Evaluate the following integrals:

Question: Evaluate the following integrals: $\int \frac{\left(x^{3}+8\right)(x-1)}{x^{2}-2 x+4} d x$ Solution: Let $I=\int \frac{\left(x^{2}+8\right)(x-1)}{x^{2}-2 x+4} d x$ We know $a^{3}+b^{3}=(a+b)\left(a^{2}-a b+b^{2}\right)$ Hence, in the numerator, we can write $x^{3}+8=x^{3}+2^{3}$ $\Rightarrow x^{3}+8=(x+2)\left(x^{2}-x \times 2+2^{2}\right)$ $\Rightarrow x^{3}+8=(x+2)\left(x^{2}-2 x+4\right)$ Therefore, we can write the integral as $I=\int \frac{(x+2)\left(x^{2}-2 x+4\right)(x-1)}{x^{2}...

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Evaluate the following integrals:

Question: Evaluate the following integrals: $\int \sin ^{-1}\left(\frac{2 \tan x}{1+\tan ^{2} x}\right) d x$ Solution: Let $I=\int \sin ^{-1}\left(\frac{2 \tan x}{1+\tan ^{2} x}\right) d x$ We know $\sin 2 \theta=\frac{2 \tan \theta}{1+\tan ^{2} \theta}$ Therefore, we can write the integral as $I=\int \sin ^{-1}(\sin 2 x) d x$ $\Rightarrow I=\int 2 x d x$ $\Rightarrow I=2 \int x d x$ Recall $\int x^{n} d x=\frac{x^{n+1}}{n+1}+c$ $\Rightarrow \mathrm{I}=2 \times \frac{\mathrm{x}^{1+1}}{1+1}+\math...

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Match the following and choose

Question: Match the following and choose the correct option Options: a. A-i, B-ii, C-iii, D-iv b. A-iv, B-iii, C-i, D-ii c. A-iii, B-i, C-iv, D-ii d. A-ii, B-i, C-iv, D-iii Solution: Option (b)A-iv, B-iii, C-i, D-ii is the answer....

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Which one of the following statements

Question: Which one of the following statements is true for cockroach? a. The number of ovarioles in each ovary is ten. b. The larval stage is called caterpillar c. Anal styles are absent in females d. They are ureotelic Solution: Option (d)They are ureotelic is the answer....

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Setae help in locomotion in earthworm

Question: Setae help in locomotion in earthworm but are not uniformly present in all the segments. They are present in a. 1st segment b. Last segment c. Clitellar segment d. 20th 22nd segment Solution: Option (b)Last segmentis the answer....

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Evaluate the following integrals:

Question: Evaluate the following integrals: $\int \cot ^{-1}\left(\frac{\sin 2 x}{1-\cos 2 x}\right) d x$ Solution: Let $I=\int \cot ^{-1}\left(\frac{\sin 2 x}{1-\cos 2 x}\right) d x$ We know $\cos 2 \theta=1-2 \sin ^{2} \theta$ Hence, in the denominator, we can write $1-\cos 2 x=2 \sin ^{2} x$ In the numerator, we have $\sin 2 x=2 \sin x \cos x$ Therefore, we can write the integral as $I=\int \cot ^{-1}\left(\frac{2 \sin x \cos x}{2 \sin ^{2} x}\right) d x$ $\Rightarrow I=\int \cot ^{-1}\left(\...

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Which one of the following is not a connective

Question: Which one of the following is not a connective tissue? a. Bone b. Cartilage c. Blood d. Muscles Solution: Option (d)Muscles is the answer....

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To which one of the following categories

Question: To which one of the following categories does adipose tissue belong? a. Epithelial b. Connective c. Muscular d. Neural Solution: Option (b)Connective is the answer....

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Which one of the following types of cell is involved

Question: Which one of the following types of cell is involved in making of the inner walls of blood vessels? a. Cuboidal epithelium b. Columnar epithelium c. Squamous epithelium d. Stratified epithelium Solution: Option (c)Squamous epithelium is the answer....

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Evaluate the following integrals:

Question: Evaluate the following integrals: $\int \cos ^{-1}(\sin x) d x$ Solution: Let $I=\int \cos ^{-1}(\sin x) d x$ We know $\sin \theta=\cos \left(90^{\circ}-\theta\right)$ Therefore, we can write the integral as $I=\int \cos ^{-1}\left[\cos \left(\frac{\pi}{2}-x\right)\right] d x$ $\Rightarrow I=\int\left(\frac{\pi}{2}-x\right) d x$ $\Rightarrow I=\int \frac{\pi}{2} d x-\int x d x$ $\Rightarrow I=\frac{\pi}{2} \int d x-\int x d x$ Recall $\int x^{n} d x=\frac{x^{n+1}}{n+1}+c$ and $\int d x...

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Distinguish between the following:

Question: Distinguish between the following: a. Exarch and endarch b. Stele and vascular bundle c. Protoxylem and metaxylem d. Interfascicular cambium and intrafascicular cambium e. Open and closed vascular bundles f. Stem hair and root hair Solution: a. Exarch is a condition where the protoxylem is present towards the periphery and metaxylem is present towards the centre. Endarch is where the protoxylem is present towards the centre and metaxylem is present towards the periphery b. A stele is t...

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Each of the following terms has some anatomical significance.

Question: Each of the following terms has some anatomical significance. What do these terms mean? Explain with the help of line diagrams. a. Plasmodesmata b. Middle lamella c. Secondary wall Solution: a. Plasmodesmata A microscopic channels between two cells through the cell wall are called Plasmodesmata. b. Middle lamella It is a layer in the cell wall and is made of calcium pectate (mainly) and magnesium pectate. c. Secondary wall this is a non-extensible layer. It is made of hemicelluloses. I...

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Evaluate the following integrals:

Question: Evaluate the following integrals: $\int \tan ^{-1}\left(\frac{\sin 2 x}{1+\cos 2 x}\right) d x$ Solution: Let I $=\int \tan ^{-1}\left(\frac{\sin 2 x}{1+\cos 2 x}\right) d x$ We know $\cos 2 \theta=2 \cos ^{2} \theta-1$ Hence, in the denominator, we can write $1+\cos 2 x=2 \cos ^{2} x$ In the numerator, we have $\sin 2 x=2 \sin x \cos x$ Therefore, we can write the integral as $\mathrm{I}=\int \tan ^{-1}\left(\frac{2 \sin \mathrm{x} \cos \mathrm{x}}{2 \cos ^{2} \mathrm{x}}\right) \math...

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Is Pinus an evergreen tree?

Question: Is Pinus an evergreen tree? Comment. Solution: Pinus does not shed its leaves during a particular season and is always evergreen. Hence Pinus is considered as an evergreen tree belongs to gymnosperms. The leaves in gymnosperms can withstand in the extreme temperatures, humidity and wind....

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Deciduous plants shed their leaves during summer or in autumn.

Question: Deciduous plants shed their leaves during summer or in autumn. This process of shedding of leaves is called abscission. Apart from physiological changes what anatomical mechanism is involved in the abscission of leaves? Solution: i. Structural: In deciduous trees, an abscission zone also known as separation zone is formed at the base of the petiole. It is composed of a top layer and bottom layer. The cells in the top layer have weak cell walls and the bottom layer expand in winter and ...

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Write the function of

Question: Write the function of a. Sieve tube b. Interfascicular cambium c. Collenchyma d. Aerenchyma Solution: a. Sieve tube They help in translocation of synthesized food throughout the plant. It is present in phloem tissue. b. Interfascicular cambium It facilitates secondary growth in dicot stem and root and also helps in the formation of the vascular cambium. c. Collenchyma It provides mechanical support to young stem. Its cells have angular thickening at corners. d. Aerenchyma It provides b...

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What is the difference between lenticels

Question: What is the difference between lenticels and stomata? Solution: Lenticels are lens-shaped openings present on the trunk or stem of the tree. Stomata are bean-shaped opening present on the lower surface of leaves....

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Trunks of some of the aged tree species appear

Question: Trunks of some of the aged tree species appear to be composed of several fused trunks. Is it a physiological or anatomical abnormality? Explain in detail. Solution: Trunks of some of the aged tree species appear to be composed of several fused trunks. This is an anatomical abnormality. It is an abnormal type of secondary growth, where a regular vascular cambium or cork cambium is not formed in its normal position....

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Evaluate the following integrals:

Question: Evaluate the following integrals: $\int \frac{1}{1-\cos 2 x} d x$ Solution: Let $I=\int \frac{1}{1-\cos 2 x} d x$ We know $\cos 2 \theta=1-2 \sin ^{2} \theta$ Hence, in the denominator, we can write $1-\cos 2 x=2 \sin ^{2} x$ Therefore, we can write the integral as $I=\int \frac{1}{2 \sin ^{2} x} d x$ $\Rightarrow I=\frac{1}{2} \int \frac{1}{\sin ^{2} x} d x$ $\Rightarrow I=\frac{1}{2} \int \operatorname{cosec}^{2} x d x$ Recall $\int \operatorname{cosec}^{2} x d x=-\cot x+c$ $\Rightar...

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A transverse section of the trunk of a tree shows

Question: A transverse section of the trunk of a tree shows concentric rings which are known as growth rings. How are these rings formed? What is the significance of these rings? Solution: The growth rings in the trunk of the tree are formed by cambial ring due to secondary growth. Each ring contains earlywood and latewood. Secondary growth occurs in dicot trees due to the activity of cambium which is a meristematic tissue. The concentric growth rings are called annual rings....

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Plants require water for their survival.

Question: Plants require water for their survival. But when watered excessively, plants die. Discuss. Solution: Anything when goes excessive it will create a harmful cause. When excess water is poured in plants, the water removes the air trapped between the soil particles. They wont get enough oxygen for respiration...

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The lawn grass (Cynodondactylon) needs to be mowed

Question: The lawn grass (Cynodondactylon) needs to be mowed frequently to prevent its overgrowth. Which tissue is responsible for its rapid growth? Solution: Meristematic tissue is responsible for the rapid growth of lawn grass....

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Epidermal cells are often modified to perform

Question: Epidermal cells are often modified to perform specialized functions in plants. Name some of them and the function they perform. Solution: i. Root Hair: It increases the surface area for absorption of water and minerals from the soils ii. Trichomes: They are also known as stem hairs. They help in water loss prevention during transpiration. ii. Bulliform cells: Bulliform cells are present in grasses. They help in closing the stomata under stressful conditions and this prevents water loss...

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What are the characteristic differences found

Question: What are the characteristic differences found in the vascular tissue of gymnosperms and angiosperms? Solution: In vascular tissues of gymnosperms, vessels are absent in Xylem instead they have tracheids. Companion cells are absent in phloem instead they have albuminous cells. In vascular tissue of angiosperms, vessels are present in xylem and lack tracheids. Companion cells are present in phloem and they lack albuminous cells....

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Evaluate the following integrals:

Question: Evaluate the following integrals: $\int \frac{1}{1+\cos 2 x} d x$ Solution: Let $I=\int \frac{1}{1+\cos 2 x} d x$ We know $\cos 2 \theta=2 \cos ^{2} \theta-1$ Hence, in the denominator, we can write $1+\cos 2 x=2 \cos ^{2} x$ Therefore, we can write the integral as $I=\int \frac{1}{2 \cos ^{2} x} d x$ $\Rightarrow I=\frac{1}{2} \int \frac{1}{\cos ^{2} x} d x$ $\Rightarrow I=\frac{1}{2} \int \sec ^{2} x d x$ Recall $\int \sec ^{2} x d x=\tan x+c$ $\therefore I=\frac{1}{2} \tan x+c$ Thus...

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