What is the maximum concentration of equimolar solutions of ferrous sulphate and sodium sulphide
Question: What is the maximum concentration of equimolar solutions of ferrous sulphate and sodium sulphide so that when mixed in equal volumes, there is no precipitation of iron sulphide? (For iron sulphide, $K_{\text {sp }}=6.3 \times 10^{-18}$ ). Solution: Let the maximum concentration of each solution be $x \mathrm{~mol} / \mathrm{L} .$ After mixing, the volume of the concentrations of each solution will be reduced to half i.e., $\frac{x}{2}$. $\therefore\left[\mathrm{FeSO}_{4}\right]=\left[\...
Read More →Express the given complex number in the form a + ib:
Question: Express the given complex number in the forma+ib:$\left(-2-\frac{1}{3} i\right)^{3}$ Solution: $\left(-2-\frac{1}{3} i\right)^{3}=(-1)^{3}\left(2+\frac{1}{3} i\right)^{3}$ $=-\left[2^{3}+\left(\frac{i}{3}\right)^{3}+3(2)\left(\frac{i}{3}\right)\left(2+\frac{i}{3}\right)\right]$ $=-\left[8+\frac{i^{3}}{27}+2 i\left(2+\frac{i}{3}\right)\right]$ $=-\left[8-\frac{i}{27}+4 i+\frac{2 i^{2}}{3}\right] \quad\left[i^{3}=-i\right]$ $=-\left[8-\frac{i}{27}+4 i-\frac{2}{3}\right] \quad\left[i^{2}=...
Read More →Write the function in the simplest form:
Question: Write the function in the simplest form: $\tan ^{-1} \frac{1}{\sqrt{x^{2}-1}},|x|1$ Solution: $\tan ^{-1} \frac{1}{\sqrt{x^{2}-1}},|x|1$ Put $x=\operatorname{cosec} \theta \Rightarrow \theta=\operatorname{cosec}^{-1} x$ $\therefore \tan ^{-1} \frac{1}{\sqrt{x^{2}-1}}=\tan ^{-1} \frac{1}{\sqrt{\operatorname{cosec}^{2} \theta-1}}$ $=\tan ^{-1}\left(\frac{1}{\cot \theta}\right)=\tan ^{-1}(\tan \theta)$ $=\theta=\operatorname{cosec}^{-1} x=\frac{\pi}{2}-\sec ^{-1} x \quad\left[\operatornam...
Read More →Express the given complex number in the form a + ib:
Question: Express the given complex number in the form a + ib: $\left(\frac{1}{3}+3 i\right)^{3}$ Solution: $\left(\frac{1}{3}+3 i\right)^{3}=\left(\frac{1}{3}\right)^{3}+(3 i)^{3}+3\left(\frac{1}{3}\right)(3 i)\left(\frac{1}{3}+3 i\right)$ $=\frac{1}{27}+27 i^{3}+3 i\left(\frac{1}{3}+3 i\right)$ $=\frac{1}{27}+27(-i)+i+9 i^{2} \quad\left[i^{3}=-i\right]$ $=\frac{1}{27}-27 i+i-9 \quad\left[i^{2}=-1\right]$ $=\left(\frac{1}{27}-9\right)+i(-27+1)$ $=\frac{-242}{27}-26 i$...
Read More →The ionization constant of benzoic acid is 6.46 × 10–5 and Ksp for silver benzoate is 2.5 × 10–13.
Question: The ionization constant of benzoic acid is $6.46 \times 10^{-5}$ and $\mathrm{Ksp}$ for silver benzoate is $2.5 \times 10^{-13}$. How many times is silver benzoate more soluble in a buffer of pH $3.19$ compared to its solubility in pure water? Solution: Since $\mathrm{pH}=3.19$. $\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]=6.46 \times 10^{-4} \mathrm{M}$ $\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COOH}+\mathrm{H}_{2} \mathrm{O} \longleftrightarrow \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{COO}^...
Read More →Express the given complex number in the form a + ib: (1 – i)4
Question: Express the given complex number in the form $a+i b:(1-i)^{4}$ Solution: $(1-i)^{4}=\left[(1-i)^{2}\right]^{2}$ $=\left[1^{2}+i^{2}-2 i\right]^{2}$ $=[1-1-2 i]^{2}$ $=(-2 i)^{2}$ $=(-2 i) \times(-2 i)$ $=4 i^{2}=-4 \quad\left[i^{2}=-1\right]$...
Read More →Write the function in the simplest form:
Question: Write the function in the simplest form: $\tan ^{-1} \frac{\sqrt{1+x^{2}}-1}{x}, x \neq 0$ Solution: $\tan ^{-1} \frac{\sqrt{1+x^{2}}-1}{x}$ Put $x=\tan \theta \Rightarrow \theta=\tan ^{-1} x$ $\therefore \tan ^{-1} \frac{\sqrt{1+x^{2}}-1}{x}=\tan ^{-1}\left(\frac{\sqrt{1+\tan ^{2} \theta}-1}{\tan \theta}\right)$ $=\tan ^{-1}\left(\frac{\sec \theta-1}{\tan \theta}\right)=\tan ^{-1}\left(\frac{1-\cos \theta}{\sin \theta}\right)$ $=\tan ^{-1}\left(\frac{2 \sin ^{2} \frac{\theta}{2}}{2 \s...
Read More →Express the given complex number in the form a + ib:
Question: Express the given complex number in the forma+ib:$\left[\left(\frac{1}{3}+i \frac{7}{3}\right)+\left(4+i \frac{1}{3}\right)\right]-\left(-\frac{4}{3}+i\right)$ Solution: $\left[\left(\frac{1}{3}+i \frac{7}{3}\right)+\left(4+i \frac{1}{3}\right)\right]-\left(\frac{-4}{3}+i\right)$ $=\frac{1}{3}+\frac{7}{3} i+4+\frac{1}{3} i+\frac{4}{3}-i$ $=\left(\frac{1}{3}+4+\frac{4}{3}\right)+i\left(\frac{7}{3}+\frac{1}{3}-1\right)$ $=\frac{17}{3}+i \frac{5}{3}$...
Read More →What is the area of the plates of a 2 F parallel plate capacitor,
Question: What is the area of the plates of a 2 F parallel plate capacitor, given that the separation between the plates is 0.5 cm? [You will realize from your answer why ordinary capacitors are in the range of F or less. However, electrolytic capacitors do have a much larger capacitance (0.1 F) because of very minute separation between the conductors.] Solution: Capacitance of a parallel capacitor,V= 2 F Distance between the two plates,d= 0.5 cm = 0.5 102m Capacitance of a parallel plate capaci...
Read More →Express the given complex number in the form a + ib:
Question: Express the given complex number in the form $a+i b:\left(\frac{1}{5}+i \frac{2}{5}\right)-\left(4+i \frac{5}{2}\right)$ Solution: $\left(\frac{1}{5}+i \frac{2}{5}\right)-\left(4+i \frac{5}{2}\right)$ $=\frac{1}{5}+\frac{2}{5} i-4-\frac{5}{2} i$ $=\left(\frac{1}{5}-4\right)+i\left(\frac{2}{5}-\frac{5}{2}\right)$ $=\frac{-19}{5}+i\left(\frac{-21}{10}\right)$ $=\frac{-19}{5}-\frac{21}{10} i$...
Read More →Express the given complex number in the form a + ib: (1 – i) – (–1 + i6)
Question: Express the given complex number in the form $a+i b:(1-i)-(-1+i 6)$ Solution: $(1-i)-(-1+i 6)=1-i+1-6 i$ $=2-7 i$...
Read More →An electrical technician requires a capacitance of 2 µF in a circuit across a potential difference of 1 kV.
Question: An electrical technician requires a capacitance of 2 F in a circuit across a potential difference of 1 kV. A large number of 1 F capacitors are available to him each of which can withstand a potential difference of not more than 400 V. Suggest a possible arrangement that requires the minimum number of capacitors. Solution: Total required capacitance,C= 2 F Potential difference,V= 1 kV = 1000 V Capacitance of each capacitor,C1= 1F Each capacitor can withstand a potential difference,V1= ...
Read More →Prove
Question: Prove $2 \tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{1}{7}=\tan ^{-1} \frac{31}{17}$ Solution: To prove: $2 \tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{1}{7}=\tan ^{-1} \frac{31}{17}$ L.H.S. $=2 \tan ^{-1} \frac{1}{2}+\tan ^{-1} \frac{1}{7}$ $=\tan ^{-1} \frac{2 \cdot \frac{1}{2}}{1-\left(\frac{1}{2}\right)^{2}}+\tan ^{-1} \frac{1}{7} \quad\left[2 \tan ^{-1} x=\tan ^{-1} \frac{2 x}{1-x^{2}}\right]$ $=\tan ^{-1} \frac{1}{\left(\frac{3}{4}\right)}+\tan ^{-1} \frac{1}{7}$ $=\tan ^{-1} \frac{4}{3}...
Read More →Express the given complex number in the form a + ib: 3(7 + i7) + i(7 + i7)
Question: Express the given complex number in the form $a+i b: 3(7+i 7)+i(7+i 7)$ Solution: $3(7+i 7)+i(7+i 7)=21+21 i+7 i+7 i^{2}$ $=21+28 i+7 \times(-1)$ $\left[\because i^{2}=-1\right]$ $=14+28 i$...
Read More →Equal volumes of 0.002 M solutions of sodium iodate and cupric chlorate are mixed together.
Question: Equal volumes of $0.002 \mathrm{M}$ solutions of sodium iodate and cupric chlorate are mixed together. Will it lead to precipitation of copper iodate? (For cupric iodate $K_{\text {sp }}=7.4 \times 10^{-8}$ ). Solution: When equal volumes of sodium iodate and cupric chlorate solutions are mixed together, then the molar concentrations of both solutions are reduced to half i.e., 0.001 M. Then, Now, the solubility equilibrium for copper iodate can be written as: Ionic product of copper io...
Read More →The solubility product constant of Ag2CrO4 and AgBr are 1.1 × 10–12 and 5.0 × 10–13 respectively.
Question: The solubility product constant of $\mathrm{Ag}_{2} \mathrm{CrO}_{4}$ and $\mathrm{AgBr}$ are $1.1 \times 10^{-12}$ and $5.0 \times 10^{-13}$ respectively, Calculate the ratio of the molarities of their saturated solutions. Solution: Let $s$ be the solubility of $\mathrm{Ag}_{2} \mathrm{CrO}_{4}$. Then, $\mathrm{Ag}_{2} \mathrm{CrO}_{4} \longleftrightarrow \mathrm{Ag}^{2+}+2 \mathrm{CrO}_{4}^{-}$ $K_{s p}=(2 s)^{2} \cdot s=4 s^{3}$ $1.1 \times 10^{-12}=4 s^{3}$ $s=6.5 \times 10^{-5} \m...
Read More →Express the given complex number in the form a + ib: i–39
Question: Express the given complex number in the form $a+i b: i^{-39}$ Solution: $i^{-39}=i^{-4 \times 9-3}=\left(i^{4}\right)^{-9} \cdot i^{-3}$ $=(1)^{-9} \cdot i^{-3} \quad\left[i^{4}=1\right]$ $=\frac{1}{i^{3}}=\frac{1}{-i} \quad\left[i^{3}=-i\right]$ $=\frac{-1}{i} \times \frac{i}{j}$ $=\frac{-i}{i^{2}}=\frac{-i}{-1}=i \quad\left[i^{2}=-1\right]$...
Read More →Express the given complex number in the form a + ib: i9 + i19
Question: Express the given complex number in the form $a+i b: i^{9}+i^{19}$ Solution: $i^{9}+i^{19}=i^{4 \times 2+1}+i^{4 \times 4+3}$ $=\left(i^{4}\right)^{2} \cdot i+\left(i^{4}\right)^{4} \cdot i^{3}$ $=1 \times i+1 \times(-i) \quad\left[i^{4}=1, i^{3}=-i\right]$ $=i+(-i)$ $=0$...
Read More →Determine the solubilities of silver chromate, barium chromate, ferric hydroxide,
Question: Determine the solubilities of silver chromate, barium chromate, ferric hydroxide, lead chloride and mercurous iodide at 298K from their solubility product constants given in Table 7.9 (page 221). Determine also the molarities of individual ions. Solution: (1) Silver chromate: $\mathrm{Ag}_{2} \mathrm{CrO}_{4} \longrightarrow 2 \mathrm{Ag}^{+}+\mathrm{CrO}_{4}{ }^{2-}$ Then, $K_{x p}=\left[\mathrm{Ag}^{+}\right]^{2}\left[\mathrm{CrO}_{4}{ }^{2-}\right]$ Let the solubility of $\mathrm{Ag...
Read More →Express the given complex number in the form a + ib:
Question: Express the given complex number in the form $a+i b:(5 i)\left(-\frac{3}{5} i\right)$ Solution: $(5 i)\left(\frac{-3}{5} i\right)=-5 \times \frac{3}{5} \times i \times i$ $=-3 i^{2}$ $=-3(-1) \quad\left[i^{2}=-1\right]$ $=3$...
Read More →Prove the following by using the principle of mathematical induction for all
Question: Prove the following by using the principle of mathematical induction for all $n \in \mathrm{N}$ : $(2 n+7)(n+3)^{2}$ Solution: Let the given statement be P(n), i.e., $\mathrm{P}(n):(2 n+7)(n+3)^{2}$ It can be observed that $P(n)$ is true for $n=1$ since $2.1+7=9(1+3)^{2}=16$, which is true. Let P(k) be true for some positive integerk, i.e., $(2 k+7)(k+3)^{2}$(1) We shall now prove that P(k+ 1) is true whenever P(k) is true. Consider $\{2(k+1)+7\}=(2 k+7)+2$ $\therefore\{2(k+1)+7\}=(2 k...
Read More →Prove $\tan ^{-1} \frac{2}{11}+\tan ^{-1} \frac{7}{24}=\tan ^{-1} \frac{1}{2}$
Question: Prove $\tan ^{-1} \frac{2}{11}+\tan ^{-1} \frac{7}{24}=\tan ^{-1} \frac{1}{2}$ Solution: To prove: $\tan ^{-1} \frac{2}{11}+\tan ^{-1} \frac{7}{24}=\tan ^{-1} \frac{1}{2}$ L.H.S. $=\tan ^{-1} \frac{2}{11}+\tan ^{-1} \frac{7}{24}$ $=\tan ^{-1} \frac{\frac{2}{11}+\frac{7}{24}}{1-\frac{2}{11} \cdot \frac{7}{24}}$ $\left[\tan ^{-1} x+\tan ^{-1} y=\tan ^{-1} \frac{x+y}{1-x y}\right]$ $=\tan ^{-1} \frac{\frac{48+77}{11 \times 24}}{\frac{11 \times 24-14}{11 \times 24}}$ $=\tan ^{-1} \frac{48+...
Read More →Prove the following by using the principle of mathematical induction for all n ∈ N: 41n – 14n is a multiple of 27.
Question: Prove the following by using the principle of mathematical induction for all $n \in N: 41^{n}-14^{n}$ is a multiple of 27 . Solution: Let the given statement be P(n), i.e., $P(n): 41^{n}-14^{n}$ is a multiple of 27 It can be observed that $P(n)$ is true for $n=1$ since $41^{1}-14^{1}=27$, which is a multiple of 27 . Let P(k) be true for some positive integerk, i.e., $41^{k}-14^{k}$ is a multiple of 27 $\therefore 41^{k}-14^{k}=27 m$, where $m \in \mathbf{N}$(1) We shall now prove that ...
Read More →Prove $3 \cos ^{-1} x=\cos ^{-1}\left(4 x^{3}-3 x\right), x \in\left[\frac{1}{2}, 1\right]$
Question: Prove $3 \cos ^{-1} x=\cos ^{-1}\left(4 x^{3}-3 x\right), x \in\left[\frac{1}{2}, 1\right]$ Solution: To prove: $3 \cos ^{-1} x=\cos ^{-1}\left(4 x^{3}-3 x\right), x \in\left[\frac{1}{2}, 1\right]$ Let $x=\cos \theta$. Then, $\cos ^{-1} x=\theta$ We have, R.H.S. $=\cos ^{-1}\left(4 x^{3}-3 x\right)$ $=\cos ^{-1}\left(4 \cos ^{3} \theta-3 \cos \theta\right)$ $=\cos ^{-1}(\cos 3 \theta)$ $=3 \theta$ $=3 \cos ^{-1} x$ $=$ L.H.S....
Read More →Prove the following by using the principle of mathematical induction for all n ∈ N: 32n + 2 – 8n – 9 is divisible by 8.
Question: Prove the following by using the principle of mathematical induction for all $n \in N: 3^{2 n+2}-8 n-9$ is divisible by 8 Solution: Let the given statement be P(n), i.e., $\mathrm{P}(n): 3^{2 n+2}-8 n-9$ is divisible by 8 It can be observed that $P(n)$ is true for $n=1$ since $3^{2} \times 1+2-8 \times 1-9=64$, which is divisible by 8 . Let $P(k)$ be true for some positive integer $k$, i.e., $3^{2 k+2}-8 k-9$ is divisible by 8 $\therefore 3^{2 k+2}-8 k-9=8 m ;$ where $m \in \mathbf{N}$...
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