The area of a rectangle remains the same if the length is increased by 7 meters and the breadth is decreased by 3 meters
Question: The area of a rectangle remains the same if the length is increased by 7 meters and the breadth is decreased by 3 meters. The area remains unaffected if the length is decreased by 7 meters and breadth in increased by 5 meters. Find the dimensions of the rectangle. Solution: Let the length and breadth of the rectangle beandunits respectively Then, area of rectangle =square units If length is increased bymeters and breadth is decreased bymeters when the area of a rectangle remains the sa...
Read More →Suppose the spheres A and B in Exercise 1.12 have identical sizes.
Question: Suppose the spheres A and B in Exercise 1.12 have identical sizes. A third sphere of the same size but uncharged is brought in contact with the first, then brought in contact with the second, and finally removed from both. What is the new force of repulsion between A and B? Solution: Distance between the spheres, A and B,r= 0.5 m Initially, the charge on each sphere, $q=6.5 \times 10^{-7} \mathrm{C}$ When sphere $A$ is touched with an uncharged sphere $C, \frac{q}{2}$ amount of charge ...
Read More →Let f: W → W be defined as f(n) = n − 1, if is odd and f(n) = n + 1,
Question: Let $f: W \rightarrow W$ be defined as $f(n)=n-1$, if is odd and $f(n)=n+1$, if $n$ is even. Show that $f$ is invertible. Find the inverse of $f$. Here, W is the set of all whole numbers. Solution: It is given that: $f: W \rightarrow W$ is defined as $f(n)= \begin{cases}n-1, \text { if } n \text { is odd } \\ n+1, \text { if } n \text { is even }\end{cases}$ One-one: Letf(n) =f(m). It can be observed that ifnis odd andmis even, then we will haven 1 =m+ 1. $\Rightarrow n-m=2$ However, t...
Read More →It has been found that the pH of a 0.01M solution of an organic acid is 4.15.
Question: It has been found that the $\mathrm{pH}$ of a $0.01 \mathrm{M}$ solution of an organic acid is $4.15$. Calculate the concentration of the anion, the ionization constant of the acid and its $\mathrm{pK}_{\mathrm{a}}$. Solution: Let the organic acid be HA. $\Rightarrow \mathrm{HA} \longleftrightarrow \mathrm{H}^{+}+\mathrm{A}^{-}$ Concentration of $\mathrm{HA}=0.01 \mathrm{M}$ $\mathrm{pH}=4.15$ $-\log \left[\mathrm{H}^{+}\right]=4.15$ $\left[\mathrm{H}^{+}\right]=7.08 \times 10^{-5}$ No...
Read More →(a) Two insulated charged copper spheres A and B have their centers separated by a distance of 50 cm.
Question: (a) Two insulated charged copper spheres $\mathrm{A}$ and $\mathrm{B}$ have their centers separated by a distance of $50 \mathrm{~cm}$. What is the mutual force of electrostatic repulsion if the charge on each is $6.5 \times 10^{-7} \mathrm{C}$ ? The radii of $A$ and $B$ are negligible compared to the distance of separation. (b)What is the force of repulsion if each sphere is charged double the above amount, and the distance between them is halved? Solution: (a) Charge on sphere $A, q_...
Read More →If in a rectangle, the length is increased and breadth reduced each by 2 units, the area is reduced by 28 square units.
Question: If in a rectangle, the length is increased and breadth reduced each by 2 units, the area is reduced by 28 square units. If, however the length is reduced by 1 unit and the breadth increased by 2 units, the area increases by 33 square units. Find the area of the rectangle. Solution: Let the length and breadth of the rectangle beandunits respectively Then, area of rectanglesquare units If length is increased and breadth reduced each byunits, then the area is reduced bysquare units $\begi...
Read More →A polythene piece rubbed with wool is found to have a negative charge of
Question: A polythene piece rubbed with wool is found to have a negative charge of $3 \times 10^{-7} \mathrm{C}$. (a)Estimate the number of electrons transferred (from which to which?) (b)Is there a transfer of mass from wool to polythene? Solution: (a)When polythene is rubbed against wool, a number of electrons get transferred from wool to polythene. Hence, wool becomes positively charged and polythene becomes negatively charged. Amount of charge on the polythene piece, $q=-3 \times 10^{-7} \ma...
Read More →Let f: R → R be defined as f(x) = 10x + 7. Find the function g: R → R such that g o f = f o g = 1R.
Question: Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be defined as $f(x)=10 x+7$. Find the function $g$ : $\mathbf{R} \rightarrow \mathbf{R}$ such that $g \circ f=f \circ g=1_{\mathbf{R}}$. Solution: It is given that $f: \mathbf{R} \rightarrow \mathbf{R}$ is defined as $f(x)=10 x+7$. One-one: Let $f(x)=f(y)$, where $x, y \in \mathbf{R}$. $\Rightarrow 10 x+7=10 y+7$ $\Rightarrow x=v$ $\therefore f$ is a one-one function Onto: For $y \in \mathbf{R}$, let $y=10 x+7$ $\Rightarrow x=\frac{y-7}{10} \i...
Read More →The ionization constant of acetic acid is 1.74 × 10–5. Calculate the degree of dissociation of acetic acid in its 0.05 M solution.
Question: The ionization constant of acetic acid is $1.74 \times 10^{-5}$. Calculate the degree of dissociation of acetic acid in its $0.05 \mathrm{M}$ solution. Calculate the concentration of acetate ion in the solution and its $\mathrm{pH}$. Solution: Method 1 Since $K aK_{w}:$ $K_{a}=\frac{(.05 \alpha)(.05 \alpha)}{(.05-0.05 \alpha)}$ $=\frac{(.05 \alpha)(0.05 \alpha)}{.05(1-\alpha)}$ $=\frac{.05 \alpha^{2}}{1-\alpha}$ $1.74 \times 10^{-5}=\frac{0.05 \alpha^{2}}{1-\alpha}$ $1.74 \times 10^{-5...
Read More →An electric dipole with dipole moment
Question: An electric dipole with dipole moment $4 \times 10^{-9} \mathrm{C} \mathrm{m}$ is aligned at $30^{\circ}$ with the direction of a uniform electric field of magnitude $5 \times 10^{4} \mathrm{~N} \mathrm{C}^{-1}$. Calculate the magnitude of the torque acting on the dipole. Solution: Electric dipole moment, $p=4 \times 10^{-9} \mathrm{C} \mathrm{m}$ Angle made by $p$ with a uniform electric field, $\theta=30^{\circ}$ Electric field. $E=5 \times 10^{4} \mathrm{~N} \mathrm{C}^{-1}$ Torque ...
Read More →A system has two charges
Question: A system has two charges $q_{A}=2.5 \times 10^{-7} C$ and $q_{B}=-2.5 \times 10^{-7} C$ located at points $A$ : $(0,0,-15 \mathrm{~cm})$ and $B$ : $(0,0,+15 \mathrm{~cm})$, respectively. What are the total charge and electric dipole moment of the system? Solution: Both the charges can be located in a coordinate frame of reference as shown in the given figure. At $A$, amount of charge, $q_{A}=2.5 \times 10^{-7} \mathrm{C}$ At $\mathrm{B}$, amount of charge, $q_{\mathrm{B}}=-2.5 \times 1...
Read More →Consider a binary operation * on N defined as a * b = a3 + b3.
Question: Consider a binary operation * on $\mathbf{N}$ defined as $a^{*} b=a^{3}+b^{3}$. Choose the correct answer. (A) Is * both associative and commutative? (B) Is * commutative but not associative? (C) Is * associative but not commutative? (D) Is * neither commutative nor associative? Solution: On $\mathbf{N}$, the operation ${ }^{*}$ is defined as $a^{*} b=a^{3}+b^{3}$. For, $a, b, \in \mathbf{N}$, we have: $a^{*} b=a^{3}+b^{3}=b^{3}+a^{3}=b^{*} a$ [Addition is commutative in $\mathbf{N}$ ]...
Read More →Two point charges
Question: Two point charges $q_{A}=3 \mu C$ and $q_{B}=-3 \mu C$ are located $20 \mathrm{~cm}$ apart in vacuum. (a)What is the electric field at the midpoint O of the line AB joining the two charges? (b) If a negative test charge of magnitude $1.5 \times 10^{-9} \mathrm{C}$ is placed at this point, what is the force experienced by the test charge? Solution: (a)The situation is represented in the given figure. O is the mid-point of line AB. Distance between the two charges, AB = 20 cm $\therefore...
Read More →Prove that: $sin x+sin 3 x+sin 5 x+sin 7 x=4 cos x cos 2 x sin 4 x$
Question: Prove that: $\sin x+\sin 3 x+\sin 5 x+\sin 7 x=4 \cos x \cos 2 x \sin 4 x$ Solution: It is known that $\sin A+\sin B=2 \sin \left(\frac{A+B}{2}\right) \cdot \cos \left(\frac{A-B}{2}\right)$. $\therefore$ L.H.S. $=\sin x+\sin 3 x+\sin 5 x+\sin 7 x$ $=(\sin x+\sin 5 x)+(\sin 3 x+\sin 7 x)$ $=2 \sin \left(\frac{x+5 x}{2}\right) \cdot \cos \left(\frac{x-5 x}{2}\right)+2 \sin \left(\frac{3 x+7 x}{2}\right) \cos \left(\frac{3 x-7 x}{2}\right)$ $=2 \sin 3 x \cos (-2 x)+2 \sin 5 x \cos (-2 x)$...
Read More →State whether the following statements are true or false. Justify.
Question: State whether the following statements are true or false. Justify. (i) For an arbitrary binary operation * on a setN,a*a=aa*N. (ii) If * is a commutative binary operation onN, thena* (b*c) = (c*b) *a Solution: (i) Define an operation * onNas: a*b=a + ba, bN Then, in particular, forb=a= 3, we have: 3 * 3 = 3 + 3 = 6 3 Therefore, statement (i) is false. (ii) R.H.S. = (c*b) *a = (b*c) *a[* is commutative] =a* (b*c) [Again, as * is commutative] = L.H.S. a* (b*c) = (c*b) *a Therefore, state...
Read More →Let A = N × N and * be the binary operation on A defined by
Question: Let $A=\mathbf{N} \times \mathbf{N}$ and ${ }^{*}$ be the binary operation on $A$ defined by $(a, b) *(c, d)=(a+c, b+d)$ Show that ${ }^{*}$ is commutative and associative. Find the identity element for * on $\mathrm{A}$, if any. Solution: A =NN * is a binary operation on A and is defined by: (a, b) * (c, d) = (a + c, b + d) Let (a, b), (c, d) A Then,a, b, c, dN We have: (a, b) * (c, d) = (a + c, b + d) (c, d) * (a, b) = (c + a, d + b) = (a + c, b + d) [Addition is commutative in the s...
Read More →(a) An electrostatic field line is a continuous curve.
Question: (a)An electrostatic field line is a continuous curve. That is, a field line cannot have sudden breaks. Why not? (b)Explain why two field lines never cross each other at any point? Solution: (a)An electrostatic field line is a continuous curve because a charge experiences a continuous force when traced in an electrostatic field. The field line cannot have sudden breaks because the charge moves continuously and does not jump from one point to the other. (b)If two field lines cross each o...
Read More →Find which of the operations given above has identity.
Question: Find which of the operations given above has identity. Solution: An elementeQwill be the identity element for the operation * if a*e = a = e*a,aQ. We are given $a^{*} b=\frac{a b}{4}$ $\Rightarrow a^{*} e=a$ $\Rightarrow \frac{a e}{4}=a$ $\Rightarrow e=4$ Similarly, it can be checked for $e^{*} a=a$, we get $e=4$ Thus, $e=4$ is the identity....
Read More →Four point charges
Question: Four point charges $q_{A}=2 \mu C, q_{B}=-5 \mu C, q_{C}=2 \mu C$, and $q_{D}=-5 \mu C$ are located at the corners of a square ABCD of side $10 \mathrm{~cm}$. What is the force on a charge of $1 \mu \mathrm{C}$ placed at the centre of the square? Solution: The given figure shows a square of side 10 cm with four charges placed at its corners. O is the centre of the square Where, (Sides) AB = BC = CD = AD = 10 cm (Diagonals) $A C=B D=10 \sqrt{2} \mathrm{~cm}$ $A O=O C=D O=O B=5 \sqrt{2} ...
Read More →When a glass rod is rubbed with a silk cloth,
Question: When a glass rod is rubbed with a silk cloth, charges appear on both. A similar phenomenon is observed with many other pairs of bodies. Explain how this observation is consistent with the law of conservation of charge Solution: Rubbing produces charges of equal magnitude but of opposite nature on the two bodies because charges are created in pairs. This phenomenon of charging is called charging by friction. The net charge on the system of two rubbed bodies is zero. This is because equa...
Read More →Let * be a binary operation on the set Q of rational numbers as follows:
Question: Let * be a binary operation on the setQof rational numbers as follows: (i) $a^{*} b=a-b$ (ii) $a^{*} b=a^{2}+b^{2}$ (iii) $a^{*} b=a+a b$ (iv) $a^{*} b=(a-b)^{2}$ (v) $a^{*} b=\frac{a b}{\text { (vi) } a^{*}} b=a b^{2}$ Find which of the binary operations are commutative and which are associative. Solution: (i) OnQ, the operation * is defined asa*b = a b. It can be observed that: $\frac{1}{2} * \frac{1}{3}=\frac{1}{2}-\frac{1}{3}=\frac{3-2}{6}=\frac{1}{6}$ and $\frac{1}{3} * \frac{1}{2...
Read More →The first ionization constant of H2S is 9.1 × 10–8. Calculate the concentration
Question: The first ionization constant of $\mathrm{H}_{2} \mathrm{~S}$ is $9.1 \times 10^{-8}$. Calculate the concentration of $\mathrm{HS}^{-}$ion in its $0.1 \mathrm{M}$ solution. How will this concentration be affected if the solution is $0.1 \mathrm{M}$ in $\mathrm{HCl}$ also? If the second dissociation constant of $\mathrm{H}_{2} \mathrm{~S}$ is $1.2 \times 10^{-13}$, calculate the concentration of $\mathrm{S}^{2-}$ under both conditions. Solution: (i)To calculate the concentration of HSio...
Read More →(a) Explain the meaning of the statement ‘electric charge of a body is quantised’.
Question: (a)Explain the meaning of the statement electric charge of a body is quantised. (b)Why can one ignore quantisation of electric charge when dealing with macroscopic i.e., large scale charges? Solution: (a)Electric charge of a body is quantized. This means that only integral (1, 2, .,n) number of electrons can be transferred from one body to the other. Charges are not transferred in fraction. Hence, a body possesses total charge only in integral multiples of electric charge. (b)In macros...
Read More →Check that the ratio
Question: Check that the ratio $k e^{2} / G m_{e} m_{p}$ is dimensionless. Look up a Table of Physical Constants and determine the value of this ratio. What does the ratio signify? Solution: The given ratio is $\frac{k e^{2}}{\mathrm{G} m_{\mathrm{e}} m_{\mathrm{p}}}$. Where, G = Gravitational constant Its unit is $\mathrm{N} \mathrm{m}^{2} \mathrm{~kg}^{-2}$.\ $m_{\mathrm{e}}$ and $m_{\mathrm{p}}=$ Masses of electron and proton. Their unit is kg. $e=$ Electric charge. Its unit is C. $k=\mathrm{...
Read More →The electrostatic force on a small sphere of charge
Question: The electrostatic force on a small sphere of charge $0.4 \mu \mathrm{C}$ due to another small sphere of charge $-0.8 \mu \mathrm{C}$ in air is $0.2 \mathrm{~N}$. (a) What is the distance between the two spheres? (b) What is the force on the second sphere due to the first? Solution: (a)Electrostatic force on the first sphere,F= 0.2 N Charge on this sphere, $q_{1}=0.4 \mu \mathrm{C}=0.4 \times 10^{-6} \mathrm{C}$ Charge on the second sphere, $q_{2}=-0.8 \mu \mathrm{C}=-0.8 \times 10^{-6}...
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