What are decomposers ?
Question: What are decomposers ? What will be the consequences of their absence in an ecosystem ? Solution: Definition:Decomposers or microconsumers are saprophytes which obtain their nourishment from organic remains by secreting digestive enzymes over the latter and absorbing the solubilised substances. Absence of Decomposers: Organic remains will pile up leaving no space for new living beings, Biogeochemical cycling will stop so that raw materials will not be available to plants for manufactur...
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 $\mathbf{n} \in \mathbf{N}:$ $\left(3^{2 n+2}-8 n-9\right)$ is divisible by 8 Solution: To Prove: $3^{2 n+2}-8 n-9$ is a divisible of 8 Let us prove this question by principle of mathematical induction (PMI) for all natural numbers Let $\mathrm{P}(\mathrm{n}):^{2 n+2}-8 n-9$ is a divisible of 8 For n = 1 P(n) is true since $3^{2 n+2}-8 n-9=3^{2+2}-8 \times 1-9=81-17=64$ Which is divisible of 8 Assume P(k...
Read More →In 8 days, the earth picks up
Question: In 8 days, the earth picks up (6.4 107) kg of dust from the atmosphere. How much dust will it pick up in 15 days? Solution: Letxkg be the required amount of dust. Then, we have: Clearly, more amount of dust will be collected in more number of days. So, this is a case of direct proportion. Now, $\frac{8}{6.4 \times 10^{7}}=\frac{15}{x}$ $\Rightarrow x=\frac{15 \times 6.4 \times 10^{7}}{8}$ $\Rightarrow x=12 \times 10^{7}$ Therefore, 12,00,00,000 kg of dust will be picked up in 15 days....
Read More →Solve this
Question: If $y=\sin \left(x^{x}\right)$, prove that $\frac{d y}{d x}=\cos \left(x^{x}\right) \cdot x^{x}(1+\log x)$ Solution: Here, $y=\sin \left(x^{x}\right) \ldots \ldots$ (i) Let $u=x^{x} \ldots \ldots$ (ii) Taking log on both sides, $\log u=\log x^{x}$ $\log u=x \log x$ Differentiating both sides with respect to $x$, $\frac{1}{\mathrm{u}} \frac{\mathrm{du}}{\mathrm{dx}}=\frac{\mathrm{d}}{\mathrm{dx}}(\mathrm{x} \log \mathrm{x})$ $=x \frac{d}{d x}(\log x)+\log x \frac{d}{d x}(x)$ $=x\left(\f...
Read More →Indicate the flow of energy in an ecosystem.
Question: Indicate the flow of energy in an ecosystem. Why is it undirectional ? Justify. (CCE 2014) Solution: An ecosystem does not have its own source of energy. It receives the same from sun. Green plants or producers trap the solar energy and change it into chemical form during synthesis of food. Herbivores obtain energy from the food they take. A lot of energy dissipates during transfer and utilization of food energy by herbivores (10% law). From herbivores the food energy passes to primary...
Read More →In a model of a ship, the mast is 9 cm high, while the mast of the actual ship is 15m high.
Question: In a model of a ship, the mast is 9 cm high, while the mast of the actual ship is 15m high. If the length of the ship is 35 metres, how long is the model ship? Solution: Let $x \mathrm{~m}$ be the length of the model of the ship. Then, we have: $1 \mathrm{~m}=100 \mathrm{~cm}$ Therefore, $15 \mathrm{~m}=1500 \mathrm{~cm}$ $35 \mathrm{~m}=3500 \mathrm{~cm}$ Length of the mast (in cm) Length of the ship (in cm) Actual ship 1500 3500 Model of the ship 9 x Clearly, if the length of the act...
Read More →We do not clean ponds or lakes
Question: We do not clean ponds or lakes but an aquarium needs to be cleaned. Why ? Solution: An aquarium is an artificial system which is also incomplete due to absence of producers, food chains and decomposers. There is no recycling and self cleaning. However, a pond or a lake is a self sustained, natural and complete ecosystem where there is perfect recycling of nutrients....
Read More →Select the mismatched pair in
Question: Select the mismatched pair in the following and correct it. (a)Biomagnification: Accumulation of chemicals at the successive trophic levels of a food chain. (b)Ecosystem: Biotic component of environment. (c)Aquarium: A man-made ecosystem. (d)Parasites: Organisms which obtain food from other living organisms. Solution: (b) Ecosystem:It is an ecological system consisting of a distinct biotic community and the physical environment (consisting of a number of abiotic factors) both interacti...
Read More →Solve this
Question: If $x^{16} y^{9}=\left(x^{2}+y\right)^{17}$, prove that $x \frac{d y}{d x}=2 y$ Solution: Here, $x^{16} y^{9}=\left(x^{2}+y\right)^{17}$ Taking log on both sides, $\log \left(x^{16} y^{9}\right)=\log \left(x^{2}+y\right)^{17}$ $16 \log x+9 \log y=17 \log \left(x^{2}+y\right)$ $\left[\right.$ Since, $\left.\log (A B)=\log A+\log B ; \log a^{b}=b \log a\right]$ Differentiating it with respect to $\mathrm{x}$ using the chain rule, $16 \frac{\mathrm{d}}{\mathrm{dx}}(\log \mathrm{x})+9 \fra...
Read More →Explain the role of decomposers
Question: Explain the role of decomposers in the environment. (CCE 2011) Solution: Decomposers are saprophytes which feed on organic remains by a process of external digestion and absorption of solubilised materials, e.g., many bacteria, fungi. In the process they perform the following functions : Cleansing the earth of organic remains and continuously creating space for newer generations of organisms. Release of minerals from organic remains. The phenomenon is called mineralisation. The release...
Read More →The cost of 15 metres of a cloth is Rs 981.
Question: The cost of 15 metres of a cloth is Rs 981. What length of this cloth can be purchased for Rs 1308? Solution: Let the length of cloth bexm. Then, we have: Length of cloth (in metres) 15 x Cost of cloth (in rupees) 981 1308 Clearly, more length of cloth can be bought by more amount of money. So, this is a case of direct proportion. Now, $\frac{15}{981}=\frac{x}{1308}$ $\Rightarrow x=\frac{15 \times 1308}{981}$ $\Rightarrow x=20$ Therefore, 20 m of cloth can be bought for Rs 1,308....
Read More →Suggest one word for each of the following
Question: Suggest one word for each of the following statements/definitions. (a)The physical and biological world where we live in. (b)Each level of food chain where transfer of energy takes place. (c)The physical factors like temperature, rainfall, wind and soil of an ecosystem. (d)Organisms which depend on the producers either directly or indirectly for food. Solution: (a)Biosphere (also environment) (b)Trophic level. (c)Abiotic factors. (d)Consumers (also heterotrophs)...
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: $\left(4^{n}+15 n-1\right)$ is divisible by 9 Solution: To Prove: $4^{n}+15 n-1$ is a divisible of 9 Let us prove this question by principle of mathematical induction (PMI) for all natural numbers Let $\mathrm{P}(\mathrm{n}): 4^{n}+15 n-1$ is a divisible of 9 For $n=1 P(n)$ is true since $4^{n}+15 n-1=4^{1}+15 \times 1-1=18$ Which is divisible of 9 Assume P(k) is true for some positive integer k ,...
Read More →Differentiate between biodegradable
Question: Differentiate between biodegradable and non-biodegradable substances. Give examples. Solution:...
Read More →If 9 kg of sugar costs ₹ 238.50,
Question: If 9 kg of sugar costs ₹ 238.50, how much sugar can be bought for ₹ 371? Solution: Let the quantity of sugar bought for ₹371 bexkg. Quantity(in kg) 9 x Price(in ₹) 238.50 371 The price increases as the quantity increases. Thus, this is a case of direct proportion. $\therefore \frac{9}{238.50}=\frac{x}{371}$ $\Rightarrow x=\frac{9 \times 371}{238.50}=14$ Thus, the quantity of sugar bought for ₹371 is 14 kg....
Read More →Solve this
Question: If $x^{13} y^{7}=(x+y)^{20}$, prove that $\frac{d y}{d x}=\frac{y}{x}$ Solution: Here, $x^{13} y^{7}=(x+y)^{20}$ Taking log on both sides, $\log \left(x^{13} y^{7}\right)=\log (x+y)^{20}$ $13 \log x+7 \log y=20 \log (x+y)$ [ Since, $\left.\log (A B)=\log A+\log B ; \log a^{b}=b \log a\right]$ Differentiating it with respect to $x$ using the chain rule, $13 \frac{\mathrm{d}}{\mathrm{dx}}(\log \mathrm{x})+7 \frac{\mathrm{d}}{\mathrm{dx}}(\log y)=20 \frac{\mathrm{d}}{\mathrm{dx}} \log (\m...
Read More →If 18 dolls cost Rs 630,
Question: If 18 dolls cost Rs 630, how many dolls can be bought for Rs 455? Solution: Let the required number of dolls bex. Then, we have: No of dolls 18 x Cost of dolls (in rupees) 630 455 Clearly, the less the amount of money, the less will be the number of dolls bought. So, this is a case of direct proportion. Now, $\frac{18}{630}=\frac{x}{455}$ $\Rightarrow \frac{1}{35}=\frac{x}{455}$ $\Rightarrow x=\frac{455}{35}$ $\Rightarrow x=13$ Therefore, 13 dolls can be bought for Rs 455....
Read More →Why are crop fields known as artificial
Question: Why are crop fields known as artificial ecosystems ? Solution: Crop fields are known as artificial ecosystems because they are raised, maintained, nourished and reaped by human beings....
Read More →What are the advantages of cloth
Question: What are the advantages of cloth bags over plastic bags during shopping ? Solution: Advantages of Cloth Bags Cloth bags are stronger and more durable as compared to plastic bags. They are washable. They are reused time and again. Cloth bags do not pollute environment. They are made of biodegradable material which can also be recycled....
Read More →Write the common food chain
Question: Write the common food chain of a pond ecosystem. Solution: Phytoplankton (and other aquatic plants) $\longrightarrow$ Zooplankton (and other small aquatic animals and larvae) $\longrightarrow$ Fish $\longrightarrow$ Bird (King Fisher)....
Read More →A loaded truck covers 16 km in 25 minutes.
Question: A loaded truck covers 16 km in 25 minutes. At the same speed, how far can it travel in 5 hours? Solution: Let the required distance bexkm. Then, we have: $1 \mathrm{~h}=60 \mathrm{~min}$ i.e., $5 \mathrm{~h}=5 \times 60=300 \mathrm{~min}$ Distance (in km) 16 x Time (in min) 25 300 Clearly, the more the time taken, the more will be the distance covered.So, this is a case of direct proportion. Now, $\frac{16}{25}=\frac{x}{300}$ $\Rightarrow x=\left(\frac{16 \times 300}{25}\right)$ $\Righ...
Read More →Find the values
Question: Find $\frac{\mathrm{dy}}{\mathrm{dx}}$, when $y=x^{\log x}+(\log x)^{x}$ Solution: Here, $y=x^{\log x}+(\log x)^{x}$ Let $u=(\log x)^{x}$, and $v=x^{\log x}$ $\therefore \mathrm{y}=\mathrm{u}+\mathrm{v}$ $\frac{d y}{d x}=\frac{d u}{d x}+\frac{d v}{d x}$ ....(1) $u=(\log x)^{x}$ $\log u=\log \left[(\log x)^{x}\right]$ $\log u=x \log (\log x)$ Differentiating both sides with respect to $x$, we get $\frac{1}{u} \cdot \frac{d u}{d x}=\frac{d}{d x}(x) \times \log (\log x)+x \frac{d}{d x}[\l...
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 $n \in N$ : $\left\{(41)^{n}-(14)^{n}\right\}$ is divisible by 27 Solution: To Prove: $41^{n}-14^{n}$ is a divisible of 27 Let us prove this question by principle of mathematical induction (PMI) for all natural numbers Let $\mathrm{P}(\mathrm{n}): 41^{n}-14^{n}$ is a divisible of 27 For $n=1 P(n)$ is true since $41^{n}-14^{n}=41^{1}-14^{1}=27$ Which is multiple of 27 Assume P(k) is true for some positive...
Read More →Why is improper disposal of wastes
Question: Why is improper disposal of wastes a curse to environment ? Solution: An improper disposal of wastes means addition of pollutants into environment air, water, soil. They will harm living beings, human assets and human beings. For example, passage of sewage into water body will cause eutrophication, stink, development of sludge, killing of animals and source of water borne pathogens....
Read More →A taxi charges a fare of Rs 2550 for a journey of 150 km.
Question: A taxi charges a fare of Rs 2550 for a journey of 150 km. How much would it charge for a journey of 124 km? Solution: Let the charge for a journey of 124 km be ₹x. Price(in ₹) 2550 x Distance(in km) 150 124 More is the distance travelled, more will be the price. So, it is a case of direct proportion. $\therefore \frac{2550}{150}=\frac{x}{124}$ $\Rightarrow x=\frac{2550 \times 124}{150}=2108$ Thus, the taxi charges ₹2,108 for the distance of 124 km....
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