If, then prove where n is any positive integer
Question: If $A=\left[\begin{array}{ll}3 -4 \\ 1 -1\end{array}\right]$, then prove $A^{n}=\left[\begin{array}{ll}1+2 n -4 n \\ n 1-2 n\end{array}\right]$ where $n$ is any positive integer Solution: It is given that $A=\left[\begin{array}{ll}3 -4 \\ 1 -1\end{array}\right]$ To prove: $\quad \mathrm{P}(n): A^{n}=\left[\begin{array}{ll}1+2 n -4 n \\ n 1-2 n\end{array}\right], n \in \mathbf{N}$ We shall prove the result by usingthe principle of mathematical induction. Forn= 1, we have: $P(1): A^{1}=\...
Read More →In a plane electromagnetic wave,
Question: In a plane electromagnetic wave, the electric field oscillates sinusoidally at a frequency of 2.0 1010Hz and amplitude 48 V m1. (a)What is the wavelength of the wave? (b)What is the amplitude of the oscillating magnetic field? (c)Show that the average energy density of theEfield equals the average energy density of theBfield. [c= 3 108m s1.] Solution: Frequency of the electromagnetic wave,= 2.0 1010Hz Electric field amplitude,E0= 48 V m1 Speed of light,c= 3 108m/s (a)Wavelength of a wa...
Read More →State as to why
Question: State as to why (a) a solution of Na2CO3is alkaline ? (b) alkali metals are prepared by electrolysis of their fused chlorides ? (c) sodium is found to be more useful than potassium ? Solution: (a)When sodium carbonate is added to water, it hydrolyses to give sodium bicarbonate and sodium hydroxide (a strong base). As a result, the solution becomes alkaline. $\mathrm{Na}_{2} \mathrm{CO}_{3}+\mathrm{H}_{2} \mathrm{O} \longrightarrow \mathrm{NaHCO}_{3}+\mathrm{NaOH}$ (b)It is not possible...
Read More →Comment on each of the following observations:
Question: Comment on each of the following observations: (a) The mobilities of the alkali metal ions in aqueous solution are Li+ Na+ K+ Rb+ Cs+ (b)Lithium is the only alkali metal to form a nitride directly. (c) $\mathrm{E}^{\circ}$ for $\mathrm{M}^{2+}(\mathrm{aq})+2 \mathrm{e}^{-} \longrightarrow \mathrm{M}(\mathrm{s})$ (where $\mathrm{M}=\mathrm{Ca}$, Sr or Ba) is nearly constant. Solution: (a)On moving down the alkali group, the ionic and atomic sizes of the metals increase. The given alkali...
Read More →In the expansion of (1 + a)m + n, prove that coefficients of am and an are equal.
Question: In the expansion of $(1+a)^{m+n}$, prove that coefficients of $a^{m}$ and $a^{n}$ are equal. Solution: It is known that $(r+1)^{\text {th }}$ term, $\left(T_{r+1}\right)$, in the binomial expansion of $(a+b)^{n}$ is given by $T_{r+1}={ }^{n} C_{r} a^{n-t} b^{z}$. Assuming that $a^{m}$ occurs in the $(r+1)^{\text {th }}$ term of the expansion $(1+a)^{m+n}$, we obtain $T_{r+1}={ }^{m+n} C_{r}(1)^{m+n-r}(a)^{r}={ }^{m+n} C_{r} a^{r}$ Comparing the indices of $a$ in $a^{m}$ and in $T_{r+1}...
Read More →The terminology of different parts of the electromagnetic spectrum is given in the text.
Question: The terminology of different parts of the electromagnetic spectrum is given in the text. Use the formulaE=h(for energy of a quantum of radiation: photon) and obtain the photon energy in units of eV for different parts of the electromagnetic spectrum. In what way are the different scales of photon energies that you obtain related to the sources of electromagnetic radiation? Solution: Energy of a photon is given as: $E=h v=\frac{h c}{\lambda}$ Where, h= Plancks constant = 6.6 1034Js c= S...
Read More →If, prove that
Question: If $A=\left[\begin{array}{lll}1 1 1 \\ 1 1 1 \\ 1 1 1\end{array}\right]$, prove that $A^{n}=\left[\begin{array}{lll}3^{n-1} 3^{n-1} 3^{n-1} \\ 3^{n-1} 3^{n-1} 3^{n-1} \\ 3^{n-1} 3^{n-1} 3^{n-1}\end{array}\right], n \in \mathbf{N}$ Solution: It is given that $A=\left[\begin{array}{lll}1 1 1 \\ 1 1 1 \\ 1 1 1\end{array}\right]$ To show: $\quad \mathrm{P}(n): A^{n}=\left[\begin{array}{lll}3^{n-1} 3^{n-1} 3^{n-1} \\ 3^{n-1} 3^{n-1} 3^{n-1} \\ 3^{n-1} 3^{n-1} 3^{n-1}\end{array}\right], n \i...
Read More →Suppose that the electric field amplitude of an electromagnetic
Question: Suppose that the electric field amplitude of an electromagnetic wave is $E_{0}=120 \mathrm{~N} / \mathrm{C}$ and that its frequency is $v=50.0$ $\mathrm{MHz}$. (a) Determine, $B_{0}, \omega, k$, and $\lambda$. (b) Find expressions for $\mathbf{E}$ and $\mathbf{B}$. Solution: Electric field amplitude,E0= 120 N/C Frequency of source,= 50.0 MHz = 50 106Hz Speed of light,c= 3 108m/s (a)Magnitude of magnetic field strength is given as: $B_{0}=\frac{E_{0}}{c}$ $=\frac{120}{3 \times 10^{8}}$ ...
Read More →Find the middle terms in the expansions of
Question: Find the middle terms in the expansions of $\left(\frac{x}{3}+9 y\right)^{10}$ Solution: It is known that in the expansion $(a+b)^{n}$, if $n$ is even, then the middle term is $\left(\frac{\mathrm{n}}{2}+1\right)^{\text {th }}$ term. Therefore, the middle term in the expansion of $\left(\frac{x}{3}+9 y\right)^{10}$ is $\left(\frac{10}{2}+1\right)^{\text {th }}=6^{\text {th }}$ term $\mathrm{T}_{6}=\mathrm{T}_{5+1}={ }^{10} \mathrm{C}_{5}\left(\frac{\mathrm{x}}{3}\right)^{10-5}(9 \mathr...
Read More →Let, show that, where I is the identity matrix of order 2 and n ∈ N
Question: Let $A=\left[\begin{array}{ll}0 1 \\ 0 0\end{array}\right]$, show that $(a I+b A)^{n}=a^{n} I+n a^{n-1} b A$, where $/$ is the identity matrix of order 2 and $n \in \mathbf{N}$ Solution: It is given that $A=\left[\begin{array}{ll}0 1 \\ 0 0\end{array}\right]$ To show: $\quad \mathrm{P}(n):(a I+b A)^{n}=a^{n} I+n a^{n-1} b A, n \in \mathbf{N}$ We shall prove the result by usingthe principle of mathematical induction. Forn= 1, we have: $\mathrm{P}(1):(a I+b A)=a I+b a^{0} A=a I+b A$ Ther...
Read More →The amplitude of the magnetic field part of a harmonic electromagnetic
Question: The amplitude of the magnetic field part of a harmonic electromagnetic wave in vacuum isB0= 510 nT. What is the amplitude of the electric field part of the wave? Solution: Amplitude of magnetic field of an electromagnetic wave in a vacuum, B0= 510 nT = 510 109T Speed of light in a vacuum,c= 3 108m/s Amplitude of electric field of the electromagnetic wave is given by the relation, E=cB0 = 3 108 510 109= 153 N/C Therefore, the electric field part of the wave is 153 N/C....
Read More →A charged particle oscillates about its mean equilibrium position with a frequency
Question: A charged particle oscillates about its mean equilibrium position with a frequency of 109Hz. What is the frequency of the electromagnetic waves produced by the oscillator? Solution: The frequency of an electromagnetic wave produced by the oscillator is the same as that of a charged particle oscillating about its mean position i.e., 109Hz....
Read More →A radio can tune in to any station in the 7.5 MHz to 12 MHz band.
Question: A radio can tune in to any station in the 7.5 MHz to 12 MHz band. What is the corresponding wavelength band? Solution: A radio can tune to minimum frequency,1= 7.5 MHz= 7.5 106Hz Maximum frequency,2= 12 MHz = 12 106Hz Speed of light,c= 3 108m/s Corresponding wavelength for1can be calculated as: $\lambda_{1}=\frac{c}{v_{1}}$ $=\frac{3 \times 10^{8}}{7.5 \times 10^{6}}=40 \mathrm{~m}$ Corresponding wavelength for2can be calculated as: $\lambda_{2}=\frac{c}{v_{2}}$ $=\frac{3 \times 10^{8}...
Read More →A plane electromagnetic wave travels in vacuum along z-direction.
Question: A plane electromagnetic wave travels in vacuum along z-direction. What can you say about the directions of its electric and magnetic field vectors? If the frequency of the wave is 30 MHz, what is its wavelength? Solution: The electromagnetic wave travels in a vacuum along the z-direction. The electric field (E) and the magnetic field (H) are in thex-yplane. They are mutually perpendicular. Frequency of the wave, = 30 MHz = 30 106s1 Speed of light in a vacuum,c= 3 108m/s Wavelength of a...
Read More →What physical quantity is the same for X-rays of wavelength
Question: What physical quantity is the same for X-rays of wavelength 1010m, red light of wavelength 6800 Å and radiowaves of wavelength 500 m? Solution: The speed of light (3 108m/s) in a vacuum is the same for all wavelengths. It is independent of the wavelength in the vacuum....
Read More →What happens when
Question: What happens when (i) sodium metal is dropped in water ? (ii) sodium metal is heated in free supply of air ? (iii) sodium peroxide dissolves in water ? Solution: (i)When Na metal is dropped in water, it reacts violently to form sodium hydroxide and hydrogen gas. The chemical equation involved in the reaction is: $2 \mathrm{Na}_{(s)}+2 \mathrm{H}_{2} \mathrm{O}_{(\prime)} \longrightarrow 2 \mathrm{NaOH}_{(a q)}+\mathrm{H}_{2(g)}$ (ii)On being heated in air, sodium reacts vigorously with...
Read More →A parallel plate capacitor (Fig. 8.7) made of circular plates each of radius R = 6.0 cm has a capacitance C = 100 pF.
Question: A parallel plate capacitor (Fig. 8.7) made of circular plates each of radiusR= 6.0 cm has a capacitanceC= 100 pF. The capacitor is connected to a 230 V ac supply with a (angular) frequency of 300 rad s1. (a)What is the rms value of the conduction current? (b)Is the conduction current equal to the displacement current? (c)Determine the amplitude ofBat a point 3.0 cm from the axis between the plates. Solution: Radius of each circular plate,R= 6.0 cm = 0.06 m Capacitance of a parallel pla...
Read More →Matrices A and B will be inverse of each other only if
Question: MatricesAandBwill be inverse of each other only if A.AB=BA C.AB= 0,BA=I B.AB=BA= 0 D.AB=BA=I Solution: Answer: D We know thatifAis a square matrix of orderm, and if there exists another square matrixBof the same orderm, such thatAB=BA=I, thenBis said to be the inverse ofA. In this case, it is clear thatAis the inverse ofB. Thus, matricesAandBwill be inverses of each other only ifAB=BA=I....
Read More →Explain the significance of sodium,
Question: Explain the significance of sodium, potassium, magnesium and calcium inbiological fluids. Solution: Importance of sodium, potassium, magnesium, and calcium in biological fluids: (i)Sodium (Na): Sodium ions are found primarily in the blood plasma. They are also found in the interstitial fluids surrounding the cells. (a)Sodium ions help in the transmission of nerve signals. (b) They help in regulating the flow of water across the cell membranes. (c) They also help in transporting sugars ...
Read More →Figure 8.6 shows a capacitor made of two circular plates each of radius 12 cm,
Question: Figure 8.6 shows a capacitor made of two circular plates each of radius 12 cm, and separated by 5.0 cm. The capacitor is being charged by an external source (not shown in the figure). The charging current is constant and equal to 0.15 A. (a)Calculate the capacitance and the rate of charge of potential difference between the plates. (b)Obtain the displacement current across the plates. (c)Is Kirchhoffs first rule (junction rule) valid at each plate of the capacitor? Explain. Solution: R...
Read More →Find the inverse of each of the matrices, if it exists.
Question: Find the inverse of each of the matrices, if it exists. $\left[\begin{array}{ccc}2 0 -1 \\ 5 1 0 \\ 0 1 3\end{array}\right]$ Solution: Let $A=\left[\begin{array}{ccc}2 0 -1 \\ 5 1 0 \\ 0 1 3\end{array}\right]$ $\therefore\left[\begin{array}{ccc}2 0 -1 \\ 5 1 0 \\ 0 1 3\end{array}\right]=\left[\begin{array}{ccc}1 0 0 \\ 0 1 0 \\ 0 0 1\end{array}\right] A$ We know thatA=IA $\therefore\left[\begin{array}{ccc}2 0 -1 \\ 5 1 0 \\ 0 1 3\end{array}\right]=\left[\begin{array}{ccc}1 0 0 \\ 0 1 0...
Read More →Find the middle terms in the expansions of
Question: Find the middle terms in the expansions of$\left(3-\frac{x^{3}}{6}\right)^{7}$ Solution: It is known that in the expansion of $(a+b)^{n}$, if $n$ is odd, then there are two middle terms, namely, $\left(\frac{\mathrm{n}+\mathrm{l}}{2}\right)^{\text {th }}$ term and $\left(\frac{\mathrm{n}+1}{2}+1\right)^{\text {th }}$ term. Therefore, the middle terms in the expansion of $\left(3-\frac{x^{3}}{6}\right)^{7}$ are $\left(\frac{7+1}{2}\right)^{\text {th }}=4^{\text {th }}$ term and $\left(\...
Read More →Why is LiF almost insoluble in water whereas
Question: Why is LiF almost insoluble in water whereas LiCl soluble not only in water but also in acetone? Solution: LiF is insoluble in water. On the contrary, LiCl is soluble not only in water, but also in acetone. This is mainly because of the greater ionic character of LiF as compared to LiCl. The solubility of a compound in water depends on the balance between lattice energy and hydration energy. Since fluoride ion is much smaller in size than chloride ion, the lattice energy of LiF is grea...
Read More →Why are lithium salts commonly hydrated
Question: Why are lithium salts commonly hydrated and those of the other alkali metal ions usually anhydrous? Solution: Lithium is the smallest in size among the alkali metals. Hence, Li+ion can polarize water molecules more easily than other alkali metals. As a result, water molecules get attached to lithium salts as water of crystallization. Hence, lithium salts such as trihydrated lithium chloride (LiCl.3H2O) are commonly hydrated. As the size of the ions increases, their polarizing power dec...
Read More →Describe the importance of the following
Question: Describe the importance of the following (i) limestone (ii) cement (iii) plaster of paris. Solution: (i)Chemically, limestone is CaCO3. Importance of limestone (a) It is used in the preparation of lime and cement. (b) It is used as a flux during the smelting of iron ores. (ii)Chemically, cement is a mixture of calcium silicate and calcium aluminate. Importance of cement (a) It is used in plastering and in construction of bridges. (b) It is used in concrete. (iii)Chemically, plaster of ...
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