Evaluate the following integrals:
Question: Evaluate the following integrals: $\int\left\{\sqrt{x}\left(a x^{2}+b x+c\right)\right\} d x$ Solution: $\int\left\{\sqrt{x}\left(a x^{2}+b x+c\right)\right\} d x$ $\Rightarrow \int\left(\sqrt{x a x}^{2}+\sqrt{x b x}+\sqrt{x c}\right) d x$ By Splitting, we get, $\Rightarrow \mathrm{a} \int \mathrm{x}^{2} \times \mathrm{x}^{\frac{1}{2}} \mathrm{dx}+\mathrm{b} \int \mathrm{x}^{1} \times \mathrm{x}^{\frac{1}{2}} \mathrm{dx}+\mathrm{c} \int \mathrm{x}^{1 / 2} \mathrm{dx}$ $\Rightarrow \mat...
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{h \rightarrow 0} \frac{(a+h)^{2} \sin (a+h)-a^{2} \sin a}{h}$ Solution: $=\lim _{h \rightarrow 0} \frac{(a+h)^{2} \sin (a+h)-a^{2} \sin a}{h}$ $=\lim _{h \rightarrow 0} \frac{a^{2}(\sin (a+h)-\sin a)+2 a h \sin (a+h)+h^{2} \sin (a+h)}{h}$ $=2 a \sin a+0+\lim _{h \rightarrow 0} \frac{a^{2} \times 2 \times \cos \left(a+\frac{h}{2}\right) \times \sin h}{h}$ $=2 a \sin a+2 a^{2} \cos a$ $=2 a^{2} \cos a+2 a \sin a$ $\therefore \lim _{h \rightarrow 0} ...
Read More →Evaluate the following limits
Question: Evaluate the following limits. $\lim _{x \rightarrow 0} \frac{(\sqrt{1+2 x}-\sqrt{1-2 x}}{\sin x}$ Solution: $=\lim _{x \rightarrow 0} \frac{(\sqrt{1+2 x}-\sqrt{1-2 x})}{\sin x}$ $=\lim _{x \rightarrow 0} \frac{(\sqrt{1+2 x}-\sqrt{1-2 x})}{\sin x} \times \frac{\sqrt{1+2 x}+\sqrt{1-2 x}}{(\sqrt{1+2 x}+\sqrt{1-2 x})}$ $=\lim _{x \rightarrow 0} \frac{1+2 x-1+2 x}{\sin x} \times \frac{1}{\sqrt{1+2 x}+\sqrt{1-2 x}}$ $=4 \times \lim _{x \rightarrow 0} \frac{x}{\sin x} \times \frac{1}{\sqrt{1...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int\left(2^{x}+\frac{5}{x}-\frac{1}{x^{1 / 3}}\right) d x$ Solution: $\int\left(2^{x}+\frac{5}{x}-\frac{1}{x^{1 / 3}}\right) d x$ By Splitting them, we get, $\Rightarrow \int 2^{\mathrm{x}} \mathrm{dx}+\int\left(\frac{5}{\mathrm{x}}\right) \mathrm{dx}-\int \frac{1}{\mathrm{x}^{1 / 3}} \mathrm{dx}$ By using the formula, $\int a^{x} d x=\frac{a^{x}}{\log a}$ $\Rightarrow \frac{2^{x}}{\log 2}+5 \int\left(\frac{1}{x}\right) d x-\int x^{-1 / 3} d x$ By us...
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow \frac{\pi}{2}}\left(\frac{\pi}{2}-x\right) \tan x$ Solution: $=\lim _{x \rightarrow \frac{\pi}{2}}\left(\frac{\pi}{2}-x\right) \tan x$ $=-1 \times \lim _{y \rightarrow 0} y \tan \left(y+\frac{\pi}{2}\right)\left[x-\frac{\pi}{2}=y\right]$ $=-1 \times \lim _{y \rightarrow 0} y \cot y \times-1$ $=\lim _{y \rightarrow 0} \frac{y}{\tan y}$ $=1$ $\therefore \lim _{x \rightarrow \frac{\pi}{2}}\left(\frac{\pi}{2}-x\right) \tan x=1$...
Read More →Evaluate the following integrals:
Question: Evaluate the following integrals: $\int(3 x \sqrt{x}+4 \sqrt{x}+5) d x$ Solution: $\int(3 x \sqrt{x}+4 \sqrt{x}+5) d x$ By Splitting, we get, $\Rightarrow \int((3 \mathrm{x} \sqrt{\mathrm{x}}) \mathrm{dx}+(4 \sqrt{\mathrm{x}}) \mathrm{dx}+5 \mathrm{dx})$ $\Rightarrow \int 3 \mathrm{x} \sqrt{\mathrm{x}} \mathrm{dx}+\int 4 \sqrt{\mathrm{x}} \mathrm{dx}+\int 5 \mathrm{dx}$ $\Rightarrow \int 3 \mathrm{x}^{\frac{3}{2}} \mathrm{dx}+\int 4 \mathrm{x}^{\left(\frac{1}{2}\right)} \mathrm{d}+\int...
Read More →Evaluate the following limits:
Question: Evaluate the following limits:. $\lim _{x \rightarrow 0} \frac{(1-\cos 2 x)}{(\cos 2 x-\cos 8 x)}$ Solution: $=\lim _{x \rightarrow 0} \frac{1-\cos 2 x}{(\cos 2 x-\cos 8 x)}$ $=\lim _{x \rightarrow 0} \frac{2 \times \sin x \times \sin x}{2 \times \sin 3 x \times \sin 5 x} \times \frac{5 x \times 3 x}{x \times x} \times \frac{1}{15}$ $=\frac{1}{15} \times 1 \times 1 \times 1 \times 1\left[\because \lim _{x \rightarrow 0} \frac{\sin \theta}{\theta}=1\right]$ $=\frac{1}{15}$ $\lim _{x \ri...
Read More →A Prothallus is
Question: A Prothallus is a. A structure in pteridophytes formed before the thallus develops b. A sporophytic free-living structure formed in pteridophytes c. A gametophyte free-living structure formed in pteridophytes d. A primitive structure formed after fertilization in pteridophytes Solution: Option (c)A gametophyte free-living structure formed in pteridophytes d. A primitive structure formed after fertilization in pteridophytesis the answer....
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow 0} \frac{[\sin (2+x)-\sin (2-x)]}{x}$ Solution: $=\lim _{x \rightarrow 0} \frac{[\sin (2+x)-\sin (2-x)]}{x}$ $=\lim _{x \rightarrow 0} \frac{\left[2 \times \cos \frac{(2+x+2-x)}{2} \times \sin \frac{(2+x-2+x)}{2}\right]}{x}$ $=\lim _{x \rightarrow 0} \frac{(2 \times \cos 2 \times \sin x)}{x}$ $=2 \cos 2 \lim _{x \rightarrow 0} \frac{\sin x}{x}$ $=2 \cos 2$ $\lim _{x \rightarrow 0} \frac{[\sin (2+x)-\sin (2-x)]}{x}=2 \cos 2$...
Read More →A plant shows thallus level of organization.
Question: A plant shows thallus level of organization. It shows rhizoids and is haploid. It needs water to complete its life cycle because the male gametes are motile. Identify the group to which it belongs to a. Pteridophytes b. Gymnosperms c. Monocots d. Bryophytes Solution: Option (d)Bryophytes is the answer....
Read More →Holdfast, stipe and frond constitute
Question: Holdfast, stipe and frond constitute the plant body in case of a. Rhodophyceae b. Chlorophyceae c. Phaeophyceae d. All of the above Solution: Option (c)Phaeophyceae is the answer....
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow 0} \frac{(\sin 3 x+\sin 5 x)}{(\sin 6 x-\sin 4 x)}$ Solution: $=\lim _{x \rightarrow 0} \frac{\left(2 \times \sin \frac{3 x+5 x}{2} \times \cos \frac{3 x-5 x}{2}\right)}{\left(2 \times \cos \frac{6 x+4 x}{2} \sin \frac{6 x-4 x}{2}\right)}$ $=\lim _{x \rightarrow 0} \frac{\sin 4 x \cos x}{\cos 5 x \sin x}$ $=\lim _{x \rightarrow 0} \frac{\sin 4 x}{\cos 5 x \times \frac{\sin x}{\cos x}} \times \frac{4 x}{4 x}$ $=4 \times \lim _{x \righ...
Read More →The fusion of two motile gametes
Question: The fusion of two motile gametes which are dissimilar in size is termed as a. Oogamy b. Isogamy c. Anisogamy d. Zoogamy Solution: Option (c)Anisogamy is the answer....
Read More →Cyanobacteria are classified under
Question: Cyanobacteria are classified under a. Protista b. Plantae c. Monera d. Algae Solution: Option (c)Monera is the answer....
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow 0} \frac{(\cos 3 x-\cos 5 x)}{x^{2}}$ Solution: $=\lim _{x \rightarrow 0} \frac{(1-\cos 5 x-(1-\cos 3 x))}{x^{2}}$ $=\lim _{x \rightarrow 0}\left(\frac{1-\cos 5 x}{x^{2}}-\frac{1-\cos 3 x}{x^{2}}\right)$ $=\left(\lim _{x \rightarrow 0} \frac{1-\cos 5 x}{x^{2}} \times \frac{25}{25}\right)-\left(\lim _{x \rightarrow 0} \frac{1-\cos 3 x}{x^{2}} \times \frac{9}{9}\right)$ $=\frac{25}{2}-\frac{9}{2}\left[\because \lim _{x \rightarrow 0} \...
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow 0} \frac{\sin 5 x-\sin 3 x}{\sin x}$ Solution: $=\lim _{x \rightarrow 0} \frac{\left(2 \sin \frac{5 x-3 x}{2} \cos \frac{5 x+3 x}{2}\right)}{\sin x}[$ Applying $\sin C-\sin D$ $\left.=2 \sin \frac{C-D}{2} \cos \frac{C+D}{2}\right]$ $=\lim _{x \rightarrow 0} 2 \cos 4 x$ $=2 \times 1$ $\therefore \lim _{x \rightarrow 0} \frac{\sin 5 x-\sin 3 x}{\sin x}=2$...
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow a} \frac{(\sin x-\sin a)}{(\sqrt{x}-\sqrt{a})}$ Solution: $=\lim _{x \rightarrow a} \frac{(\sin x-\sin a)}{(\sqrt{x}-\sqrt{a})}$ $=\lim _{x \rightarrow a} \frac{(\sin x-\sin a)}{(\sqrt{x}-\sqrt{a})} \times \frac{(\sqrt{x}+\sqrt{a})}{(\sqrt{x}+\sqrt{a})}[$ Multiply and divide by $\sqrt{x}-\sqrt{a}]$ $=\lim _{x \rightarrow a} \frac{(\sin x-\sin a) \times(\sqrt{x}+\sqrt{a})}{(x-a)}$ $=\cos a \times \lim _{x \rightarrow a}(\sqrt{x}+\sqrt...
Read More →Biological classification is a dynamic
Question: Biological classification is a dynamic and ever-evolving phenomenon which keeps changing with our understanding of life forms. Justify the statement taking any two examples. Solution: C. Linnaeus classified the organisms based on morphological characters like Plantae and Animalia. But it was not sufficient as there were certain organisms claimed by both botanist and zoologist. Later R.H. Whittaker proposed five kingdom classification system. Euglena and slime moulds were given place un...
Read More →‘Peat’ is an important source of domestic fuel in several countries.
Question: Peat is an important source of domestic fuel in several countries. How is peat formed in nature? Solution: Peat is formed from the wetland vegetation especially from the Sphagnum (bog moss). Sphagnum likes to grow in acidic nature in the bog, due to the acidity the complete degradation of the cell could not take place due to the presence of less number of bacteria. Sphagnum uptakes water from the dead hyaline pores and holds a great amount of water. Due to the fossilization of Sphagnum...
Read More →Make a list of algae and fungi that have commercial
Question: Make a list of algae and fungi that have commercial value as a source of food, chemicals, medicines and fodder. Solution: 1. Importance of algae Algae is used as food in many species like Porphyra, laminaria etc. Chlorella and spirulina are unicellular algae rich in protein Some kind of marine algae produce a large number of hydrocolloids and used commercially. Example like agar obtained from delirium and gracilaria are used to grow microbes in ice-creams. 2. Importance of fungi There ...
Read More →Apart from chlorophyll,
Question: Apart from chlorophyll, algae have several other pigments in their chloroplast. What pigments are found in blue-green, red and brown algae that are responsible for their characteristic colours? Solution: Blue-green algae phycocyanin and r-phycoerythrin pigments are present beside chlorophyll a Brown algae possess chlorophyll a, c, carotenoids and xanthophylls. They are olive green in colour. Red algae possess chlorophyll a, d and phycoerythrin in their body...
Read More →Algae are known to reproduce asexually by a variety
Question: Algae are known to reproduce asexually by a variety of spores under different environmental conditions. Name these spores and the conditions under which they are produced. Solution: Zoospores are thin-walled spores that are motile and flagellated. They are produced under favourable conditions Aplanospores are thin-walled non-motile spores and produced under unfavourable condition. Hypnospores are thick-walled spores and produced under unfavourable condition. Akinites are from the veget...
Read More →Fungi are cosmopolitan.
Question: Fungi are cosmopolitan. Write the role of fungi in your daily life. Solution: Used in the brewery and baking industry: Saccharomyces cerivesea is used. Under anaerobic conditions process of fermentation takes place which releases ethanol and CO2. The release of CO2 helps the dough to rise. Fungi are used as biological control agents: Soil inhabiting fungus Trichoderma kills Pythium fungus known to cause root rot. LONG ANSWER TYPE QUESTIONS...
Read More →Evaluate the following limits:
Question: Evaluate the following limits: $\lim _{x \rightarrow a} \frac{(\sin x-\sin a)}{(x-a)}$ Solution: $=\lim _{x \rightarrow a} \frac{(\sin x-\sin a)}{(x-a)}$ $=\lim _{x \rightarrow a} \frac{\left(2 \times \cos \frac{x+a}{2} \sin \frac{x-a}{2}\right)}{(x-a)}\left[\because \sin x-\sin a=2 \times \cos \frac{x+a}{2} \sin \frac{x-a}{2}\right]$ $=1 \times \lim _{x \rightarrow a} \cos \frac{x+a}{2}\left[\because \lim _{x \rightarrow a} \frac{\sin \theta}{\theta}=1\right]$ $=\cos \frac{a+a}{2}$ $\...
Read More →What observable features in Trypanosoma
Question: What observable features in Trypanosoma would make you classify it under kingdom Protista? Solution: Trypanosoma is a unicellular eukaryotic organism. Locomotion is facilitated by flagella. Method of reproduction is asexual. The reserve food material is present in the form of granules....
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