A silver wire has a resistance of 2.1 Ω at 27.5 °C,
Question: A silver wire has a resistance of 2.1 Ω at 27.5 C, and a resistance of 2.7 Ω at 100 C. Determine the temperature coefficient of resistivity of silver. Solution: Temperature,T1= 27.5C Resistance of the silver wire atT1,R1= 2.1 Ω Temperature,T2= 100C Resistance of the silver wire atT2,R2= 2.7 Ω Temperature coefficient of silver = It is related with temperature and resistance as $\alpha=\frac{R_{2}-R_{1}}{R_{1}\left(T_{2}-T_{1}\right)}$ $=\frac{2.7-2.1}{2.1(100-27.5)}=0.0039^{\circ} \math...
Read More →Write the formulae for the following compounds:
Question: Write the formulae for the following compounds: (a) Mercury(II) chloride (b) Nickel(II) sulphate (c) Tin(IV) oxide (d) Thallium(I) sulphate (e) Iron(III) sulphate (f) Chromium(III) oxide Solution: (a) Mercury (II) chloride: $\mathrm{HgCl}_{2}$ (b) Nickel (II) sulphate: $\mathrm{NiSO}_{4}$ (c) Tin (IV) oxide: $\mathrm{SnO}_{2}$ (d) Thallium (I) sulphate: $\mathrm{Tl}_{2} \mathrm{SO}_{4}$ (e) Iron (III) sulphate: $\mathrm{Fe}_{2}\left(\mathrm{SO}_{4}\right)_{3}$ (f) Chromium (III) oxide:...
Read More →A negligibly small current is passed through a wire of length 15 m and uniform cross-section
Question: A negligibly small current is passed through a wire of length 15 m and uniform cross-section 6.0 107m2, and its resistance is measured to be 5.0 Ω. What is the resistivity of the material at the temperature of the experiment? Solution: Length of the wire,l=15 m Area of cross-section of the wire,a= 6.0 107m2 Resistance of the material of the wire,R= 5.0 Ω Resistivity of the material of the wire = Resistance is related with the resistivity as $R=\rho \frac{l}{A}$ $\rho=\frac{R A}{l}$ $=\...
Read More →Find $\sin \frac{x}{2}, \cos \frac{x}{2}$ and $\tan \frac{x}{2}$ for $\cos x=-\frac{1}{3}, x$ in quadrant III
[question] Question. Find $\sin \frac{x}{2}, \cos \frac{x}{2}$ and $\tan \frac{x}{2}$ for $\cos x=-\frac{1}{3}, x$ in quadrant III [/question] [solution] solution: Here, $x$ is in quadrant III. $\Rightarrow \frac{\pi}{2}<\frac{x}{2}<\frac{3 \pi}{4}$ Therefore, $\cos \frac{x}{2}$ and $\tan \frac{x}{2}$ are negative, whereas $\sin \frac{x}{2}$ is positive. It is given that $\cos \mathrm{x}=-\frac{1}{3}$. $\cos x=1-2 \sin ^{2} \frac{x}{2}$ $\Rightarrow \sin ^{2} \frac{x}{2}=\frac{1-\cos x}{2}$ $\Ri...
Read More →Calculate the oxidation number of sulphur, chromium and nitrogen in
Question: Calculate the oxidation number of sulphur, chromium and nitrogen in $\mathrm{H}_{2} \mathrm{SO}_{5}, \mathrm{C}_{2} \mathrm{O}_{7}^{2-}$ and $\mathrm{NO}_{3}^{-}$-. Suggest structure of these compounds. Count for the fallacy. Solution: $2(+1)+1(x)+5(-2)=0$ $\Rightarrow 2+x-10=0$ $\Rightarrow x=+8$ However, the O.N. of S cannot be +8. S has six valence electrons. Therefore, the O.N. of S cannot be more than +6. The structure of $\mathrm{H}_{2} \mathrm{SO}_{5}$ is shown as follows: Now, ...
Read More →Reduce to the standard form.
Question: Reduce $\left(\frac{1}{1-4 i}-\frac{2}{1+i}\right)\left(\frac{3-4 i}{5+i}\right)$ to the standard form. Solution: $\left(\frac{1}{1-4 i}-\frac{2}{1+i}\right)\left(\frac{3-4 i}{5+i}\right)=\left[\frac{(1+i)-2(1-4 i)}{(1-4 i)(1+i)}\right]\left[\frac{3-4 i}{5+i}\right]$ $=\left[\frac{1+i-2+8 i}{1+i-4 i-4 i^{2}}\right]\left[\frac{3-4 i}{5+i}\right]=\left[\frac{-1+9 i}{5-3 i}\right]\left[\frac{3-4 i}{5+i}\right]$ $=\left[\frac{-3+4 i+27 i-36 i^{2}}{25+5 i-15 i-3 i^{2}}\right]=\frac{33+31 i}...
Read More →At room temperature (27.0 °C) the resistance of a heating element is 100 Ω.
Question: At room temperature $\left(27.0^{\circ} \mathrm{C}\right)$ the resistance of a heating element is $100 \Omega$. What is the temperature of the element if the resistance is found to be $117 \Omega$, given that the temperature coefficient of the material of the resistor is $1.70 \times 10^{-4}{ }^{\circ} \mathrm{C}^{-1}$ Solution: Room temperature,T= 27C Resistance of the heating element atT,R= 100 Ω LetT1is the increased temperature of the filament. Resistance of the heating element atT...
Read More →For any two complex numbers z1 and z2, prove that
Question: For any two complex numbers $z_{1}$ and $z_{2}$, prove that $\operatorname{Re}\left(z_{1} z_{2}\right)=\operatorname{Re} z_{1} \operatorname{Re} z_{2}-\operatorname{Im} z_{1} \mid m z_{2}$ Solution: Let $z_{1}=x_{1}+i y_{1}$ and $z_{2}=x_{2}+i y_{2}$ $\therefore z_{1} z_{2}=\left(x_{1}+i y_{1}\right)\left(x_{2}+i y_{2}\right)$ $=x_{1}\left(x_{2}+i y_{2}\right)+i y_{1}\left(x_{2}+i y_{2}\right)$ $=x_{1} x_{2}+i x_{1} y_{2}+i y_{1} x_{2}+i^{2} y_{1} y_{2}$ $=x_{1} x_{2}+i x_{1} y_{2}+i y...
Read More →Find the values of
Question: Find the values of $\tan ^{-1} \sqrt{3}-\cot ^{-1}(-\sqrt{3})$ is equal to (A) $\pi$ (B) $-\frac{\pi}{2}$ (C) 0 (D) $2 \sqrt{3}$ Solution: Let $\tan ^{-1} \sqrt{3}=x$. Then, $\tan x=\sqrt{3}=\tan \frac{\pi}{3}$ where $\frac{\pi}{3} \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. We know that the range of the principal value branch of $\tan ^{-1}$ is $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$. $\therefore \tan ^{-1} \sqrt{3}=\frac{\pi}{3}$ Let $\cot ^{-1}(-\sqrt{3})=y$ Then, $\cot y=-...
Read More →(a) Three resistors 2 Ω, 4 Ω and 5 Ω are combined in parallel.
Question: (a)Three resistors 2 Ω, 4 Ω and 5 Ω are combined in parallel. What is the total resistance of the combination? (b)If the combination is connected to a battery of emf 20 V and negligible internal resistance, determine the current through each resistor, and the total current drawn from the battery. Solution: (a)There are three resistors of resistances, R1= 2 Ω,R2= 4 Ω, andR3= 5 Ω They are connected in parallel. Hence, total resistance (R) of the combination is given by, $\frac{1}{R}=\fra...
Read More →Evaluate:
Question: Evaluate: $\left[i^{18}+\left(\frac{1}{i}\right)^{25}\right]^{3}$ Solution: $\left[i^{18}+\left(\frac{1}{i}\right)^{25}\right]^{3}$ $=\left[i^{4 \times 4+2}+\frac{1}{i^{4 \times 6+1}}\right]^{3}$ $=\left[\left(i^{4}\right)^{4} \cdot i^{2}+\frac{1}{\left(i^{4}\right)^{6} \cdot i}\right]^{3}$ $=\left[i^{2}+\frac{1}{i}\right]^{3} \quad\left[i^{4}=1\right]$ $=\left[-1+\frac{1}{i} \times \frac{i}{i}\right]^{3} \quad\left[i^{2}=-1\right]$ $=\left[-1+\frac{i}{i^{2}}\right]^{3}$ $=[-1-i]^{3}$ ...
Read More →Fluorine reacts with ice and results in the change:
Question: Fluorine reacts with ice and results in the change: $\mathrm{H}_{2} \mathrm{O}(\mathrm{s})+\mathrm{F}_{2}(\mathrm{~g}) \rightarrow \mathrm{HF}(\mathrm{g})+\mathrm{HOF}(\mathrm{g})$ Justify that this reaction is a redox reaction. Solution: Let us write the oxidation number of each atom involved in the given reaction above its symbol as: Here, we have observed that the oxidation number of $\mathrm{F}$ increases from 0 in $\mathrm{F}_{2}$ to $+1$ in $\mathrm{HOF}$. Also, the oxidation num...
Read More →(a) Three resistors 1 Ω, 2 Ω, and 3 Ω are combined in series.
Question: (a)Three resistors 1 Ω, 2 Ω, and 3 Ω are combined in series. What is the total resistance of the combination? (b)If the combination is connected to a battery of emf 12 V and negligible internal resistance, obtain the potential drop across each resistor. Solution: (a)Three resistors of resistances 1 Ω, 2 Ω, and 3 Ω are combined in series. Total resistance of the combination is given by the algebraic sum of individual resistances. Total resistance = 1 + 2 + 3 = 6 Ω (b)Current flowing thr...
Read More →Solve the equation
Question: Solve the equation $x^{2}+\frac{x}{\sqrt{2}}+1=0$ Solution: The given quadratic equation is $x^{2}+\frac{x}{\sqrt{2}}+1=0$ This equation can also be written as $\sqrt{2} x^{2}+x+\sqrt{2}=0$ On comparing this equation with $a x^{2}+b x+c=0$, we obtain $a=\sqrt{2}, b=1$, and $c=\sqrt{2}$ $\therefore$ Discriminant $(D)=b^{2}-4 a c=1^{2}-4 \times \sqrt{2} \times \sqrt{2}=1-8=-7$ Therefore, the required solutions are $\frac{-b \pm \sqrt{D}}{2 a}=\frac{-1 \pm \sqrt{-7}}{2 \sqrt{2}}=\frac{-1 ...
Read More →Find the values of
Question: Find the values of $\sin \left(\frac{\pi}{3}-\sin ^{-1}\left(-\frac{1}{2}\right)\right)$ is equal to (A) $\frac{1}{2}$ (B) $\frac{1}{3}$ (C) $\frac{1}{4}$ (D) 1 Solution: Let $\sin ^{-1}\left(\frac{-1}{2}\right)=x$. Then, $\sin x=\frac{-1}{2}=-\sin \frac{\pi}{6}=\sin \left(\frac{-\pi}{6}\right)$ We know that the range of the principal value branch of $\sin ^{-1}$ is $\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]$. $\therefore \sin ^{-1}\left(\frac{-1}{2}\right)=\frac{-\pi}{6}$ $\therefore...
Read More →Justify that the following reactions are redox reactions:
Question: Justify that the following reactions are redox reactions: (a) $\mathrm{CuO}(\mathrm{s})+\mathrm{H}_{2}(\mathrm{~g}) \rightarrow \mathrm{Cu}(\mathrm{s})+\mathrm{H}_{2} \mathrm{O}(\mathrm{g})$ (b) $\mathrm{Fe}_{2} \mathrm{O}_{3}(\mathrm{~s})+3 \mathrm{CO}(\mathrm{g}) \rightarrow 2 \mathrm{Fe}(\mathrm{s})+3 \mathrm{CO}_{2}(\mathrm{~g})$ (c) $4 \mathrm{BCl}_{3}(\mathrm{~g})+3 \mathrm{LiAlH}_{4}(\mathrm{~s}) \rightarrow 2 \mathrm{~B}_{2} \mathrm{H}_{6}(\mathrm{~g})+3 \mathrm{LiCl}(\mathrm{s...
Read More →(a) Three resistors 1 Ω, 2 Ω, and 3 Ω are combined in series.
Question: (a)Three resistors 1 Ω, 2 Ω, and 3 Ω are combined in series. What is the total resistance of the combination? (b)If the combination is connected to a battery of emf 12 V and negligible internal resistance, obtain the potential drop across each resistor. Solution: (a)Three resistors of resistances 1 Ω, 2 Ω, and 3 Ω are combined in series. Total resistance of the combination is given by the algebraic sum of individual resistances. Total resistance = 1 + 2 + 3 = 6 Ω (b)Current flowing thr...
Read More →Solve the equation
Question: Solve the equation $x^{2}+x+\frac{1}{\sqrt{2}}=0$ Solution: The given quadratic equation is $x^{2}+x+\frac{1}{\sqrt{2}}=0$ This equation can also be written as $\sqrt{2} x^{2}+\sqrt{2} x+1=0$ On comparing this equation with $a x^{2}+b x+c=0$, we obtain $a=\sqrt{2}, b=\sqrt{2}$, and $c=1$ $\therefore$ Discri min ant $(D)=b^{2}-4 a c=(\sqrt{2})^{2}-4 \times(\sqrt{2}) \times 1=2-4 \sqrt{2}$ Therefore, the required solutions are $\frac{-b \pm \sqrt{D}}{2 a}=\frac{-\sqrt{2} \pm \sqrt{2-4 \s...
Read More →Solve the equation
Question: Solve the equation $\sqrt{3} x^{2}-\sqrt{2} x+3 \sqrt{3}=0$ Solution: The given quadratic equation is $\sqrt{3} x^{2}-\sqrt{2} x+3 \sqrt{3}=0$ On comparing the given equation with $a x^{2}+b x+c=0$, we obtain $a=\sqrt{3}, b=-\sqrt{2}$, and $c=3 \sqrt{3}$ Therefore, the discriminant of the given equation is $D=b^{2}-4 a c=(-\sqrt{2})^{2}-4(\sqrt{3})(3 \sqrt{3})=2-36=-34$ Therefore, the required solutions are $\frac{-b \pm \sqrt{\mathrm{D}}}{2 a}=\frac{-(-\sqrt{2}) \pm \sqrt{-34}}{2 \tim...
Read More →A battery of emf 10 V and internal resistance 3 Ω is connected to a resistor.
Question: A battery of emf 10 V and internal resistance 3 Ω is connected to a resistor. If the current in the circuit is 0.5 A, what is the resistance of the resistor? What is the terminal voltage of the battery when the circuit is closed? Solution: Emf of the battery,E= 10 V Internal resistance of the battery,r= 3 Ω Current in the circuit,I= 0.5 A Resistance of the resistor =R The relation for current using Ohms law is, $I=\frac{E}{R+r}$ $R+r=\frac{E}{I}$ $=\frac{10}{0.5}=20 \Omega$ $\therefore...
Read More →Solve the equation
Question: Solve the equation $\sqrt{2} x^{2}+x+\sqrt{2}=0$ Solution: The given quadratic equation is $\sqrt{2} x^{2}+x+\sqrt{2}=0$ On comparing the given equation with $a x^{2}+b x+c=0$, we obtain $a=\sqrt{2}, b=1$, and $c=\sqrt{2}$ Therefore, the discriminant of the given equation is $D=b^{2}-4 a c=1^{2}-4 \times \sqrt{2} \times \sqrt{2}=1-8=-7$ Therefore, the required solutions are $\frac{-b \pm \sqrt{\mathrm{D}}}{2 a}=\frac{-1 \pm \sqrt{-7}}{2 \times \sqrt{2}}=\frac{-1 \pm \sqrt{7} i}{2 \sqrt...
Read More →Solve the equation x2 – x + 2 = 0
Question: Solve the equation $x^{2}-x+2=0$ Solution: The given quadratic equation is $x^{2}-x+2=0$ On comparing the given equation with $a x^{2}+b x+c=0$, we obtain $a=1, b=-1$, and $c=2$ Therefore, the discriminant of the given equation is $D=b^{2}-4 a c=(-1)^{2}-4 \times 1 \times 2=1-8=-7$ Therefore, the required solutions are $\frac{-b \pm \sqrt{D}}{2 a}=\frac{-(-1) \pm \sqrt{-7}}{2 \times 1}=\frac{1 \pm \sqrt{7} i}{2}$ $[\sqrt{-1}=i]$...
Read More →The storage battery of a car has an emf of 12 V.
Question: The storage battery of a car has an emf of 12 V. If the internal resistance of the battery is 0.4Ω, what is the maximum current that can be drawn from the battery? Solution: Emf of the battery,E= 12 V Internal resistance of the battery,r= 0.4 Ω Maximum current drawn from the battery =I According to Ohms law, $E=I r$ $I=\frac{E}{r}$ $=\frac{12}{0.4}=30 \mathrm{~A}$ The maximum current drawn from the given battery is 30 A....
Read More →Find the values of
Question: Find the values of $\cos ^{-1}\left(\cos \frac{7 \pi}{6}\right)$ is equal to (A) $\frac{7 \pi}{6}$ (B) $\frac{5 \pi}{6}$ (C) $\frac{\pi}{3}$ (D) $\frac{\pi}{6}$ Solution: We know that $\cos ^{-1}(\cos x)=x$ if $x \in[0, \pi]$, which is the principal value branch of $\cos ^{-1} x$. Here, $\frac{7 \pi}{6} \notin x \in[0, \pi]$. Now, $\cos ^{-1}\left(\cos \frac{7 \pi}{6}\right)$ can be written as: $\cos ^{-1}\left(\cos \frac{7 \pi}{6}\right)=\cos ^{-1}\left[\cos \left(\pi+\frac{\pi}{6}\ri...
Read More →Solve the equation x2 + 3x + 5 = 0
Question: Solve the equation $x^{2}+3 x+5=0$ Solution: The given quadratic equation is $x^{2}+3 x+5=0$ On comparing the given equation with $a x^{2}+b x+c=0$, we obtain $a=1, b=3$, and $c=5$ Therefore, the discriminant of the given equation is $\mathrm{D}=b^{2}-4 a c=3^{2}-4 \times 1 \times 5=9-20=-11$ Therefore, the required solutions are $\frac{-b \pm \sqrt{D}}{2 a}=\frac{-3 \pm \sqrt{-11}}{2 \times 1}=\frac{-3 \pm \sqrt{11} i}{2}$ $[\sqrt{-1}=i]$...
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