The magnetic moment vectors
Question: The magnetic moment vectorssand lassociated with the intrinsic spin angular momentumSand orbital angular momentuml, respectively, of an electron are predicted by quantum theory (and verified experimentally to a high accuracy) to be given by: s= (e/m)S, l=(e/2m)l Which of these relations is in accordance with the result expectedclassically? Outline the derivation of the classical result. Solution: The magnetic moment associated with the orbital angular momentum is valid with the classic...
Read More →Find the transpose of each of the following matrices:
Question: Find the transpose of each of the following matrices: (i) $\left[\begin{array}{c}5 \\ \frac{1}{2} \\ -1\end{array}\right]$ (ii) $\left[\begin{array}{rr}1 -1 \\ 2 3\end{array}\right]$ (iii) $\left[\begin{array}{ccc}-1 5 6 \\ \sqrt{3} 5 6 \\ 2 3 -1\end{array}\right]$ Solution: (i) Let $A=\left[\begin{array}{c}5 \\ \frac{1}{2} \\ -1\end{array}\right]$, then $A^{\mathrm{T}}=\left[\begin{array}{lll}5 \frac{1}{2} -1\end{array}\right]$ (ii) Let $A=\left[\begin{array}{rr}1 -1 \\ 2 3\end{array}...
Read More →Write chemical reactions to justify that hydrogen peroxide can function
Question: Write chemical reactions to justify that hydrogen peroxide can function as an oxidizing as well as reducing agent. Solution: Hydrogen peroxide, H2O2acts as an oxidizing as well as a reducing agent in both acidic and alkaline media. Reactions involving oxidizing actions are: Reactions involving reduction actions are:...
Read More →Assume X, Y, Z, W and P are matrices of order, and respectively.
Question: Assume $X, Y, Z, W$ and $P$ are matrices of order $2 \times n, 3 \times k, 2 \times p, n \times 3$, and $p \times k$ respectively. If $n=p$, then the order of the matrix $7 X-5 Z$ is $\mathbf{A} p \times 2 \mathbf{B} 2 \times n \mathbf{C} n \times 3 \mathbf{D} p \times n$ Solution: The correct answer is B. MatrixXis of the order 2n. Therefore, matrix 7Xis also of the same order. MatrixZis of the order 2p, i.e., 2n[Sincen=p] Therefore, matrix 5Zis also of the same order. Now, both the m...
Read More →How many words, with or without meaning,
Question: How many words, with or without meaning, can be formed using all the letters of the word EQUATION, using each letter exactly once? Solution: There are 8 different letters in the word EQUATION. Therefore, the number of words that can be formed using all the letters of the word EQUATION, using each letter exactly once, is the number of permutations of 8 different objects taken 8 at a time, Which is ${ }^{8} \mathrm{P}_{8}=8 !$ Thus, required number of words that can be formed $=8 !=40320...
Read More →A Rowland ring of mean radius 15 cm has 3500 turns of wire wound on a ferromagnetic core of relative permeability 800.
Question: A Rowland ring of mean radius 15 cm has 3500 turns of wire wound on a ferromagnetic core of relative permeability 800. What is the magnetic field Bin the core for a magnetising current of 1.2 A? Solution: Mean radius of a Rowland ring,r= 15 cm = 0.15 m Number of turns on a ferromagnetic core,N= 3500 Relative permeability of the core material, $\mu_{r}=800$ Magnetising current,I= 1.2 A The magnetic field is given by the relation: $B=\frac{\mu_{r} \mu_{0} I N}{2 \pi r}$ Where, 0= Permeab...
Read More →Assume X, Y, Z, W and P are matrices of order, and respectively. The restriction on n, k and p so that will be defined are:
Question: AssumeX,Y,Z,WandPare matrices of order, andrespectively. The restriction onn,kandpso thatwill be defined are: A.k= 3,p=n B.kis arbitrary,p= 2 C.pis arbitrary,k= 3 D.k= 2,p= 3 Solution: MatricesPandYare of the orderspkand 3 krespectively. Therefore, matrixPYwill be defined ifk= 3. Consequently,PYwill be of the orderpk. MatricesWandYare of the ordersn 3 and 3 krespectively. Since the number of columns inWis equal to the number of rows inY, matrixWYis well-defined and is of the ordernk. M...
Read More →Find r if (i) (ii) .
Question: Find $r$ if (i) ${ }^{5} \mathrm{P}_{r}=2^{6} \mathrm{P}_{r-1}$ (ii) ${ }^{3} \mathrm{P}_{n}={ }^{6} \mathrm{P}_{r-1}$ Solution: (i) ${ }^{5} \mathrm{P}_{r}=2^{6} \mathrm{P}_{r-1}$ $\Rightarrow \frac{5 !}{(5-r) !}=2 \times \frac{6 !}{(6-r+1) !}$ $\Rightarrow \frac{5 !}{(5-r) !}=\frac{2 \times 6 !}{(7-r) !}$ $\Rightarrow \frac{5 !}{(5-r) !}=\frac{2 \times 6 \times 5 !}{(7-r)(6-r)(5-r) !}$ $\Rightarrow 1=\frac{2 \times 6}{(7-r)(6-r)}$ $\Rightarrow(7-r)(6-r)=12$ $\Rightarrow 42-6 r-7 r+r^...
Read More →Write chemical reactions to show the amphoteric nature of water.
Question: Write chemical reactions to show the amphoteric nature of water. Solution: The amphoteric nature of water can be described on the basis of the following reactions: 1) Reaction with H2S The reaction takes place as: In the forward reaction, $\mathrm{H}_{2} \mathrm{O}_{(i)}$ accepts a proton from $\mathrm{H}_{2} \mathrm{~S}_{(a q)}$. Hence, it acts as a Lewis base. 2) Reaction with NH3 The reaction takes place as: In the forward reaction, $\mathrm{H}_{2} \mathrm{O}_{(t)}$ denotes its prot...
Read More →The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books,
Question: The bookshop of a particular school has 10 dozen chemistry books, 8 dozen physics books, 10 dozen economics books. Their selling prices are Rs 80, Rs 60 and Rs 40 each respectively. Find the total amount the bookshop will receive from selling all the books using matrix algebra. Solution: The bookshop has 10 dozen chemistry books, 8 dozen physics books, and 10 dozen economics books. Theselling prices of a chemistry book, a physics book, and an economics book are respectively given as Rs...
Read More →A sample of paramagnetic salt contains
Question: A sample ofparamagnetic salt contains 2.0 1024atomic dipoles each of dipole moment 1.5 1023J T1. The sample is placed under a homogeneous magnetic field of 0.64 T, and cooled to a temperature of 4.2 K. The degree of magnetic saturation achieved is equal to 15%. What is the total dipole moment of the sample for a magnetic field of 0.98 T and a temperature of 2.8 K? (Assume Curies law) Solution: Number of atomic dipoles,n= 2.0 1024 Dipole moment of each atomic dipole,M= 1.5 1023J T1 When...
Read More →A trust fund has Rs 30,000 that must be invested in two different types of bonds.
Question: A trust fund has Rs 30,000 that must be invested in two different types of bonds. The first bond pays 5% interest per year, and the second bond pays 7% interest per year. Using matrix multiplication, determine how to divide Rs 30,000 among the two types of bonds. If the trust fund must obtain an annual total interest of: (a) Rs 1,800 (b) Rs 2,000 Solution: (a)Let Rsxbe invested in the first bond. Then, the sum of money invested in the second bond will be Rs (30000 x). It is given that ...
Read More →Discuss the principle and method of softening of hard water by synthetic ion-exchange resins.
Question: Discuss the principle and method of softening of hard water by synthetic ion-exchange resins. Solution: The process of treating permanent hardness of water using synthetic resins is based on the exchange of cations (e.g., $\mathrm{Na}^{+}, \mathrm{Ca}^{2+}, \mathrm{Mg}^{2+}$ etc) and anions (e.g., $\mathrm{Cl}^{-}, \mathrm{SO}_{4}^{2-}, \mathrm{HCO}_{3}^{-}$etc) present in water by $\mathrm{H}^{+}$and $\mathrm{OH}^{-}$ions respectively. Synthetic resins are of two types:. 1) Cation exc...
Read More →A monoenergetic (18 keV) electron beam initially
Question: A monoenergetic (18 keV) electron beam initially in the horizontal direction is subjected to a horizontal magnetic field of 0.04 G normal to the initial direction. Estimate the up or down deflection of the beam over a distance of 30 cm (me= 9.11 1019C). [Note:Datain this exercise are so chosen that the answer will give you an idea of the effect of earths magnetic field on the motion of the electron beam from the electron gun to the screen in a TV set.] Solution: Energy of an electron b...
Read More →If and I is the identity matrix of order 2, show that
Question: If $A=\left[\begin{array}{lc}0 -\tan \frac{\alpha}{2} \\ \tan \frac{\alpha}{2} 0\end{array}\right]$ and $/$ is the identity matrix of order 2, show that $I+A=(I-A)\left[\begin{array}{ll}\cos \alpha -\sin \alpha \\ \sin \alpha \cos \alpha\end{array}\right]$ Solution: On the L.H.S. $I+A$ $=\left[\begin{array}{ll}1 0 \\ 0 1\end{array}\right]+\left[\begin{array}{cc}0 -\tan \frac{\alpha}{2} \\ \tan \frac{\alpha}{2} 0\end{array}\right]$ $=\left[\begin{array}{cc}1 -\tan \frac{\alpha}{2} \\ \t...
Read More →Find n if
Question: Find $n$ if ${ }^{n-1} \mathrm{P}_{3}:{ }^{\pi} \mathrm{P}_{4}=1: 9$ Solution: ${ }^{n-1} \mathrm{P}_{3}:{ }^{n} \mathrm{P}_{4}=1: 9$ $\Rightarrow \frac{{ }^{n-1} P_{3}}{{ }^{n} P_{4}}=\frac{1}{9}$ $\Rightarrow \frac{\left[\frac{(\mathrm{n}-1) !}{(\mathrm{n}-1-3) !}\right]}{\left[\frac{\mathrm{n} !}{(\mathrm{n}-4) !}\right]}=\frac{1}{9}$ $\Rightarrow \frac{(n-1) !}{(n-4) !} \times \frac{(n-4) !}{n !}=\frac{1}{9}$ $\Rightarrow \frac{(n-1) !}{n \times(n-1) !}=\frac{1}{9}$ $\Rightarrow \f...
Read More →rom a committee of 8 persons, in how many ways can we choose a chairman
Question: rom a committee of 8 persons, in how many ways can we choose a chairman and a vice chairman assuming one person cannot hold more than one position? Solution: From a committee of 8 persons, a chairman and a vice chairman are to be chosen in such a way that one person cannot hold more than one position. Here, the number of ways of choosing a chairman and a vice chairman is the permutation of 8 different objects taken 2 at a time. Thus, required number of ways = ${ }^{8} \mathrm{P}_{2}=\f...
Read More →Find the number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, 5 if no digit is repeated. How many of these will be even?
Question: Find the number of 4-digit numbers that can be formed using the digits 1, 2, 3, 4, 5 if no digit is repeated. How many of these will be even? Solution: 4-digit numbers are to be formed using the digits, 1, 2, 3, 4, and 5. There will be as many 4-digit numbers as there are permutations of 5 different digits taken 4 at a time. Therefore, required number of 4 digit numbers $={ }^{5} \mathrm{P}_{4}=\frac{5 !}{(5-4) !}=\frac{5 !}{1 !}$ $=1 \times 2 \times 3 \times 4 \times 5=120$ Among the ...
Read More →If and, find k so that
Question: If $A=\left[\begin{array}{ll}3 -2 \\ 4 -2\end{array}\right]$ and $I=\left[\begin{array}{ll}1 0 \\ 0 1\end{array}\right]$, find $k$ so that $A^{2}=k A-2 I$ Solution: $\begin{aligned} A^{2}=A \cdot A =\left[\begin{array}{ll}3 -2 \\ 4 -2\end{array}\right]\left[\begin{array}{ll}3 -2 \\ 4 -2\end{array}\right] \\ =\left[\begin{array}{ll}3(3)+(-2)(4) 3(-2)+(-2)(-2) \\ 4(3)+(-2)(4) 4(-2)+(-2)(-2)\end{array}\right]=\left[\begin{array}{ll}1 -2 \\ 4 -4\end{array}\right] \end{aligned}$ Now $A^{2}=...
Read More →A magnetic dipole is under the influence of two magnetic fields.
Question: A magnetic dipole is under the influence of two magnetic fields. The angle between the field directions is 60, and one of the fields has a magnitude of 1.2 102T. If the dipole comes to stable equilibrium at an angle of 15 with this field, what is the magnitude of the other field? Solution: Magnitude of one of the magnetic fields,B1= 1.2 102T Magnitude of the other magnetic field =B2 Angle between the two fields,= 60 At stable equilibrium, the angle between the dipole and fieldB1,1= 15 ...
Read More →How many 3-digit even numbers can be made using the digits 1, 2, 3, 4, 6, 7, if no digit is repeated?
Question: How many 3-digit even numbers can be made using the digits 1, 2, 3, 4, 6, 7, if no digit is repeated? Solution: 3-digit even numbers are to be formed using the given six digits, 1, 2, 3, 4, 6, and 7, without repeating the digits. Then, units digits can be filled in 3 ways by any of the digits, 2, 4, or 6. Since the digits cannot be repeated in the 3-digit numbers and units place is already occupied with a digit (which is even), the hundreds and tens place is to be filled by the remaini...
Read More →If, prove that
Question: If $A=\left[\begin{array}{lll}1 0 2 \\ 0 2 1 \\ 2 0 3\end{array}\right]$, prove that $A^{3}-6 A^{2}+7 A+2 I=O$ Solution: $A^{2}=A A=\left[\begin{array}{lll}1 0 2 \\ 0 2 1 \\ 2 0 3\end{array}\right]\left[\begin{array}{lll}1 0 2 \\ 0 2 1 \\ 2 0 3\end{array}\right]$ $=\left[\begin{array}{ccc}1+0+4 0+0+0 2+0+6 \\ 0+0+2 0+4+0 0+2+3 \\ 2+0+6 0+0+0 4+0+9\end{array}\right]=\left[\begin{array}{ccc}5 0 8 \\ 2 4 5 \\ 8 0 13\end{array}\right]$ Now $A^{3}=A^{2} \cdot A$ $=\left[\begin{array}{ccc}5 ...
Read More →How many 4-digit numbers are there with no digit repeated?
Question: How many 4-digit numbers are there with no digit repeated? Solution: The thousands place of the 4-digit number is to be filled with any of the digits from 1 to 9 as the digit 0 cannot be included. Therefore, the number of ways in which thousands place can be filled is 9. The hundreds, tens, and units place can be filled by any of the digits from 0 to 9. However, the digits cannot be repeated in the 4-digit numbers and thousands place is already occupied with a digit. The hundreds, tens...
Read More →A compass needle free to turn in a horizontal plane is placed at the centre of circular coil of 30 turns and radius 12 cm.
Question: A compass needle free to turn in a horizontal plane is placed at the centre of circular coil of 30 turns andradius 12 cm. The coil is in a vertical plane making an angle of 45 with the magnetic meridian. When the current in the coil is 0.35 A, the needle points west to east. (a)Determine the horizontal component of the earths magnetic field at the location. (b)The current in the coil is reversed, and the coil is rotated about its vertical axis by an angle of 90 in the anticlockwise sen...
Read More →How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit is repeated?
Question: How many 3-digit numbers can be formed by using the digits 1 to 9 if no digit is repeated? Solution: 3-digit numbers have to be formed using the digits 1 to 9. Here, the order of the digits matters. Therefore, there will be as many 3-digit numbers as there are permutations of 9 different digits taken 3 at a time. Therefore, required number of 3-digit numbers $={ }^{9} \mathrm{P}_{3}=\frac{9 !}{(9-3) !}=\frac{9 !}{6 !}$ $=\frac{9 \times 8 \times 7 \times 6 !}{6 !}=9 \times 8 \times 7=50...
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