The degree of ionization of a 0.1M bromoacetic acid solution is 0.132
Question: The degree of ionization of a 0.1M bromoacetic acid solution is 0.132. Calculate the pH of the solution and thepKaof bromoacetic acid. Solution: Degree of ionization, = 0.132 Concentration,c= 0.1 M Thus, the concentration of $\mathrm{H}_{3} \mathrm{O}^{+}=c . a$ = 0.1 0.132 = 0.0132 $\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right]$ $=-\log (0.0132)$ $=1.879: 1.88$ Now, $K_{a}=C \alpha^{2}$ $=0.1 \times(0.132)^{2}$ $K_{a}=.0017$ $p K_{a}=2.75$...
Read More →Prove the following by using the principle of mathematical induction for all n ∈ N:
Question: Prove the following by using the principle of mathematical induction for allnN: $1+3+3^{2}+\ldots+3^{n-1}=\frac{\left(3^{n}-1\right)}{2}$ Solution: Let the given statement be $P(n)$, i.e., $P(n): 1+3+3^{2}+\ldots+3^{n-1}=\frac{\left(3^{n}-1\right)}{2}$ For $n=1$, we have $P(1): 1=\frac{\left(3^{1}-1\right)}{2}=\frac{3-1}{2}=\frac{2}{2}=1$, which is true. Let $P(k)$ be true for some positive integer $k$, i.e., $1+3+3^{2}+\ldots+3^{k-1}=\frac{\left(3^{k}-1\right)}{2}$ We shall now prove ...
Read More →An oil drop of 12 excess electrons is held stationary under a constant electric field of
Question: An oil drop of 12 excess electrons is held stationary under a constant electric field of $2.55 \times 10^{4} \mathrm{~N} \mathrm{C}^{-1}$ in Millikan's oil drop experiment. The density of the oil is $1.26 \mathrm{~g} \mathrm{~cm}^{-3}$. Estimate the radius of the drop. $\left(\mathrm{g}=9.81 \mathrm{~m} \mathrm{~s}^{-2} ; e=1.60 \times 10^{-19} \mathrm{C}\right)$. Solution: Excess electrons on an oil drop,n= 12 Electric field intensity,E= 2.55 104N C1 Density of oil,= 1.26 gm/cm3= 1.26...
Read More →There are two examination rooms A and B. If 10 candidates are sent from A to B, the number of students in each room is same
Question: There are two examination rooms A and B. If 10 candidates are sent from A to B, the number of students in each room is same. If 20 candidates are sent from B to A, the number of students in A is double the number of students in B. Find the number of students in each room. Solution: Let us take the A examination room will be x and the B examination room will be y If 10 candidates are sent from A to B, the number of students in each room is same. According to the above condition equation...
Read More →Calculate the pH of the following solutions:
Question: Calculate the pH of the following solutions: a) $2 \mathrm{~g}$ of $\mathrm{TlOH}$ dissolved in water to give 2 litre of solution. b) $0.3 \mathrm{~g}$ of $\mathrm{Ca}(\mathrm{OH})_{2}$ dissolved in water to give $500 \mathrm{~mL}$ of solution. c) $0.3 \mathrm{~g}$ of $\mathrm{NaOH}$ dissolved in water to give $200 \mathrm{~mL}$ of solution. d) $1 \mathrm{~mL}$ of $13.6 \mathrm{M} \mathrm{HCl}$ is diluted with water to give 1 litre of solution. Solution: (a) For $2 \mathrm{~g}$ of $\ma...
Read More →Let S = {a, b, c} and T = {1, 2, 3}.
Question: Let $S=\{a, b, c\}$ and $T=\{1,2,3\} .$ Find $\mathrm{F}^{-1}$ of the following functions $\mathrm{F}$ from $S$ to $T$, if it exists. (i) $F=\{(a, 3),(b, 2),(c, 1)\}$ (ii) $F=\{(a, 2),(b, 1),(c, 1)\}$ Solution: S= {a,b,c},T= {1, 2, 3} (i) F:STis defined as: F = {(a, 3), (b, 2), (c, 1)} $\Rightarrow F(a)=3, F(b)=2, F(c)=1$ Therefore, $\mathrm{F}^{-1}: T \rightarrow S$ is given by $\mathrm{F}^{-1}=\{(3, a),(2, b),(1, c)\}$ (ii) $F: S \rightarrow T$ is defined as: $F=\{(a, 2),(b, 1),(c, 1...
Read More →Two large, thin metal plates are parallel and close to each other.
Question: Two large, thin metal plates are parallel and close to each other. On their inner faces, the plates have surface charge densities of opposite signs and of magnitude $17.0 \times 10^{-22} \mathrm{C} / \mathrm{m}^{2}$. What is $\mathrm{E}$ : (a) in the outer region of the first plate, (b) in the outer region of the second plate, and (c) between the plates? Solution: The situation is represented in the following figure. A and B are two parallel plates close to each other. Outer region of ...
Read More →A and B each has some money. If A gives Rs 30 to B, then B will have twice the money left with A. But, if B gives Rs 10 to A, then A will have thrice as much as is left with B. How much money does each have?
Question: A and B each has some money. If A gives Rs 30 to B, then B will have twice the money left with A. But, if B gives Rs 10 to A, then A will have thrice as much as is left with B. How much money does each have? Solution: Let the money with A be Rs x and the money with B be Rs y. If A gives Rs 30 to B, Then B will have twice the money left with A, According to the condition we have, $y+30=2(x-30)$ $y+30=2 x-60$ $0=2 x-y-60-30$ $0=2 x-y-90 \cdots(i)$ If B gives Rs 10 to A, then A will have ...
Read More →Find the number of all onto functions from the set {1, 2, 3, … , n) to itself.
Question: Find the number of all onto functions from the set {1, 2, 3, ,n) to itself. Solution: Onto functions from the set $\{1,2,3, \ldots, n\}$ to itself is simply a permutation on $n$ symbols $1,2, \ldots, n$. Thus, the total number of onto maps from $\{1,2, \ldots, n\}$ to itself is the same as the total number of permutations on $n$ symbols $1,2, \ldots, n$, which is $n !$....
Read More →An infinite line charge produces a field of
Question: An infinite line charge produces a field of $9 \times 10^{4} \mathrm{~N} / \mathrm{C}$ at a distance of $2 \mathrm{~cm}$. Calculate the linear charge density. Solution: Electric field produced by the infinite line charges at a distancedhaving linear charge densityis given by the relation, $E=\frac{\lambda}{2 \pi \in_{0} d}$ $\lambda=2 \pi \in_{0} d E$ Where, d= 2 cm = 0.02 m $E=9 \times 10^{4} \mathrm{~N} / \mathrm{C}$ $\epsilon_{0}=$ Permittivity of free space $\frac{1}{4 \pi \epsilon...
Read More →Given a non-empty set X, consider the binary operation *: P(X) × P(X) → P(X) given by A * B = A ∩ B &mnForE;
Question: Given a non-empty set $X$, consider the binary operation *: $P(X) \times P(X) \rightarrow P(X)$ given by $A$ * $B=A \cap B$ \mnForE; $A, B$ in $P(X)$ is the power set of $X$. Show that $X$ is the identity element for this operation and $X$ is the only invertible element in $\mathrm{P}(X)$ with respect to the operation*. Solution: It is given that $*: \mathrm{P}(X) \times \mathrm{P}(X) \rightarrow \mathrm{P}(X)$ is defined as $A^{*} B=A \cap B \forall A, B \in \mathrm{P}(X)$. We know th...
Read More →A uniformly charged conducting sphere of 2.4 m
Question: A uniformly charged conducting sphere of $2.4 \mathrm{~m}$ diameter has a surface charge density of $80.0 \mu \mathrm{C} / \mathrm{m}^{2}$. (a) Find the charge on the sphere. (b) What is the total electric flux leaving the surface of the sphere? Solution: (a)Diameter of the sphere,d= 2.4 m Radius of the sphere,r= 1.2 m Surface charge density, $\sigma=80.0 \mu \mathrm{C} / \mathrm{m}^{2}=80 \times 10^{-6} \mathrm{C} / \mathrm{m}^{2}$ Total charge on the surface of the sphere, Q= Charge ...
Read More →Given a non empty set X, consider P(X) which is the set of all subsets of X.
Question: Given a non empty setX, consider P(X) which is the set of all subsets ofX. Define the relation R in P(X) as follows: For subsetsA,Bin P(X),ARBif and only ifAB. Is R an equivalence relation on P(X)? Justify you answer: Solution: Since every set is a subset of itself, $A R A$ for all $A \in \mathrm{P}(X)$. R is reflexive. Let $A R B \Rightarrow A \subset B$. This cannot be implied to $B \subset A$. For instance, ifA= {1, 2} andB= {1, 2, 3}, then it cannot be implied thatBis related toA. ...
Read More →A conducting sphere of radius 10 cm has an unknown charge.
Question: A conducting sphere of radius $10 \mathrm{~cm}$ has an unknown charge. If the electric field $20 \mathrm{~cm}$ from the centre of the sphere is $1.5 \times 10^{3} \mathrm{~N} / \mathrm{C}$ and points radially inward, what is the net charge on the sphere? Solution: Electric field intensity (E) at a distance (d) from the centre of a sphere containing net chargeqis given by the relation, $E=\frac{q}{4 \pi \in_{0} d^{2}}$ Where, $q=$ Net charge $=1.5 \times 10^{3} \mathrm{~N} / \mathrm{C}$...
Read More →Given examples of two functions f: N → N and g: N → N such that gof is onto but f is not onto.
Question: Given examples of two functions $t: \mathbf{N} \rightarrow \mathbf{N}$ and $g: \mathbf{N} \rightarrow \mathbf{N}$ such that got is onto but $f$ is not onto. (Hint: Consider $f(x)=x+1$ and $g(x)= \begin{cases}x-1 \text { if } x1 \\ 1 \text { if } x=1\end{cases}$ Solution: Definef:NNby, f(x) =x+ 1 And,g:NNby, $g(x)= \begin{cases}x-1 \text { if } x1 \\ 1 \text { if } x=1\end{cases}$ We first show thatgis not onto. For this, consider element 1 in co-domainN. It is clear that this element i...
Read More →A point charge causes an electric flux of
Question: A point charge causes an electric flux of $-1.0 \times 10^{3} \mathrm{Nm}^{2} / \mathrm{C}$ to pass through a spherical Gaussian surface of $10.0 \mathrm{~cm}$ radius centered on the charge. (a) If the radius of the Gaussian surface were doubled, how much flux would pass through the surface? (b) What is the value of the point charge? Solution: (a) Electric flux, $\Phi=-1.0 \times 10^{3} \mathrm{~N} \mathrm{~m}^{2} / \mathrm{C}$ Radius of the Gaussian surface, r= 10.0 cm Electric flux p...
Read More →A point charge of 2.0 μC is at the centre of a cubic Gaussian surface 9.0 cm on edge.
Question: A point charge of 2.0 C is at the centre of a cubic Gaussian surface 9.0 cm on edge. What is the net electric flux through the surface? Solution: Net electric flux $\left(\Phi_{\text {Net }}\right)$ through the cubic surface is given by, $\phi_{\text {Net }}=\frac{q}{\epsilon_{0}}$ Where, $\epsilon_{0}=$ Permittivity of free space $=8.854 \times 10^{-12} \mathrm{~N}^{-1} \mathrm{C}^{2} \mathrm{~m}^{-2}$ $q=$ Net charge contained inside the cube $=2.0 \mu \mathrm{C}=2 \times 10^{-6} \ma...
Read More →Give examples of two functions f: N → Z and g:
Question: Give examples of two functions $t: \mathbf{N} \rightarrow \mathbf{Z}$ and $g: \mathbf{Z} \rightarrow \mathbf{Z}$ such that $g \circ f$ is injective but $g$ is not injective. (Hint: Consider $f(x)=x$ and $g(x)=|x|$ ) Solution: Define $f: \mathbf{N} \rightarrow \mathbf{Z}$ as $f(x)=x$ and $g: \mathbf{Z} \rightarrow \mathbf{Z}$ as $g(x)=|x|$. We first show thatgis not injective. It can be observed that: $g(-1)=|-1|=1$ $g(1)=|1|=1$ $\therefore g(-1)=g(1)$, but $-1 \neq 1 .$ $\therefore g$ ...
Read More →A point charge
Question: A point charge $+10 \mu \mathrm{C}$ is a distance $5 \mathrm{~cm}$ directly above the centre of a square of side $10 \mathrm{~cm}$, as shown in Fig. 1.34. What is the magnitude of the electric flux through the square? (Hint: Think of the square as one face of a cube with edge $10 \mathrm{~cm}$.) Solution: The square can be considered as one face of a cube of edge 10 cm with a centre where chargeqis placed. According to Gausss theorem for a cube, total electric flux is through all its s...
Read More →Careful measurement of the electric field at the surface of a black box indicates
Question: Careful measurement of the electric field at the surface of a black box indicates that the net outward flux through the surface of the box is $8.0 \times$ $10^{3} \mathrm{~N} \mathrm{~m}^{2} / \mathrm{C}$. (a) What is the net charge inside the box? (b) If the net outward flux through the surface of the box were zero, could you conclude that there were no charges inside the box? Why or Why not? Solution: (a) Net outward flux through the surface of the box, $\Phi=8.0 \times 10^{3} \mathr...
Read More →$\tan x=-\frac{4}{3}, x$ in quadrant II
[question] Question. $\tan x=-\frac{4}{3}, x$ in quadrant II [/question] [solution] solution: Here, x is in quadrant II. $\Rightarrow \frac{\pi}{4}<\frac{x}{2}<\frac{\pi}{2}$ Therefore, $\sin \frac{x}{2}, \cos \frac{x}{2}$ and $\tan \frac{x}{2}$ are all positive. It is given that $\tan x=-\frac{4}{3}$. $\sec ^{2} x=1+\tan ^{2} x=1+\left(\frac{-4}{3}\right)^{2}=1+\frac{16}{9}=\frac{25}{9}$ $\therefore \cos ^{2} x=\frac{9}{25}$ $\Rightarrow \cos x=\pm \frac{3}{5}$ As $x$ is in quadrant II, $\cos x...
Read More →Prove that: $\sin 3 x+\sin 2 x-\sin x=4 \sin x \cos \frac{x}{2} \cos \frac{3 x}{2}$
[question] Question. Prove that: $\sin 3 x+\sin 2 x-\sin x=4 \sin x \cos \frac{x}{2} \cos \frac{3 x}{2}$ [/question] [solution] solution: L.H.S. $=\sin 3 x+\sin 2 x-\sin x$ $=\sin 3 x+(\sin 2 x-\sin x)$ $=\sin 3 x+\left[2 \cos \left(\frac{2 x+x}{2}\right) \sin \left(\frac{2 x-x}{2}\right)\right] \quad\left[\sin A-\sin B=2 \cos \left(\frac{A+B}{2}\right) \sin \left(\frac{A-B}{2}\right)\right]$ $=\sin 3 x+\left[2 \cos \left(\frac{3 x}{2}\right) \sin \left(\frac{x}{2}\right)\right]$ $=\sin 3 x+2 \c...
Read More →Prove that: $\frac{(\sin 7 x+\sin 5 x)+(\sin 9 x+\sin 3 x)}{(\cos 7 x+\cos 5 x)+(\cos 9 x+\cos 3 x)}=\tan 6 x$
[question] Question. Prove that: $\frac{(\sin 7 x+\sin 5 x)+(\sin 9 x+\sin 3 x)}{(\cos 7 x+\cos 5 x)+(\cos 9 x+\cos 3 x)}=\tan 6 x$ [/question] [solution] solution: It is known that $\sin \mathrm{A}+\sin \mathrm{B}=2 \sin \left(\frac{\mathrm{A}+\mathrm{B}}{2}\right) \cdot \cos \left(\frac{\mathrm{A}-\mathrm{B}}{2}\right), \cos \mathrm{A}+\cos \mathrm{B}=2 \cos \left(\frac{\mathrm{A}+\mathrm{B}}{2}\right) \cdot \cos \left(\frac{\mathrm{A}-\mathrm{B}}{2}\right)$ L.H.S. $=\frac{(\sin 7 \mathrm{x}+\...
Read More →Prove that: $(\cos x-\cos y)^{2}+(\sin x-\sin y)^{2}=4 \sin ^{2} \frac{x-y}{2}$
[question] Question. Prove that: $(\cos x-\cos y)^{2}+(\sin x-\sin y)^{2}=4 \sin ^{2} \frac{x-y}{2}$ [/question] [solution] solution: L.H.S. $=(\cos x-\cos y)^{2}+(\sin x-\sin y)^{2}$ $=\cos ^{2} x+\cos ^{2} y-2 \cos x \cos y+\sin ^{2} x+\sin ^{2} y-2 \sin x \sin y$ $=\left(\cos ^{2} x+\sin ^{2} x\right)+\left(\cos ^{2} y+\sin ^{2} y\right)-2[\cos x \cos y+\sin x \sin y]$ $=1+1-2[\cos (x-y)] \quad[\cos (A-B)=\cos A \cos B+\sin A \sin B]$ $=2[1-\cos (x-y)]$ $=2\left[1-\left\{1-2 \sin ^{2}\left(\f...
Read More →Prove that: $(\cos x+\cos y)^{2}+(\sin x-\sin y)^{2}=4 \cos ^{2} \frac{x+y}{2}$
[question] Question. Prove that: $(\cos x+\cos y)^{2}+(\sin x-\sin y)^{2}=4 \cos ^{2} \frac{x+y}{2}$ [/question] [solution] solution: L.H.S. $=(\cos x+\cos y)^{2}+(\sin x-\sin y)^{2}$ $=\cos ^{2} x+\cos ^{2} y+2 \cos x \cos y+\sin ^{2} x+\sin ^{2} y-2 \sin x \sin y$ $=\left(\cos ^{2} x+\sin ^{2} x\right)+\left(\cos ^{2} y+\sin ^{2} y\right)+2(\cos x \cos y-\sin x \sin y)$ $=1+1+2 \cos (x+y) \quad[\cos (A+B)=(\cos A \cos B-\sin A \sin B)]$ $=2+2 \cos (x+y)$ $=2[1+\cos (x+y)]$ $=2\left[1+2 \cos ^{...
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