Find the 13th term in the expansion of.
Question: Find the $13^{\text {th }}$ term in the expansion of $\left(9 x-\frac{1}{3 \sqrt{x}}\right)^{18}, x \neq 0$ Solution: It is known that $(r+1)^{\text {th }}$ term, $\left(T_{r+1}\right)$, in the binomial expansion of $(a+b)^{n}$ is given by $T_{r+1}={ }^{n} C_{r} a^{n-r} b^{r}$. Thus, $13^{\text {th }}$ term in the expansion of $\left(9 x-\frac{1}{3 \sqrt{x}}\right)^{18}$ is $\mathrm{T}_{13}=\mathrm{T}_{12+1}={ }^{18} \mathrm{C}_{12}(9 \mathrm{x})^{18-12}\left(-\frac{1}{3 \sqrt{\mathrm{...
Read More →Find the inverse of each of the matrices, if it exists.
Question: Find the inverse of each of the matrices, if it exists. $\left[\begin{array}{llr}1 3 -2 \\ -3 0 -5 \\ 2 5 0\end{array}\right]$ Solution: Let $A=\left[\begin{array}{llr}1 3 -2 \\ -3 0 -5 \\ 2 5 0\end{array}\right]$ We know thatA=IA $\therefore\left[\begin{array}{llr}1 3 -2 \\ -3 0 -5 \\ 2 5 0\end{array}\right]=\left[\begin{array}{lll}1 0 0 \\ 0 1 0 \\ 0 0 1\end{array}\right] A$ Applying $R_{2} \rightarrow R_{2}+3 R_{1}$ and $R_{3} \rightarrow R_{3}-2 R_{1}$, we have: $\left[\begin{array...
Read More →The hydroxides and carbonates of sodium and potassium
Question: The hydroxides and carbonates of sodium and potassium are easily soluble in water while the corresponding salts of magnesium and calcium are sparingly soluble in water. Explain. Solution: The atomic size of sodium and potassium is larger than that of magnesium and calcium. Thus, the lattice energies of carbonates and hydroxides formed by calcium and magnesium are much more than those of sodium and potassium. Hence, carbonates and hydroxides of sodium and potassium dissolve readily in w...
Read More →Find the 4th term in the expansion of (x – 2y)12 .
Question: Find the $4^{\text {th }}$ term in the expansion of $(x-2 y)^{12}$ Solution: It is known that $(r+1)^{\text {th }}$ term, $\left(T_{r+1}\right)$, in the binomial expansion of $(a+b)^{n}$ is given by $\mathrm{T}_{\mathrm{r}+1}={ }^{\mathrm{n}} \mathrm{C}_{\mathrm{r}} \mathrm{a}^{\mathrm{n}-\mathrm{r}} \mathrm{b}^{\mathrm{r}}$. Thus, the $4^{\text {th }}$ term in the expansion of $(x-2 y)^{12}$ is $\mathrm{T}_{4}=\mathrm{T}_{3+1}={ }^{12} \mathrm{C}_{3}(\mathrm{x})^{12-3}(-2 \mathrm{y})^...
Read More →Draw the structure of
Question: Draw the structure of (i) BeCl2(vapour) (ii) BeCl2(solid). Solution: (a)Structure of BeCl2(solid) BeCl2exists as a polymer in condensed (solid) phase. In the vapour state, BeCl2exists as a monomer with a linear structure....
Read More →Write the general term in the expansion of (x2 – yx)12, x ≠ 0
Question: Write the general term in the expansion of $\left(x^{2}-y x\right)^{12}, x \neq 0$ Solution: It is known that the general term $T_{r+1}$ \{which is the $(r+1)^{\text {th }}$ term $\}$ in the binomial expansion of $(a+b)^{n}$ is given by $T_{r+1}={ }^{n} C_{r} a^{n-r} b^{r}$. Thus, the general term in the expansion of $\left(x^{2}-y x\right)^{12}$ is $\mathrm{T}_{\mathrm{r}+1}={ }^{12} \mathrm{C}_{\mathrm{r}}\left(\mathrm{x}^{2}\right)^{12-\mathrm{t}}(-\mathrm{yx})^{\mathrm{r}}=(-1)^{\m...
Read More →Find the inverse of each of the matrices, if it exists.
Question: Find the inverse of each of the matrices, if it exists. $\left[\begin{array}{rrr}2 -3 3 \\ 2 2 3 \\ 3 -2 2\end{array}\right]$ Solution: Let $A=\left[\begin{array}{rrr}2 -3 3 \\ 2 2 3 \\ 3 -2 2\end{array}\right]$ We know thatA=IA $\therefore\left[\begin{array}{rrr}2 -3 3 \\ 2 2 3 \\ 3 -2 2\end{array}\right]=\left[\begin{array}{lll}1 0 0 \\ 0 1 0 \\ 0 0 1\end{array}\right] A$ $\Rightarrow\left[\begin{array}{rrr}2 -3 3 \\ 0 5 0 \\ 3 -2 2\end{array}\right]=\left[\begin{array}{lll}1 0 0 \\ ...
Read More →Write the general term in the expansion of (x2 – y)6
Question: Write the general term in the expansion of $\left(x^{2}-y\right)^{6}$ Solution: It is known that the general term $T_{r+1}\left\{\right.$ which is the $(r+1)^{\text {th }}$ term $\}$ in the binomial expansion of $(a+b)^{n}$ is given by $T_{r+1}={ }^{n} C_{r} a^{n-r} b^{r}$. Thus, the general term in the expansion of $\left(x^{2}-y^{6}\right)$ is $T_{r+1}={ }^{6} C_{r}\left(x^{2}\right)^{6-r}(-y)^{r}=(-1)^{r}{ }^{6} C_{r} \cdot x^{12-2 r} \cdot y^{r}$...
Read More →Describe two important uses of each of the following
Question: Describe two important uses of each of the following: (i) caustic soda (ii) sodium carbonate (iii) quicklime. Solution: (i)Uses of caustic soda (a)It is used in soap industry. (b)It is used as a reagent in laboratory. (ii)Uses of sodium carbonate (a)It is generally used in glass and soap industry. (b)It is used as a water softener. (iii)Uses of quick lime (a)It is used as a starting material for obtaining slaked lime. (b)It is used in the manufacture of glass and cement....
Read More →Do the same exercise as above with the replacement of the earlier transformer
Question: Do the same exercise as above with the replacement of the earlier transformer by a 40,000-220 V step-down transformer (Neglect, as before, leakage losses though this may not be a good assumption any longer because of the very high voltage transmission involved). Hence, explain why high voltage transmission is preferred? Solution: The rating of a step-down transformer is 40000 V220 V. Input voltage,V1= 40000 V Output voltage,V2= 220 V Total electric power required,P= 800 kW = 800 103W S...
Read More →Find the coefficient of a5b7 in (a – 2b)12
Question: Find the coefficient of $a^{5} b^{7}$ in $(a-2 b)^{12}$ Solution: It is known that $(r+1)^{\text {th }}$ term,$\left(T_{r+1}\right)$, in the binomial expansion of $(a+b)^{n}$ is given by $\mathrm{T}_{\mathrm{r}+1}={ }^{n} \mathrm{C}_{\mathrm{r}} \mathrm{a}^{\mathrm{n}-\mathrm{r}} \mathrm{b}^{\mathrm{r}}$. Assuming that $a^{5} b^{7}$ occurs in the $(r+1)^{\text {th }}$ term of the expansion $(a-2 b)^{12}$, we obtain $\mathrm{T}_{r+1}={ }^{12} \mathrm{C}_{r}(\mathrm{a})^{12-r}(-2 \mathrm...
Read More →What happens when
Question: What happens when (i) magnesium is burnt in air (ii) quick lime is heated with silica (iii) chlorine reacts with slaked lime (iv) calcium nitrate is heated ? Solution: (i) Magnesium burns in air with a dazzling light to form $\mathrm{MgO}$ and $\mathrm{Mg}_{3} \mathrm{~N}_{2}$. $2 \mathrm{Mg}+\mathrm{O}_{2} \stackrel{\text { Buming }}{\longrightarrow} 2 \mathrm{MgO}$ $3 \mathrm{Mg}+\mathrm{N}_{2} \stackrel{\text { Buming }}{\longrightarrow} \mathrm{Mg}_{3} \mathrm{~N}_{2}$ (ii) Quick l...
Read More →Find the inverse of each of the matrices, if it exists.
Question: Find the inverse of each of the matrices, if it exists. $\left[\begin{array}{ll}2 1 \\ 4 2\end{array}\right]$ Solution: Let $A=\left[\begin{array}{ll}2 1 \\ 4 2\end{array}\right]$ We know that $A=I A$ $\therefore\left[\begin{array}{ll}2 1 \\ 4 2\end{array}\right]=\left[\begin{array}{ll}1 0 \\ 0 1\end{array}\right] A$ Applying $R_{1} \rightarrow R_{1}-\frac{1}{2} R_{2}$, we have: $\left[\begin{array}{ll}0 0 \\ 4 2\end{array}\right]=\left[\begin{array}{cc}1 -\frac{1}{2} \\ 0 1\end{array}...
Read More →Find the coefficient of x5 in (x + 3)8
Question: Find the coefficient of $x^{5}$ in $(x+3)^{8}$ Solution: It is known that $(r+1)^{\text {th }}$ term,$\left(T_{r+1}\right)$, in the binomial expansion of $(a+b)^{n}$ is given by $T_{r+1}={ }^{n} C_{r} a^{n-r} b^{r}$. Assuming that $x^{5}$ occurs in the $(r+1)^{\text {th }}$ term of the expansion $(x+3)^{8}$, we obtain $T_{r+1}={ }^{8} C_{r}(x)^{8-r}(3)^{r}$ Comparing the indices of $x$ in $x^{5}$ and in $T_{r+1}$, we obtain $r=3$ Thus, the coefficient of $x^{5}$ is $^{8} \mathrm{C}_{3}...
Read More →A small town with a demand of 800 kW of electric power
Question: A small town with a demand of 800 kW of electric power at 220 V is situated 15 km away from an electric plant generating power at 440 V. The resistance of the two wire line carrying power is 0.5 Ω per km. The town gets power from the line through a 4000-220 V step-down transformer at a sub-station in the town. (a)Estimate the line power loss in the form of heat. (b)How much power must the plant supply, assuming there is negligible power loss due to leakage? (c)Characterise the step up ...
Read More →Prove that $sum_{r=0}^{n} 3^{r ~}{ }^{n} C_{r}=4^{n}$.
Question: Prove that $\sum_{r=0}^{n} 3^{r ~}{ }^{n} C_{r}=4^{n}$. Solution: ByBinomial Theorem, $\sum_{r=0}^{n}{ }^{n} C_{r} a^{n-r} b^{r}=(a+b)^{n}$ By puttingb= 3 anda= 1 in the above equation, we obtain $\sum_{r=0}^{n}{ }^{n} C_{r}(1)^{n-r}(3)^{r}=(1+3)^{n}$ $\Rightarrow \sum_{r=0}^{n} 3^{r^{n}} C_{r}=4^{n}$ Hence, proved....
Read More →At a hydroelectric power plant, the water pressure head is at a height of 300 m
Question: At a hydroelectric power plant, the water pressure head is at a height of 300 m and the water flow available is 100 m3s1. If the turbine generator efficiency is 60%, estimate the electric power available from the plant (g= 9.8 m s2). Solution: Height of water pressure head,h= 300 m Volume of water flow per second,V= 100 m3/s Efficiency of turbine generator,n= 60% = 0.6 Acceleration due to gravity, g = 9.8 m/s2 Density of water,= 103kg/m3 Electric power available from the plant =hgV = 0...
Read More →Find the inverse of each of the matrices, if it exists.
Question: Find the inverse of each of the matrices, if it exists. $\left[\begin{array}{ll}2 1 \\ 4 2\end{array}\right]$ Solution: Let $A=\left[\begin{array}{ll}2 1 \\ 4 2\end{array}\right]$ We know thatA=IA Applying $R_{1} \rightarrow R_{1}-\frac{1}{2} R_{2}$, we have: $\left[\begin{array}{ll}0 0 \\ 4 2\end{array}\right]=\left[\begin{array}{cc}1 -\frac{1}{2} \\ 0 1\end{array}\right] A$ Now, in the above equation, we can see all the zeros in the first row of the matrix on the L.H.S. Therefore, $A...
Read More →A power transmission line feeds input power at 2300 V to a stepdown transformer with its primary windings having 4000 turns.
Question: A power transmission line feeds input power at 2300 V to a stepdown transformer with its primary windings having 4000 turns. What should be the number of turns in the secondary in order to get output power at 230 V? Solution: Input voltage,V1= 2300 Number of turns in primary coil,n1= 4000 Output voltage,V2= 230 V Number of turns in secondary coil =n2 Voltage is related to the number of turns as: $\frac{V_{1}}{V_{2}}=\frac{n_{1}}{n_{2}}$ $\frac{2300}{230}=\frac{4000}{n_{2}}$ $n_{2}=\fra...
Read More →Starting with sodium chloride how would you proceed to prepare
Question: Starting with sodium chloride how would you proceed to prepare (i) sodium metal (ii) sodium hydroxide (iii) sodium peroxide (iv) sodium carbonate? Solution: (a)Sodium can be extracted from sodium chloride by Downs process. This process involves the electrolysis of fused NaCl (40%) and CaCl2(60 %) at a temperature of 1123 K in Downs cell. Steel is the cathode and a block of graphite acts as the anode. Metallic Na and Ca are formed at cathode. Molten sodium is taken out of the cell and c...
Read More →Show that is divisible by 64, whenever n is a positive integer.
Question: Show that $9^{n+1}-8 n-9$ is divisible by 64, whenever $n$ is a positive integer. Solution: In order to show that $9^{n+1}-8 n-9$ is divisible by 64 , it has to be proved that, $9^{n+1}-8 n-9=64 k$, where $k$ is some natural number By Binomial Theorem, $(1+a)^{m}={ }^{m} C_{0}+{ }^{m} C_{1} a+{ }^{m} C_{2} a^{2}+\ldots+{ }^{m} C_{m} a^{m}$ For $a=8$ and $m=n+1$, we obtain $(1+8)^{n+1}={ }^{n+1} C_{0}+{ }^{n+1} C_{1}(8)+{ }^{n+1} C_{2}(8)^{2}+\ldots+{ }^{n+1} C_{n+1}(8)^{n+1}$ $\Rightar...
Read More →Answer the following questions:
Question: Answer the following questions: (a)In any ac circuit, is the applied instantaneous voltage equal to the algebraic sum of the instantaneous voltages across the series elements of the circuit? Is the same true for rms voltage? (b)A capacitor is used in the primary circuit of an induction coil. (c)An applied voltage signal consists of a superposition of a dc voltage and an ac voltage of high frequency. The circuit consists of an inductor and a capacitor in series. Show that the dc signal ...
Read More →Obtain the resonant frequency and Q-factor of a series LCR circuit with
Question: Obtain the resonant frequency andQ-factor of a seriesLCRcircuit withL= 3.0 H,C= 27 F, andR= 7.4 Ω. It is desired to improve the sharpness of the resonance of the circuit by reducing its full width at half maximum by a factor of 2. Suggest a suitable way. Solution: Inductance,L= 3.0 H Capacitance,C= 27 F = 27 106F Resistance,R= 7.4 Ω At resonance, angular frequency of the source for the givenLCRseries circuit is given as: $\omega_{r}=\frac{1}{\sqrt{L C}}$ $=\frac{1}{\sqrt{3 \times 27 \t...
Read More →Find (x + 1)6 + (x – 1)6. Hence or otherwise evaluate.
Question: Find $(x+1)^{6}+(x-1)^{6}$. Hence or otherwise evaluate $(\sqrt{2}+1)^{6}+(\sqrt{2}-1)^{6}$. Solution: Using Binomial Theorem, the expressions, $(x+1)^{6}$ and $(x-1)^{6}$, can be expanded as $(x+1)^{6}={ }^{6} \mathrm{C}_{0} x^{6}+{ }^{6} \mathrm{C}_{1} x^{5}+{ }^{6} \mathrm{C}_{2} x^{4}+{ }^{6} \mathrm{C}_{3} x^{3}+{ }^{6} \mathrm{C}_{4} x^{2}+{ }^{6} \mathrm{C}_{5} x+{ }^{6} \mathrm{C}_{6}$ $(x-1)^{6}={ }^{6} \mathrm{C}_{0} x^{6}-{ }^{6} \mathrm{C}_{1} x^{5}+{ }^{6} \mathrm{C}_{2} x...
Read More →Find the inverse of each of the matrices, if it exists.
Question: Find the inverse of each of the matrices, if it exists. $\left[\begin{array}{lr}2 -3 \\ -1 2\end{array}\right]$ Solution: Let $A=\left[\begin{array}{lr}2 -3 \\ -1 2\end{array}\right]$ We know thatA=IA $\therefore\left[\begin{array}{lr}2 -3 \\ -1 2\end{array}\right]=\left[\begin{array}{ll}1 0 \\ 0 1\end{array}\right] A$ $\Rightarrow\left[\begin{array}{lr}1 -1 \\ -1 2\end{array}\right]=\left[\begin{array}{ll}1 1 \\ 0 1\end{array}\right] A \quad\left(\mathrm{R}_{1} \rightarrow \mathrm{R}_...
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