x_{2}$ Clearly $x_{1}$ has more than one imag...
Read More →Evaluate the integral:
Question: Evaluate the integral: $\int e^{x} \sqrt{e^{2 x}+1} d x$ 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 ...
Read More →Let R0 be the set of all nonzero real numbers.
Question: Let $R_{0}$ be the set of all nonzero real numbers. Then, show that the function $f: R_{0} \rightarrow R_{0}: f(x)=\frac{1}{x}$ is one- one and onto. Solution: To prove: function is one-one and onto Given: $f: R_{0} \rightarrow R_{0}: f(x)=\frac{1}{x}$ We have, $f(x)=\frac{1}{x}$ For, $f\left(x_{1}\right)=f\left(x_{2}\right)$ $\Rightarrow \frac{1}{x_{1}}=\frac{1}{x_{2}}$ $\Rightarrow x_{1}=x_{2}$ When, $f\left(x_{1}\right)=f\left(x_{2}\right)$ then $x_{1}=x_{2}$ $\Rightarrow y=\frac{1}...
Read More →Assertion : ([Fe(CN)6]3– ion shows magnetic moment
Question: Assertion : ([Fe(CN)6]3 ion shows magnetic moment corresponding to two unpaired electrons. Reason: Because it has d2sp3 type hybridisation. (i) Assertion and reason both are true, the reason is the correct explanation of assertion. (ii) Assertion and reason both are true but the reason is not the correct explanation of assertion. (iii) An assertion is true, the reason is false. (iv) The assertion is false, the reason is true. Solution: Option (iv)The assertion is false, the reason is t...
Read More →Assertion: Linkage isomerism arises in coordination
Question: Assertion: Linkage isomerism arises in coordination compounds containing ambidentate ligand. Reason: Ambidentate ligand has two different donor atoms. (i) Assertion and reason both are true, the reason is the correct explanation of assertion. (ii) Assertion and reason both are true but the reason is not the correct explanation of assertion. (iii) An assertion is true, the reason is false. (iv) The assertion is false, the reason is true. Solution: Option (i)Assertion and reason both are...
Read More →Show that the function
Question: Show that the function $f: Z \rightarrow Z: f(x)=x^{3}$ is one-one and into. Solution: To prove: function is one-one and into Given: $f: Z \rightarrow Z: f(x)=x^{3}$ Solution: We have, $f(x)=x^{3}$ For, $f\left(x_{1}\right)=f\left(x_{2}\right)$ $\Rightarrow \mathrm{x}_{1}^{3}=\mathrm{x}_{2}^{3}$ $\Rightarrow \mathrm{x}_{1}=\mathrm{x}_{2}$ When, $f\left(x_{1}\right)=f\left(x_{2}\right)$ then $x_{1}=x_{2}$ $\therefore f(x)$ is one-one $f(x)=x^{3}$ Let $f(x)=y$ such that $y \in Z$ $\Right...
Read More →Assertion: Complexes of MX6 and MX5L
Question: Assertion: Complexes of MX6 and MX5L type (X and L are unidentate) do not show geometrical isomerism. Reason: Geometrical isomerism is not shown by complexes of coordination number 6. (i) Assertion and reason both are true, the reason is the correct explanation of assertion. (ii) Assertion and reason both are true but the reason is not the correct explanation of assertion. (iii) An assertion is true, the reason is false. (iv) The assertion is false, the reason is true. Solution: Option...
Read More →Assertion : [Cr(H2O)6]Cl2 and [Fe(H2O)6]Cl2
Question: Assertion : [Cr(H2O)6]Cl2 and [Fe(H2O)6]Cl2 are reducing in nature. Reason: Unpaired electrons are present in their d-orbitals. (i) Assertion and reason both are true, the reason is the correct explanation of assertion. (ii) Assertion and reason both are true but the reason is not the correct explanation of assertion. (iii) An assertion is true, the reason is false. (iv) The assertion is false, the reason is true. Solution: Option (ii)Assertion and reason both are true but the reason i...
Read More →Show that the function
Question: Show that the function $f: R \rightarrow R: f(x)=x^{4}$ is neither one-one nor onto. Solution: To prove: function is neither one-one nor onto Given: $f: R \rightarrow R: f(x)=x^{4}$ We have, $f(x)=x^{4}$ For, $f\left(x_{1}\right)=f\left(x_{2}\right)$ $\Rightarrow x_{1}^{4}=x_{2}^{4}$ $\Rightarrow\left(x_{1}^{4}-x_{2}^{4}\right)=0$ $\Rightarrow\left(x_{1}^{2}-x_{2}^{2}\right)\left(x_{1}^{2}+x_{2}^{2}\right)=0$ $\Rightarrow\left(x_{1}-x_{2}\right)\left(x_{1}+x_{2}\right)\left(x_{1}^{2}+x...
Read More →Assertion: Toxic metal ions are removed
Question: Assertion: Toxic metal ions are removed by the chelating ligands. Reason: Chelate complexes tend to be more stable. (i) Assertion and reason both are true, the reason is the correct explanation of assertion. (ii) Assertion and reason both are true but the reason is not the correct explanation of assertion. (iii) An assertion is true, the reason is false. (iv) The assertion is false, the reason is true. Solution: Option (i)Assertion and reason both are true, the reason is the correct ex...
Read More →Evaluate the integral:
Question: Evaluate the integral: $\int \cos x \sqrt{4-\sin ^{2} x} d x$ Solution: Key points to solve the problem: - Such problems require the use of the method of substitution along with a method of integration by parts. By the 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{d} \mathrm{x}\right) \ma...
Read More →Show that the function
Question: Show that the function $f: N \rightarrow N: f(x)=x^{2}$ is one-one and into. Solution: To prove: function is one-one and into Given: $f: N \rightarrow N: f(x)=x^{2}$ Solution: We have, $f(x)=x^{2}$ For, $f\left(x_{1}\right)=f\left(x_{2}\right)$ $\Rightarrow \mathrm{x}_{1}^{2}=\mathrm{x}_{2}^{2}$ $\Rightarrow \mathrm{x}_{1}=\mathrm{x}_{2}$ Here we can't consider $x_{1}=-x_{2}$ as $x \in N$, we can't have negative values $\therefore f(x)$ is one-one $f(x)=x^{2}$ Let $f(x)=y$ such that $y...
Read More →Match the compounds given in Column I with
Question: Match the compounds given in Column I with the oxidation state of cobalt present in it (given in Column II) and assign the correct code. Code : (i) A (1) B (2) C (4) D (5) (ii) A (4) B (3) C (2) D (1) (iii) A (5) B (1) C (4) D (2) (iv) A (4) B (1) C (2) D (3) Solution: Option (ii)A (4) B (3) C (2) D (1)is the answer....
Read More →Evaluate the integral:
Question: Evaluate the integral: $\int \sqrt{1+x-2 x^{2}} d x$ 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 $\int f(x) g(x) d x=f(x) \int g(x) d x-\int f^{\prime}(x)\left(\int g(x) d x\right) d x$ - To solve the integrals of the form: $\int \sqrt{a x^{2}+b x+c} d x$ after applying substitution and integration by parts we have direct formulae as described...
Read More →Match the complex species given in Column
Question: Match the complex species given in Column I with the possible isomerism given in Column II and assign the correct code : Code : (i) A (1) B (2) C (4) D (5) (ii) A (4) B (3) C (2) D (1) (iii) A (4) B (1) C (5) D (3) (iv) A (4) B (1) C (2) D (3) Solution: Option (iv)A (4) B (1) C (2) D (3) is the answer...
Read More →Show that the function
Question: Show that the function $f: R \rightarrow R: f(x)=x^{2}$ is neither one-one nor onto. Solution: To prove: function is neither one-one nor onto Given: $f: R \rightarrow R: f(x)=x^{2}$ Solution: We have, $f(x)=x^{2}$ For, $f\left(x_{1}\right)=f\left(x_{2}\right)$ $\Rightarrow \mathrm{x}_{1}^{2}=\mathrm{x}_{2}^{2}$ $\Rightarrow \mathrm{x}_{1}=\mathrm{x}_{2}$ or, $\mathrm{x}_{1}=-\mathrm{x}_{2}$ Since $x_{1}$ doesn't has unique image $\therefore f(x)$ is not one-one $f(x)=x^{2}$ Let $f(x)=y...
Read More →Match the complex ions given in Column
Question: Match the complex ions given in Column I with the hybridisation and number of unpaired electrons given in Column II and assign the correct code : Code : (i) A (3) B (1) C (5) D (2) (ii) A (4) B (3) C (2) D (1) (iii) A (3) B (2) C (4) D (1) (iv) A (4) B (1) C (2) D (3) Solution: Option (i)A (3) B (1) C (5) D (2)is the answer...
Read More →Match the coordination compounds given
Question: Match the coordination compounds given in Column I with the central metal atoms given in Column II and assign the correct code : Code : (i) A (5) B (4) C (1) D (2) (ii) A (3) B (4) C (5) D (1) (iii) A (4) B (3) C (2) D (1) (iv) A (3) B (4) C (1) D (2) Solution: Option (i)A (5) B (4) C (1) D (2) is the answer....
Read More →Evaluate the integral:
Question: Evaluate the integral: $\int \sqrt{x-x^{2}} d x$ Solution: Key points to solve the problem: - Such problems require the use of the method of substitution along with a method of integration by parts. By the 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 th...
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