A truck covers a distance of 510 km in 34 litres of diesel.
Question: A truck covers a distance of 510 km in 34 litres of diesel. How much distance would it cover in 20 litres of diesel? Solution: Let the required distance be xkm. Then, we have: Quantity of diesel (in litres) 34 20 Distance (in km) 510 x Clearly, the less the quantity of diesel consumed, the less is the distance covered. So, this is a case of direct proportion. Now, $\frac{34}{510}=\frac{20}{x}$ $\Rightarrow \frac{1}{15}=\frac{20}{x}$ $\Rightarrow x \times 1=20 \times 15=300$ Therefore, ...
Read More →Find the values
Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$, when $y=x^{x}+x^{1 / x}$ Solution: Here, $y=x^{x}+x^{1 / x}$ $=e^{\log x^{x}}+e^{\log x^{\frac{1}{x}}}$ $y=e^{x \log x}+e^{\left(\frac{1}{x} \log x\right)}$ [ Sincelog $a^{b}=b \log a$ ] Differentiating it with respect to $x$ using the chain rule and product rule, $\frac{d y}{d x}=\frac{d}{d x}\left(e^{x \log x}\right)+\frac{d}{d x}\left(e^{\frac{1}{x} \log x}\right)$ $=e^{x \log x}+\frac{d}{d x}(x \log x)+e^{\frac{1}{x} \log x} \frac{d}{d x}\lef...
Read More →If x and y are directly proportional,
Question: Ifxandyare directly proportional, find the values ofx1,x2andy1in the table given below: Solution: Since $x$ and $y$ are directly propotional, we have: $\frac{3}{72}=\frac{x_{1}}{120}=\frac{x_{2}}{192}=\frac{10}{y_{1}}$ Now, $\frac{3}{72}=\frac{x_{1}}{120}$ $\Rightarrow x_{1}=\frac{120 \times 3}{72}=5$ And, $\frac{3}{72}=\frac{x_{2}}{192}$ $\quad \Rightarrow \quad x_{2}=\frac{3 \times 192}{72}=8$ And, $\frac{3}{72}=\frac{10}{y_{1}}$ $\Rightarrow y_{1}=\frac{72 \times 10}{3}=240$ Therefo...
Read More →What are the limitation in obtaining
Question: What are the limitation in obtaining energy from wind ? Solution: We cannot depend upon wind energy as it is available only when air is in motion. The appliances or machines operating with wind energy stop working as soon as wind stops. The minimum speed of wind to operate generator to produce electricity is about 15 km/h. As soon as the speed of the wind becomes less than 15 km/h, the generator stops working. There are certain regions where wind is not available, so the use of wind en...
Read More →What is biomass ?
Question: What is biomass ? What can be done to obtain bio-energy using biomass ? Solution: A material which contains carbon and other combustible materials is called biomass. The waste of plants and animals is the example of bio mass. (CBSE Papers)...
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 $n \in N:$ $\left(x^{2 n}-1\right)-1$ is divisible by $(x-y)$, where $x \neq 1$ Solution: To Prove: $x^{2 n-1}-1$ is divisible by $x-1$ Let us prove this question by principle of mathematical induction (PMI) for all natural numbers Let $\mathrm{P}(\mathrm{n}):^{2 n-1}-1$ is divisible by $x-1$ For n = 1 $\mathrm{P}(\mathrm{n})$ is true since $x^{2 n-1}-1=x^{2-1}-1=(x-1)$ Which is divisible by x - 1 Assume...
Read More →Mention three advantages of a solar cell ?
Question: Mention three advantages of a solar cell ? Solution: Advantages of Solar Cells They directly convert solar energy into electrical energy. They are environment-friendly i.e. they do not cause pollution. They are used to operate electric bulbs and tubes in remote areas where hydroelectricity is not available....
Read More →Observe the tables given below and in each one find whether x and y are proportional:
Question: Observe the tables given below and in each one find whetherxandyare proportional: (i) Solution: (i) Clearly, $\frac{x}{y}=\frac{3}{9}=\frac{5}{15}=\frac{8}{24}=\frac{11}{33}=\frac{26}{78}=\frac{1}{3}($ constant $)$ Therefore, $\mathrm{x}$ and $\mathrm{y}$ are proportional. (ii) Clearly, $\frac{x}{y}=\frac{2.5}{10}=\frac{4}{16}=\frac{7.5}{30}=\frac{10}{40}=\frac{1}{4}$, while $\frac{14}{42}=\frac{1}{3}$ i. e., $\frac{2.5}{10}=\frac{4}{16}=\frac{7.5}{30}=\frac{10}{40}$ is not equal to $\...
Read More →Solve this
Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$, when $y=(\tan x)^{\log x}+\cos ^{2}\left(\frac{\pi}{4}\right)$ Solution: Let $y=(\tan x)^{\log x}+\cos ^{2}\left(\frac{\pi}{4}\right)$ $\Rightarrow y=a+b$ where, $a=(\tan x)^{\log x} ; b=\cos ^{2}\left(\frac{\pi}{4}\right)$ $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{da}}{\mathrm{dx}}+\frac{\mathrm{db}}{\mathrm{dx}}$ $\left\{\right.$ Using chain rule, $\frac{\mathrm{d}(\mathrm{u}+\mathrm{a})}{\mathrm{dx}}=\frac{\mathrm{du}}{\mathrm{dx}}+\frac{...
Read More →What is the role of a plane mirror
Question: What is the role of a plane mirror and a glass sheet in a solar cooker ? Solution: A solar cooker covered by a plane glass slab will be more efficient. This is because glass slab does not allow the heat radiation to escape from the solar cooker and hence the temperature of the solar cooker covered with glass slab increases more than the temperature of the solar cooker which is left open....
Read More →What steps would you suggest to minimise
Question: What steps would you suggest to minimise environmental pollution caused by burning of fossile fuel ? Solution: We can minimize environmental pollution caused by the burning of fossil fuel by growing more and more trees, Using smokeless chulahs and smokeless chimneys in thermal power plants...
Read More →Write two different ways of harnessing
Question: Write two different ways of harnessing energy from occean. Solution: Tidal energy. Ocean Thermal energy (OTEC)....
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 $n \in N:$ Solution: To Prove: $x^{2 n}-y^{2 n}$ is divisible by $x+y$ Let us prove this question by principle of mathematical induction (PMI) for all natural numbers Let $\mathrm{P}(\mathrm{n}): x^{2 n}-y^{2 n}$ is divisible by $x+y$ For $\mathrm{n}=1 \mathrm{P}(\mathrm{n})$ is true since $^{x^{2 n}}-y^{2 n}=x^{2}-y^{2}=(x+y) \times(x-y)$ Which is divisible by x + y Assume P(k) is true for some positive...
Read More →Why is there a need to harness non-convential
Question: Why is there a need to harness non-convential sources of energy ? Give two main reasons. Solution: Non-conventional sources of energy are pollution free, whereas fossil fuels cause lot of pollution. Non-conventional sources of energy are in exhaustible, whereas fossil fuels are limited. Our demand of energy is increasing day by day....
Read More →Choose the incorrect statement
Question: Choose the incorrect statement (a)We are encouraged to plant more trees so as to ensure clean environment and also provide bio-mass fuel i (b)Gobar-gas is produced when crops, vegetable wastes etc., decompose in the absence of oxygen (c)The main ingredient of bio-gas is ethane and it gives a lot of smoke and also produces a log of residual ash (d)Bio-mass is a renewable source of energy Solution: (c)The main ingredient of bio-gas is ethane and it gives a lot of smoke and also produces ...
Read More →Choose the incorrect statement regarding wind power
Question: Choose the incorrect statement regarding wind power (a)It is expected to harness wind power to minimum in open space (b)The potential energy content of wind blowing at high altitudes is the source of wind power (c)Wind hitting at the blades of a windmill causes them to rotate. The rotation thus achieved can be ultilised further (d)One possible method of ultilising the energy of rotational motion of the blades of a windmill is to run the turbine of an electric generator Solution: (b)The...
Read More →Solve this
Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$, when $y=x^{x}+(\sin x)^{x}$ Solution: let $y=x^{x}+(\sin x)^{x}$ $\Rightarrow y=a+b$ where $a=x^{x} ; b=(\sin x)^{x}$ $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{da}}{\mathrm{dx}}+\frac{\mathrm{db}}{\mathrm{dx}}$ $\left\{\right.$ 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 $\}$ $a=x^{x}$ Taking log...
Read More →In a hydroelectric power plant more electrical power
Question: In a hydroelectric power plant more electrical power can be generated if water falls from a greater height because (a)its temperature increases (b)larger amount of potential energy is converted into kinetic energy (c)the electricity content of water increases with height (d)more water molecules dissociate into ions Solution: (b)larger amount of potential energy is converted into kinetic energy...
Read More →Fill in the blanks:
Question: Fill in the blanks: (i) $A=P\left(1+\frac{\ldots \ldots \ldots}{100}\right)^{n}$. (ii) (Amount) - (Principal) = ......... (iii) If the value of a machine is RsPand it depreciates atR% per annum, then its value after 2 years is ......... (iv) If the populationPof a town increases atR% per annum, then its population after 5 years is ......... Solution: (i) $A=P\left(1+\frac{R}{100}\right)^{n}$ (ii) Compound interest (iii) $A=P\left(1-\frac{R}{100}\right)^{2}$, where $A$ is the value of t...
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: n ( n + 1 ) ( n + 2 ) is multiple of 6 Solution: To Prove: $n \times(n+1) \times(n+2)$ is multiple of 6 Let us prove this question by principle of mathematical induction (PMI) for all natural numbers $n \times(n+1) \times(n+2)$ is multiple of 6 Let $P(n): n \times(n+1) \times(n+2)$, which is multiple of 6 For $n=1 P(n)$ is true since $1 \times(1+1) \times(1+2)=6$, which is multiple of 6 Assume P(k...
Read More →Choose the correct statement
Question: Choose the correct statement (a)Sun can be taken as an inexhaustible source of energy (b)There is infinite storage of fossil fuel inside the earth (c)Hydro and wind energy plants are non polluting sources of energy (d)Waste from a nuclear power plant can be easily disposed off Solution: (a)Sun can be taken as an inexhaustible source of energy...
Read More →Mark (✓) against the correct answer:
Question: Mark (✓) against the correct answer: If the compound interest on a certain sum for 2 years at 10% per annum is Rs 1050, the sum is (a) Rs 3000 (b) Rs 4000 (c) Rs 5000 (d) Rs 6000 Solution: (c) Rs 5000 Here, $A=P \times\left(1+\frac{R}{100}\right)^{n}$ $=P \times\left(1+\frac{10}{100}\right)^{2}$ $=P \times\left(\frac{110}{100}\right)^{2}$ $=P \times\left(\frac{11}{10}\right) \times\left(\frac{11}{10}\right)$ Now, $C I=A-P$ $\Rightarrow$ Rs. $1050=\frac{121 p}{100}-P$ $=\frac{121 P-100 ...
Read More →The power generated in a windmill
Question: The power generated in a windmill (a)is more in rainy season since damp air would mean more air mass hitting the blades (b)depends on the height of the tower (c)depends on wind velocity (d)can be increased by planting tall trees close to the tower Solution: (c)depends on wind velocity...
Read More →The main constituent of biogas is
Question: The main constituent of biogas is (a)methane (b)carbon dioxide (c)hydrogen (d)hydrogen sulphide Solution: (a)methane...
Read More →Mark (✓) against the correct answer:
Question: Mark (✓) against the correct answer: If the simple interest on a sum of money at 10% per annum for 3 years is Rs 1500, then the compound interest on the same sum at the same rate for the same period is (a) Rs 1655 (b) Rs 1155 (c) Rs 1555 (d) Rs 1855 Solution: (a) Rs 1655 Here, SI $=\frac{P \times R \times T}{100}$ $\Rightarrow$ Rs. $1500=\frac{P \times 10 \times 3}{100}$ $\Rightarrow P=\frac{1500 \times 100}{10 \times 3}=$ Rs. 5000 Now, $A=P \times\left(1+\frac{R}{100}\right)^{n}$ $=$ ...
Read More →