The Mn3+ ion is unstable in solution and undergoes disproportionation to give Mn2+,
Question: The $\mathrm{Mn}^{3+}$ ion is unstable in solution and undergoes disproportionation to give $\mathrm{Mn}^{2+}, \mathrm{MnO}_{2}$, and $\mathrm{H}^{+}$ion. Write a balanced ionic equation for the reaction. Solution: The given reaction can be represented as: The oxidation half equation is: The oxidation number is balanced by adding one electron as: The charge is balanced by adding 4H+ions as: The O atoms and H+ions are balanced by adding 2H2O molecules as: The reduction half equation is:...
Read More →For a circular coil of radius R and N turns carrying current I,
Question: For a circular coil of radiusRandNturns carrying currentI, the magnitude of the magnetic field at a point on its axis at a distancexfrom its centre is given by, $B=\frac{\mu_{0} I R^{2} N}{2\left(x^{2}+R^{2}\right)^{\frac{3}{2}}}$ (a)Show that this reduces to the familiar result for field at the centre of the coil. (b)Consider two parallel co-axial circular coils of equal radiusR, and number of turnsN, carrying equal currents in the same direction, and separated by a distanceR. Show th...
Read More →The longest side of a triangle is 3 times the shortest side and the third side is 2 cm shorter than the longest side.
Question: The longest side of a triangle is 3 times the shortest side and the third side is 2 cm shorter than the longest side. If the perimeter of the triangle is at least 61 cm, find the minimum length of the shortest side. Solution: Let the length of the shortest side of the triangle be $x \mathrm{~cm}$. Then, length of the longest side $=3 x \mathrm{~cm}$ Length of the third side $=(3 x-2) \mathrm{cm}$ Since the perimeter of the triangle is at least $61 \mathrm{~cm}$, $x \mathrm{~cm}+3 x \ma...
Read More →Find all pairs of consecutive even positive integers,
Question: Find all pairs of consecutive even positive integers, both of which are larger than 5 such that their sum is less than 23. Solution: Letxbe the smaller of the two consecutive even positive integers. Then, the other integer isx+ 2. Since both the integers are larger than 5, $x5 \ldots$ (1) Also, the sum of the two integers is less than 23 . $x+(x+2)23$ $\Rightarrow 2 x+223$ $\Rightarrow 2 x23-2$ $\Rightarrow 2 x21$ $\Rightarrow x\frac{21}{2}$ $\Rightarrow x10.5$ $\ldots$ (2) From (1) an...
Read More →What sorts of informations can you draw from the following reaction ?
Question: What sorts of informations can you draw from the following reaction ? $(\mathrm{CN})_{2(g)}+2 \mathrm{OH}_{(2 q)}^{-} \longrightarrow \mathrm{CN}_{((q)}^{-}+\mathrm{CNO}_{(a q)}^{-}+\mathrm{H}_{2} \mathrm{O}_{(l)}$ Solution: The oxidation numbers of carbon in $(\mathrm{CN})^{2}, \mathrm{CN}^{-}$and $\mathrm{CNO}^{-}$are $+3,+2$ and $+4$ respectively. These are obtained as shown below: Let the oxidation number of C bex. $(\mathrm{CN})_{2}$ $2(x-3)=0$ $\therefore x=3$ $\mathrm{CN}^{-}$ $...
Read More →Find all pairs of consecutive odd positive integers both of which are smaller
Question: Find all pairs of consecutive odd positive integers both of which are smaller than 10 such that their sum is more than 11. Solution: Letxbe the smaller of the two consecutive odd positive integers. Then, the other integer isx+ 2. Since both the integers are smaller than 10, $x+210$ $\Rightarrow x10-2$ $\Rightarrow x8 \ldots$ (i) Also, the sum of the two integers is more than 11. $\therefore x+(x+2)11$ $\Rightarrow 2 x+211$ $\Rightarrow 2 x11-2$ $\Rightarrow 2 x9$ $\Rightarrow x\frac{9}...
Read More →A magnetic field of 100 G
Question: A magnetic field of 100 G (1 G = 104T) is required which is uniform in a region of linear dimension about 10 cm and area of cross-section about 103m2. The maximum current-carrying capacity of a given coil of wire is 15 A and the number of turns per unit length that can be wound round a core is at most 1000 turns m1. Suggest some appropriate design particulars of a solenoid for the required purpose. Assume the core is not ferromagnetic Solution: Magnetic field strength,B= 100 G = 100 10...
Read More →To receive Grade ‘A’ in a course,
Question: To receive Grade A in a course, one must obtain an average of 90 marks or more in five examinations (each of 100 marks). If Sunitas marks in first four examinations are 87, 92, 94 and 95, find minimum marks that Sunita must obtain in fifth examination to get grade A in the course. Solution: Letxbe the marks obtained by Sunita in the fifth examination. In order to receive grade A in the course, she must obtain an average of 90 marks or more in five examinations. Therefore, $\frac{87+92+...
Read More →Two concentric circular coils X and Y of radii 16 cm and 10 cm,
Question: Two concentric circular coils X and Y of radii 16 cm and 10 cm, respectively, lie in the same vertical plane containing the north to south direction. Coil X has 20 turns and carries a current of 16 A; coil Y has 25 turns and carries a current of 18 A. The sense of the current in X is anticlockwise, and clockwise in Y, for an observer looking at the coils facing west. Give the magnitude and direction of the net magnetic field due to the coils at their centre. Solution: Radius of coil X,...
Read More →Ravi obtained 70 and 75 marks in first two unit test. Find the minimum marks he should get in the third
Question: Ravi obtained 70 and 75 marks in first two unit test. Find the minimum marks he should get in the third test to have an average of at least 60 marks. Solution: Letxbe the marks obtained by Ravi in the third unit test. Since the student should have an average of at least 60 marks, $\frac{70+75+x}{3} \geq 60$ $\Rightarrow 145+x \geq 180$ $\Rightarrow x \geq 180-145$ $\Rightarrow x \geq 35$ Thus, the student must obtain a minimum of 35 marks to have an average of at least 60 marks....
Read More →Two concentric circular coils X and Y of radii 16 cm and 10 cm,
Question: Two concentric circular coils X and Y of radii 16 cm and 10 cm, respectively, lie in the same vertical plane containing the north to south direction. Coil X has 20 turns and carries a current of 16 A; coil Y has 25 turns and carries a current of 18 A. The sense of the current in X is anticlockwise, and clockwise in Y, for an observer looking at the coils facing west. Give the magnitude and direction of the net magnetic field due to the coils at their centre. Solution: Radius of coil X,...
Read More →Solve the given inequality and show the graph of the solution on number line:
Question: Solve the given inequality and show the graph of the solution on number line:$\frac{x}{2} \geq \frac{(5 x-2)}{3}-\frac{(7 x-3)}{5}$ Solution: $\frac{x}{2} \geq \frac{(5 x-2)}{3}-\frac{(7 x-3)}{5}$ $\Rightarrow \frac{x}{2} \geq \frac{5(5 x-2)-3(7 x-3)}{15}$ $\Rightarrow \frac{x}{2} \geq \frac{25 x-10-21 x+9}{15}$ $\Rightarrow \frac{x}{2} \geq \frac{4 x-1}{15}$$\Rightarrow 15 x \geq 2(4 x-1)$ $\Rightarrow 15 x \geq 8 x-2$ $\Rightarrow 15 x-8 x \geq 8 x-2-8 x$ $\Rightarrow 7 x \geq-2$ $\R...
Read More →Solve the given inequality and show the graph of the solution on number line: 3(1 – x) < 2 (x + 4)
Question: Solve the given inequality and show the graph of the solution on number line: $3(1-x)2(x+4)$ Solution: $3(1-x)2(x+4)$ $\Rightarrow 3-3 x2 x+8$ $\Rightarrow 3-82 x+3 x$ $\Rightarrow-55 x$ $\Rightarrow \frac{-5}{5}\frac{5 x}{5}$ $\Rightarrow-1x$ The graphical representation of the solutions of the given inequality is as follows....
Read More →Balance the following equations in basic medium by ion-electron method and oxidation
Question: Balance the following equations in basic medium by ion-electron method and oxidation number methods and identify the oxidising agent and the reducing agent. (a) $\mathrm{P}_{4(s)}+\mathrm{OH}_{\text {(aq) }} \longrightarrow \mathrm{PH}_{3(g)}+\mathrm{HPO}_{2}^{-}$(aq) (b) $\mathrm{N}_{2} \mathrm{H}_{4(())}+\mathrm{ClO}_{3(a q)}^{-} \longrightarrow \mathrm{NO}_{(g)}+\mathrm{Cl}_{(g)}^{-}$ (c) $\mathrm{Cl}_{2} \mathrm{O}_{7(g)}+\mathrm{H}_{2} \mathrm{O}_{2(a q)} \longrightarrow \mathrm{C...
Read More →(a) A circular coil of 30 turns and radius 8.0 cm carrying a current of 6.0 A is suspended vertically in a uniform horizontal magnetic field of magnitude 1.0 T.
Question: (a)A circular coil of 30 turns and radius 8.0 cm carrying a current of 6.0 A is suspended vertically in a uniform horizontal magnetic field of magnitude 1.0 T. The field lines make an angle of 60 with the normal of the coil. Calculate the magnitude of the counter torque that must be applied to prevent the coil from turning. (b)Would your answer change, if the circular coil in (a) were replaced by a planar coil of some irregular shape that encloses the same area? (All other particulars ...
Read More →Solve the given inequality and show the graph of the solution on number line: 5x – 3 ≥ 3x – 5
Question: Solve the given inequality and show the graph of the solution on number line: $5 x-3 \geq 3 x-5$ Solution: $5 x-3 \geq 3 x-5$ $\Rightarrow 5 x-3 x \geq-5+3$ $\Rightarrow 2 x \geq-2$ $\Rightarrow \frac{2 x}{2} \geq \frac{-2}{2}$ $\Rightarrow x \geq-1$ The graphical representation of the solutions of the given inequality is as follows....
Read More →Construct a 3 × 4 matrix, whose elements are given by
Question: Construct a $3 \times 4$ matrix, whose elements are given by (i) $a_{i j}=\frac{1}{2}|-3 i+j|$ (ii) $a_{i j}=2 i-j$ Solution: In general, a $3 \times 4$ matrix is given by $A=\left[\begin{array}{llll}a_{11} a_{12} a_{13} a_{14} \\ a_{21} a_{22} a_{23} a_{24} \\ a_{31} a_{32} a_{33} a_{34}\end{array}\right]$ (i) $a_{i j}=\frac{1}{2}|-3 i+j|, i=1,2,3$ and $j=1,2,3,4$ $\therefore a_{11}=\frac{1}{2}|-3 \times 1+1|=\frac{1}{2}|-3+1|=\frac{1}{2}|-2|=\frac{2}{2}=1$ $a_{21}=\frac{1}{2}|-3 \tim...
Read More →In Exercise 4.11 obtain the frequency of revolution of the electron in its circular orbit.
Question: In Exercise 4.11 obtain the frequency of revolution of the electron in its circular orbit. Does the answer depend on the speed of the electron? Explain. Solution: Magnetic field strength,B= 6.5 104T Charge of the electron,e= 1.6 1019C Mass of the electron,me= 9.1 1031kg Velocity of the electron,v= 4.8 106m/s Radius of the orbit,r= 4.2 cm = 0.042 m Frequency of revolution of the electron = Angular frequency of the electron == 2 Velocity of the electron is related to the angular frequenc...
Read More →Solve the given inequality and show the graph of the solution on number line: 3x – 2 < 2x +1
Question: Solve the given inequality and show the graph of the solution on number line: $3 x-22 x+1$ Solution: $3 x-22 x+1$ $\Rightarrow 3 x-2 x1+2$ $\Rightarrow x3$ The graphical representation of the solutions of the given inequality is as follows....
Read More →Solve the given inequality for real x:
Question: Solve the given inequality for realx:$\frac{(2 x-1)}{3} \geq \frac{(3 x-2)}{4}-\frac{(2-x)}{5}$ Solution: $\frac{(2 x-1)}{3} \geq \frac{(3 x-2)}{4}-\frac{(2-x)}{5}$ $\Rightarrow \frac{(2 x-1)}{3} \geq \frac{5(3 x-2)-4(2-x)}{20}$ $\Rightarrow \frac{(2 x-1)}{3} \geq \frac{15 x-10-8+4 x}{20}$ $\Rightarrow \frac{(2 x-1)}{3} \geq \frac{19 x-18}{20}$ $\Rightarrow 20(2 x-1) \geq 3(19 x-18)$ $\Rightarrow 40 x-20 \geq 57 x-54$ $\Rightarrow-20+54 \geq 57 x-40 x$ $\Rightarrow 34 \geq 17 x$ $\Righ...
Read More →In a chamber, a uniform magnetic field of
Question: In a chamber, a uniform magnetic field of $6.5 \mathrm{G}\left(1 \mathrm{G}=10^{-4} \mathrm{~T}\right)$ is maintained. An electron is shot into the field with a speed of $4.8 \times 10^{6} \mathrm{~m} \mathrm{~s}^{-1}$ normal to the field. Explain why the path of the electron is a circle. Determine the radius of the circular orbit. $\left(e=1.6 \times 10^{-19} \mathrm{C}, m_{\mathrm{e}}=9.1 \times 10^{-31}\right.$ $\mathrm{kg}$ ) Solution: Magnetic field strength,B= 6.5 G = 6.5 104T Sp...
Read More →Solve the given inequality for real x:
Question: Solve the given inequality for realx:$\frac{x}{4}\frac{(5 x-2)}{3}-\frac{(7 x-3)}{5}$ Solution: $\frac{x}{4}\frac{(5 x-2)}{3}-\frac{(7 x-3)}{5}$ $\Rightarrow \frac{x}{4}\frac{5(5 x-2)-3(7 x-3)}{15}$ $\Rightarrow \frac{x}{4}\frac{25 x-10-21 x+9}{15}$ $\Rightarrow \frac{x}{4}\frac{4 x-1}{15}$ $\Rightarrow 15 x4(4 x-1)$ $\Rightarrow 15 x16 x-4$ $\Rightarrow 416 x-15 x$ $\Rightarrow 4x$ Thus, all real numbers $x$, which are greater than 4, are the solutions of the given inequality. Hence, ...
Read More →Solve the given inequality for real x: 37 – (3x + 5) ≥ 9x – 8(x – 3)
Question: Solve the given inequality for real $x: 37-(3 x+5) \geq 9 x-8(x-3)$ Solution: $37-(3 x+5) \geq 9 x-8(x-3)$ $\Rightarrow 37-3 x-5 \geq 9 x-8 x+24$ $\Rightarrow 32-3 x \geq x+24$ $\Rightarrow 32-24 \geq x+3 x$ $\Rightarrow 8 \geq 4 x$ $\Rightarrow 2 \geq x$ Thus, all real numbers $x$, which are less than or equal to 2 , are the solutions of the given inequality. Hence, the solution set of the given inequality is $(-\infty, 2]$....
Read More →Two moving coil meters,
Question: Two moving coil meters, M1and M2have the following particulars: R1= 10 Ω,N1= 30, A1= 3.6 103m2,B1= 0.25 T R2= 14 Ω,N2= 42, A2= 1.8 103m2,B2= 0.50 T (The spring constants are identical for the two meters). Determine the ratio of (a) current sensitivity and (b) voltage sensitivity of M2and M1. Solution: For moving coil meter M1: Resistance,R1= 10 Ω Number of turns,N1= 30 Area of cross-section,A1= 3.6 103m2 Magnetic field strength,B1= 0.25 T Spring constantK1=K For moving coil meter M2: R...
Read More →Construct a 2 × 2 matrix,
Question: Construct a $2 \times 2$ matrix, $A=\left[a_{i j}\right]$, whose elements are given by: (i) $a_{i j}=\frac{(i+j)^{2}}{2}$ (ii) $a_{i j}=\frac{i}{j}$ (iii) $a_{i j}=\frac{(i+2 j)^{2}}{2}$ Solution: In general, a $2 \times 2$ matrix is given by $A=\left[\begin{array}{ll}a_{11} a_{12} \\ a_{21} a_{22}\end{array}\right]$ (i) $a_{i j}=\frac{(i+j)^{2}}{2} ; i, j=1,2$ $\therefore a_{11}=\frac{(1+1)^{2}}{2}=\frac{4}{2}=2$ $a_{12}=\frac{(1+2)^{2}}{2}=\frac{9}{2}$ $a_{21}=\frac{(2+1)^{2}}{2}=\fr...
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