Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \cot ^{3} x \operatorname{cosec}^{2} x d x$ Solution: Assume $\cot x=t$ $\Rightarrow \mathrm{d}(\cot \mathrm{x})=\mathrm{dt}$ $\Rightarrow-\operatorname{cosec}^{2} \mathrm{x} \cdot \mathrm{d} \mathrm{x}=\mathrm{dt}$ $\Rightarrow \mathrm{dt}=\frac{-\mathrm{dt}}{\csc ^{2} \mathrm{x}}$ $\therefore$ Substituting $\mathrm{t}$ and $\mathrm{dt}$ in the given equation we get $\Rightarrow \int \mathrm{t}^{3} \csc ^{2} \mathrm{x} \cdot \frac{-\mathrm{dt}}{...
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Question: Differentiate $\left(\frac{e^{x}+\sin x}{1+\log x}\right)$ Solution: To find: Differentiation of $\left(\frac{e^{x}+\sin x}{1+\log x}\right)$ Formula used: $(i)\left(\frac{u}{v}\right)^{\prime}=\frac{u^{\prime} v-u v^{\prime}}{v^{2}}$ where $v \neq 0$ (Quotient rule) (ii) $\frac{d \sin x}{d x}=\cos x$ (iii) $\frac{d \log x}{d x}=\frac{1}{x}$ (iv) $\frac{d e^{x}}{d x}=e^{x}$ Let us take $u=\left(e^{x}+\sin x\right)$ and $v=(1+\log x)$ $u^{\prime}=\frac{d u}{d x}=\frac{d\left(e^{x}+\sin ...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{e^{x}}{\left(1+e^{x}\right)^{2}} d x$ Solution: Assume $1+\mathrm{e}^{\mathrm{x}}=\mathrm{t}$ $\Rightarrow \mathrm{d}\left(1+\mathrm{e}^{\mathrm{x}}\right)=\mathrm{dt}$ $\Rightarrow \mathrm{e}^{\mathrm{x}} \mathrm{dx}=\mathrm{dt}$ $\therefore$ Substituting $t$ and $d t$ in given equation we get $\Rightarrow \int \frac{1}{t^{2}} d t$ $\Rightarrow \int t^{-2} \cdot d t$ $\Rightarrow \frac{-1}{t}+c$ But $1+e^{x}=t$ $\Rightarrow \frac{-1}{1+e^{...
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Question: Differentiate $\left(\frac{\sec x-\tan x}{\sec x+\tan x}\right)$ Solution: To find: Differentiation of $\left(\frac{\sec x-\tan x}{\sec x+\tan x}\right)$ Formula used: (i) $\left(\frac{u}{v}\right)^{\prime}=\frac{u^{\prime} v-u v^{\prime}}{v^{2}}$ where $v \neq 0$ (Quotient rule) (ii) $\frac{d \sec x}{d x}=\sec x \tan x$ (iii) $\frac{d \tan x}{d x}=\sec ^{2} x$ Let us take $u=(\sec x-\tan x)$ and $v=(\sec x+\tan x)$ $u^{\prime}=\frac{d u}{d x}=\frac{d(\sec x-\tan x)}{d x}=\left(\sec x ...
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Question: Differentiate $\left(\frac{\sin x-\cos x}{\sin x+\cos x}\right)$ Solution: To find: Differentiation of $\left(\frac{\sin x-\cos x}{\sin x+\cos x}\right)$ Formula used: (i) $\left(\frac{u}{v}\right)^{\prime}=\frac{u^{\prime} v-u v^{\prime}}{v^{2}}$ where $v \neq 0$ (Quotient rule) (ii) $\frac{d \sin x}{d x}=\cos x$ (iii) $\frac{d \cos x}{d x}=-\sin x$ Let us take $u=(\sin x-\cos x)$ and $v=(\sin x+\cos x)$ $u^{\prime}=\frac{d u}{d x}=\frac{d(\sin x-\cos x)}{d x}=(\cos x+\sin x)$ $v^{\pr...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \sqrt[3]{\cos ^{2} x} \sin x d x$ Solution: Assume $\cos x=t$ $\Rightarrow \mathrm{d}(\cos x)=\mathrm{dt}$ $\Rightarrow-\sin x \mathrm{~d} x=\mathrm{dt}$ $\Rightarrow \mathrm{dx}=\frac{-\mathrm{dt}}{\sin \mathrm{x}}$ $\therefore$ Substituting $t$ and $d t$ in the given equation we get $\Rightarrow \int \sqrt[3]{t^{2}} \sin x \cdot \frac{d t}{\sin x}$ $\Rightarrow \int t^{3 / 2} \cdot d t$ $\Rightarrow \frac{2 t^{\frac{3}{2}}}{3}+c$ But $\cos x=t$...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \sqrt{1+e^{x}} e^{x} d x$ Solution: Assume $1+e^{x}=t$ $\Rightarrow \mathrm{d}\left(1+\mathrm{e}^{\mathrm{x}}\right)=\mathrm{dt}$ $\Rightarrow \mathrm{e}^{\mathrm{x}} \mathrm{dx}=\mathrm{dt}$ $\therefore$ Substituting $\mathrm{t}$ and dt in given equation we get $\Rightarrow \int \sqrt{t} \cdot d t$ $\Rightarrow \int t^{1 / 2} \cdot d t$ $\Rightarrow \frac{2 t^{\frac{3}{2}}}{3}+c$ But $1+e^{x}=t$ $\Rightarrow \frac{2\left(1+\mathrm{e}^{\mathrm{x}...
Read More →Growth is one of the characteristics of all living organisms?
Question: Growth is one of the characteristics of all living organisms? Do unicellular organisms also grow? If so, what are the parameters? Solution: Yes, unicellular organisms also grow. Characteristics of growth increase in the mass of the cell and an increase in cell number. It grows by dividing its cells or undergoing cell division by an increase in cell mass as well as an increase in several cells....
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Question: Differentiate $\left(\frac{1-\cos x}{1+\cos x}\right)$ Solution: To find: Differentiation of $\left(\frac{1-\cos x}{1+\cos x}\right)$ Formula used: (i) $\left(\frac{u}{v}\right)^{\prime}=\frac{u^{\prime} v-u v^{\prime}}{v^{2}}$ where $v \neq 0$ (Quotient rule) (ii) $\frac{d \cos x}{d x}=-\sin x$ Let us take $u=(1-\cos x)$ and $v=(1+\cos x)$ $u^{\prime}=\frac{d u}{d x}=\frac{d(1-\cos x)}{d x}=\sin x$ $v^{\prime}=\frac{d v}{d x}=\frac{d(1+\cos x)}{d x}=-\sin x$ Putting the above obtained...
Read More →In the figure of sigmoid growth curve given below,
Question: In the figure of sigmoid growth curve given below, label segments 1, 2 and 3. Solution: 1: Lag phase 2: Log phase 3: Stationary phase...
Read More →Light plays an important role in the life of all organisms.
Question: Light plays an important role in the life of all organisms. Name any three physiological processes in plants which are affected by light. Solution: (a) Photosynthesis: The process of photosynthesis occurs when green plants use light energy to convert carbon dioxide into carbohydrates. (b) Photoperiodism: In photoperiodism, flowering is regulated in response to the day length exposure time or photoperiod....
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{(1+\sqrt{x})^{2}}{\sqrt{x}} d x$ Solution: Assume $1+\sqrt{x}=t$ $\Rightarrow d(1+\sqrt{x})=d t$ $\Rightarrow \frac{1}{2 \sqrt{x}} \mathrm{dx}=\mathrm{dt}$ $\Rightarrow \frac{1}{\sqrt{x}} \mathrm{dx}=2 \mathrm{dt}$ $\therefore$ Substituting $t$ and $d t$ in the given equation we get $\Rightarrow \int 2 t^{2} \cdot d t$ $\Rightarrow 2 \int t^{2} \cdot d t$ $\Rightarrow \frac{2 t^{3}}{3}+c$ But $1+\sqrt{x}=t$ $\Rightarrow \frac{2(1+\sqrt{x})^...
Read More →In botanical gardens and tea gardens,
Question: In botanical gardens and tea gardens, gardeners trim the plants regularly so that they remain bushy. Does this practice have any scientific explanation? Solution: They are removing the shoot tips that result in the growth of lateral buds. This lateral bud growth is suppressed by the apical buds if the tip is not removed....
Read More →Where are the following hormones synthesized in plants?
Question: Where are the following hormones synthesized in plants? a. IAA b. Gibberellins c. Cytokinins Solution: a. IAA: in the tips of the shoot and apical buds b. Gibberellins: in young leaves and tips of roots c. Cytokinins: meristematic zones of the root...
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Question: Differentiate $\left(\frac{1+\sin x}{1-\sin x}\right)$ Solution: To find: Differentiation of $\left(\frac{1+\sin x}{1-\sin x}\right)$ Formula used: (i) $\left(\frac{u}{v}\right)^{\prime}=\frac{u^{\prime} v-u v^{\prime}}{v^{2}}$ where $v \neq 0$ (Quotient rule) (ii) $\frac{d \sin x}{d x}=\cos x$ Let us take $u=(1+\sin x)$ and $v=(1-\sin x)$ $u^{\prime}=\frac{d u}{d x}=\frac{d(1+\sin x)}{d x}=\cos x$ $v^{\prime}=\frac{d v}{d x}=\frac{d(1-\sin x)}{d x}=-\cos x$ Putting the above obtained ...
Read More →A farmer grows cucumber plants in his field.
Question: A farmer grows cucumber plants in his field. He wants to increase the number of female flowers in them. Which plant growth regulator can be applied to achieve this? Solution: The plant growth regulator that can be applied to achieve this is ethylene (C2H4)....
Read More →Classify the following plants into Long-Day Plants (LDP),
Question: Classify the following plants into Long-Day Plants (LDP), Short Day Plants (SDP) and Day Neutral Plants (DNP) Xanthium, Henbane (Hyoscyamusniger), Spinach, Rice, Strawberry, Bryophyllum, Sunflower, Tomato, Maize. Solution: Henbane: Long Day Plant (LDP) Spinach: Long Day Plant (LDP) Rice: Short Day Plant (SDP) Strawberry: Short Day Plant (SDP) Bryophyllum: Long Short Day plant (LSDP) Sunflower: Day Neutral Plant (DNP) Tomato: Day Neutral Plant (DNP) Maize: Day Neutral Plant (DNP)...
Read More →Gibberellins promote the formation of ______ flowers
Question: Gibberellins promote the formation of ______ flowers on genetically ____ plants in Cannabis whereas ethylene promotes formation of _______ flowers on genetically ______ plants. Solution: Gibberellins promote the formation of male flowers on genetically female plants in Cannabis whereas ethylene promotes the formation of female flowers on genetically male plants....
Read More →Gibberellins were first discovered in Japan when rice plants were suffering
Question: Gibberellins were first discovered in Japan when rice plants were suffering from balance (the foolish seedling disease) caused by a fungus Gibberellafujikuroi. a. Give two functions of this phytohormone. b. Which property of Gibberellin caused foolish seedling disease in rice? Solution: (a) Gibberellins cause fruits to elongate and improve in its shape. It is also responsible to delay senescence. Spraying sugarcane crop with gibberellins increases the length of the stem, thus increasin...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int \frac{\log \left(1+\frac{1}{x}\right)}{x(1+x)} d x$ Solution: Assume $\log \left(1+\frac{1}{\mathrm{x}}\right)=\mathrm{t}$ $\Rightarrow \mathrm{d}\left(\log \left(1+\frac{1}{\mathrm{x}}\right)\right)=\mathrm{dt}$ $\Rightarrow \frac{1}{1+\frac{1}{\mathrm{x}}} \times \frac{-1}{\mathrm{x}^{2}} \mathrm{dx}=\mathrm{dt}$ $\Rightarrow \frac{\mathrm{x}}{\mathrm{x}+1} \times \frac{-1}{\mathrm{x}^{2}} \mathrm{~d} \mathrm{x}=\mathrm{dt}$ $\Rightarrow \frac{...
Read More →A primary root grows from 5 cm to 19 cm in a week.
Question: A primary root grows from 5 cm to 19 cm in a week. Calculate the growth rate and relative growth rate over the period. Solution: L1 = L0 + rt 19 5 = r x 1 14cm per week Relative growth rate = Change/Initial*100 19-5/5 x 100 = 14/5 x 100 =280%...
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Question: Differentiate $\frac{\sin x}{(1+\cos x)}$ Solution: To find: Differentiation of $\frac{\sin x}{(1+\cos x)}$ Formula used: (i) $\left(\frac{u}{v}\right)^{\prime}=\frac{u^{\prime} v-u v^{\prime}}{v^{2}}$ where $v \neq 0$ (Quotient rule) (ii) $\frac{d \cos x}{d x}=-\sin x$ (iii) $\frac{d \sin x}{d x}=\cos x$ Let us take $u=(\sin x)$ and $v=(1+\cos x)$ $u^{\prime}=\frac{d u}{d x}=\frac{d(\sin x)}{d x}=\cos x$ $v^{\prime}=\frac{d v}{d x}=\frac{d(1+\cos x)}{d x}=-\sin x$ Putting the above ob...
Read More →Plant growth substances (PGS) have innumerable
Question: Plant growth substances (PGS) have innumerable practical applications. Name the PGS you should use to a. Increase the yield of sugar cane. b. Promote lateral shoot growth. c. Cause sprouting of potato tuber. d. Inhibit seed germination. Solution: a. Increase the yield of sugar cane: Gibberellins b. Promote lateral shoot growth: Cytokinins c. Cause sprouting of potato tuber: Ethylene d. Inhibit seed germination: Abscisic acid...
Read More →Fill in the places with appropriate word/ words.
Question: Fill in the places with appropriate word/ words. a. A phase of growth which is maximum and fastest is __________. b. Apical dominance as expressed in dicotyledonous plants is due to the presence of more ________ in the apical bud than in the lateral ones. c. In addition to auxin, a ________ must be supplied to the culture medium to obtain a good callus in plant tissue culture. d. ________ of vegetative plants are the sites of photoperiodic perception Solution: (a) Log phase or exponent...
Read More →The photoperiod in plants is perceived at
Question: The photoperiod in plants is perceived at a. Meristem b. Flower c. Floral buds d. Leaves Solution: Option (d)Leaves is the answer....
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