Although coal and petroleum are produced
Question: Although coal and petroleum are produced by degradation of biomass, yet we need to conserve them. Why ? (CCE 2012) Solution: Coal and petroleum have been produced from large amounts of biomass entrapped inside the earth under high temperature, pressure and anaerobic conditions. Such a situation develops only rarely like big upheavals on earth. At present no more coal or petroleum is being formed. All that is available has been formed millions of years ago. Being rich source of energy, ...
Read More →If x and y are inversely proportional,
Question: Ifxandyare inversely proportional, find the values ofx1,x2,y1andy2in the table given below: Solution: Since $x$ and $y$ are inversely proportional, $x y$ must be a constant. Therefore, $8 \times y_{1}=x_{1} \times 4=16 \times 5=x_{2} \times 2=80 \times y_{2}$ Now, $16 \times 5=8 \times y_{1}$ $\Rightarrow \frac{80}{8}=y_{1}$ $\therefore y_{1}=10$ $16 \times 5=x_{1} \times 4$ $\Rightarrow \frac{80}{4}=x_{1}$ $\therefore x_{1}=20$ $16 \times 5=x_{2} \times 2$ $\Rightarrow \frac{80}{2}=x_...
Read More →Solve this
Question: If $x^{m} y^{n}=1$, prove that $\frac{d y}{d x}=-\frac{m y}{n x}$ Solution: Here, $x^{m} y^{n}=1$ Taking log on both sides, $\log \left(x^{m} y^{n}\right)=\log 1$ $m \log x+n \log y=\log 1\left[\right.$ Since, $\left.\log (A B)=\log A+\log B ; \log a^{b}=b \log a\right]$ Differentiating with respect to $x$ $\frac{\mathrm{d}}{\mathrm{dx}}(\mathrm{m} \log \mathrm{x})+\frac{\mathrm{d}}{\mathrm{dx}}(\mathrm{nlogy})=\frac{\mathrm{d}}{\mathrm{dx}}(\log (1))$ $\frac{\mathrm{m}}{\mathrm{x}}+\f...
Read More →What measures would you take
Question: What measures would you take to conserve electricity in your house ? (CCE 2012) Solution: Judicious use of electricity by switching off lights and electrical appliances not required, Replacement of incandescent bulbs with fluorescent, compact fluorescent ones and LED bulbs. Replacement of electricity or gas operated geysers with solar water heaters, Replacement of electricity generating sets with solar light, Having more natural light and ventilation with design supporting warming duri...
Read More →In a village in Karnataka people started cultivating
Question: In a village in Karnataka people started cultivating crops all around a lake which was always filled with water. They added fertilizers to their field in order to enhance the yield. Soon they discovered that the water body was completely covered with green floating plants and fishes started dying in large numbers. Analyse the situation and give reasons for excessive growth of plants and death of the fish in the lake. Solution: Fertilizer rich run off from fields must have passed into t...
Read More →Observe the tables given below and in each case find whether x and y are inversely proportional:
Question: Observe the tables given below and in each case find whetherxandyare inversely proportional: (i) Solution: (i) Clearly, $6 \times 9 \neq 10 \times 15 \neq 14 \times 21 \neq 16 \times 24$ Therefore, $x$ and $y$ are not inversely proportional. (ii) Clearly, $5 \times 18=9 \times 10=15 \times 6=3 \times 30=45 \times 2=90=($ consant $)$ Therefore, $x$ and $y$ are inversely proportional. (iii) Clearly, $9 \times 4=3 \times 12=36 \times 1=36$, while $6 \times 9=54$ i. e., $9 \times 4=3 \time...
Read More →List the advantages associated with
Question: List the advantages associated with water harvesting at the community level. (CCE 2012) Solution: Water harvesting at the community level is capturing, collection and storage of rain water and surface run off for filling either small water bodies or recharging ground water. This is carried out through water shed management, check dams, earthen dams, roof top harvesting and filter wells in flood drains. Benefits: It ensures water availability in non-rainy season, It reduces the chances ...
Read More →Solve this
Question: If $x^{y}+y^{x}=(x+y)^{x+y}$, find $\frac{d y}{d x}$ Solution: Here, $x^{y}+y^{x}=(x+y)^{x+y}$ $e^{\log x^{y}}+e^{\log y^{x}}=e^{\log (x+y)^{x+y}}$ $e^{y \log x}+e^{x \log y}=e^{(x+y) \log (x+y)}$ Differentiating it with respect to $\mathrm{x}$ using chain rule, product rule, $\frac{d}{d x} e^{y \log x}+\frac{d}{d x} e^{x \log y}=\frac{d}{d x} e^{(x+y) \log (x+y)}$ $e^{y \log x}\left[y \frac{d}{d x}(\log x)+\log x \frac{d y}{d x}\right]+e^{x \log y}\left[x \frac{d}{d x}(\operatorname{l...
Read More →Reenu types 540 words during half an hour.
Question: Reenu types 540 words during half an hour. How many words would she type in 8 minutes? Solution: Let Reenu typexwords in 8 minutes. No. of words 540 x Time taken (in min) 30 8 Clearly, less number of words will be typed in less time. So, it is a case of direct proportion. Now, $\frac{540}{30}=\frac{x}{8}$ $\Rightarrow x=\frac{540 \times 8}{30}$ v Therefore, Reenu will type 144 words in 8 minutes....
Read More →Prepare a list of items that you use daily
Question: Prepare a list of items that you use daily in the school. Identify from the list five such items that can be recycled. Solution: Items. Rexin bag, steel lunch box, steel spoon, steel compass, steel dividers, paper, plastic box, pen, pencil, blade, eraser, handkerchief. Recycleable Items. Steel lunch box, steel spoon, steel compass, steel dividers, blade, paper, plastic box....
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: $3^{n} \geq 2^{n}$ Solution: To Prove: $3^{n} \geq 2^{n}$ Let us prove this question by principle of mathematical induction (PMI) for all natural numbers Let $\mathrm{P}(\mathrm{n}): 3^{n} \geq 2^{n}$ For $n=1 P(n)$ is true since $3^{n} \geq 2^{n} i \times e \times 3 \geq 2$, which is true Assume P(k) is true for some positive integer k , ie $=3^{k} \geq 2^{k} \ldots(1)$ We will now prove that P(k...
Read More →How many men should be employed for digging 27-metre-long trench of the same type in one day?
Question: 11 men can dig $6 \frac{3}{4}$-metre-long trench in one day. How many men should be employed for digging 27-metre-long trench of the same type in one day? Solution: Let $x$ be the required number of men. Now, $6 \frac{3}{4} \mathrm{~m}=\frac{27}{4} \mathrm{~m}$ Then, we have: Clearly, the longer the trench, the greater will be the number of men required. So, it is a case of direct proportion. Now, $\frac{11}{\frac{27}{4}}=\frac{x}{27}$ $\Rightarrow \frac{11 \times 4}{27}=\frac{x}{27}$ ...
Read More →How many men should be employed for digging 27-metre-long trench of the same type in one day?
Question: 11 men can dig $6 \frac{3}{4}$-metre-long trench in one day. How many men should be employed for digging 27-metre-long trench of the same type in one day? Solution: Let $x$ be the required number of men. Now, $6 \frac{3}{4} \mathrm{~m}=\frac{27}{4} \mathrm{~m}$ Then, we have: Number of men 11 x Length of trench (in metres) 274 27 Clearly, the longer the trench, the greater will be the number of men required. So, it is a case of direct proportion. Now, $\frac{11}{\frac{27}{4}}=\frac{x}{...
Read More →Solve this
Question: If $x^{x}+y^{x}=1$, find $\frac{d y}{d x}=-\frac{y(y+x \log y)}{x(y \log x+x)}$ Solution: Let $x^{x}=u$ and $y^{x}=v$ Taking log on both sides we get, $x \log x=\log u \ldots \ldots(1)$ $x \log y=\log v \ldots \ldots(2)$ Using $\log a^{b}=b \log a$ Differentiating both sides of equation (1) we get, $\mathrm{x} \times \frac{1}{\mathrm{x}}+\log \mathrm{x}=\frac{1}{\mathrm{u}} \frac{\mathrm{du}}{\mathrm{dx}}$ $\frac{d u}{d x}=x^{x}(1+\log x)$ ....(3) Differentiating both sides of equation...
Read More →Explain some harmful effects of agricultural
Question: Explain some harmful effects of agricultural practices on the environment. Solution: Soil: Fertilizer added to soil not only changes the chemistry of the soil but also kills many useful microbes. Ground Water: A part of fertilizer always leaches down into soil and reaches ground water. It raises the salt content of ground water. Eutrophication: Run-off from fields sprayed with fertilizer reaches water bodies. It results in their eutrophication. Pesticides: Pesticides sprayed over crops...
Read More →What are the by products of fertilizer
Question: What are the by products of fertilizer industries ? How do they affect the environment. Solution: The most common by product of fertilizer industries are axides of nitrogen and sulphur. They pass into atmosphere and spread to all nearby places. The gases have a corrosive effect on several items besides being harmful to living beings. They also give rise to acid rain. Acid rain is highly destructive to forests, crops and aquatic biota....
Read More →If the thickness of a pile of 12 cardboards is 65 mm,
Question: If the thickness of a pile of 12 cardboards is 65 mm, find the thickness of a pile of 312 such cardboards. Solution: Letxmm be the required thickness. Then, we have: Thickness of cardboard (in mm) 65 x No. of cardboards 12 312 Clearly, when the number of cardboard is more, the thickness will also be more. So, it is a case of direct proportion. Now, $\frac{65}{12}=\frac{x}{312}$ $\Rightarrow x=\frac{65 \times 312}{12}$ $\Rightarrow x=1690$ Therefore, the thickness of the pile of 312 car...
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 $\mathbf{n} \in \mathbf{N}:$ $\left(2^{3 n}-1\right)$ is a multiple of 7 Solution: To Prove $2^{3 n}-1$, which is multiple of 7 Let us prove this question by principle of mathematical induction (PMI) for all natural numbers $2^{3 n}-1$ is multiple of 7 Let $P(n): 2^{3 n}-1$, which is multiple of 7 For $n=1 P(n)$ is true since $2^{3}-1=8-1=7$, which is multiple of 7 Assume P(k) is true for some positive i...
Read More →Suggest suitable mechanisms for waste
Question: Suggest suitable mechanisms for waste management in fertilizer industries. Solution: Fertilizer industries produce mainly two types of wastes (a) Gaseous (b) Effluents. Additionally they may release heat and fuel waste if coal is being used as a source of energy. Gaseous Wastes:They are passed through wet scrubbers to dissolve the pollutant gases. Effluents:The effluents of the industry are tested for chemicals present in them. The same can be retrieved and made available to the market...
Read More →Solve this
Question: If $x^{x}+y^{x}=1$, prove that $\frac{d y}{d x}=-\left\{\frac{x^{x}(1+\log x)+y^{x} \cdot \log y}{x \cdot y^{(x-1)}}\right\}$ Solution: Here $x^{x}+y^{x}=1$ $\mathrm{e}^{\log x^{x}}+\mathrm{e}^{\log y^{x}}=1$ $\mathrm{e}^{\mathrm{xlogx}}+\mathrm{e}^{\mathrm{xlog} y}=1$ $\left[\right.$ Since $\left.\mathrm{e}^{\log a}=\mathrm{a} \cdot \log \mathrm{a}^{\mathrm{b}}=\mathrm{b} \log \mathrm{a}\right]$ Differentiating it with respect to $x$ using chain rule and product rule, $\frac{d}{d x} e...
Read More →Ravi walks at the uniform rate of 5 km/hr.
Question: Ravi walks at the uniform rate of 5 km/hr. What distance would he cover in 2 hours 24 minutes? Solution: Letxkm be the required distance covered by Ravi in 2 h 24 min. Then, we have: $1 \mathrm{~h}=60 \mathrm{~min}$ i. e., $2 \mathrm{~h} 24 \mathrm{~min}=(120+24) \mathrm{min}=144 \mathrm{~min}$ Distance covered (in km) 5 x Time (in min) 60 144 Clearly, more distance will be covered in more time. So, this is a case of direct proportion. Now,$\frac{5}{60}=\frac{x}{144}$ $\Rightarrow x=\f...
Read More →Name the wastes which are generated
Question: Name the wastes which are generated in your house daily. What measures would you take for their disposal ? Solution: Wastes : Vegetable and fruit peels and rind, stale food, food leftovers, used tea leaves. Milk pouches, polythene bags, empty cartons. Waste paper (newspaper, bags, envelopes), packing paper, empty bottles, torn cloth pieces, etc. Dust and other sweepings. Disposal : Separation into biodegradable and non-biodegradable, recyclable and non-recyclable wastes. Recyclable was...
Read More →Give two differences between
Question: Give two differences between food chain and food web. Solution:...
Read More →A car is travelling at the average speed of 50 km/hr.
Question: A car is travelling at the average speed of 50 km/hr. How much distance would it travel in 1 hour 12 minutes? Solution: Letxkm be the required distance. Then, we have: $1 \mathrm{~h}=60 \mathrm{~min}$ i. e., $1 \mathrm{~h} 12 \mathrm{~min}=(60+12) \mathrm{min}=72 \mathrm{~min}$ Distance covered (in km) 50 x Time (in min) 60 72 Clearly, more distance will be covered in more time. So, this is a case of direct proportion. Now, $\frac{50}{60}=\frac{x}{72}$ $\Rightarrow x=\frac{50 \times 72...
Read More →Suggest any four activities in daily
Question: Suggest any four activities in daily life which are ecofriendly. Solution: Use of cloth bags instead of polythene or plastic bags. Separation of biodegradable and non-biodegradable in green and blue coloured bins. Use of compact fluorescent lamps instead of incandescent lamps. Harvesting of rain water and preventing wastage of resources....
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