Find the inverse of each of the matrices, if it exists.
Question: Find the inverse of each of the matrices, if it exists. $\left[\begin{array}{ll}2 3 \\ 5 7\end{array}\right]$ Solution: Let $A=\left[\begin{array}{ll}2 3 \\ 5 7\end{array}\right]$ We know thatA=IA $\therefore\left[\begin{array}{ll}2 3 \\ 5 7\end{array}\right]=\left[\begin{array}{ll}1 0 \\ 0 1\end{array}\right] A$ $\Rightarrow\left[\begin{array}{ll}1 \frac{3}{2} \\ 5 7\end{array}\right]=\left[\begin{array}{ll}\frac{1}{2} 0 \\ 0 1\end{array}\right] A \quad\left(\mathrm{R}_{1} \rightarrow...
Read More →How many 6-digit numbers can be formed from the digits 0, 1, 3, 5, 7 and 9 which
Question: How many 6-digit numbers can be formed from the digits 0, 1, 3, 5, 7 and 9 which are divisible by 10 and no digit is repeated? Solution: A number is divisible by 10 if its units digits is 0. Therefore, 0 is fixed at the units place. Therefore, there will be as many ways as there are ways of filling 5 vacant placesin succession by the remaining 5 digits (i.e., 1, 3, 5, 7 and 9). The 5 vacant places can be filled in 5! ways. Hence, required number of 6-digit numbers = 5! = 120...
Read More →If the different permutations of all the letter of the word EXAMINATION are listed as in a dictionary,
Question: If the different permutations of all the letter of the word EXAMINATION are listed as in a dictionary, how many words are there in this list before the first word starting with E? Solution: In the given word EXAMINATION, there are 11 letters out of which, A, I, and N appear 2 times and all the other letters appear only once. The words that will be listed before the words starting with E in a dictionary will be the words that start with A only. Therefore, to get the number of words star...
Read More →Figure 7.21 shows a series LCR circuit connected to a variable frequency 230 V source.
Question: Figure 7.21 shows a seriesLCRcircuit connected to a variable frequency 230 V source.L= 5.0 H,C= 80F,R= 40 Ω (a)Determine the source frequency which drives the circuit in resonance. (b)Obtain the impedance of the circuit and the amplitude of current at the resonating frequency. (c)Determine the rms potential drops across the three elements of the circuit. Show that the potential drop across theLCcombination is zero at the resonating frequency. Solution: Inductance of the inductor,L= 5.0...
Read More →Discuss the general characteristics and gradation in properties of alkaline earth metals.
Question: Discuss the general characteristics and gradation in properties of alkaline earth metals. Solution: General characteristics of alkaline earth metals are as follows. (i)The general electronic configuration of alkaline earth metals is [noble gas]ns2. (ii)These metals lose two electrons to acquire the nearest noble gas configuration. Therefore, their oxidation state is +2. (iii)These metals have atomic and ionic radii smaller than that of alkali metals. Also, when moved down the group, th...
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. $\left[\begin{array}{ll}1 3 \\ 2 7\end{array}\right]$ Solution: Let $A=\left[\begin{array}{ll}1 3 \\ 2 7\end{array}\right]$ We know that $A=I A$ $\therefore\left[\begin{array}{ll}1 3 \\ 2 7\end{array}\right]=\left[\begin{array}{ll}1 0 \\ 0 1\end{array}\right] A$ $\Rightarrow\left[\begin{array}{ll}1 3 \\ 0 1\end{array}\right]=\left[\begin{array}{ll}1 0 \\ -2 1\end{array}\right] A \quad\left(\mathrm{R}_{2} \rightarrow \mathrm{R}_{2}...
Read More →A committee of 7 has to be formed from 9 boys and 4 girls.
Question: A committee of 7 has to be formed from 9 boys and 4 girls. In how many ways can this be done when the committee consists of: (i) exactly 3 girls? (ii) atleast 3 girls? (iii) atmost 3 girls? Solution: A committee of 7 has to be formed from 9 boys and 4 girls. Since exactly 3 girls are to be there in every committee, each committee must consist of (7 3) = 4 boys only Thus, in this case, required number of ways$={ }^{4} \mathrm{C}_{3} \times{ }^{9} \mathrm{C}_{4}=\frac{4 !}{3 ! 1 !} \time...
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 at a time so that the vowels and consonants occur together? Solution: In the word EQUATION, there are 5 vowels, namely, A, E, I, O, and U, and 3 consonants, namely, Q, T, and N. Since all the vowels and consonants have to occur together, both (AEIOU) and (QTN) can be assumed as single objects. Then, the permutations of these 2 objects taken all at a time are counted. This number would be $...
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. $\left[\begin{array}{ll}2 1 \\ 1 1\end{array}\right]$ Solution: Let $A=\left[\begin{array}{ll}2 1 \\ 1 1\end{array}\right]$ We know thatA=IA $\therefore\left[\begin{array}{ll}2 1 \\ 1 1\end{array}\right]=\left[\begin{array}{ll}1 0 \\ 0 1\end{array}\right] A$ $\Rightarrow\left[\begin{array}{ll}1 0 \\ 1 1\end{array}\right]=\left[\begin{array}{cc}1 -1 \\ 0 1\end{array}\right] A \quad\left(\mathrm{R}_{1} \rightarrow \mathrm{R}_{1}-\ma...
Read More →A radio can tune over the frequency range of a portion of MW broadcast band:
Question: A radio can tune over the frequency range of a portion of MW broadcast band: (800 kHz to 1200 kHz). If itsLCcircuit has an effective inductance of 200 H, what must be the range of its variable capacitor? [Hint:For tuning, the natural frequency i.e., the frequency of free oscillations of theLCcircuit should be equal to the frequency of the radiowave.] Solution: The range of frequency () of a radio is 800 kHz to 1200 kHz. Lower tuning frequency, 1= 800 kHz = 800 103Hz Upper tuning freque...
Read More →How many words, with or without meaning,
Question: How many words, with or without meaning, each of 2 vowels and 3 consonants can be formed from the letters of the word DAUGHTER? Solution: In the word DAUGHTER, there are 3 vowels namely, A, U, and E, and 5 consonants namely, D, G, H, T, and R. Number of ways of selecting 2 vowels out of 3 vowels $={ }^{3} \mathrm{C}_{2}=3$ Number of ways of selecting 3 consonants out of 5 consonants $={ }^{5} \mathrm{C}_{3}=10$ Therefore, number of combinations of 2 vowels and 3 consonants = 3 10 = 30 ...
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. $\left[\begin{array}{rr}1 -1 \\ 2 3\end{array}\right]$ Solution: Let $A=\left[\begin{array}{rr}1 -1 \\ 2 3\end{array}\right]$ We know that $A=I A$ $\therefore\left[\begin{array}{rr}1 -1 \\ 2 3\end{array}\right]=\left[\begin{array}{ll}1 0 \\ 0 1\end{array}\right] A$ $\Rightarrow\left[\begin{array}{rr}1 -1 \\ 0 5\end{array}\right]=\left[\begin{array}{ll}1 0 \\ -2 1\end{array}\right] A \quad\left(\mathrm{R}_{2} \rightarrow \mathrm{R}...
Read More →In how many ways can a student choose a programme of 5 courses if 9
Question: In how many ways can a student choose a programme of 5 courses if 9 courses are available and 2 specific courses are compulsory for every student? Solution: There are 9 courses available out of which, 2 specific courses are compulsory for every student. Therefore, every student has to choose 3 courses out of the remaining 7 courses. This can be chosen in ${ }^{7} \mathrm{C}_{3}$ ways. Thus, required number of ways of choosing the programme $={ }^{7} \mathrm{C}_{3}=\frac{7 !}{3 ! 4 !}=\...
Read More →A series LCR circuit with R = 20 Ω,
Question: A series $L C R$ circuit with $R=20 \Omega, L=1.5 \mathrm{H}$ and $C=35 \mu \mathrm{F}$ is connected to a variable-frequency $200 \mathrm{~V}$ ac supply. When the frequency of the supply equals the natural frequency of the circuit, what is the average power transferred to the circuit in one complete cycle? Solution: At resonance, the frequency of the supply power equals the natural frequency of the givenLCRcircuit. Resistance,R= 20 Ω Inductance,L= 1.5 H Capacitance,C= 35 F = 30 106F AC...
Read More →A bag contains 5 black and 6 red balls.
Question: A bag contains 5 black and 6 red balls. Determine the number of ways in which 2 black and 3 red balls can be selected. Solution: There are 5 black and 6 red balls in the bag. 2 black balls can be selected out of 5 black balls in ${ }^{5} \mathrm{C}_{2}$ ways and 3 red balls can be selected out of 6 red balls in ${ }^{6} \mathrm{C}_{3}$ ways. Thus, by multiplication principle, required number of ways of selecting 2 black and 3 red balls $={ }^{5} \mathrm{C}_{2} \times{ }^{6} \mathrm{C}_...
Read More →What are the common physical and chemical features of alkali metals?
Question: What are the common physical and chemical features of alkali metals? Solution: Physical properties of alkali metalsare as follows. (1)They are quite soft and can be cut easily. Sodium metal can be easily cut using a knife. (2)They are light coloured and are mostly silvery white in appearance. (3)They have low density because of the large atomic sizes. The density increases down the group from Li to Cs. The only exceptionto this isK, which has lower density than Na. (4)The metallic bond...
Read More →Suppose the initial charge on the capacitor in Exercise 7.7 is 6 mC.
Question: Suppose the initial charge on the capacitor in Exercise 7.7 is 6 mC. What is the total energy stored in the circuit initially? What is the total energy at later time? Solution: Capacitance of the capacitor,C= 30 F = 30106F Inductance of the inductor,L= 27 mH = 27 103H Charge on the capacitor,Q= 6 mC = 6 103C Total energy stored in the capacitor can be calculated by the relation, $E=\frac{1}{2} \frac{Q^{2}}{C}$ $=\frac{1}{2} \times \frac{\left(6 \times 10^{-3}\right)^{2}}{30 \times 10^{...
Read More →If, then, if the value of α is
Question: If $A=\left[\begin{array}{cc}\cos \alpha -\sin \alpha \\ \sin \alpha \cos \alpha\end{array}\right]$, then $A+A^{*}=I$, if the value of $\alpha$ is A. $\frac{\pi}{6}$ B. $\frac{\pi}{3}$ C. D. $\frac{3 \pi}{2}$ Solution: The correct answer is B. $A=\left[\begin{array}{lr}\cos \alpha -\sin \alpha \\ \sin \alpha \cos \alpha\end{array}\right]$ $\Rightarrow A^{\prime}=\left[\begin{array}{ll}\cos \alpha \sin \alpha \\ -\sin \alpha \cos \alpha\end{array}\right]$ Now, $A+A^{\prime}=I$ $\therefo...
Read More →In how many ways can one select a cricket team of eleven from
Question: In how many ways can one select a cricket team of eleven from 17 players in which only 5 players can bowl if each cricket team of 11 must include exactly 4 bowlers? Solution: Out of 17 players, 5 players are bowlers. A cricket team of 11 players is to be selected in such a way that there are exactly 4 bowlers. 4 bowlers can be selected in ${ }^{5} \mathrm{C}_{4}$ ways and the remaining 7 players can be selected out of the 12 players in ${ }^{12} \mathrm{C}_{7}$ ways. Thus, by multiplic...
Read More →A charged 30 μF capacitor is connected to a 27 mH inductor.
Question: A charged 30 F capacitor is connected to a 27 mH inductor. What is the angular frequency of free oscillations of the circuit? Solution: Capacitance,C= 30F = 30106F Inductance,L= 27 mH = 27 103H Angular frequency is given as: $\omega_{r}=\frac{1}{\sqrt{L C}}$ $=\frac{1}{\sqrt{27 \times 10^{-3} \times 30 \times 10^{-6}}}=\frac{1}{9 \times 10^{-4}}=1.11 \times 10^{3} \mathrm{rad} / \mathrm{s}$ Hence, the angular frequency of free oscillations of the circuit is 1.11 103rad/s....
Read More →If A, B are symmetric matrices of same order, then AB − BA is a
Question: IfA,Bare symmetric matrices of same order, thenABBAis a A.Skew symmetric matrixB.Symmetric matrix C.Zero matrixD.Identity matrix Solution: The correct answer is A. AandBare symmetric matrices, therefore, we have: $A^{\prime}=A$ and $B^{\prime}=B$ .........(1) $\begin{aligned} \text { Consider }(A B-B A)^{\prime} =(A B)^{\prime}-(B A)^{\prime} \left[(A-B)^{\prime}=A^{\prime}-B^{\prime}\right] \\ =B^{\prime} A^{\prime}-A^{\prime} B^{\prime} \left[(A B)^{\prime}=B^{\prime} A^{\prime}\righ...
Read More →Obtain the resonant frequency
Question: Obtain the resonant frequencyrof a seriesLCRcircuit withL= 2.0 H,C= 32 F andR= 10 Ω. What is theQ-value of this circuit? Solution: Inductance,L= 2.0 H Capacitance,C= 32 F = 32 106F Resistance,R= 10 Ω Resonant frequency is given by the relation, $\omega_{\mathrm{r}}=\frac{1}{\sqrt{L C}}$ $=\frac{1}{\sqrt{2 \times 32 \times 10^{-6}}}=\frac{1}{8 \times 10^{-3}}=125 \mathrm{~s}^{-1}$ Now,Q-value of the circuit is given as: $Q=\frac{1}{R} \sqrt{\frac{L}{C}}$ $=\frac{1}{10} \sqrt{\frac{2}{32...
Read More →Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly one ace in each combination.
Question: Determine the number of 5 card combinations out of a deck of 52 cards if there is exactly one ace in each combination. Solution: In a deck of 52 cards, there are 4 aces. A combination of 5 cards have to be made in which there is exactly one ace. Then, one ace can be selected in ${ }^{4} \mathrm{C}_{1}$ ways and the remaining 4 cards can be selected out of the 48 cards in ${ }^{48} \mathrm{C}_{4}$ ways. $={ }^{48} \mathrm{C}_{4} \times{ }^{4} \mathrm{C}_{1}=\frac{48 !}{4 ! 44 !} \times ...
Read More →In Exercises 7.3 and 7.4,
Question: In Exercises 7.3 and 7.4, what is the net power absorbed by each circuit over a complete cycle. Explain your answer. Solution: In the inductive circuit, Rms value of current,I= 15.92 A Rms value of voltage,V= 220 V Hence, the net power absorbed can be obtained by the relation, P=VIcosΦ Where, Φ = Phase difference betweenVandI For a pure inductive circuit, the phase difference between alternating voltage and current is 90 i.e.,Φ= 90. Hence,P= 0 i.e., the net power is zero. In the capaci...
Read More →Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
Question: Express the following matrices as the sum of a symmetric and a skew symmetric matrix: (i) $\left[\begin{array}{rr}3 5 \\ 1 -1\end{array}\right]$ (ii) $\left[\begin{array}{rrr}6 -2 2 \\ -2 3 -1 \\ 2 -1 3\end{array}\right]$ (iii) $\left[\begin{array}{rrr}3 3 -1 \\ -2 -2 1 \\ -4 -5 2\end{array}\right]$ (iv) $\left[\begin{array}{rr}1 5 \\ -1 2\end{array}\right]$ Solution: (i) Let $A=\left[\begin{array}{cc}3 5 \\ 1 -1\end{array}\right]$, then $A^{\prime}=\left[\begin{array}{cc}3 1 \\ 5 -1\e...
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