Evaluate, when
Question: Evaluate $\frac{n !}{(n-r) !^{2}}$, when (i)n= 6,r= 2 (ii)n= 9,r= 5 Solution: (i) When $n=6, r=2, \frac{n !}{(n-r) !}=\frac{6 !}{(6-2) !}=\frac{6 !}{4 !}=\frac{6 \times 5 \times 4 !}{4 !}=30$ (ii) When $n=9, r=5, \frac{n !}{(n-r) !}=\frac{9 !}{(9-5) !}=\frac{9 !}{4 !}=\frac{9 \times 8 \times 7 \times 6 \times 5 \times 4 !}{4 !}$ $=9 \times 8 \times 7 \times 6 \times 5=15120$...
Read More →Find if
Question: Find $A^{2}-5 A+6 I$ if $A=\left[\begin{array}{ccc}2 0 1 \\ 2 1 3 \\ 1 -1 0\end{array}\right]$ Solution: We have $A^{2}=A \times A$ $A^{2}=A A=\left[\begin{array}{ccc}2 0 1 \\ 2 1 3 \\ 1 -1 0\end{array}\right]\left[\begin{array}{ccc}2 0 1 \\ 2 1 3 \\ 1 -1 0\end{array}\right]$ $=\left[\begin{array}{lll}2(2)+0(2)+1(1) 2(0)+0(1)+1(-1) 2(1)+0(3)+1(0) \\ 2(2)+1(2)+3(1) 2(0)+1(1)+3(-1) 2(1)+1(3)+3(0) \\ 1(2)+(-1)(2)+0(1) 1(0)+(-1)(1)+0(-1) 1(1)+(-1)(3)+0(0)\end{array}\right]$ $=\left[\begin{...
Read More →A telephone cable at a place has four long straight horizontal wires carrying a current of 1.0 A in the same direction east to west.
Question: A telephone cable at a place has four long straight horizontal wires carrying a current of 1.0 A in the same direction east to west. The earths magnetic field at the place is 0.39 G, and the angle of dip is 35. The magnetic declination is nearly zero. What are the resultant magnetic fields at points 4.0 cm below the cable? Solution: Number of horizontal wires in the telephone cable,n= 4 Current in each wire,I= 1.0 A Earths magnetic field at a location,H= 0.39 G = 0.39 104T Angle of dip...
Read More →A telephone cable at a place has four long straight horizontal wires carrying a current of 1.0 A in the same direction east to west.
Question: A telephone cable at a place has four long straight horizontal wires carrying a current of 1.0 A in the same direction east to west. The earths magnetic field at the place is 0.39 G, and the angle of dip is 35. The magnetic declination is nearly zero. What are the resultant magnetic fields at points 4.0 cm below the cable? Solution: Number of horizontal wires in the telephone cable,n= 4 Current in each wire,I= 1.0 A Earths magnetic field at a location,H= 0.39 G = 0.39 104T Angle of dip...
Read More →If , find x.
Question: If $\frac{1}{6 !}+\frac{1}{7 !}=\frac{x}{8 !}$, find $x$ Solution: $\frac{1}{6 !}+\frac{1}{7 !}=\frac{x}{8 !}$ $\Rightarrow \frac{1}{6 !}+\frac{1}{7 \times 6 !}=\frac{x}{8 \times 7 \times 6 !}$ $\Rightarrow \frac{1}{6 !}\left(1+\frac{1}{7}\right)=\frac{x}{8 \times 7 \times 6 !}$ $\Rightarrow 1+\frac{1}{7}=\frac{x}{8 \times 7}$ $\Rightarrow \frac{8}{7}=\frac{x}{8 \times 7}$ $\Rightarrow x=\frac{8 \times 8 \times 7}{7}$ $\therefore x=64$...
Read More →Compute
Question: Compute$\frac{8 !}{6 ! \times 2 !}$ Solution: $\frac{8 !}{6 ! \times 2 !}=\frac{8 \times 7 \times 6 !}{6 ! \times 2 \times 1}=\frac{8 \times 7}{2}=28$...
Read More →What causes the temporary and permanent hardness of water?
Question: What causes the temporary and permanent hardness of water? Solution: Temporary hardness of water is due to the presence of soluble salts of magnesium and calcium in the form of hydrogen carbonates (MHCO3, where M = Mg, Ca) in water. Permanent hardness of water is because of the presence of soluble salts of calcium and magnesium in the form of chlorides in water....
Read More →Is 3! + 4! = 7!?
Question: Is 3! + 4! = 7!? Solution: $3 !=1 \times 2 \times 3=6$ $4 !=1 \times 2 \times 3 \times 4=24$ $\therefore 3 !+4 !=6+24=30$ $7 !=1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7=5040$ $\therefore 3 !+4 ! \neq 7 !$...
Read More →Describe the structure of the common form of ice.
Question: Describe the structure of the common form of ice. Solution: Ice is the crystalline form of water. It takes a hexagonal form if crystallized at atmospheric pressure, but condenses to cubic form if the temperature is very low. The three-dimensional structure of ice is represented as: The structure is highly ordered and has hydrogen bonding. Each oxygen atom is surrounded tetrahedrally by four other oxygen atoms at a distance of 276 pm. The structure also contains wide holes that can hold...
Read More →A long straight horizontal cable carries a current of 2.5 A in the direction 10º south of west to 10° north of east.
Question: A long straight horizontal cable carries a current of 2.5 A in the direction 10 south of west to 10north of east. The magnetic meridian of the place happens to be 10 west of the geographic meridian. The earths magnetic field at the location is 0.33 G, and the angle of dip is zero. Locate the line of neutral points (ignore the thickness of the cable). (Atneutral points, magnetic field due to a current-carrying cable is equal and opposite to the horizontal component of earths magnetic fi...
Read More →Evaluate (i) 8!
Question: Evaluate (i) 8! (ii) 4! 3! Solution: (i) 8! = 1 2 3 4 5 6 7 8 = 40320 (ii) 4! = 1 2 3 4 = 24 3! = 1 2 3 = 6 4! 3! = 24 6 = 18...
Read More →Given 5 flags of different colours,
Question: Given 5 flags of different colours, how many different signals can be generated if each signal requires the use of 2 flags, one below the other? Solution: Each signal requires the use of 2 flags. There will be as many flags as there are ways of filling in 2 vacant placesin succession by the given 5 flags of different colours. The upper vacant place can be filled in 5 different ways by any one of the 5 flags following which, the lower vacant place can be filled in 4 different ways by an...
Read More →A coin is tossed 3 times and the outcomes are recorded. How many possible outcomes are there?
Question: A coin is tossed 3 times and the outcomes are recorded. How many possible outcomes are there? Solution: When a coin is tossed once, the number of outcomes is 2 (Head and tail) i.e., in each throw, the number of ways of showing a different face is 2. Thus, by multiplication principle, the required number of possible outcomes is 2 2 2 = 8...
Read More →Answer the following questions:
Question: Answer the following questions: (a)Explain qualitatively on the basis of domain picture the irreversibility in the magnetisation curve of a ferromagnet. (b)The hysteresis loop of a soft iron piece has a much smaller area than that of a carbon steel piece. If the material is to go through repeated cycles of magnetisation, which piece will dissipate greater heat energy? (c)A system displaying a hysteresis loop such as a ferromagnet, is a device for storing memory? Explain the meaning of ...
Read More →How many 5–digit telephone numbers can be constructed using the digits 0 to 9 if each number starts with 67 and no digit appears more than once?
Question: How many 5digit telephone numbers can be constructed using the digits 0 to 9 if each number starts with 67 and no digit appears more than once? Solution: It is given that the 5-digit telephone numbers always start with 67. Therefore, there will be as many phone numbers as there are ways of filling 3 vacant placesby the digits 0 9, keeping in mind that the digits cannot be repeated. The units place can be filled by any of the digits from 0 9, except digits 6 and 7. Therefore, the units ...
Read More →How many 4-letter code can be formed using the first 10 letters of the English alphabet, if no letter can be repeated?
Question: How many 4-letter code can be formed using the first 10 letters of the English alphabet, if no letter can be repeated? Solution: There are as many codes as there are ways of filling 4 vacant placesin succession by the first 10 letters of the English alphabet, keeping in mind that the repetition of letters is not allowed. The first place can be filled in 10 different ways by any of the first 10 letters of the English alphabet following which, the second place can be filled in by any of ...
Read More →Answer the following questions:
Question: Answer the following questions: (a)Why does a paramagnetic sample display greater magnetisation (for the same magnetising field) when cooled? (b)Why is diamagnetism, in contrast, almost independent of temperature? (c)If a toroid uses bismuth for its core, will the field in the core be (slightly) greater or (slightly) less than when the core is empty? (d)Is the permeability of a ferromagnetic material independent of the magnetic field? If not, is it more for lower or higher fields? (e)M...
Read More →How many 3-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 if the digits can be repeated?
Question: How many 3-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 if the digits can be repeated? Solution: There will be as many ways as there are ways of filling 3 vacant places in succession by the given six digits. In this case, the units place can be filled by 2 or 4 or 6 only i.e., the units place can be filled in 3 ways. The tens place can be filled by any of the 6 digits in 6 different ways and also the hundreds place can be filled by any of the 6 digits in 6 differen...
Read More →Complete the following chemical reactions.
Question: Complete the following chemical reactions. Classify the above into (a) hydrolysis, (b) redox and (c) hydration reactions. Solution: H2O2is acting as an oxidizing agent in the reaction. Hence, it is a redox reaction. $\mathrm{H}_{2} \mathrm{O}_{2(a q)}$ is acting as a reducing agent in the acidic medium, thereby oxidizing $\mathrm{MnO}_{4(a q)}^{-} .$Hence, the given reaction is a redox reaction. The reactions in which a compound reacts with water to produce other compounds are called h...
Read More →A short bar magnet of magnetic moment
Question: A short bar magnet of magnetic moment 5.25 102J T1is placed with its axis perpendicular to the earths field direction. At what distance from the centre of the magnet, the resultant field is inclined at 45 with earths field on (a) its normal bisector and (b) its axis. Magnitude of the earths field at the place is given to be 0.42 G. Ignore the length of the magnet in comparison to the distances involved. Solution: Magnetic moment of the bar magnet, $M=5.25 \times 10^{-2} \mathrm{~J} \ma...
Read More →How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that
Question: How many 3-digit numbers can be formed from the digits 1, 2, 3, 4 and 5 assuming that (i) repetition of the digits is allowed? (ii) repetition of the digits is not allowed? Solution: (i) There will be as many ways as there are ways of filling 3 vacant places in succession by the given five digits. In this case, repetition of digits is allowed. Therefore, the units place can be filled in by any of the given five digits. Similarly, tens and hundreds digits can be filled in by any of the ...
Read More →Show that
Question: Show that (i) $\left[\begin{array}{rr}5 -1 \\ 6 7\end{array}\right]\left[\begin{array}{ll}2 1 \\ 3 4\end{array}\right] \neq\left[\begin{array}{rr}2 1 \\ 3 4\end{array}\right]\left[\begin{array}{rr}5 -1 \\ 6 7\end{array}\right]$ (ii) $\left[\begin{array}{lll}1 2 3 \\ 0 1 0 \\ 1 1 0\end{array}\right]\left[\begin{array}{rrr}-1 1 0 \\ 0 -1 1 \\ 2 3 4\end{array}\right] \neq\left[\begin{array}{rrr}-1 1 0 \\ 0 -1 1 \\ 2 3 4\end{array}\right]\left[\begin{array}{lll}1 2 3 \\ 0 1 0 \\ 1 1 0\end{...
Read More →If the bar magnet in exercise 5.13 is turned around by 180º,
Question: If the bar magnet in exercise 5.13 is turned around by 180, wherewill the new null points be located? Solution: The magnetic field on the axis of the magnet at a distance $d_{1}=14 \mathrm{~cm}$, can be written as: $B_{1}=\frac{\mu_{0} 2 M}{4 \pi\left(d_{1}\right)^{3}}=H$ ...(i) Where, M= Magnetic moment $\mu_{0}=$ Permeability of free space H= Horizontal component of the magnetic field atd1 If the bar magnet is turned through 180, then the neutral point will lie on the equatorial line...
Read More →A merchant has 120 litres of oil of one kind, 180 litres of another kind and 240 litres of third kind. He wants to sell the oil by filling the three kinds of oil in tins of equal capacity. What should be the greatest capacity of such a tin?
Question: A merchant has 120 litres of oil of one kind, 180 litres of another kind and 240 litres of third kind. He wants to sell the oil by filling the three kinds of oil in tins of equal capacity. What should be the greatest capacity of such a tin? Solution: The merchant has 3 different oils of 120 liters, 180 liters and 240 liters respectively. So the greatest capacity of the tin for filling three different types of oil is given by the H.C.F. of 120,180 and 240. So first we will calculate H.C...
Read More →A short bar magnet placed in a horizontal plane has its axis aligned along the magnetic north-south direction.
Question: A short bar magnet placed in a horizontal plane has its axis alignedalong the magnetic north-south direction. Null points are found on the axis of the magnet at 14 cm from the centre of the magnet. The earths magnetic field at the place is 0.36 G and the angle of dip is zero. What is the total magnetic field on the normal bisector of the magnet at the same distance as the nullpoint (i.e., 14 cm) from the centre of the magnet? (Atnull points, field due to a magnet is equal and opposite ...
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