Find the coordinates of the foci and the vertices, the eccentricity,
Question: Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola $49 y^{2}-16 x^{2}=784$ Solution: The given equation is $49 y^{2}-16 x^{2}=784$. It can be written as $49 y^{2}-16 x^{2}=784$ Or, $\frac{y^{2}}{16}-\frac{x^{2}}{49}=1$ Or, $\frac{y^{2}}{4^{2}}-\frac{x^{2}}{7^{2}}=1$ $\ldots(1)$ On comparing equation (1) with the standard equation of hyperbola i.e., $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1$, we obtain $a=4$ and $b=7...
Read More →Discuss briefly the following:
Question: Discuss briefly the following: (a)Radioactive wastes (b)Defunct ships and e-wastes (c)Municipal solid wastes Solution: (a) Radioactive wastes:- Radioactive wastes are generated during the process of generating nuclear energy from radioactive materials. Nuclear waste is rich in radioactive materials that generate large quantities of ionizing radiations such as gamma rays. These rays cause mutation in organisms, which often results in skin cancer. At high dosage, these rays can be lethal...
Read More →If, show that
Question: If $e^{y}(x+1)=1$, show that $\frac{d^{2} y}{d x^{2}}=\left(\frac{d y}{d x}\right)^{2}$ Solution: The given relationship is $e^{y}(x+1)=1$ $e^{y}(x+1)=1$ $\Rightarrow e^{y}=\frac{1}{x+1}$ Taking logarithm on both the sides, we obtain $y=\log \frac{1}{(x+1)}$ Differentiating this relationship with respect to x, we obtain $\frac{d y}{d x}=(x+1) \frac{d}{d x}\left(\frac{1}{x+1}\right)=(x+1) \cdot \frac{-1}{(x+1)^{2}}=\frac{-1}{x+1}$ $\therefore \frac{d^{2} y}{d x^{2}}=-\frac{d}{d x}\left(...
Read More →What measures, as an individual, you would take to reduce environmental pollution?
Question: What measures, as an individual, you would take to reduce environmental pollution? Solution: The following initiatives can be taken to prevent environmental pollution: Measures for preventing Air pollution: (i)Planting more trees (ii)Use of clean and renewable energy sources such as CNG and bio-fuels (iii)Reducing the use of fossil fuels (iv)Use of catalytic converters in automobiles Measures for preventing water pollution:- (i)Optimizing the use of water (ii)Using kitchen waste water ...
Read More →Find the coordinates of the foci and the vertices, the eccentricity,
Question: Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola $5 y^{2}-9 x^{2}=36$ Solution: The given equation is $5 y^{2}-9 x^{2}=36$. $\Rightarrow \frac{y^{2}}{\left(\frac{36}{5}\right)}-\frac{x^{2}}{4}=1$ $\Rightarrow \frac{y^{2}}{\left(\frac{6}{\sqrt{5}}\right)^{2}}-\frac{x^{2}}{2^{2}}=1$ $\ldots(1)$ On comparing equation (1) with the standard equation of hyperbola i.e., $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1$, we obta...
Read More →Discuss the role of women and communities in protection and conservation of forests.
Question: Discuss the role of women and communities in protection and conservation of forests. Solution: Women and communities have played a major role in environmental conservation movements. (i) Case study of the bishnoi community:TheBishnoi community in Rajasthan strictly believes in the concept of living peacefully with nature. In 1731, the king of Jodhpur ordered his ministers to arrange wood for the construction of his new palace. For this purpose, the minister and the workers went to bish...
Read More →If, show that
Question: If $y=500 e^{7 x}+600 e^{-7 x}$, show that $\frac{d^{2} y}{d x^{2}}=49 y$ Solution: It is given that, $y=500 e^{7 x}+600 e^{-7 x}$ Then, $\frac{d y}{d x}=500 \cdot \frac{d}{d x}\left(e^{7 x}\right)+600 \cdot \frac{d}{d x}\left(e^{-7 x}\right)$ $=500 \cdot e^{7 x} \cdot \frac{d}{d x}(7 x)+600 \cdot e^{-7 x} \cdot \frac{d}{d x}(-7 x)$ $=3500 e^{7 x}-4200 e^{-7 x}$ $\therefore \frac{d^{2} y}{d x^{2}}=3500 \cdot \frac{d}{d x}\left(e^{7 x}\right)-4200 \cdot \frac{d}{d x}\left(e^{-7 x}\right...
Read More →Why ozone hole forms over Antarctica? How will enhanced ultraviolet Radiations affect us?
Question: Why ozone hole forms over Antarctica? How will enhanced ultraviolet Radiations affect us? Solution: The ozone hole is more prominent over the region of Antarctica. It is formed due to an increased concentration of chlorine in the atmosphere. Chlorine is mainly released from chlorofluorocarbons (CFCs) widely used as refrigerants. The CFCs magnate from the troposphere to the stratosphere, where they release chlorine atoms by the action of UV rays on them. The release of Chlorine atoms ca...
Read More →Find the coordinates of the foci and the vertices, the eccentricity,
Question: Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola $16 x^{2}-9 y^{2}=576$ Solution: The given equation is $16 x^{2}-9 y^{2}=576$ It can be written as $16 x^{2}-9 y^{2}=576$ $\Rightarrow \frac{x^{2}}{36}-\frac{y^{2}}{64}=1$ $\Rightarrow \frac{x^{2}}{6^{2}}-\frac{y^{2}}{8^{2}}=1$ $\ldots(1)$ On comparing equation (1) with the standard equation of hyperbola i.e., $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$, we obtain $a...
Read More →Write critical notes on the following:
Question: Write critical notes on the following: (a)Eutrophication (b)Biological magnification (c)Groundwater depletion and ways for its replenishment Solution: (a) Eutrophication:-It is the natural ageing process of a lake caused due to nutrient enrichment. It is brought down by the runoff of nutrients such as animal wastes, fertilizers, and sewage from land which leads to an increased fertility of the lake. As a result, it causes a tremendous increase in the primary productivity of the ecosyst...
Read More →Find the coordinates of the foci and the vertices, the eccentricity,
Question: Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola $9 y^{2}-4 x^{2}=36$ Solution: The given equation is $9 y^{2}-4 x^{2}=36$. It can be written as $9 y^{2}-4 x^{2}=36$ Or, $\frac{y^{2}}{4}-\frac{x^{2}}{9}=1$ $\mathrm{Or}, \frac{y^{2}}{2^{2}}-\frac{x^{2}}{3^{2}}=1$ .(1) Or, $\frac{y^{2}}{2^{2}}-\frac{x^{2}}{3^{2}}=1$ (1) On comparing equation (1) with the standard equation of hyperbola i.e., $\frac{y^{2}}{a^{2}}-\frac...
Read More →If show that
Question: If $y=A e^{m x}+B e^{n x}$, show that $\frac{d^{2} y}{d x^{2}}-(m+n) \frac{d y}{d x}+m n y=0$ Solution: It is given that, $y=A e^{m x}+B e^{n x}$ Then, $\frac{d y}{d x}=A \cdot \frac{d}{d x}\left(e^{m x}\right)+B \cdot \frac{d}{d x}\left(e^{m x}\right)=A \cdot e^{m x} \cdot \frac{d}{d x}(m x)+B \cdot e^{n x} \cdot \frac{d}{d x}(n x)=A m e^{m x}+B n e^{n x}$ $\frac{d^{2} y}{d x^{2}}=\frac{d}{d x}\left(A m e^{m x}+B n e^{n x}\right)=A m \cdot \frac{d}{d x}\left(e^{m x}\right)+B n \cdot \...
Read More →Match the items given in column A and B:
Question: Match the items given in column A and B: Solution:...
Read More →Discuss the causes and effects of global warming.
Question: Discuss the causes and effects of global warming. What measures need to be taken to control global warming? Solution: Global warming is defined as an increase in the average temperature of the Earths surface. Causes of global warming:Global warming occurs as a result of the increased concentration of greenhouse gases in the atmosphere. Greenhouse gases include carbon dioxide, methane, and water vapour. These gases trap solar radiations released back by the Earth. This helps in keeping ...
Read More →List all the wastes that you generate, at home, school or during your Trips to other places,
Question: List all the wastes that you generate, at home, school or during your Trips to other places, could you very easily reduce? Which would be Difficult or rather impossible to reduce? Solution: Wastes generated at homeinclude plastic bags, paper napkin, toiletries, kitchen wastes (such as peelings of vegetables and fruits, tea leaves), domestic sewage, glass, etc. Wastes generated at schoolsinclude waste paper, plastics, vegetable and fruit peels, food wrappings, sewage etc. Wastes generat...
Read More →What are the various constituents of domestic sewage?
Question: What are the various constituents of domestic sewage? Discuss the effects of sewage discharge on a river. Solution: Domestic sewage is the waste originating from the kitchen, toilet, laundry, and other sources. It contains impurities such as suspended solid (sand, salt, clay), colloidal material (fecal matter, bacteria, plastic and cloth fiber), dissolved materials (nitrate, phosphate, calcium, sodium, ammonia), and disease-causing microbes. When organic wastes from the sewage enter th...
Read More →If, show that
Question: If $y=3 \cos (\log x)+4 \sin (\log x)$, show that $x^{2} y_{2}+x y_{1}+y=0$ Solution: It is given that, $y=3 \cos (\log x)+4 \sin (\log x)$ Then, $y_{1}=3 \cdot \frac{d}{d x}[\cos (\log x)]+4 \cdot \frac{d}{d x}[\sin (\log x)]$ $=3 \cdot\left[-\sin (\log x) \cdot \frac{d}{d x}(\log x)\right]+4 \cdot\left[\cos (\log x) \cdot \frac{d}{d x}(\log x)\right]$ $\therefore y_{1}=\frac{-3 \sin (\log x)}{x}+\frac{4 \cos (\log x)}{x}=\frac{4 \cos (\log x)-3 \sin (\log x)}{x}$ $\therefore y_{2}=\f...
Read More →Can you think of a situation where we deliberately want to make a species extinct? How would you justify it?
Question: Can you think of a situation where we deliberately want to make a species extinct? How would you justify it? Solution: Yes, there are various kinds of parasites and disease-causing microbes that we deliberately want to eradicate from the Earth. Since these micro-organisms are harmful to human beings, scientists are working hard to fight against them. Scientists have been able to eliminate small pox virus from the world through the use of vaccinations. This shows that humans deliberatel...
Read More →The species diversity of plants (22 per cent) is much less than that of animals (72 per cent).
Question: The species diversity of plants (22 per cent) is much less than that of animals (72 per cent). What could be the explanations to how animals achieved greater diversification? Solution: More than 70 percent of species recorded on the Earth are animals and only 22 percent species are plants. There is quiet a large difference in their percentage. This is because animals have adapted themselves to ensure their survival in changing environments in comparison to plants. For example, insects ...
Read More →Among the ecosystem services are control of floods and soil erosion.
Question: Among the ecosystem services are control of floods and soil erosion. How is this achieved by the biotic components of the ecosystem? Solution: The biotic components of an ecosystem include the living organisms such as plants and animals. Plants play a very important role in controlling floods and soil erosion. The roots of plants hold the soil particles together, thereby preventing the top layer of the soil to get eroded by wind or running water. The roots also make the soil porous, th...
Read More →What are sacred groves? What is their role in conservation?
Question: What are sacred groves? What is their role in conservation? Solution: Sacred groves are tracts of forest which are regenerated around places of worship. Sacred groves are found in Rajasthan, Western Ghats of Karnataka,and Maharashtra, Meghalaya, and Madhya Pradesh. Sacred groves help in the protection of many rare, threatened, and endemic species of plants and animals found in an area. The process of deforestation is strictly prohibited in this region by tribals. Hence, the sacred grov...
Read More →Find the coordinates of the foci and the vertices, the eccentricity,
Question: Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola $\frac{y^{2}}{9}-\frac{x^{2}}{27}=1$ Solution: The given equation is $\frac{y^{2}}{9}-\frac{x^{2}}{27}=1$ or $\frac{y^{2}}{3^{2}}-\frac{x^{2}}{(\sqrt{27})^{2}}=1$. On comparing this equation with the standard equation of hyperbola i.e., $\frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1$, we obtain $a=3$ and $b=\sqrt{27}$. We know that $a^{2}+b^{2}=c^{2}$. $\therefore c^{2}=...
Read More →How is biodiversity important for ecosystem functioning?
Question: How is biodiversity important for ecosystem functioning? Solution: An ecosystem with high species diversity is much more stable than an ecosystem with low species diversity. Also, high biodiversity makes the ecosystem more stable in productivity and more resistant towards disturbances such as alien species invasions and floods. If an ecosystem is rich in biodiversity, then the ecological balance would not get affected. As we all know, various trophic levels are connected through food c...
Read More →If findin terms of y alone.
Question: If $y=\cos ^{-1} x$, find $\frac{d^{2} y}{d x^{2}}$ in terms of $y$ alone. Solution: It is given that, $y=\cos ^{-1} x$ Then, $\frac{d y}{d x}=\frac{d}{d x}\left(\cos ^{-1} x\right)=\frac{-1}{\sqrt{1-x^{2}}}=-\left(1-x^{2}\right)^{\frac{-1}{2}}$ $\frac{d^{2} y}{d x^{2}}=\frac{d}{d x}\left[-\left(1-x^{2}\right)^{\frac{-1}{2}}\right]$ $=-\left(-\frac{1}{2}\right) \cdot\left(1-x^{2}\right)^{\frac{-3}{2}} \cdot \frac{d}{d x}\left(1-x^{2}\right)$ $=\frac{1}{2 \sqrt{\left(1-x^{2}\right)^{3}}...
Read More →What are the major causes of species losses in a geographical region?
Question: What are the major causes of species losses in a geographical region? Solution: Biodiversity is the variety of living forms present in various ecosystems. It includes variability among life forms from all sources including land, air, and water. Biodiversity around the world is declining at a very fast pace. The following are the major causes for the loss of biodiversity around the world. (i)Habitat loss and fragmentation:Habitats of various organisms are altered or destroyed by uncontr...
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