Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \sqrt{\frac{1-\sin 2 x}{1+\sin 2 x}} d x$ Solution: We know $\cos ^{2} x+\sin ^{2} x=1$ Also, $2 \sin x \cos x=\sin 2 x$ $1+\sin 2 x=\cos ^{2} x+\sin ^{2} x+2 \sin x \cos x=(\cos x+\sin x)^{2}$ $1-\sin 2 x=\cos ^{2} x+\sin ^{2} x-2 \sin x \cos x=(\cos x-\sin x)^{2}$ $\therefore$ The equation becomes $\Rightarrow \int \sqrt{\frac{(\cos x-\sin x)^{2}}{(\cos x+\sin x)^{2}} d x}$ $\Rightarrow \int \frac{(\cos x-\sin x)}{(\cos x+\sin x)} d x$ Assume $...
Read More →Find the derivation of each of the following from the first principle:
Question: Find the derivation of each of the following from the first principle: $\tan ^{2} x$ Solution: Let $f(x)=\tan ^{2} x$ We need to find the derivative of f(x) i.e. f(x) We know that, $\mathrm{f}^{\prime}(\mathrm{x})=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{f}(\mathrm{x}+\mathrm{h})-\mathrm{f}(\mathrm{x})}{\mathrm{h}}$ (i) $f(x)=\tan ^{2} x$ $f(x+h)=\tan ^{2}(x+h)$ Putting values in (i), we get $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{\tan ^{2}(x+h)-\tan ^{2} x}{h}$ $=\lim _{h \...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\cos x}{\cos (x-a)} d x$ Solution: Add and subtract a from $x$ in the numerator $\therefore$ The equation becomes $\Rightarrow \int \frac{\cos (x-a+a)}{\cos (x-a)}$ Numerator is of the form $\cos (A+B)=\cos A \cos B-\sin A \sin B$ Where $A=x-a ; B=a$ $\Rightarrow \int \frac{\cos (x-a) \cos a}{\cos (x-a)} d x-\int \frac{\sin (x-a) \sin a}{\cos (x-a)} d x$ $\Rightarrow \cos a \int d x-\sin a \int \tan (x-a) d x$ As $\int \tan x=\ln |\sec x|+c...
Read More →Sugar crystals do not dissolve
Question: Sugar crystals do not dissolve easily in ice-cold water. Explain Solution: The dissolution or diffusion of the solid crystal of sugar does not occur since the number of striking water molecules is less when water is ice cold....
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: Solution: Convert $\tan x$ in form of $\sin x$ and $\cos x$. $\Rightarrow \tan x=\frac{\sin x}{\cos x}$ $\therefore$ The equation now becomes $\Rightarrow \int \frac{1+\frac{\sin x}{\cos x}}{1-\frac{\sin x}{\cos x}} \mathrm{dx}$ $\Rightarrow \int \frac{\frac{\cos x+\sin x}{\cos x}}{\frac{\cos x}{x-\sin x}} \mathrm{dx}$ $\Rightarrow \int \frac{\cos x+\sin x}{\cos x-\sin x} \mathrm{dx}$ Let $\cos x-\sin x=t$ $\therefore \frac{\mathrm{d}(\cos \mathrm{x}-\...
Read More →When a freshly collected Spirogyra filament is kept in a 10%
Question: When a freshly collected Spirogyra filament is kept in a 10% potassium nitrate solution, it is observed that the protoplasm shrinks in size: a. What is this phenomenon called? b. What will happen if the filament is replaced in distilled water? Solution: a. This phenomenon is called exosmosis b. If the filament is replaced in distilled water, then the protoplasm again becomes turgid as water flows in as the shrunken protoplasm swells up or gets deplasmolysed....
Read More →Find the derivation of each of the following from the first principle:
Question: Find the derivation of each of the following from the first principle: $\sqrt{\sec x}$ Solution: Let $f(x)=\sqrt{\sec x}$ We need to find the derivative of f(x) i.e. f(x) We know that, $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ (i) $f(x)=\sqrt{\sec x}$ $\mathrm{f}(\mathrm{x}+\mathrm{h})=\sqrt{\sec (\mathrm{x}+\mathrm{h})}$ Putting values in (i), we get $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{\sqrt{\sec (x+h)}-\sqrt{\sec x}}{h}$ Now rationalizing the numerator by...
Read More →Keep some freshly cut flowers in a solution of food colour.
Question: Keep some freshly cut flowers in a solution of food colour. Wait for some time for the dye to rise in the flower, when the stem of the flower is held up in light, coloured strands can be seen inside. Can this experiment demonstrate which tissue is conducting water up the stem? Solution: This experiment can demonstrate which tissue is conducting water up the stem. Coloured strands are indicative that the food colour has been conducted upwards through a conducting vessel which is the xyl...
Read More →Will the ascent of sap be possible without
Question: Will the ascent of sap be possible without the cohesion and adhesion of the water molecules? Explain. Solution: No. it is not possible because these are responsible for the same. The continuous water column in the xylem vessels for the sap ascent is maintained by the adhesive and the cohesive forces of water. Absence of which would have led to cavitation of xylem vessels or breakage of the water column or the sap movement upwards....
Read More →How can plants be grown under limited
Question: How can plants be grown under limited water supply without compromising on metabolic activities? Solution: Plants usually shrink when the rate of transpiration exceeds the rate of absorption of water by roots. So if we use anti-transpirants like Abscisic acid then it checks the rate of transpiration....
Read More →Various types of transport mechanisms
Question: Various types of transport mechanisms are needed to fulfil the mineral requirements of a plant. Why are they not fulfilled by diffusion alone? Solution: Plants also require mineral nutrients and charged ions like potassium, sodium, etc. which cannot be transported via simple diffusion because the cell membrane which is selectively permeable does not allow large, polar and charged particles through....
Read More →In a girdled plant,
Question: In a girdled plant, when water is supplied to the leaves above the girdle, leaves may remain green for some time then wilt and ultimately die. What does it indicate? Solution: In a girdled plant, when water is supplied to the leaves above the girdle, leaves may remain green for some time because leaves have the capability of synthesizing their food by photosynthesis using water and CO2 from the air....
Read More →Cut pieces of beetroot do not leave
Question: Cut pieces of beetroot do not leave colour in cold water but do so in hot water. Explain. Solution: Beetroot loses colour only when it is placed in hot water as the plasma membrane of the cells denatures at high temperature...
Read More →How are the intracellular levels of ions maintained
Question: How are the intracellular levels of ions maintained higher than extracellular levels in animal cells? Solution: Animal cells have ATP powered pumps like the Na+- K+ pump where 3Na+ is sent out in exchange of 2 K+ ions thus increasing the level of K+ ions in the intracellular region. This is how via the pumps, ion concentration is maintained higher intracellularly than in the extracellular region....
Read More →Water is indispensable for life.
Question: Water is indispensable for life. What properties of water make it useful for all biological processes on the earth? (a) Water is a polar solvent due to which it is regarded as the universal solvent. (b) Water has a high specific heat capacity. (c) Water has low viscosity and high surface tension. (d) The density of water decreases below 4◦C. (e) Capillarity of water Solution: (a) All the essential substances in living organisms are solutes dissolved in water and transported from one pl...
Read More →Do different species of plants growing
Question: Do different species of plants growing in the same area show the same rate of transpiration at a particular time? Justify your answer. Solution: The rate of transpiration also depends on internal factors like the morphological characteristics of a plant. For eg: Leaf area, the thickness of cuticle....
Read More →From your knowledge of physiology
Question: From your knowledge of physiology can you think of some method of increasing the life of cut plants in a vase? Solution: Using a plant growth hormone called cytokinins that help delay senescence or ageing of plant parts. Placing the cut stem immersed in water in the vase to not allow water cavitation to occur by air....
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\sin (\mathrm{x}-\alpha)}{\sin (\mathrm{x}+\alpha)} \mathrm{d} \mathrm{x}$ Solution: Add and subtract $\alpha$ in the numerator $\Rightarrow \int \frac{\sin (x-\alpha+\alpha-\alpha)}{\sin (x+\alpha)} d x$ $\Rightarrow \int \frac{\sin (x+\alpha-2 \alpha)}{\sin (x+\alpha)}$ Numerator is of the form $\sin (A-B)=\sin A \cos B-\cos A \sin B$ Where $A=x+\alpha ; B=2 \alpha$ $\Rightarrow \int \frac{\sin (x+\alpha) \cos (2 \alpha)-\cos (x+\alpha) \...
Read More →If one wants to find minerals and in the form,
Question: If one wants to find minerals and in the form, they are mobilised in the plant, how will an analysis of the exudate help? Solution: Exudates analysis would help to find minerals and in the form that they are mobilised in the plant because exudates are often planted sap which contains xylem sap as well as phloem sap....
Read More →Minerals absorbed by the roots travel up the xylem.
Question: Minerals absorbed by the roots travel up the xylem. How do they reach the parts where they are needed most? Do all the parts of the plant get the same amount of the minerals? Solution: Minerals are taken by the roots from the soil and are transported to various parts of the plants wherever needed via the xylem vessels. The mineral requirement is not the same for all plants and neither for all plant parts so all the parts of the plants do not get the same amount of minerals....
Read More →Correct the statements
Question: Correct the statements a. Cells shrink in hypotonic solutions and swell in hypertonic solutions. b. Imbibition is a special type of diffusion when water is absorbed by living cells. c. Most of the water flow in the roots occurs via the symplast. Solution: (a) Cells shrink in a hypertonic solution and swell in hypotonic solutions. (b) Imbibition is a special type of diffusion when water is absorbed by dead or living cells. (c) Most of the water flow in roots occurs via the apoplast path...
Read More →Identify a type of molecular movement
Question: Identify a type of molecular movement which is highly selective and requires special membrane proteins, but does not require energy Solution: The type of molecular movement which is highly selective and requires special membrane proteins but does not require energy is passive diffusion....
Read More →A gardener forgot to water a potted plant
Question: A gardener forgot to water a potted plant for a day during summer, what will happen to the plant? Do you think it is reversible? If yes, how? Solution: The rate of transpiration is more due to the higher temperature than the absorption of water so this will lead to wilting or losing the plants turgidity....
Read More →Describe briefly the three physical properties of
Question: Describe briefly the three physical properties of water which helps in the ascent of water in the xylem. Solution: (i) Adhesion is the attraction between water molecules and other molecules (ii) Cohesion is the force due to which molecules of water stay linked together due to hydrogen bonding in between them. (iii)Surface tension is the property of the surface of a liquid to resist an external force applied owing to the cohesive nature of the molecules of the liquid. Capillary action i...
Read More →Transpiration is a necessary evil in plants.
Question: Transpiration is a necessary evil in plants. Explain. Solution: Transpiration is the process by which plants lose water in the form of water vapour from aerial parts of the plant. When there is a huge amount of transpiration that takes place, it leads to a huge amount of water loss thus causing wilting of the plant....
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