Solve the given inequality for real x: 2(2x + 3) – 10 < 6 (x – 2)
Question: Solve the given inequality for real $x: 2(2 x+3)-106(x-2)$ Solution: $2(2 x+3)-106(x-2)$ $\Rightarrow 4 x+6-106 x-12$ $\Rightarrow 4 x-46 x-12$ $\Rightarrow-4+126 x-4 x$ $\Rightarrow 82 x$ $\Rightarrow 4x$ Thus, all real numbers $x$, which are greater than or equal to 4, are the solutions of the given inequality. Hence, the solution set of the given inequality is $(4, \infty)$....
Read More →Solve the given inequality for real x:
Question: Solve the given inequality for realx:$\frac{1}{2}\left(\frac{3 x}{5}+4\right) \geq \frac{1}{3}(x-6)$ Solution: $\frac{1}{2}\left(\frac{3 x}{5}+4\right) \geq \frac{1}{3}(x-6)$ $\Rightarrow 3\left(\frac{3 x}{5}+4\right) \geq 2(x-6)$ $\Rightarrow \frac{9 x}{5}+12 \geq 2 x-12$ $\Rightarrow 12+12 \geq 2 x-\frac{9 x}{5}$ $\Rightarrow 24 \geq \frac{10 x-9 x}{5}$ $\Rightarrow 24 \geq \frac{x}{5}$ $\Rightarrow 120 \geq x$ Thus, all real numbers $x$, which are less than or equal to 120 , are the...
Read More →A square coil of side 10 cm consists of 20 turns and carries a current of 12 A.
Question: A square coil of side 10 cm consists of 20 turns and carries a current of 12 A. The coil is suspended vertically and the normal to the plane of the coil makes an angle of 30 with the direction of a uniform horizontal magnetic field of magnitude 0.80 T. What is the magnitude of torque experienced by the coil? Solution: Length of a side of the square coil,l= 10 cm = 0.1 m Current flowing in the coil,I= 12 A Number of turns on the coil,n= 20 Angle made by the plane of the coil with magnet...
Read More →Show that the square of any positive integer cannot be of the form $3 m+2$, where $m$ is a natural number..
Question: Show that the square of any positive integer cannot be of the form $3 m+2$, where $m$ is a natural number.. Solution: By Euclid's lemma, $b=a q+r, 0 \leq ra$. Here,bis a positive integer anda= 3. $\therefore b=3 q+r$, for $0 \leq r3$ This must be in the form $3 q, 3 q+1$ or $3 q+2$. Now, $(3 q)^{2}=9 q^{2}=3 m$, where $m=3 q^{2}$ $(3 q+1)^{2}=9 q^{2}+6 q+1=3\left(3 q^{2}+2 q\right)+1=3 m+1$, where $m=3 q^{2}+2 q$ $(3 q+2)^{2}=9 q^{2}+12 q+4=3\left(3 q^{2}+4 q+1\right)+1=3 m+1$, where $...
Read More →Solve the given inequality for real x:
Question: Solve the given inequality for realx: $\frac{3(x-2)}{5} \leq \frac{5(2-x)}{3}$ Solution: $\frac{3(x-2)}{5} \leq \frac{5(2-x)}{3}$ $\Rightarrow 9(x-2) \leq 25(2-x)$ $\Rightarrow 9 x-18 \leq 50-25 x$ $\Rightarrow 9 x-18+25 x \leq 50$ $\Rightarrow 34 x-18 \leq 50$ $\Rightarrow 34 x \leq 50+18$ $\Rightarrow 34 x \leq 68$ $\Rightarrow \frac{34 x}{34} \leq \frac{68}{34}$ $\Rightarrow x \leq 2$ Thus, all real numbers $x$, which are less than or equal to 2, are the solutions of the given inequ...
Read More →If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements?
Question: If a matrix has 18 elements, what are the possible orders it can have? What, if it has 5 elements? Solution: We know that if a matrix is ofthe ordermn, it hasmnelements. Thus, to find all the possible orders of a matrix having 18 elements, we have to find all the ordered pairs of natural numbers whose product is 18. Theordered pairs are: (1, 18), (18, 1), (2, 9), (9, 2), (3, 6,), and (6, 3) Hence, the possible orders of a matrix having 18 elements are: 118, 181, 29, 92, 36, and 63 (1, ...
Read More →A closely wound solenoid 80 cm long has 5 layers of windings of 400 turns each.
Question: A closely wound solenoid 80 cm long has 5 layers of windings of 400 turns each. The diameter of the solenoid is 1.8 cm. If the current carried is 8.0 A, estimate the magnitude ofBinside the solenoid near its centre. Solution: Length of the solenoid,l= 80 cm = 0.8 m There are five layers of windings of 400 turns each on the solenoid. $\therefore$ Total number of turns on the solenoid, $N=5 \times 400=2000$ Diameter of the solenoid,D= 1.8 cm = 0.018 m Current carried by the solenoid,I= 8...
Read More →If a matrix has 24 elements, what are the possible order it can have? What, if it has 13 elements?
Question: If a matrix has 24 elements, what are the possible order it can have? What, if it has 13 elements? Solution: Weknow that if a matrix is of the ordermn, it hasmnelements. Thus, to find all the possible orders of a matrix having 24 elements, we have to find all the ordered pairs of natural numbers whose product is 24. Theordered pairs are: (1, 24), (24, 1), (2, 12), (12, 2), (3, 8), (8, 3), (4, 6), and(6, 4) Hence, the possible orders of a matrix having 24 elements are: 124, 241, 212, 12...
Read More →Two long and parallel straight wires A and B carrying currents
Question: Two long and parallel straight wires A and B carrying currents of 8.0 A and 5.0 A in the same direction are separated by a distance of 4.0 cm. Estimate the force on a 10 cm section of wire A. Solution: Current flowing in wire A,IA= 8.0 A Current flowing in wire B,IB= 5.0 A Distance between the two wires,r= 4.0 cm = 0.04 m Length of a section of wire A,l= 10 cm = 0.1 m Force exerted on lengthldue to the magnetic field is given as: $B=\frac{\mu_{0} 2 I_{\mathrm{A}} I_{\mathrm{B}} l}{4 \p...
Read More →Solve the given inequality for real x:
Question: Solve the given inequality for real x:$\frac{x}{3}\frac{x}{2}+1$ Solution: $\frac{x}{3}\frac{x}{2}+1$ $\Rightarrow \frac{x}{3}-\frac{x}{2}1$ $\Rightarrow \frac{2 x-3 x}{6}1$ $\Rightarrow-\frac{x}{6}1$ $\Rightarrow-x6$ $\Rightarrow x-6$ Thus, all real numbers $x$, which are less than $-6$, are the solutions of the given inequality. Hence, the solution set of the given inequality is $(-\infty,-6)$....
Read More →In the matrix
Question: In the matrix $A=\left[\begin{array}{cccc}2 5 19 -7 \\ 35 -2 \frac{5}{2} 12 \\ \sqrt{3} 1 -5 17\end{array}\right]$, write: (i) The order of the matrix (ii) The number of elements, (iii) Write the elements $a_{13}, a_{21}, a_{33}, a_{24}, a_{23}$ Solution: (i)In the given matrix, the number of rows is 3 and the number of columns is 4. Therefore, the order of the matrix is 34. (ii)Since the order of the matrix is 34, there are 34 = 12 elements in it. (iii) $a_{13}=19, a_{21}=35, a_{33}=-...
Read More →A 3.0 cm wire carrying a current of 10 A is placed inside a solenoid perpendicular to its axis.
Question: A 3.0 cm wire carrying a current of 10 A is placed inside a solenoid perpendicular to its axis. The magnetic field inside the solenoid is given to be 0.27 T. What is the magnetic force on the wire? Solution: Length of the wire,l= 3 cm = 0.03 m Current flowing in the wire,I= 10 A Magnetic field,B= 0.27 T Angle between the current and magnetic field,= 90 Magnetic force exerted on the wire is given as: F=BIlsin $=0.27 \times 10 \times 0.03 \sin 90^{\circ}$ $=8.1 \times 10^{-2} \mathrm{~N}...
Read More →A positive integer is of the form $3 q+1, q$ being a anatural number. Can you write its square in any form other than $3 m+1,3 m$ or $3 m+2$ for some integer $m ?$ Justify your answer.
Question: A positive integer is of the form $3 q+1, q$ being a anatural number. Can you write its square in any form other than $3 m+1,3 m$ or $3 m+2$ for some integer $m ?$ Justify your answer. Solution: By Euclid's lemma, $b=a q+r, 0 \leq ra$. Here,bis a positive integer anda= 3. $\therefore b=3 q+r$, for $0 \leq r3$ This must be in the form $3 q, 3 q+1$ or $3 q+2$. Now, $(3 q)^{2}=9 q^{2}=3 m$, where $m=3 q^{2}$ $(3 q+1)^{2}=9 q^{2}+6 q+1=3\left(3 q^{2}+2 q\right)+1=3 m+1$, where $m=3 q^{2}+2...
Read More →Solve the given inequality for real x:
Question: Solve the given inequality for real x:$x+\frac{x}{2}+\frac{x}{3}11$ Solution: $x+\frac{x}{2}+\frac{x}{3}11$ $\Rightarrow x\left(1+\frac{1}{2}+\frac{1}{3}\right)11$ $\Rightarrow x\left(\frac{6+3+2}{6}\right)11$ $\Rightarrow \frac{11 x}{6}11$ $\Rightarrow \frac{11 x}{6 \times 11}\frac{11}{11}$ $\Rightarrow \frac{x}{6}1$ $\Rightarrow x6$ Thus, all real numbers $x$, which are less than 6, are the solutions of the given inequality. Hence, the solution set of the given inequality is $(-\inft...
Read More →What is the magnitude of magnetic force per unit length
Question: What is the magnitude of magnetic force per unit length on a wire carrying a current of 8 A and making an angle of 30 with the direction of a uniform magnetic field of 0.15 T? Solution: Current in the wire,I= 8 A Magnitude of the uniform magnetic field,B= 0.15 T Angle between the wire and magnetic field,= 30. Magnetic force per unit length on the wire is given as: $f=B / \sin \theta$ $=0.15 \times 8 \times 1 \times \sin 30^{\circ}$ $=0.6 \mathrm{~N} \mathrm{~m}^{-1}$ Hence, the magneti...
Read More →Solve the given inequality for real x: 3(2 – x) ≥ 2(1 – x)
Question: Solve the given inequality for real $x: 3(2-x) \geq 2(1-x)$ Solution: $3(2-x) \geq 2(1-x)$ $\Rightarrow 6-3 x \geq 2-2 x$ $\Rightarrow 6-3 x+2 x \geq 2-2 x+2 x$ $\Rightarrow 6-x \geq 2$ $\Rightarrow 6-x-6 \geq 2-6$ $\Rightarrow-x \geq-4$ $\Rightarrow x \leq 4$ Thus, all real numbers $x$, which are less than or equal to 4 , are the solutions of the given inequality. Hence, the solution set of the given inequality is $(-\infty, 4]$....
Read More →A horizontal overhead power line carries a current of 90 A in east to west direction.
Question: A horizontal overhead power line carries a current of 90 A in east to west direction. What is the magnitude and direction of the magnetic field due to the current 1.5 m below the line? Solution: Current in the power line,I= 90 A Point is located below the power line at distance,r= 1.5 m Hence, magnetic field at that point is given by the relation, $B=\frac{\mu_{0} 2 I}{4 \pi r}$ Where, $\mu_{0}=$ Permeability of free space $=4 \pi \times 10^{-7} \mathrm{Tm} \mathrm{A}^{-1}$ $B=\frac{4 ...
Read More →Solve the given inequality for real x: 3(x – 1) ≤ 2 (x – 3)
Question: Solve the given inequality for real $x: 3(x-1) \leq 2(x-3)$ Solution: $3(x-1) \leq 2(x-3)$ $\Rightarrow 3 x-3 \leq 2 x-6$ $\Rightarrow 3 x-3+3 \leq 2 x-6+3$ $\Rightarrow 3 x \leq 2 x-3$ $\Rightarrow 3 x-2 x \leq 2 x-3-2 x$ $\Rightarrow x \leq-3$ Thus, all real numbers $x$, which are less than or equal to $-3$, are the solutions of the given inequality. Hence, the solution set of the given inequality is $(-\infty,-3]$....
Read More →Solve
Question: Solve $\tan ^{-1}\left(\frac{x}{y}\right)-\tan ^{-1} \frac{x-y}{x+y}$ is equal to (A) $\frac{\pi}{2}$ (B). $\frac{\pi}{3}$ (C) $\frac{\pi}{4}$ (D) $\frac{-3 \pi}{4}$ Solution: $\tan ^{-1}\left(\frac{x}{y}\right)-\tan ^{-1} \frac{x-y}{x+y}$ $=\tan ^{-1}\left[\frac{\frac{x}{y}-\frac{x-y}{x+y}}{1+\left(\frac{x}{y}\right)\left(\frac{x-y}{x+y}\right)}\right]$ $\left[\tan ^{-1} y-\tan ^{-1} y=\tan ^{-1} \frac{x-y}{1+x y}\right]$ $=\tan ^{-1}\left[\frac{\frac{x(x+y)-y(x-y)}{y(x+y)}}{\frac{y(x...
Read More →Solve the given inequality for real x: 3x – 7 > 5x – 1
Question: Solve the given inequality for real $x: 3 x-75 x-1$ Solution: $3 x-75 x-1$ $\Rightarrow 3 x-7+75 x-1+7$ $\Rightarrow 3 x5 x+6$ $\Rightarrow 3 x-5 x5 x+6-5 x$ $\Rightarrow-2 x6$ $\Rightarrow \frac{-2 x}{-2}\frac{6}{-2}$ $\Rightarrow x-3$ Thus, all real numbers $x$, which are less than $-3$, are the solutions of the given inequality. Hence, the solution set of the given inequality is $(-\infty,-3)$....
Read More →A long straight wire in the horizontal plane carries a current of 50 A in north to south direction.
Question: A long straight wire in the horizontal plane carries a current of 50 A in north to south direction. Give the magnitude and direction ofBat a point 2.5 m east of the wire. Solution: Current in the wire,I= 50 A A point is 2.5 m away from the East of the wire. $\therefore$ Magnitude of the distance of the point from the wire, $r=2.5 \mathrm{~m}$. Where, $\mu_{0}=$ Permeability of free space $=4 \pi \times 10^{-7} \mathrm{~T} \mathrm{~m} \mathrm{~A}^{-1}$ $B=\frac{4 \pi \times 10^{-7} \tim...
Read More →Show that the square of an odd positive integer can be of the form 6q+1 or 6q+3 for some integer q.
Question: Show that the square of an odd positive integer can be of the form 6q+1 or 6q+3 for some integer q. Solution: It is known that any positive integer can be written in the form of $6 m, 6 m+1,6 m+2,6 m+3,6 m+4,6 m+5$ for some integer $m$. Thus, an odd positive integer can be of the form $6 m+1,6 m+3,6 m+5$. We have, $(6 m+1)^{2}=36 m^{2}+12 m+1=6\left(6 m^{2}+2 m\right)+1=6 q+1$, where $q=6 m^{2}+2 m$ is an integer $(6 m+3)^{2}=36 m^{2}+36 m+9=6\left(6 m^{2}+6 m+1\right)+3=6 q+3$, where ...
Read More →Solve the given inequality for real x: 4x + 3 < 5x + 7
Question: Solve the given inequality for real $x: 4 x+35 x+7$ Solution: $4 x+35 x+7$ $\Rightarrow 4 x+3-75 x+7-7$ $\Rightarrow 4 x-45 x$ $\Rightarrow 4 x-4-4 x5 x-4 x$ $\Rightarrow-4x$ Thus, all real numbers $x$, which are greater than $-4$, are the solutions of the given inequality. Hence, the solution set of the given inequality is $(-4, \infty)$....
Read More →A long straight wire in the horizontal plane carries a current of 50 A in north to south direction.
Question: A long straight wire in the horizontal plane carries a current of 50 A in north to south direction. Give the magnitude and direction ofBat a point 2.5 m east of the wire. Solution: Current in the wire,I= 50 A A point is 2.5 m away from the East of the wire. $\therefore$ Magnitude of the distance of the point from the wire, $r=2.5 \mathrm{~m}$. Where, $\mu_{0}=$ Permeability of free space $=4 \pi \times 10^{-7} \mathrm{~T} \mathrm{~m} \mathrm{~A}^{-1}$ $\mu_{0}=$ Permeability of free sp...
Read More →Solve
Question: Solve $\sin ^{-1}(1-x)-2 \sin ^{-1} x=\frac{\pi}{2}$, then $x$ is equal to (A) $0, \frac{1}{2}$ (B) $1, \frac{1}{2}$ (C) 0 (D) $\frac{1}{2}$ Solution: $\sin ^{-1}(1-x)-2 \sin ^{-1} x=\frac{\pi}{2}$ $\Rightarrow-2 \sin ^{-1} x=\frac{\pi}{2}-\sin ^{-1}(1-x)$ $\Rightarrow-2 \sin ^{-1} x=\cos ^{-1}(1-x) \quad \ldots .(1)$ Let $\sin ^{-1} x=\theta \Rightarrow \sin \theta=x \Rightarrow \cos \theta=\sqrt{1-x^{2}}$. $\therefore \theta=\cos ^{-1}\left(\sqrt{1-x^{2}}\right)$ $\therefore \sin ^{-...
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