Consider the reactions:

Question: Consider the reactions: (a) $\mathrm{H}_{3} \mathrm{PO}_{2}$ (aq) $+4 \mathrm{AgNO}_{3}(\mathrm{aq})+2 \mathrm{H}_{2} \mathrm{O}(\mathrm{I}) \rightarrow \mathrm{H}_{3} \mathrm{PO}_{4}(\mathrm{aq})+4 \mathrm{Ag}(\mathrm{s})+4 \mathrm{HNO}_{3}(\mathrm{aq})$ (b) $\mathrm{H}_{3} \mathrm{PO}_{2}$ (aq) $+2 \mathrm{CuSO}_{4}$ (aq) $+2 \mathrm{H}_{2} \mathrm{O}$ (I) $\rightarrow \mathrm{H}_{3} \mathrm{PO}_{4}$ (aq) $+2 \mathrm{Cu}$ (s) $+\mathrm{H}_{2} \mathrm{SO}_{4}$ (aq) (c) $\mathrm{C}_{6}...

Prove

Question: Prove $\tan ^{-1} \sqrt{x}=\frac{1}{2} \cos ^{-1}\left(\frac{1-x}{1+x}\right), x \in[0,1]$ Solution: Let $x=\tan ^{2} \theta$. Then, $\sqrt{x}=\tan \theta \Rightarrow \theta=\tan ^{-1} \sqrt{x}$. $\therefore \frac{1-x}{1+x}=\frac{1-\tan ^{2} \theta}{1+\tan ^{2} \theta}=\cos 2 \theta$ Now, we have: R.H.S. $=\frac{1}{2} \cos ^{-1}\left(\frac{1-x}{1+x}\right)=\frac{1}{2} \cos ^{-1}(\cos 2 \theta)=\frac{1}{2} \times 2 \theta=\theta=\tan ^{-1} \sqrt{x}=$ L.H.S....

(a) Given n resistors each of resistance R, how will you combine them to get the

Question: (a)Givennresistors each of resistanceR, how will you combine them to get the (i) maximum (ii) minimum effective resistance? What is the ratio of the maximum to minimum resistance? (b)Given the resistances of 1 Ω, 2 Ω, 3 Ω, how will be combine them to get an equivalent resistance of (i) (11/3) Ω (ii) (11/5) Ω, (iii) 6 Ω, (iv) (6/11) Ω? (c)Determine the equivalent resistance of networks shown in Fig. 3.31. Solution: (a)Total number of resistors =n Resistance of each resistor =R (i)Whennr...

Question: Find the modulus of $\frac{1+i}{1-i}-\frac{1-i}{1+i}$ Solution: $\frac{1+i}{1-i}-\frac{1-i}{1+i}=\frac{(1+i)^{2}-(1-i)^{2}}{(1-i)(1+i)}$ $=\frac{1+i^{2}+2 i-1-i^{2}+2 i}{1^{2}+1^{2}}$ $=\frac{4 i}{2}=2 i$ $\therefore\left|\frac{1+i}{1-i}-\frac{1-i}{1+i}\right|=|2 i|=\sqrt{2^{2}}=2$...

Why does the following reaction occur?

Question: Why does the following reaction occur? $\mathrm{XeO}_{6}^{4-}(\mathrm{aq})+2 \mathrm{~F}^{-}(\mathrm{aq})+6 \mathrm{H}^{+}(\mathrm{aq}) \rightarrow \mathrm{XeO}_{3}(\mathrm{~g})+\mathrm{F}_{2}(\mathrm{~g})+3 \mathrm{H}_{2} \mathrm{O}(\mathrm{l})$ What conclusion about the compound $\mathrm{Na}_{4} \mathrm{XeO}_{6}$ (of which $\mathrm{XeO}_{6}^{4-}$ is a part) can be drawn from the reaction. Solution: The given reaction occurs because $\mathrm{XeO}_{6}^{4}$ oxidises $\mathrm{F}^{-}$and ...

Prove

Question: Prove $\tan ^{-1} \frac{1}{5}+\tan ^{-1} \frac{1}{7}+\tan ^{-1} \frac{1}{3}+\tan ^{-1} \frac{1}{8}=\frac{\pi}{4}$ Solution: L.H.S. $=\tan ^{-1} \frac{1}{5}+\tan ^{-1} \frac{1}{7}+\tan ^{-1} \frac{1}{3}+\tan ^{-1} \frac{1}{8}$ $=\tan ^{-1}\left(\frac{\frac{1}{5}+\frac{1}{7}}{1-\frac{1}{5} \times \frac{1}{7}}\right)+\tan ^{-1}\left(\frac{\frac{1}{3}+\frac{1}{8}}{1-\frac{1}{3} \times \frac{1}{8}}\right) \quad\left[\tan ^{-1} x+\tan ^{-1} y=\tan ^{-1} \frac{x+y}{1-x y}\right]$ $=\tan ^{-1}...

(a) Given n resistors each of resistance R, how will you combine them to get the

Question: (a)Givennresistors each of resistanceR, how will you combine them to get the (i) maximum (ii) minimum effective resistance? What is the ratio of the maximum to minimum resistance? (b)Given the resistances of 1 Ω, 2 Ω, 3 Ω, how will be combine them to get an equivalent resistance of (i) (11/3) Ω (ii) (11/5) Ω, (iii) 6 Ω, (iv) (6/11) Ω? (c)Determine the equivalent resistance of networks shown in Fig. 3.31. Solution: (a)Total number of resistors =n Resistance of each resistor =R (i)Whennr...

Is zero a rational number? Justify.

Question: Is zero a rational number? Justify. Solution: Yes, 0 is a rational number.0 can be expressed in the form of the fraction p/q, wherep=0andqcan be any integer except 0....

Prove

Question: Prove $\tan ^{-1} \frac{63}{16}=\sin ^{-1} \frac{5}{13}+\cos ^{-1} \frac{3}{5}$ Solution: Let $\sin ^{-1} \frac{5}{13}=x$. Then, $\sin x=\frac{5}{13} \Rightarrow \cos x=\frac{12}{13}$. $\therefore \tan x=\frac{5}{12} \Rightarrow x=\tan ^{-1} \frac{5}{12}$ $\therefore \sin ^{-1} \frac{5}{13}=\tan ^{-1} \frac{5}{12}$.....(1) Let $\cos ^{-1} \frac{3}{5}=y$. Then, $\cos y=\frac{3}{5} \Rightarrow \sin y=\frac{4}{5}$. $\therefore \tan y=\frac{4}{3} \Rightarrow y=\tan ^{-1} \frac{4}{3}$ $\the...

Justify giving reactions that among halogens, fluorine is the best oxidant and among hydrohalic compounds,

Question: Justify giving reactions that among halogens, fluorine is the best oxidant and among hydrohalic compounds, hydroiodic acid is the best reductant. Solution: $\mathrm{F}_{2}$ can oxidize $\mathrm{Cl}^{-}$to $\mathrm{Cl}_{2}$, Br to $\mathrm{Br}_{2}$, and $\mathrm{I}^{-}$to $\mathrm{I}_{2}$ as: $\mathrm{F}_{2(\text { av })}+2 \mathrm{Cl}_{(s)}^{-} \longrightarrow 2 \mathrm{~F}_{(\text {aq })}^{-}+\mathrm{Cl}_{(g)}$ $\mathrm{F}_{2(a q)}+2 \mathrm{Br}_{(a q)}^{-} \longrightarrow 2 \mathrm{~...

Find the real numbers x and y if (x – iy) (3 + 5i) is the conjugate of –6 – 24i.

Question: Find the real numbers $x$ and $y$ if $(x-i y)(3+5 i)$ is the conjugate of $-6-24 i$. Solution: Let $z=(x-i y)(3+5 i)$ $z=3 x+5 x i-3 y i-5 y i^{2}=3 x+5 x i-3 y i+5 y=(3 x+5 y)+i(5 x-3 y)$ $\therefore \bar{z}=(3 x+5 y)-i(5 x-3 y)$ It is given that, $\bar{z}=-6-24 i$ $\therefore(3 x+5 y)-i(5 x-3 y)=-6-24 i$ Equating real and imaginary parts, we obtain $3 x+5 y=-6$ (i) $5 x-3 y=24$ $\ldots$ (ii) Multiplying equation (i) by 3 and equation (ii) by 5 and then adding them, we obtain $9 x+15 ...

Consider the reactions:

Question: Consider the reactions: $2 \mathrm{~S}_{2} \mathrm{O}_{3}^{2-}(\mathrm{aq})+\mathrm{I}_{2}(\mathrm{~s}) \rightarrow \mathrm{S}_{4} \mathrm{O}_{6}^{2-}(\mathrm{aq})+2 \mathrm{I}^{-}(\mathrm{aq})$ $\left.\mathrm{S}_{2} \mathrm{O}_{3}^{2-} \mathrm{aq}\right)+2 \mathrm{Br}_{2}(\mathrm{l})+5 \mathrm{H}_{2} \mathrm{O}(\mathrm{I}) \rightarrow 2 \mathrm{SO}_{4}^{2-}(\mathrm{aq})+4 \mathrm{Br}(\mathrm{aq})+10 \mathrm{H}^{+}(\mathrm{aq})$ Why does the same reductant, thiosulphate react different...

Prove

Question: Prove $\cos ^{-1} \frac{12}{13}+\sin ^{-1} \frac{3}{5}=\sin ^{-1} \frac{56}{65}$ Solution: Let $\sin ^{-1} \frac{3}{5}=x$. Then, $\sin x=\frac{3}{5} \Rightarrow \cos x=\sqrt{1-\left(\frac{3}{5}\right)^{2}}=\sqrt{\frac{16}{25}}=\frac{4}{5}$. $\therefore \tan x=\frac{3}{4} \Rightarrow x=\tan ^{-1} \frac{3}{4}$ $\therefore \sin ^{-1} \frac{3}{5}=\tan ^{-1} \frac{3}{4}$.....(1) Now, let $\cos ^{-1} \frac{12}{13}=y$. Then, $\cos y=\frac{12}{13} \Rightarrow \sin y=\frac{5}{13}$. $\therefore ...

Choose the correct alternative:

Question: Choose the correct alternative: (a)Alloys of metals usually have (greater/less) resistivity than that of their constituent metals. (b)Alloys usually have much (lower/higher) temperature coefficients of resistance than pure metals. (c)The resistivity of the alloy manganin is nearly independent of/increases rapidly with increase of temperature. (d)The resistivity of a typical insulator (e.g., amber) is greater than that of a metal by a factor of the order of (1022/103) Solution: (a)Alloy...

Answer the following questions:

Question: Answer the following questions: (a)A steady current flows in a metallic conductor of non-uniform cross- section. Which of these quantities is constant along the conductor: current, current density, electric field, drift speed? (b)Is Ohms law universally applicable for all conducting elements? If not, give examples of elements which do not obey Ohms law. (c)A low voltage supply from which one needs high currents must have very low internal resistance. Why? (d)A high tension (HT) supply ...

Find the modulus and argument of the complex number.

Question: Find the modulus and argument of the complex number$\frac{1+2 i}{1-3 i}$ Solution: Let $z=\frac{1+2 i}{1-3 i}$, then $z=\frac{1+2 i}{1-3 i} \times \frac{1+3 i}{1+3 i}=\frac{1+3 i+2 i+6 i^{2}}{1^{2}+3^{2}}=\frac{1+5 i+6(-1)}{1+9}$ $=\frac{-5+5 i}{10}=\frac{-5}{10}+\frac{5 i}{10}=\frac{-1}{2}+\frac{1}{2} i$ Let $z=r \cos \theta+i r \sin \theta$ i.e., $r \cos \theta=\frac{-1}{2}$ and $r \sin \theta=\frac{1}{2}$ On squaring and adding, we obtain $r^{2}\left(\cos ^{2} \theta+\sin ^{2} \thet...

What conclusion can you draw from the following observations on a resistor made of alloy manganin?

Question: What conclusion can you draw from the following observations on a resistor made of alloy manganin? Solution: It can be inferred from the given table that the ratio of voltage with current is a constant, which is equal to 19.7. Hence, manganin is an ohmic conductor i.e., the alloy obeys Ohms law. According to Ohms law, the ratio of voltage with current is the resistance of the conductor. Hence, the resistance of manganin is 19.7 Ω....

Let . Find

Question: Let $z_{1}=2-i, z_{2}=-2+i$. Find (i) $\operatorname{Re}\left(\frac{\mathrm{z}_{1} \mathrm{z}_{2}}{\overline{\mathrm{z}}_{1}}\right)$, (ii) $\operatorname{Im}\left(\frac{1}{z_{1} \bar{z}_{1}}\right)$ Solution: $z_{1}=2-i, z_{2}=-2+i$ (i) $\mathrm{z}_{1} \mathrm{z}_{2}=(2-\mathrm{i})(-2+\mathrm{i})=-4+2 \mathrm{i}+2 \mathrm{i}-\mathrm{i}^{2}=-4+4 \mathrm{i}-(-1)=-3+4 \mathrm{i}$ $\bar{z}_{1}=2+i$ $\therefore \frac{\mathrm{z}_{1} \mathrm{z}_{2}}{\overline{\mathrm{z}}_{1}}=\frac{-3+4 \mat...

Two wires of equal length, one of aluminium and the other of copper have the same resistance.

Question: Two wires of equal length, one of aluminium and the other of copper have the same resistance. Which of the two wires is lighter? Hence explain why aluminium wires are preferred for overhead power cables. $\left(\rho_{A l}=2.63 \times 10^{-8} \Omega \mathrm{m}, \rho_{C u}=1.72 \times 10^{-8} \Omega \mathrm{m}\right.$, Relative density of Al $=$ $2.7$, of $\mathrm{Cu}=8.9$.) Solution: Resistivity of aluminium,Al= 2.63 108Ω m Relative density of aluminium,d1= 2.7 Letl1be the length of alu...

Identify the substance oxidised, reduced, oxidising agent and reducing agent for each of the following reactions:

Question: Identify the substance oxidised, reduced, oxidising agent and reducing agent for each of the following reactions: (a) $2 \mathrm{AgBr}(\mathrm{s})+\mathrm{C}_{6} \mathrm{H}_{6} \mathrm{O}_{2}(\mathrm{aq}) \rightarrow 2 \mathrm{Ag}(\mathrm{s})+2 \mathrm{HBr}(\mathrm{aq})+\mathrm{C}_{6} \mathrm{H}_{4} \mathrm{O}_{2}(\mathrm{aq})$ (b) $\mathrm{HCHO}(\mathrm{I})+2\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}\right]^{+}(\mathrm{aq})+3 \mathrm{OH}^{-}(\mathrm{aq}) \rightarrow 2 \mathrm{A...

Prove

Question: Prove $\cos ^{-1} \frac{4}{5}+\cos ^{-1} \frac{12}{13}=\cos ^{-1} \frac{33}{65}$ Solution: Let $\cos ^{-1} \frac{4}{5}=x$. Then, $\cos x=\frac{4}{5} \Rightarrow \sin x=\sqrt{1-\left(\frac{4}{5}\right)^{2}}=\frac{3}{5}$. $\therefore \tan x=\frac{3}{4} \Rightarrow x=\tan ^{-1} \frac{3}{4}$ $\therefore \cos ^{-1} \frac{4}{5}=\tan ^{-1} \frac{3}{4}$....(1) Now, let $\cos ^{-1} \frac{12}{13}=y .$ Then, $\cos y=\frac{12}{13} \Rightarrow \sin y=\frac{5}{13}$. $\therefore \tan y=\frac{5}{12} \...

If a + ib =, prove that a2 + b2 =

Question: If $a+i b=\frac{(x+i)^{2}}{2 x^{2}+1}$, prove that $a^{2}+b^{2}=\frac{\left(x^{2}+1\right)^{2}}{(2 x+1)^{2}}$ Solution: $a+i b=\frac{(x+i)^{2}}{2 x^{2}+1}$ $=\frac{x^{2}+i^{2}+2 x i}{2 x^{2}+1}$ $=\frac{x^{2}-1+i 2 x}{2 x^{2}+1}$ $=\frac{x^{2}-1}{2 x^{2}+1}+i\left(\frac{2 x}{2 x^{2}+1}\right)$ On comparing real and imaginary parts, we obtain $a=\frac{x^{2}-1}{2 x^{2}+1}$ and $b=\frac{2 x}{2 x^{2}+1}$ $\therefore a^{2}+b^{2}=\left(\frac{x^{2}-1}{2 x^{2}+1}\right)^{2}+\left(\frac{2 x}{2 ...

(a) Six lead-acid type of secondary cells each of emf

Question: (a)Six lead-acid type of secondary cells each of emf 2.0 V and internal resistance 0.015 Ω are joined in series to provide a supply to a resistance of 8.5 Ω. What are the current drawn from the supply and its terminal voltage? (b)A secondary cell after long use has an emf of 1.9 V and a large internal resistance of 380 Ω. What maximum current can be drawn from the cell? Could the cell drive the starting motor of a car? Solution: (a)Number of secondary cells,n= 6 Emf of each secondary c...

How do you count for the following observations?

Question: How do you count for the following observations? (a) Though alkaline potassium permanganate and acidic potassium permanganate both are used as oxidants, yet in the manufacture of benzoic acid from toluene we use alcoholic potassium permanganate as an oxidant. Why? Write a balanced redox equation for the reaction. (b) When concentrated sulphuric acid is added to an inorganic mixture containing chloride, we get colourless pungent smelling gas HCl, but if the mixture contains bromide then...

If find .

Question: If $z_{1}=2-i, z_{2}=1+i$, find $\left|\frac{z_{1}+z_{2}+1}{z_{1}-z_{2}+1}\right|$. Solution: $z_{1}=2-i, z_{2}=1+i$ $\therefore\left|\frac{z_{1}+z_{2}+1}{z_{1}-z_{2}+1}\right|=\left|\frac{(2-i)+(1+i)+1}{(2-i)-(1+i)+1}\right|$ $=\left|\frac{4}{2-2 i}\right|=\left|\frac{4}{2(1-i)}\right|$ $=\left|\frac{2}{1-i} \times \frac{1+i}{1+i}\right|=\left|\frac{2(1+i)}{1^{2}-i^{2}}\right|$ $=\left|\frac{2(1+i)}{1+1}\right| \quad\left[i^{2}=-1\right]$ $=\left|\frac{2(1+i)}{2}\right|$ $=|1+i|=\sqrt...