Find the principal value of tan−1 (−1)
Question: Find the principal value of $\tan ^{-1}(-1)$ Solution: Let $\tan ^{-1}(-1)=y .$ Then, $\tan y=-1=-\tan \left(\frac{\pi}{4}\right)=\tan \left(-\frac{\pi}{4}\right)$ We know that the range of the principal value branch of $\tan ^{-1}$ is $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ and $\tan \left(-\frac{\pi}{4}\right)=-1$ Therefore, the principal value of $\tan ^{-1}(-1)$ is $-\frac{\pi}{4}$....
Read More →Find the principal value of tan−1 (−1)
Question: Find the principal value of $\tan ^{-1}(-1)$ Solution: Let $\tan ^{-1}(-1)=y .$ Then, $\tan y=-1=-\tan \left(\frac{\pi}{4}\right)=\tan \left(-\frac{\pi}{4}\right)$ We know that the range of the principal value branch of $\tan ^{-1}$ is $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ and $\tan \left(-\frac{\pi}{4}\right)=-1$ Therefore, the principal value of $\tan ^{-1}(-1)$ is $-\frac{\pi}{4}$....
Read More →Find the principal value of
Question: Find the principal value of $\cos ^{-1}\left(-\frac{1}{2}\right)$ Solution: Let $\cos ^{-1}\left(-\frac{1}{2}\right)=y .$ Then, $\cos y=-\frac{1}{2}=-\cos \left(\frac{\pi}{3}\right)=\cos \left(\pi-\frac{\pi}{3}\right)=\cos \left(\frac{2 \pi}{3}\right) .$ We know that the range of the principal value branch of $\cos ^{-1}$ is $[0, \pi]$ and $\cos \left(\frac{2 \pi}{3}\right)=-\frac{1}{2}$ Therefore, the principal value of $\cos ^{-1}\left(-\frac{1}{2}\right)$ is $\frac{2 \pi}{3}$....
Read More →Three capacitors of capacitances 2 pF, 3 pF and 4 pF are connected in parallel.
Question: Three capacitors of capacitances 2 pF, 3 pF and 4 pF are connected in parallel. (a)What is the total capacitance of the combination? (b)Determine the charge on each capacitor if the combination is connected to a 100 V supply. Solution: (a)Capacitances of the given capacitors are $C_{1}=2 \mathrm{pF}$ $C_{2}=3 \mathrm{pF}$ $C_{3}=4 \mathrm{pF}$ For the parallel combination of the capacitors, equivalent capacitor $C^{\prime}$ is given by the algebraic sum, $C^{\prime}=2+3+4=9 \mathrm{pF}...
Read More →Find the principal value of
Question: Find the principal value of $\tan ^{-1}(-\sqrt{3})$ Solution: Let $\tan ^{-1}(-\sqrt{3})=y .$ Then, $\tan y=-\sqrt{3}=-\tan \frac{\pi}{3}=\tan \left(-\frac{\pi}{3}\right)$. We know that the range of the principal value branch of $\tan ^{-1}$ is $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ and $\tan \left(-\frac{\pi}{3}\right)$ is $-\sqrt{3}$. Therefore, the principal value of $\tan ^{-1}(\sqrt{3})$ is $-\frac{\pi}{3}$....
Read More →Prove the following by using the principle of mathematical induction for all $n in N:
Question: Prove the following by using the principle of mathematical induction for allnN:$\frac{1}{2.5}+\frac{1}{5.8}+\frac{1}{8.11}+\ldots+\frac{1}{(3 n-1)(3 n+2)}=\frac{n}{(6 n+4)}$ Solution: Let the given statement be P(n), i.e., $\mathrm{P}(n): \frac{1}{2.5}+\frac{1}{5.8}+\frac{1}{8.11}+\ldots+\frac{1}{(3 n-1)(3 n+2)}=\frac{n}{(6 n+4)}$ Forn= 1, we have $\mathrm{P}(1)=\frac{1}{2.5}=\frac{1}{10}=\frac{1}{6.1+4}=\frac{1}{10}$, which is true. Let $\mathrm{P}(k)$ be true for some positive intege...
Read More →Three capacitors each of capacitance 9 pF are connected in series.
Question: Three capacitors each of capacitance 9 pF are connected in series. (a)What is the total capacitance of the combination? (b)What is the potential difference across each capacitor if the combination is connected to a 120 V supply? Solution: (a)Capacitance of each of the three capacitors,C= 9 pF Equivalent capacitance (C) of the combination of the capacitors is given by the relation, $\frac{1}{C^{\prime}}=\frac{1}{C}+\frac{1}{C}+\frac{1}{C}$ $\Rightarrow \frac{1}{C^{\prime}}=\frac{1}{9}+\...
Read More →Find the principal value of cosec−1 (2)
Question: Find the principal value of $\operatorname{cosec}^{-1}$ (2) Solution: Let $\operatorname{cosec}^{-1}(2)=y .$ Then, $\operatorname{cosec} y=2=\operatorname{cosec}\left(\frac{\pi}{6}\right)$ We know that the range of the principal value branch of $\operatorname{cosec}^{-1}$ is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]-\{0\}$.Therefore, the principal value of $\operatorname{cosec}^{-1}$ (2) is $\frac{\pi}{6}$....
Read More →Three capacitors each of capacitance 9 pF are connected in series.
Question: Three capacitors each of capacitance 9 pF are connected in series. (a)What is the total capacitance of the combination? (b)What is the potential difference across each capacitor if the combination is connected to a 120 V supply? Solution: (a)Capacitance of each of the three capacitors,C= 9 pF Equivalent capacitance (C) of the combination of the capacitors is given by the relation, $\frac{1}{C^{\prime}}=\frac{1}{C}+\frac{1}{C}+\frac{1}{C}$ $\Rightarrow \frac{1}{C^{\prime}}=\frac{1}{9}+\...
Read More →Find the principal value of
Question: Find the principal value of $\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)$ Solution: Let $\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)=y$. Then, $\cos y=\frac{\sqrt{3}}{2}=\cos \left(\frac{\pi}{6}\right)$. We know that the range of the principal value branch of $\cos ^{-1}$ is $[0, \pi]$ and $\cos \left(\frac{\pi}{6}\right)=\frac{\sqrt{3}}{2}$. Therefore, the principal value of $\cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)$ is $\frac{\pi}{6}$....
Read More →If 0.561 g of KOH is dissolved in water to give 200 mL of solution at 298 K.
Question: If $0.561 \mathrm{~g}$ of KOH is dissolved in water to give $200 \mathrm{~mL}$ of solution at $298 \mathrm{~K}$. Calculate the concentrations of potassium, hydrogen and hydroxyl ions. What is its pH? Solution: $\left[\mathrm{KOH}_{a q}\right]=\frac{0.561}{\frac{1}{5}} \mathrm{~g} / L$ $=2.805 \mathrm{~g} / L$ $=2.805 \times \frac{1}{56.11} \mathrm{M}$ $=.05 \mathrm{M}$ $\mathrm{KOH}_{(a q)} \rightarrow \mathrm{K}_{(a q)}^{+}+\mathrm{OH}^{-}{ }_{(a q)}$ $\left[\mathrm{OH}^{-}\right]=.05...
Read More →Find the principal value of
Question: Find the principal value of $\sin ^{-1}\left(-\frac{1}{2}\right)$ Solution: Let $\sin ^{-1}\left(-\frac{1}{2}\right)=y$. Then $\sin y=-\frac{1}{2}=-\sin \left(\frac{\pi}{6}\right)=\sin \left(-\frac{\pi}{6}\right)$. We know that the range of the principal value branch of $\sin ^{-1}$ is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ and $\sin \left(-\frac{\pi}{6}\right)=-\frac{1}{2}$ Therefore, the principal value of $\sin ^{-1}\left(-\frac{1}{2}\right)$ is $-\frac{\pi}{6}$....
Read More →A parallel plate capacitor with air between the plates has a capacitance of
Question: A parallel plate capacitor with air between the plates has a capacitance of $8 \mathrm{pF}\left(1 \mathrm{pF}=10^{-12} \mathrm{~F}\right)$. What will be the capacitance if the distance between the plates is reduced by half, and the space between them is filled with a substance of dielectric constant $6 ?$ Solution: Capacitance between the parallel plates of the capacitor, C = 8 pF Initially, distance between the parallel plates wasdand it was filled with air. Dielectric constant of air...
Read More →Prove the following by using the principle of mathematical induction for all $n in N: rac{1}{2}+ rac{1}{4}+ rac{1}{8}+$
Question: Prove the following by using the principle of mathematical induction for allnN:$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots+\frac{1}{2^{n}}=1-\frac{1}{2^{n}}$ Solution: Let the given statement be P(n), i.e., $\mathrm{P}(n): \frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots+\frac{1}{2^{n}}=1-\frac{1}{2^{n}}$ For $n=1$, we have $P(1): \frac{1}{2}=1-\frac{1}{2^{1}}=\frac{1}{2}$, which is true. Let $P(k)$ be true for some positive integer $k$, i.e., $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots .+\f...
Read More →A spherical conductor of radius 12 cm has a charge of
Question: A spherical conductor of radius 12 cm has a charge of 1.6 107C distributed uniformly on its surface. What is the electric field (a)Inside the sphere (b)Just outside the sphere (c)At a point 18 cm from the centre of the sphere? Solution: (a)Radius of the spherical conductor,r= 12 cm = 0.12 m Charge is uniformly distributed over the conductor,q= 1.6 107C Electric field inside a spherical conductor is zero. This is because if there is field inside the conductor, then charges will move to ...
Read More →Two charges 2 μC and −2 µC are placed at points A and B 6 cm apart.
Question: Two charges 2 C and 2 C are placed at points A and B 6 cm apart. (a)Identify an equipotential surface of the system. (b)What is the direction of the electric field at every point on this surface? Solution: (a)The situation is represented in the given figure. An equipotential surface is the plane on which total potential is zero everywhere. This plane is normal to line AB. The plane is located at the mid-point of line AB because the magnitude of charges is the same. (b)The direction of ...
Read More →The pH of milk, black coffee, tomato juice, lemon juice and egg white are 6.8, 5.0, 4.2, 2.2 and 7.8 respectively.
Question: The $\mathrm{pH}$ of milk, black coffee, tomato juice, lemon juice and egg white are $6.8,5.0,4.2,2.2$ and $7.8$ respectively. Calculate corresponding hydrogen ion concentration in each. Solution: The hydrogen ion concentration in the given substances can be calculated by using the given relation: $\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right]$ (i) $\mathrm{pH}$ of milk $=6.8$ Since, $\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right]$ $6.8=-\log \left[\mathrm{H}^{+}\right]$ $\log \left[\ma...
Read More →Prove the following by using the principle of mathematical induction for all n ∈ N: 1.2 + 2.22 + 3.22
Question: Prove the following by using the principle of mathematical induction for allnN:$1.2+2.2^{2}+3.2^{2}+\ldots+n .2^{n}=(n-1) 2^{n+1}+2$ Solution: Let the given statement be P(n), i.e., $P(n): 1.2+2.2^{2}+3.2^{2}+\ldots+n \cdot 2^{n}=(n-1) 2^{n+1}+2$ For $n=1$, we have $P(1): 1.2=2=(1-1) 2^{1+1}+2=0+2=2$, which is true. Let $\mathrm{P}(k)$ be true for some positive integer $k$, i.e., $1.2+2.2^{2}+3.2^{2}+\ldots+k .2^{k}=(k-1) 2^{k+1}+2 \ldots$ (i) We shall now prove that $P(k+1)$ is true. ...
Read More →A regular hexagon of side 10 cm has a charge 5 µC at each of its vertices.
Question: A regular hexagon of side 10 cm has a charge 5 C at each of its vertices. Calculate the potential at the centre of the hexagon. Solution: The given figure shows six equal amount of charges,q, at the vertices of a regular hexagon. Where, Charge, $q=5 \mu \mathrm{C}=5 \times 10^{-6} \mathrm{C}$ Side of the hexagon,l= AB = BC = CD = DE = EF = FA = 10 cm Distance of each vertex from centre O,d= 10 cm Electric potential at point O, $V=\frac{6 \times q}{4 \pi \epsilon_{0} d}$ Where, $\epsilo...
Read More →Two charges
Question: Two charges $5 \times 10^{-8} \mathrm{C}$ and $-3 \times 10^{-8} \mathrm{C}$ are located $16 \mathrm{~cm}$ apart. At what point(s) on the line joining the two charges is the electric potential zero? Take the potential at infinity to be zero. Solution: There are two charges, $q_{1}=5 \times 10^{-8} \mathrm{C}$ $q_{2}=-3 \times 10^{-8} \mathrm{C}$ Distance between the two charges,d= 16 cm = 0.16 m Consider a point P on the line joining the two charges, as shown in the given figure. r= Di...
Read More →Number of binary operations on the set {a, b} are
Question: Number of binary operations on the set {a,b} are (A) 10 (B) 16 (C) 20 (D) 8 Solution: A binary operation * on $\{a, b\}$ is a function from $\{a, b\} \times\{a, b\} \rightarrow\{a, b\}$ i.e., ${ }^{*}$ is a function from $\{(a, a),(a, b),(b, a),(b, b)\} \rightarrow\{a, b\}$. Hence, the total number of binary operations on the set $\{a, b\}$ is $2^{4}$ i.e., 16 . The correct answer is B....
Read More →Prove the following by using the principle of mathematical induction for all n ∈ N: 1.2 + 2.22 + 3.22 + … + n.2n = (n – 1) 2n+1 + 2
Question: Prove the following by using the principle of mathematical induction for all $n \in N: 1.2+22^{2}+3.2^{2}+\ldots+n \cdot 2^{n}=(n-1) 2^{n+1}+2$ Solution: Let the given statement be $P(n)$, i.e., $P(n): 1.2+2.2^{2}+3.2^{2}+\ldots+n .2^{n}=(n-1) 2^{n+1}+2$ For $n=1$, we have $P(1): 1.2=2=(1-1) 2^{1+1}+2=0+2=2$, which is true. Let $P(k)$ be true for some positive integer $k$, i.e., $1.2+2.2^{2}+3.2^{2}+\ldots+k \cdot 2^{k}=(k-1) 2^{k+1}+2 \ldots$ (i) We shall now prove that $\mathrm{P}(\m...
Read More →Let f: R → R be the Signum Function defined as
Question: Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be the Signum Function defined as $f(x)=\left\{\begin{aligned} 1, x0 \\ 0, x=0 \\-1, x0 \end{aligned}\right.$ and $g: \mathbf{R} \rightarrow \mathbf{R}$ be the Greatest Integer Function given by $g(x)=[x]$, where $[x]$ is greatest integer less than or equal to $x$. Then does fog and gof coincide in $(0,1]$ ? Solution: It is given that, $f: \mathbf{R} \rightarrow \mathbf{R}$ is defined as $f(x)=\left\{\begin{array}{rr}1, x0 \\ 0, x=0 \\ -1, x0\...
Read More →Prove the following by using the principle of mathematical induction for all $n in N: 1.3+3.5+5.7+ldots+(2 n-1)(2 n+1)=$
Question: Prove the following by using the principle of mathematical induction for allnN:$1.3+3.5+5.7+\ldots+(2 n-1)(2 n+1)=\frac{n\left(4 n^{2}+6 n-1\right)}{3}$ Solution: Let the given statement be P(n), i.e., $\mathrm{P}(n): 1.3+3.5+5.7+\ldots+(2 n-1)(2 n+1)=\frac{n\left(4 n^{2}+6 n-1\right)}{3}$ Forn= 1, we have $P(1): 1.3=3=\frac{1\left(4.1^{2}+6.1-1\right)}{3}=\frac{4+6-1}{3}=\frac{9}{3}=3$, which is true. Let P(k) be true for some positive integerk, i.e., $1.3+3.5+5.7+\ldots \ldots+(2 k-1...
Read More →Calculate the hydrogen ion concentration in the following biological fluids whose pH are given below:
Question: Calculate the hydrogen ion concentration in the following biological fluids whose pH are given below: (a) Human muscle-fluid, 6.83 (b) Human stomach fluid, 1.2 (c) Human blood, 7.38 (d) Human saliva, 6.4. Solution: (a) Human muscle fluid 6.83: $\mathrm{pH}=6.83$ $\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right]$ $\therefore 6.83=-\log \left[\mathrm{H}^{+}\right]$ $\left[\mathrm{H}^{+}\right]=1.48 \times 10^{-7} \mathrm{M}$ (b) Human stomach fluid, 1.2: $\mathrm{pH}=1.2$ $1.2=-\log \left[\...
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