Anand obtained a loan of Rs 125000 from the Allahabad Bank for buying computers.
Question: Anand obtained a loan of Rs 125000 from the Allahabad Bank for buying computers. The bank charges compound interest at 8% per annum, compounded annually. What amount wil he have to pay after 3 years to clear the debt? Solution: Principal amount, $P=$ Rs. 125000 Rate of interest, $R=8 \%$ p. a. Time, $n=3$ year $s$ The amount including the compound interest is calculated using the formula, $A=$ Rs. $\mathrm{P}\left(1+\frac{R}{100}\right)^{n}$ $=$ Rs. $125000\left(1+\frac{8}{100}\right)^...
Read More →A child has drawn the electric circuit to study
Question: A child has drawn the electric circuit to study Ohms law as shown in figure. His teacher told that the circuit diagram needs correction. Study the circuit and redraw it after making all corrections. Solution: Ammeter is connected in series in an electric circuit and voltmeter is connected in parallel to the resistor R. Positive terminals of ammeter and voltmeter are connected with the positive terminal of battery and negative terminals of ammeter and voltmeter are connected with the ne...
Read More →Find the amount of Rs 8000 for 2 years compounded annually and the rates being 9%
Question: Find the amount of Rs 8000 for 2 years compounded annually and the rates being 9% per annum during the first year and 10% per annum during the second year. Solution: Principal amount, $P=$ Rs. 8000 Rate of interest for the first year, $p=9 \%$ p. a. Rate of interest for the second year, $q=10 \%$ p. a. Time, $n=2$ years. Formula for the amount including the compound interest for the first year: $A=$ Rs. $\left\{\mathrm{P} \times\left(1+\frac{p}{100}\right) \times\left(1+\frac{q}{100}\r...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $x^{x^{2}-3}+(x-3)^{x^{2}}$ Solution: Let $y=x^{x^{2}-3}+(x-3)^{x^{2}}$ $\Rightarrow y=a+b$ where $a=x^{x^{2}-3} ; b=(x-3)^{x^{2}}$ $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{da}}{\mathrm{dx}}+\frac{\mathrm{db}}{\mathrm{dx}}$ $\left\{\right.$ Using chain rule, $\frac{d(u+a)}{d x}=\frac{d u}{d x}+\frac{d a}{d x}$ where a and $u$ are any variables $\}$ $a=x^{x^{2}-3}$ Taking log both the sides: $\Rightarrow \log a=\log x^{x^...
Read More →Unit of electric power may also be expressed as
Question: Unit of electric power may also be expressed as (a)Volt ampere (b)kilowatt hour (c)watt second (d)joule second Solution: (a).Explanation :P = VI....
Read More →Using the principle of mathematical induction, prove each of the following
Question: Using the principle of mathematical induction, prove each of the following for all n ϵ N: $1^{2}+3^{2}+5^{2}+7^{2}+\ldots+(2 n-1)^{2}=$ $\frac{\mathrm{n}(2 \mathrm{n}-1)(2 \mathrm{n}+1)}{3}$ Solution: To Prove $1^{2}+3^{2}+5^{2}+7^{2}+\ldots+(2 n-1)^{2}=\frac{n(2 n-1)(2 n+1)}{3}$ Steps to prove by mathematical induction: Let $P(n)$ be a statement involving the natural number $n$ such that (i) $P(1)$ is true (ii) $P(k+1)$ is true, whenever $P(k)$ is true Then $P(n)$ is true for all $n \...
Read More →By using the formula, find the amount and compound interest on:
Question: By using the formula, find the amount and compound interest on:Rs 9000 for 2 years 4 months at 10% per annum compounded annually. Solution: Principal amount, $P=$ Rs. 9000 Rate of interest, $R=10 \%$ p. a. Time, $n=2$ years 4 months $=2 \frac{1}{3}$ years $=\frac{7}{3}$ years The formula for the amount including the compound interest is given below: $A=$ Rs. $P \times\left(1+\frac{R}{100}\right)^{\mathrm{n}}$ $=$ Rs. $\left(9000 \times\left(1+\frac{10}{100}\right)^{2} \times\left(1+\fr...
Read More →Two resistors of resistance 2Ω and 4Ω
Question: Two resistors of resistance 2Ω and 4Ω when connected to a battery will have (a)same current flowing through them when connected in parallel (b)same current flowing through them when connected in series. (c)same potential difference across them when connected in series (d)different potential difference across them when connected in parallel Solution: (b) Explanation :Resistors are said to be connected in series if same current flows through them as series combination has a single path f...
Read More →By using the formula, find the amount and compound interest on:
Question: By using the formula, find the amount and compound interest on: Rs 10240 for 3 years at $12 \frac{1}{2} \%$ per annum compounded annually. Solution: Principal amount, $P=$ Rs 62500 Rate of interest, $R=12 \%$ p. a. Time, $n=2$ years 6 months $=\frac{5}{2}=2 \frac{1}{2}$ years The formula for the amount including the compound interest is given below : $A=$ Rs. $P\left(1+\frac{R}{100}\right)^{\mathrm{n}}$ $\Rightarrow A=$ Rs. $62500\left(1+\frac{12}{100}\right)^{2} \times\left(1+\frac{\f...
Read More →An electric kettle consumes 1 kW
Question: An electric kettle consumes 1 kW of electric power when operated at 220 V. A fuse wire of what rating must be used for it ? (a)1 A (b)2 A (c)4 A (d)5 A Solution: (d). Explanation : $\mathrm{P}=\mathrm{VI}$ or $\mathrm{I}=\frac{\mathrm{P}}{\mathrm{V}}=\frac{1 \mathrm{~kW}}{220 \mathrm{~V}}$ $=\frac{1000 \mathrm{~W}}{220 \mathrm{~A}}=4.55 \Omega=5 \mathrm{~A}$...
Read More →In an electrical circuit two resistors of 2Ω and 4Ω
Question: In an electrical circuit two resistors of 2Ω and 4Ω respectively are connected in series to a 6V battery. The heat dissipated by 4Ω resistor in 5 s will be (a)5 J (b)10 J (c)20 J (d)30 J Solution: (c). Explantion: Total resistance of the circuit, $\mathrm{R}$ $=2 \Omega+4 \Omega=6 \Omega$ Voltage, $\mathrm{V}=6 \mathrm{~V}$ $\therefore$ Current in the circuit, $\mathrm{I}=\frac{\mathrm{V}}{\mathrm{R}}=\frac{6}{6}=1 \mathrm{~A}$ Heat dissipated by $4 \Omega$ resistor in $5 s=I^{2} R t$ ...
Read More →By using the formula, find the amount and compound interest on:
Question: By using the formula, find the amount and compound interest on: Rs 10240 for 3 years at $12 \frac{1}{2} \%$ per annum compounded annually. Solution: Principal amount, $P=$ Rs. 10240 Rate of interest, $R=12 \frac{1}{2} \%$ p. a. Time, $n=3$ years The formula for the amount including the compound interest is given below : $A=$ Rs. $P\left(1+\frac{R}{100}\right)^{n}$ $\Rightarrow A=$ Rs. $10240\left(1+\frac{25}{100 \times 2}\right)^{3}$ $\Rightarrow A=$ Rs. $10240\left(1+\frac{25}{200}\ri...
Read More →By using the formula, find the amount and compound interest on:
Question: By using the formula, find the amount and compound interest on:Rs 31250 for 3 years at 8% per annum compounded annually. Solution: Principal amount, $P=$ Rs. 31250 Rate of interest, $R=8 \%$ per annum. Time, $n=3$ years. The formula for the amount including the compound interest is given below : $A=$ Rs. $P\left(1+\frac{R}{100}\right)^{\mathrm{n}}$ $\Rightarrow A=$ Rs. $31250\left(1+\frac{8}{100}\right)^{3}$ $\Rightarrow A=$ Rs. $31250\left(\frac{100+8}{100}\right)^{3}$ $\Rightarrow A=...
Read More →In an electrical circuit, three incandescent bulbs A,
Question: In an electrical circuit, three incandescent bulbs A, B and C of rating 40 W, 60 W and 100 W respectively are connected in parallel to an electric source. Which of the following is likely to happen regarding their brightness ? (a)Brightness of bulb A will be maximum (b)Brightness of bulb B will be more than that of A (c)Brightness of bulb C will be less than that of B (d)Brightness of all the bulbs will be same Solution: (b). Explanation :Brightness of bulb depends upon the heat produc...
Read More →By using the formula, find the amount and compound interest on:
Question: By using the formula, find the amount and compound interest on:Rs 10000 for 2 years at 11% per annum compounded annually. Solution: Principal amount, $P=$ Rs. 10000 Rate of interest, $R=11 \%$ per annum. Time, $n=2$ years. The formula for the amount including the compound interest is given below : $A=$ Rs. $P\left(1+\frac{\mathrm{R}}{100}\right)^{\mathrm{n}}$ $\Rightarrow A=$ Rs. $10000\left(1+\frac{11}{100}\right)^{2}$ $\Rightarrow A=$ Rs. $10000\left(\frac{100+11}{100}\right)^{2}$ $\...
Read More →Differentiate the following functions with respect to x :
Question: Differentiate the following functions with respect to $x$ : $(\cos x)^{x}+(\sin x)^{1 / x}$ Solution: Let $y=(\cos x)^{x}+(\sin x)^{\frac{1}{x}}$ $\Rightarrow y=a+b$ where $a=(\cos x)^{x} ; b=(\sin x)^{\frac{1}{x}}$ $\frac{d y}{d x}=\frac{d a}{d x}+\frac{d b}{d x}$ $\left\{\right.$ Using chain rule, $\frac{\mathrm{d}(\mathrm{u}+\mathrm{a})}{\mathrm{dx}}=\frac{\mathrm{du}}{\mathrm{dx}}+\frac{\mathrm{da}}{\mathrm{dx}}$ where $\mathrm{a}$ and $\mathrm{u}$ are any variables $\}$ $a=(\cos x...
Read More →The resistivity does not change if
Question: The resistivity does not change if (a)the material is changed (b)the temperature is changed (c)the shape of the resistor is changed (d)both material and temperature are changed. Solution: (c). Explanation :Resistivity of material does not depend on its dimensions and shape. However, resistivity depends upon the temperature and the nature of material....
Read More →By using the formula, find the amount and compound interest on:
Question: By using the formula, find the amount and compound interest on:Rs 6000 for 2 years at 9% per annum compounded annually. Solution: Principal amount, $\mathrm{P}=$ Rs 6000 Rate of interest, $\mathrm{R}=9 \%$ per annum Time, $n=2$ years. The formula for the amount including the compound interest is given below : $A=$ Rs. $P\left(1+\frac{R}{100}\right)^{\mathrm{n}}$ $\Rightarrow A=$ Rs. $6000\left(1+\frac{9}{100}\right)^{2}$ $\Rightarrow A=$ Rs. $6000\left(\frac{100+9}{100}\right)^{2}$ $\R...
Read More →Using the principle of mathematical induction, prove each of the following
Question: Using the principle of mathematical induction, prove each of the following for all n ϵ N: $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots . .+\frac{1}{2^{\mathrm{n}}}=\left(1-\frac{1}{2^{\mathrm{n}}}\right)$ Solution: To Prove: $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\ldots .+\frac{1}{2^{n}}=\left(1-\frac{1}{2^{n}}\right)$ Steps to prove by mathematical induction: Let $P(n)$ be a statement involving the natural number $n$ such that (i) $P(1)$ is true (ii) $P(k+1)$ is true, whenever $P(k)$ is ...
Read More →If the current I through a resistor is increased by 100%
Question: If the current I through a resistor is increased by 100% (assume that temperature remains unchanged), the increase in power dissipated will be (a)100 % (b)200% (c)300% (d)400% Solution: (c). Explanation: Power dissipated, $\mathrm{P}=\mathrm{I}^{2} \mathrm{R}$ When $\mathrm{I}^{\prime}=\mathrm{I}+100 \%$ of $\mathrm{I}=2 \mathrm{I}$ Power dissipated, $\mathrm{P}^{\prime}=(2 \mathrm{I})^{2} \mathrm{R}=4 \mathrm{I}^{2} \mathrm{R}$ $\therefore$ Increase in power dissipated $=\frac{P^{\pri...
Read More →Michael borrowed Rs 16000 from a finance company at $10 %$ per annum, compounded half-yearly.
Question: Michael borrowed Rs 16000 from a finance company at $10 \%$ per annum, compounded half-yearly. What amount of money will discharge his debt after $1 \frac{1}{2}$ years? Solution: Principal amount $=$ Rs. 16000 Rate of interest $=10 \%$ per annum $=5 \%$ for half year Time $=1 \frac{1}{2}$ years $=3$ half years Principal for the first half year $=$ Rs. 16000 Interest for the first half year $=$ Rs. $\left(\frac{16000 \times 5 \times 1}{100}\right)=$ Rs. 800 Now, amount at the end of the...
Read More →Divakaran deposited a sum of Rs 6250 in the Allahabad Bank for 1 year,
Question: Divakaran deposited a sum of Rs 6250 in the Allahabad Bank for 1 year, compounded half-yearly at 8% per annum. Find the compound interest he gets. Solution: Principal amount $=$ Rs. 6250 Rate of interest $=8 \%$ per annum $=4 \%$ for half year Time $=1$ year $=2$ half years Principal for the first half year $=$ Rs. 6250 Interest for the first half year $=$ Rs. $\left(\frac{6250 \times 4 \times 1}{100}\right)=$ Rs. 250 Now, amount at the end of the first half year $=$ Rs. $(6250+250)=$ ...
Read More →A student carries out an experiment and
Question: A student carries out an experiment and plots the VI graph of three samples of nichrome wire with resistances R1, R2and R3respectively as shown in figure. Which of the following is true ? (a)R1= R2= R3 (b)R1 R2 R3 (c)R3 R2 R1 (d)R2 R3 R1 Solution: (c). Explanation: Resistance of wire $\propto \frac{1}{\text { Slope of V-Igraph }}$ Since slope of V-I graph for $R_{3}$ is smallest, so $R_{3}$ is the highest....
Read More →Manoj deposited a sum of Rs 64000 in a post office for 3 years,
Question: Manoj deposited a sum of Rs 64000 in a post office for 3 years, compounded annually at $7 \frac{1}{2} \%$ per annum. What amount will he get on maturity? Solution: Principal for the first year $=$ Rs. 64000 Interest for the first year $=$ Rs. $\left(\frac{64000 \times 15 \times 1}{100 \times 2}\right)=$ Rs. 4800 Now, amount at the end of the first year $=$ Rs. $(64000+4800)=$ Rs. 68800 Principal for the second year $=$ Rs. 68800 Interest for the second year $=$ Rs. $\left(\frac{68800 \...
Read More →Harpreet borrowed Rs 20000 from her friend at 12% per annum simple interest.
Question: Harpreet borrowed Rs 20000 from her friend at 12% per annum simple interest. She lent it to Alam at the same rate but compounded annually. Find her gain after 2 years. Solution: Principal amount $=$ Rs. 20000 Simple interest $=$ Rs. $\left(\frac{20000 \times 2 \times 12}{100}\right)=$ Rs. 4800 The compound interest can be calculated as follows : Principal for the first year $=$ Rs. 20000 Interest for the first year $=$ Rs. $\left(\frac{20000 \times 12 \times 1}{100}\right)=$ Rs. 2400 N...
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