Find the number of ways of selecting 9 balls from 6 red balls,

Question: Find the number of ways of selecting 9 balls from 6 red balls, 5 white balls and 5 blue balls if each selection consists of 3 balls of each colour. Solution: There are a total of 6 red balls, 5 white balls, and 5 blue balls. 9 balls have to be selected in such a way that each selection consists of 3 balls of each colour. Here, 3 balls can be selected from 6 red balls in ${ }^{6} \mathrm{C}_{3}$ ways. 3 balls can be selected from 5 white balls in ${ }^{5} \mathrm{C}_{3}$ ways. 3 balls c...

A 60 μF capacitor is connected to a 110 V, 60 Hz ac supply.

Question: A 60 F capacitor is connected to a 110 V, 60 Hz ac supply. Determine the rms value of the current in the circuit. Solution: Capacitance of capacitor,C= 60 F = 60 106F Supply voltage,V= 110 V Frequency,= 60 Hz Angular frequency, $\omega=2 \pi v$ Capacitive reactance $X_{\mathrm{c}}=\frac{1}{\omega C}$ $=\frac{1}{2 \pi v C}$ $=\frac{1}{2 \times 3.14 \times 60 \times 60 \times 10^{-6}} \Omega^{-1}$ Rms value of current is given as: $I=\frac{v}{X_{\mathrm{c}}}$ $=110 \times 2 \times 3.14 \...

In how many ways can a team of 3 boys and 3 girls be selected from 5 boys and 4 girls?

Question: In how many ways can a team of 3 boys and 3 girls be selected from 5 boys and 4 girls? Solution: A team of 3 boys and 3 girls is to be selected from 5 boys and 4 girls. 3 boys can be selected from 5 boys in ${ }^{5} \mathrm{C}_{3}$ ways. 3 girls can be selected from 4 girls in ${ }^{4} \mathrm{C}_{3}$ ways. Therefore, by multiplication principle, number of ways in which a team of 3 boys and 3 girls can be selected $={ }^{5} \mathrm{C}_{3} \times{ }^{4} \mathrm{C}_{3}=\frac{5 !}{3 ! 2 !...

What do you understand by the terms:

Question: What do you understand by the terms: (i) hydrogen economy (ii) hydrogenation (iii) syngas (iv) water-gas shift reaction (v) fuel-cell ? Solution: (i) Hydrogen economy Hydrogen economy is a technique of using dihydrogen in an efficient way. It involves transportation and storage of dihydrogen in the form of liquid or gas. Dihydrogen releases more energy than petrol and is more ecofriendly. Hence, it can be used in fuel cells to generate electric power. Hydrogen economy is about the tran...

A 44 mH inductor is connected to 220 V, 50 Hz ac supply.

Question: A 44 mH inductor is connected to 220 V, 50 Hz ac supply. Determine the rms value of the current in the circuit. Solution: Inductance of inductor,L= 44 mH = 44 103H Supply voltage,V= 220 V Frequency,= 50 Hz Angular frequency, $\omega=2 \pi v$ Inductive reactance, $X_{\mathrm{L}}=\omega L=2 \pi v L=2 \pi \times 50 \times 44 \times 10^{-3} \Omega$ Rms value of current is given as: $I=\frac{v}{X_{\mathrm{L}}}$ $=\frac{220}{2 \pi \times 50 \times 44 \times 10^{-3}}=15.92 \mathrm{~A}$ Hence,...

How many chords can be drawn through 21 points on a circle?

Question: How many chords can be drawn through 21 points on a circle? Solution: For drawing one chord on a circle, only 2 points are required. To know the number of chords that can be drawn through the given 21 points on a circle, the number of combinations have to be counted. Therefore, there will be as many chords as there are combinations of 21 points taken 2 at a time. Thus, required number of chords = ${ }^{21} \mathrm{C}_{2}=\frac{21 !}{2 !(21-2) !}=\frac{21 !}{2 ! 19 !}=\frac{21 \times 20...

(a) The peak voltage of an ac supply is 300 V.

Question: (a)The peak voltage of an ac supply is 300 V. What is the rms voltage? (b)The rms value of current in an ac circuit is 10 A. What is the peak current? Solution: (a)Peak voltage of the ac supply,V0= 300 V Rms voltage is given as: $V=\frac{V_{0}}{\sqrt{2}}$ $=\frac{300}{\sqrt{2}}=212.1 \mathrm{~V}$ (b)Therms value of current is given as: I= 10 A Now, peak current is given as: $I_{0}=\sqrt{2} I$ $=10 \sqrt{2}=14.1 \mathrm{~A}$...

Determine n if

Question: Determinenif (i) ${ }^{2 n} \mathrm{C}_{3}:{ }^{n} \mathrm{C}_{3}=12: 1$ (ii) ${ }^{2 \mathrm{n}} \mathrm{C}_{3}:{ }^{\mathrm{n}} \mathrm{C}_{3}=11: 1$ Solution: (i) $\frac{{ }^{2 n} \mathrm{C}_{3}}{{ }^{n} \mathrm{C}_{3}}=\frac{12}{1}$ $\Rightarrow \frac{(2 n) !}{3 !(2 n-3) !} \times \frac{3 !(n-3) !}{n !}=\frac{12}{1}$ $\Rightarrow \frac{(2 n)(2 n-1)(2 n-2)(2 n-3) !}{(2 n-3) !} \times \frac{(n-3) !}{n(n-1)(n-2)(n-3) !}=12$ $\Rightarrow \frac{2(2 n-1)(2 n-2)}{(n-1)(n-2)}=12$ $\Rightar...

A 100 Ω resistor is connected to a 220 V, 50 Hz ac supply.

Question: A 100 Ω resistor is connected to a 220 V, 50 Hz ac supply. (a)What is the rms value of current in the circuit? (b)What is the net power consumed over a full cycle? Solution: Resistance of the resistor,R= 100 Ω Supply voltage,V= 220 V Frequency,=50 Hz (a)The rms value of current in the circuit is given as: $I=\frac{V}{R}$ $=\frac{220}{100}=2.20 \mathrm{~A}$ (b)The net power consumed over a full cycle is given as: P=VI = 220 2.2 = 484 W...

How does H2O2 behave as a bleaching agent?

Question: How does $\mathrm{H}_{2} \mathrm{O}_{2}$ behave as a bleaching agent? Solution: $\mathrm{H}_{2} \mathrm{O}_{2}$ or hydrogen peroxide acts as a strong oxidizing agent both in acidic and basic media. When added to a cloth, it breaks the chemical bonds of the chromophores (colour producing agents). Hence, the visible light is not absorbed and the cloth gets whitened....

A line charge λ per unit length is lodged uniformly onto the rim of a wheel of mass M and radius R.

Question: A line charge per unit length is lodged uniformly onto the rim of a wheel of massMand radiusR. The wheel has light non-conducting spokes and is free to rotate without friction about its axis (Fig. 6.22). A uniform magnetic field extends over a circular region within the rim. It is given by, B= B0k(ra;aR) = 0 (otherwise) What is the angular velocity of the wheel after the field is suddenly switched off? Solution: Line charge per unit length $=\lambda=\frac{\text { Total charge }}{\text ...

If, find.

Question: If ${ }^{\prime \prime} \mathrm{C}_{8}={ }^{n} \mathrm{C}_{2}$, find ${ }^{\text {" }} \mathrm{C}_{2}$ Solution: It is known that, ${ }^{n} C_{a}={ }^{n} C_{b} \Rightarrow a=b$ or $n=a+b$ Therefore, ${ }^{n} \mathrm{C}_{8}={ }^{n} \mathrm{C}_{2} \Rightarrow \mathrm{n}=8+2=10$ $\therefore{ }^{\mathrm{n}} \mathrm{C}_{2}={ }^{10} \mathrm{C}_{2}=\frac{10 !}{2 !(10-2) !}=\frac{10 !}{2 ! 8 !}=\frac{10 \times 9 \times 8 !}{2 \times 1 \times 8 !}=45$...

Find and, when

Question: Find $\frac{1}{2}\left(A+A^{\prime}\right)$ and $\frac{1}{2}\left(A-A^{\prime}\right)$, when $A=\left[\begin{array}{ccc}0 a b \\ -a 0 c \\ -b -c 0\end{array}\right]$ Solution: The given matrix is $A=\left[\begin{array}{ccc}0 a b \\ -a 0 c \\ -b -c 0\end{array}\right]$ $\therefore A^{\prime}=\left[\begin{array}{ccc}0 -a -b \\ a 0 -c \\ b c 0\end{array}\right]$ $A+A^{\prime}=\left[\begin{array}{ccc}0 a b \\ -a 0 c \\ -b -c 0\end{array}\right]+\left[\begin{array}{ccc}0 -a -b \\ a 0 -c \\ ...

Obtain an expression for the mutual inductance between a long straight wire and a square loop of side a as shown in Fig. 6.21.

Question: (a)Obtain an expression for the mutual inductance between a long straight wire and a square loop of sideaas shown in Fig. 6.21. (b)Now assume that the straight wire carries a current of 50 A and the loop is moved to the right with a constant velocity,v= 10 m/s. Calculate the induced emf in the loop at the instant whenx= 0.2 m. Takea= 0.1 m and assume that the loop has a large resistance. Solution: (a)Take a small elementdyin the loop at a distanceyfrom the long straight wire (as shown ...

An air-cored solenoid with length 30 cm,

Question: An air-cored solenoid with length 30 cm, area of cross-section 25 cm2and number of turns 500, carries a current of 2.5 A. The current is suddenly switched off in a brief time of 103s. How much is the average back emf induced across the ends of the open switch in the circuit? Ignore the variation in magnetic field near the ends of the solenoid. Solution: Length of the solenoid,l= 30 cm = 0.3 m Area of cross-section,A= 25 cm2= 25 104m2 Number of turns on the solenoid,N= 500 Current in th...

Question: Figure 6.20 shows a metal rod PQ resting on the smooth rails AB and positioned between the poles of a permanent magnet. The rails, the rod, and the magnetic field are in three mutual perpendicular directions. A galvanometer G connects the rails through a switch K. Length of the rod = 15 cm,B= 0.50 T, resistance of the closed loop containing the rod = 9.0 mΩ. Assume the field to be uniform. (a)Suppose K is open and the rod is moved with a speed of 12 cm s1in the direction shown. Give th...

It is desired to measure the magnitude of field between the poles of a powerful loud speaker magnet.

Question: It is desired to measure the magnitude of field between the poles of a powerful loud speaker magnet. A small flat search coil of area 2 cm2with 25 closely wound turns, is positioned normal to the field direction, and then quickly snatched out of the field region. Equivalently, one can give it a quick 90 turn to bring its plane parallel to the field direction). The total charge flown in the coil (measured by a ballistic galvanometer connected to coil) is 7.5 mC. The combined resistance ...

A square loop of side 12 cm with its sides parallel to X and Y axes is moved with a velocity

Question: A square loop of side 12 cm with its sides parallel to X and Y axes is moved with a velocity of 8 cm s1in the positivex-direction in an environment containing a magnetic field in the positivez-direction. The field is neither uniform in space nor constant in time. It has a gradient of 103T cm1along the negativex-direction (that is it increases by 10 3T cm1as one moves in the negativex-direction), and it is decreasing in time at the rate of 103T s1. Determine the direction and magnitude ...

Suppose the loop in Exercise 6.4 is stationary but the current feeding

Question: Suppose the loop in Exercise 6.4 is stationary but the current feeding the electromagnet that produces the magnetic field is gradually reduced so that the field decreases from its initial value of 0.3 T at the rate of 0.02 T s1. If the cut is joined and the loop has a resistance of 1.6 Ω how much power is dissipated by the loop as heat? What is the source of this power? Solution: Sides of the rectangular loop are 8 cm and 2 cm. Hence, area of the rectangular wire loop, A= length width ...

A jet plane is travelling towards west at a speed of 1800 km/h.

Question: A jet plane is travelling towards west at a speed of 1800 km/h. What is the voltage difference developed between the ends of the wing having a span of 25 m, if the Earths magnetic field at the location has a magnitude of 5 104T and the dip angle is 30. Solution: Speed of the jet plane,v= 1800 km/h = 500 m/s Wing spanof jet plane,l= 25 m Earths magnetic field strength,B= 5.0 104T Angle of dip, $\delta=30^{\circ}$ Vertical component of Earths magnetic field, $B_{V}=B \sin \delta$ $=5 \ti...

A pair of adjacent coils has a mutual inductance of 1.5 H.

Question: A pair of adjacent coils has a mutual inductance of 1.5 H. If the current in one coil changes from 0 to 20 A in 0.5 s, what is the change of flux linkage with the other coil? Solution: Mutual inductance of a pair of coils,= 1.5 H Initial current,I1= 0 A Final currentI2= 20 A Change in current, $d I=I_{2}-I_{1}=20-0=20 \mathrm{~A}$ Time taken for the change, $t=0.5 \mathrm{~s}$ Induced emf, $e=\frac{d \phi}{d t}$ ...(1) Where $d \phi$ is the change in the flux linkage with the coil. Emf...

For the matrix, verify that

Question: For the matrix $A=\left[\begin{array}{ll}1 5 \\ 6 7\end{array}\right]$, verify that (i) $\left(A+A^{\prime}\right)$ is a symmetric matrix (ii) $\left(A-A^{\prime}\right)$ is a skew symmetric matrix Solution: $A^{\prime}=\left[\begin{array}{ll}1 6 \\ 5 7\end{array}\right]$ (i) $A+A^{\prime}$$=\left[\begin{array}{ll}1 5 \\ 6 7\end{array}\right]+\left[\begin{array}{ll}1 6 \\ 5 7\end{array}\right]=\left[\begin{array}{ll}2 11 \\ 11 14\end{array}\right]$ $\therefore\left(A+A^{\prime}\right)^...

Current in a circuit falls from 5.0 A to 0.0 A in 0.1 s.

Question: Current in a circuit falls from 5.0 A to 0.0 A in 0.1 s. If an average emf of 200 V induced, give an estimate of the self-inductance of the circuit. Solution: Initial current, $I_{1}=5.0 \mathrm{~A}$ Final current, $l_{2}=0.0 \mathrm{~A}$ Change in current, $d I=I_{1}-I_{2}=5 \mathrm{~A}$ Time taken for the change, $t=0.1 \mathrm{~s}$ Average emf,e= 200 V For self-inductance (L)of the coil, we have the relation for average emf as: $e=L \frac{d i}{d t}$ $L=\frac{e}{\left(\frac{d i}{d t}...

(i) Show that the matrix is a symmetric matrix

Question: (i) Show that the matrix $A=\left[\begin{array}{ccc}1 -1 5 \\ -1 2 1 \\ 5 1 3\end{array}\right]$ is a symmetric matrix (ii) Show that the matrix $A=\left[\begin{array}{ccc}0 1 -1 \\ -1 0 1 \\ 1 -1 0\end{array}\right]$ is a skew symmetric matrix Solution: (i)We have: $A^{\prime}=\left[\begin{array}{ccc}1 -1 5 \\ -1 2 1 \\ 5 1 3\end{array}\right]=A$ $\therefore A^{\prime}=A$ Hence, $A$ is a symmetric matrix (ii) We have: $A^{\prime}=\left[\begin{array}{ccc}0 -1 1 \\ 1 0 -1 \\ -1 1 0\end{...

A horizontal straight wire 10 m long extending from east to west is falling with a speed

Question: A horizontal straight wire 10 m long extending from east to west is falling with a speed of 5.0 m s1, at right angles to the horizontal component of the earths magnetic field, 0.30 104Wb m2. (a)What is the instantaneous value of the emf induced in the wire? (b)What is the direction of the emf? (c)Which end of the wire is at the higher electrical potentia Solution: Length of the wire,l= 10 m Falling speed of the wire,v= 5.0 m/s Magnetic field strength,B= 0.3 104Wb m2 (a)Emf induced in t...