Evaluate:
Question: Evaluate: $\int \frac{x^{3}}{x-2} d x$ Solution: By doing long division of the given equation we get Quotient $=x^{2}+2 x+4$ Remainder $=8$ $\therefore$ We can write the above equation as $\Rightarrow x^{2}+2 x+4+\frac{8}{x-2}$ $\therefore$ The above equation becomes $\Rightarrow \int \mathrm{x}^{2}+2 \mathrm{x}+4+\frac{8}{\mathrm{x}-2} \mathrm{dx}$ $\Rightarrow \int \mathrm{x}^{2} \mathrm{dx}+2 \int \mathrm{x} \mathrm{dx}+4 \int \mathrm{dx}+8 \int \frac{1}{\mathrm{x}-2} \mathrm{dx}$ W...
Read More →prove that
Question: If $y=\left(\sqrt{\frac{x}{a}}+\sqrt{\frac{a}{x}}\right)$, prove that $(2 x y)\left(\frac{d y}{d x}\right)=\left(\frac{x}{a}-\frac{a}{x}\right)$ Solution: To prove: $(2 x y)\left(\frac{d y}{d x}\right)=\left(\frac{x}{a}-\frac{a}{x}\right)$ Differentiating $y$ with respect to $x$ $\frac{d y}{d x}=\frac{d}{d x}\left(\sqrt{\frac{x}{a}}+\sqrt{\frac{a}{x}}\right)=\frac{1}{2 \sqrt{a x}}-\frac{\sqrt{a}}{2 x^{\frac{3}{2}}}$ Now, $\mathrm{LHS}=(2 \mathrm{xy})\left(\frac{\mathrm{dy}}{\mathrm{dx}...
Read More →Is there a species-specific or region-specific
Question: Is there a species-specific or region-specific type of plastids? How does one distinguish one from the other? Solution: Plastids are species-specific or region-specific. They bear some specific pigments, thus imparting specific colours to the plants. According to the types of pigments, plastids are divided into three. chloroplasts, chromoplasts and leucoplasts. The chloroplasts impart green colour as they contain chlorophyll and carotenoid pigments The chromoplasts contain fat-soluble ...
Read More →Justify the statement, “Mitochondria
Question: Justify the statement, Mitochondria are powerhouses of the cell Solution: There are two membranes for each mitochondrion. They are Inner and outer membrane. The inner membrane is called the matrix. They have their specific enzymes associated with their functions. They are the sites for aerobic respiration and produce cellular energy in the form of ATP and are called powerhouses of a cell....
Read More →The genomic content of the nucleus is constant for a given
Question: The genomic content of the nucleus is constant for a given species whereas the extrachromosomal DNA is found to be variable among the members of a population. Explain. Solution: Consider a human cell. It has 46 chromosomes and will be constant for all human beings. But the mitochondria present in the human will have a different amount of extrachromosomal DNA. As like in human beings, bacteria will have the same amount of genomic DNA but the extra genomic DNA contained in plastids and m...
Read More →show that
Question: If $y=\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)$, show that $2 x \cdot \frac{d y}{d x}+y=2 \sqrt{x}$ Solution: To show: $2 x \cdot \frac{d y}{d x}+y=2 \sqrt{x}$ Differentiating with respect to $\mathrm{x}$ $\frac{d y}{d x}=\frac{d}{d x}\left(\sqrt{x}+\frac{1}{\sqrt{x}}\right)=\frac{1}{2 \sqrt{x}}-\frac{1}{2 x^{\frac{3}{2}}}$ Now, $\mathrm{LHS}=2 \mathrm{x} \cdot \frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}$ $L H S=2 x \times\left(\frac{1}{2 \sqrt{x}}-\frac{1}{2 x^{\frac{3}{2}}}\right)+\sq...
Read More →Evaluate:
Question: Evaluate: $\int \frac{x^{2}+5 x+2}{x+2} d x$ Solution: By doing long division of the given equation we get Quotient $=x+3$ Remainder $=-4$ $\therefore$ We can write the above equation as $\Rightarrow x+3-\frac{4}{x+2}$ $\therefore$ The above equation becomes $\Rightarrow \int \mathrm{x}+3-\frac{4}{\mathrm{x}+2} \mathrm{dx}$ $\Rightarrow \int \mathrm{x} \mathrm{dx}+3 \int \mathrm{dx}-4 \int \frac{1}{\mathrm{x}+2} \mathrm{dx}$ We know $\int x d x=\frac{x^{n}}{n+1} ; \int \frac{1}{x} d x=...
Read More →Eukaryotic cells have organelles which may a.
Question: Eukaryotic cells have organelles which may a. not be bound by a membrane b. bound by a single membrane c. bound by a double membrane Group the various subcellular organelles into these three categories. Solution: a. Non-membrane bound cell organelles Ribosomes, Centriole, nucleolus, Cytoskeletal structures b. Single membrane-bound cell organelles Endoplasmic reticulum, Golgi bodies, lysosomes, vacuoles, Thylakoid. c. Double membrane-bound cell organelles Plastids, Mitochondria and nucl...
Read More →Evaluate:
Question: Evaluate: $\int \frac{x^{2}+5 x+2}{x+2} d x$ Solution: By doing long division of the given equation we get Quotient $=x+3$ Remainder $=-4$ $\therefore$ We can write the above equation as $\Rightarrow x+3-\frac{4}{x+2}$ $\therefore$ The above equation becomes $\Rightarrow \int \mathrm{x}+3-\frac{4}{\mathrm{x}+2} \mathrm{dx}$ $\Rightarrow \int \mathrm{x} \mathrm{dx}+3 \int \mathrm{dx}-4 \int \frac{1}{\mathrm{x}+2} \mathrm{dx}$ We know $\int x d x=\frac{x^{n}}{n+1} ; \int \frac{1}{x} d x=...
Read More →Structure and function are correlatable
Question: Structure and function are correlatable in living organisms. Can you justify this by taking the plasma membrane as an example? Solution: It is made up of proteins, carbohydrates and lipids. The lipids arranged with their hydrophilic polar head directed outwards and non-polar hydrophobic tails directed inwards providing fluidity to the membrane. This provides the cell membrane with its selective permeability....
Read More →Is extra genomic DNA present in prokaryotes
Question: Is extra genomic DNA present in prokaryotes and eukaryotes? If yes, indicate their location in both the types of organisms. Solution: Yes, extra genomic DNA is present in prokaryotes and eukaryotes. In organisms like bacteria in addition to the genomic DNA, there is a small circular DNA outside the genomic DNA. They are called plasmids. This is a unique phenotypic character to such bacteria. In eukaryotes, the extra genomic DNA is present in two organelles- Mitochondria and plastids....
Read More →Briefly give the contributions of the following scientists
Question: Briefly give the contributions of the following scientists in formulating the cell theory a. Robert Virchow b. Scheilden and Schwann Solution: Rudolf Virchows modern theory of cell is the cell divides and new cells are formed from the [pre-existing cells. He modified the hypothesis of Schleiden and Schwann to give the cell theory to its final. 1. All living organism is composed of cells and products of cells 2. All cells arise from pre-existing cells In 1839, a British Zoologist Theodo...
Read More →Evaluate:
Question: Evaluate: $\int \frac{1}{\cos ^{2} x(1-\tan x)^{2}} d x$ Solution: Let $I=\int \frac{1}{\cos ^{2} x(1-\tan x)^{2}} d x$ $=\int \frac{1}{\cos ^{2} x(1-\tan x)^{2}} d x$ $=\int \frac{1}{\cos ^{2} x\left(1-\frac{\sin x}{\cos x}\right)^{2}} d x$ $=\int \frac{1}{(\cos x-\sin x)^{2}} d x$ $=\int \frac{1}{1-\sin 2 x} d x$ $=\int \frac{1}{1+\cos \left(\frac{\pi}{2}+2 x\right)} d x$ $=\int \frac{1}{2 \cos ^{2}\left(\frac{\pi}{4}+x\right)} d x$ $=\frac{1}{2} \int \sec ^{2}\left(\frac{\pi}{4}+x\r...
Read More →Differentiate the following functions:
Question: Differentiate the following functions: (i) If $y=6 x^{5}-4 x^{4}-2 x^{2}+5 x-9$, find $\frac{d y}{d x}$ at $x=-1$ (ii) If $y=(\sin x+\tan x)$, find $\frac{d y}{d x}$ at $x=\frac{\pi}{3}$. (iii) If $y=\frac{(2-3 \cos x)}{\sin x}$, find $\frac{d y}{d x}$ at $x=\frac{\pi}{4}$. Solution: Formulae: $\frac{d}{d x} x^{n}=n x^{n-1}$ $\frac{d}{d x} \cot x=-\operatorname{cosec}^{2} x$ $\frac{d}{d x} \operatorname{cosec} x=-\operatorname{cosec} x \cot x$ $\frac{d}{d x} \tan x=\sec ^{2} x$ $\frac{...
Read More →What structural and functional attributes
Question: What structural and functional attributes must a cell have to be called a living cell? Solution: A cell must have the following structural and functional attributes to be a called a living cell (a) Cell membrane It is the outer membrane of the cell which is made up of phospholipids primarily. (b) Cytoplasm The liquid containing the controlled concentration of organic and inorganic compounds. They provide energy and material for reproduction. (c) Nucleus It contains DNA, the nuclear mat...
Read More →What are histones?
Question: What are histones? What are their functions? Solution: Histones are highly alkaline proteins that are found in nuclei of eukaryotes. They package and order DNA into structural units. They also function as gene regulation....
Read More →What are plasmids?
Question: What are plasmids? Describe their role in bacteria? Solution: Plasmids are smaller DNA present in the bacteria outside the genomic DNA. he plasmid DNA grants certain phenotypic characters like resistance to antibiotics to such bacteria. DNA is used to monitor bacterial transformation with foreign DNA....
Read More →If the solve the problem
Question: $\int \tan ^{2}(2 x-3) d x$ Solution: Let $I=\int \tan ^{2}(2 x-3) d x$ $=\int \tan ^{2}(2 x-3) d x$ $=\int \sec ^{2}(2 x-3)-1 d x$ Let $2 x-3=t d x=d t / 2$ $=\frac{1}{2} \int \sec ^{2} t-1 d t$ $=\frac{1}{2} \tan t-x$ Substitute the value of $t$ Hence, $I=\frac{1}{2} \tan (2 x-3)-x+C$...
Read More →Give the biochemical composition of the plasma
Question: Give the biochemical composition of the plasma membrane. How are lipid molecules arranged in the membrane? Solution: The plasma membrane is composed of proteins, lipids, water and carbohydrates. The ratio of lipids and proteins varies considerably in different types of cells. The major lipids are phospholipids that are arranged in a bilayer. They also contain cholesterol....
Read More →Differentiate between Rough Endoplasmic
Question: Differentiate between Rough Endoplasmic Reticulum (RER) and Smooth Endoplasmic Reticulum (SER). Solution:...
Read More →Evaluate:
Question: Evaluate: $\int \frac{1}{\sqrt{x+3}-\sqrt{x+2}} d x$ Solution: Let $I=\int \frac{1}{\sqrt{x+3}-\sqrt{x+2}} \mathrm{dx}$ $=\int \frac{1}{\sqrt{x+3}-\sqrt{x+2}} d x$ Now, Multiply with the conjugate $=\int \frac{1}{\sqrt{x+3}-\sqrt{x+2}} \times \frac{\sqrt{x+3}+\sqrt{x+2}}{\sqrt{x+3}+\sqrt{x+2}} d x$ $=\int \frac{\sqrt{x+3}+\sqrt{x+2}}{(\sqrt{x+3})^{2}-(\sqrt{x+2})^{2}} d x$ $=\int \frac{\sqrt{x+3}+\sqrt{x+2}}{x+3-x-2} d x$ $=\int(\mathrm{x}+3)^{\frac{1}{2}}+(\mathrm{x}+2)^{\frac{1}{2}} ...
Read More →Briefly describe the cell theory.
Question: Briefly describe the cell theory. Solution: Cell theory was given by Schleiden and Schwann which stated (a) Bodies of all the plants and animals are made up of cells. (b) The cells are structural and functional units of life. The modern cell theory by Rudolf Virchow is all living organisms are composed of cells and products of cells and all arise from pre-existing cells....
Read More →Comment on the cartwheel structure
Question: Comment on the cartwheel structure of centriole. Solution: The centromere is an organelle usually containing two cylindrical structures called centrioles. The centrioles present in a centrosome lie perpendicular to each other in which each as an organization like the cartwheel. Each of the peripheral fibrils is a triplet. The adjacent triplets are also linked....
Read More →Explain the association of carbohydrate
Question: Explain the association of carbohydrate to the plasma membrane and its significance. Solution: By glycosylation carbohydrates forms glycoproteins and glycolipids. They are biochemical involved in cell recognition and adhesion....
Read More →Discuss briefly the role of nucleolus
Question: Discuss briefly the role of nucleolus in the cells actively involved in protein synthesis. Solution: It contains small spherical bodies which are known as the nucleolus. It is not a membrane-bound structure. It is a site for active ribosomal RNA synthesis. It also receives and stores ribosomal proteins formed in the cytoplasm....
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