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: $\frac{1}{x^{3}}$ Solution: Let $f(x)=\frac{1}{x^{3}}$ 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) $f(x)=\frac{1}{x^{3}}$ $\mathrm{f}(\mathrm{x}+\mathrm{h})=\frac{1}{(\mathrm{x}+\mathrm{h})^{3}}$ Putting values in (i), we get $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{\frac{1}{(x+h)^{3}}-\frac{1}{x^{3}}}{h}$ $=\lim _{h \rightarrow...
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Question: Evaluate: $\int x \sqrt{x+2} d x$ Solution: Here Add and subtract 2 from $x$ We get $\Rightarrow \int(x+2-2) \sqrt{x+2} d x$ $\Rightarrow \int(x+2)^{\frac{3}{2}} d x-\int 2 \sqrt{x+2} d x$ $\Rightarrow \frac{2(x+2)^{\frac{5}{2}}}{5}-\frac{4(x+2)^{\frac{3}{2}}}{3}+c$...
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: $\mathbf{x}^{8}$ Solution: Let $f(x)=x^{8}$ 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) $f(x)=x^{8}$ $f(x+h)=(x+h)^{8}$ Putting values in (i), we get $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{(x+h)^{8}-x^{8}}{h}$ $=\lim _{h \rightarrow 0} \frac{(x+h)...
Read More →A homopolymer has only one type of building
Question: A homopolymer has only one type of building block called monomer repeated n number of times. A heteropolymer has more than one type of monomer. Proteins are heteropolymers usually made ofa. a. 20 types of monomers b. 40 types of monomers c. 30 types of monomers d. only one type of monomer Solution: Option (a)20 types of monomers is the answer...
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Question: Evaluate: $\int \frac{\mathrm{x}+1}{\sqrt{2 \mathrm{x}+3}} \mathrm{dx}$ Solution: In these questions, little manipulation makes the questions easier to solve Here multiply and divide by 2 we get $\Rightarrow \frac{1}{2} \int \frac{2 x+2}{\sqrt{2 x+3}} d x$ Add and subtract 1 from the numerator $\Rightarrow \frac{1}{2} \int \frac{2 x+2+1-1}{\sqrt{2 x+3}} d x$ $\Rightarrow \frac{1}{2} \int \frac{2 x+3-1}{\sqrt{2 x+3}} d x$ $\Rightarrow \frac{1}{2} \int \frac{2 x+3}{\sqrt{2 x+3}} d x-\fra...
Read More →The most abundant component of living
Question: The most abundant component of living organisms is a. Protein b. Water c. Sugar d. Nucleic acid Solution: Option (b)Water is the answer....
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Question: Evaluate: $\int \frac{2 x-1}{(x-1)^{2}} d x$ Solution: In this question degree of denominator is larger than that of numerator so we need to manipulate numerator. $\Rightarrow \int \frac{2 x+2-2-1}{(x-1)^{2}}$ $\Rightarrow \int \frac{2(x-1)-1}{(x-1)^{2}}$ $\Rightarrow \int \frac{2}{x-1} d x-\frac{1}{(x-1)^{2}} d x$ We know $\int x d x=\frac{x^{n}}{n+1} ; \int \frac{1}{x} d x=\ln x$ $\Rightarrow 2 \ln (x-1)-\int(x-1)^{-2} d x$ $\Rightarrow 2 \ln (\mathrm{x}-1)-\frac{1}{\mathrm{x}-1}+\ma...
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: $x^{3}-2 x^{2}+x+3$ Solution: Let $f(x)=x^{3}-2 x^{2}+x+3$ 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) $f(x)=x^{3}-2 x^{2}+x+3$ $f(x+h)=(x+h)^{3}-2(x+h)^{2}+(x+h)+3$ Putting values in (i), we get $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{(x+h)^{3}-2(x+h)^{2}+(x+h)+3-\left[x^{3}-2 x^{2}+x+3\right]}{h}$ $=\lim _{h \rightarrow 0} \fr...
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Question: Evaluate: $\int \frac{\mathrm{x}^{2}+3 \mathrm{x}-1}{(\mathrm{x}+1)^{2}} \mathrm{dx}$ Solution: $\Rightarrow \int \frac{x^{2}+x+2 x-1}{(x+1)^{2}} d x$ $\Rightarrow \int \frac{x(x+1)+2 x-1}{(x+1)^{2}} d x$ $\Rightarrow \int \frac{x(x+1)}{(x+1)^{2}} d x+\int \frac{2 x-1}{(x+1)^{2}} d x$ $\Rightarrow \int \frac{x}{x+1} d x+\int \frac{2 x+2-2-1}{(x+1)^{2}} d x$ $\Rightarrow \int \frac{x+1-1}{x+1} d x+\int \frac{2(x+1)-3}{(x+1)^{2}} d x$ $\Rightarrow \int d x-\int \frac{1}{x+1} d x+\int \fr...
Read More →When we homogenise any tissue
Question: When we homogenise any tissue in acid the acid-soluble pool represents a. Cytoplasm b. Cell membrane c. Nucleus d. Mitochondria Solution: Option (a)Cytoplasm is the answer....
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Question: Evaluate: $\int \frac{2 \mathrm{x}+3}{(\mathrm{x}-1)^{2}} \mathrm{dx}$ Solution: The above equation can be written as $\Rightarrow \int \frac{2 x-2+2+3}{(x-1)^{2}}$ $\Rightarrow \int \frac{2(x-1)+5}{(x-1)^{2}}$ $\Rightarrow 2 \int \frac{1 . d x}{(x-1)}+5 \int \frac{1 . d x}{(x-1)^{2}}$ We know $\int \mathrm{x} \mathrm{dx}=\frac{x^{\mathrm{n}}}{\mathrm{n}+1} ; \int \frac{1}{\mathrm{x}} \mathrm{dx}=\ln \mathrm{x}$ $\Rightarrow 2 \ln (\mathrm{x}-1)+5 \int(\mathrm{x}-1)^{-2} \mathrm{dx}$ $...
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: $3 x^{2}+2 x-5$ Solution: Let $f(x)=3 x^{2}+2 x-5$ 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) $f(x)=3 x^{2}+2 x-5$ $f(x+h)=3(x+h)^{2}+2(x+h)-5$ $=3\left(x^{2}+h^{2}+2 x h\right)+2 x+2 h-5$ $\left[\because(a+b)^{2}=a^{2}+b^{2}+2 a b\right]$ $=3 x^{2}+3 h^{2}+6 x h+2 x+2 h-5$ Putting values in (i), we get $f^{\prime}(x)=\lim _{h \rig...
Read More →When you take cells or tissue pieces and grind
Question: When you take cells or tissue pieces and grind them with an acid in a mortar and pestle, all the small biomolecules dissolve in the acid. Proteins, polysaccharides and nucleic acids are insoluble in mineral acid and get precipitated. The acid-soluble compounds include amino acids, nucleosides, small sugars etc. When one adds a phosphate group to a nucleoside, one gets another acid-soluble biomolecule called a. Nitrogen base b. Adenine c. Sugar phosphate d. Nucleotide Solution: Option (...
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Question: Evaluate: $\int \frac{x^{2}+x+5}{3 x+2} d x$ Solution: By doing long division of the given equation we get Quotient $=\frac{x}{3}+\frac{1}{9}$ Remainder $=\frac{43}{9}$ $\therefore$ We can write the above equation as $\Rightarrow \frac{x}{3}+\frac{1}{9}+\frac{43}{9}\left(\frac{1}{3 x+2}\right)$ $\Rightarrow \frac{x}{3}+\frac{1}{9}+\frac{43}{9}\left(\frac{1}{3 x+2}\right)$ $\therefore$ The above equation becomes $\Rightarrow \int \frac{x}{3}+\frac{1}{9}+\frac{43}{9}\left(\frac{1}{3 x+2}...
Read More →Sugars are technically called carbohydrates,
Question: Sugars are technically called carbohydrates, referring to the fact that their formulae are only multiple of C(H2O). Hexoses, therefore, have six carbons, twelve hydrogens and six oxygen atoms. Glucose is a hexose. Choose from among the following another hexose. a. Fructose b. Erythrose c. Ribulose d. Ribose Solution: Option (a)Fructose is the answer....
Read More →An amino acid under certain conditions
Question: An amino acid under certain conditions has both positive and negative charges simultaneously in the same molecule. Such a form of amino acid is called a. Acidic form b. Basic form c. Aromatic form d. Zwitterionic form Solution: Option (d)Zwitterionic formis the answer....
Read More →Aminoacids have both an amino group
Question: Aminoacids have both an amino group and a carboxyl group in their structure. Which one of the following is an amino acid? a. Formic acid b. Glycerol c. Glycolic Acid d. Glycine Solution: Option (d) Glycine is the answer....
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: $\left(a x^{2}+\frac{b}{x}\right)$ Solution: Let $\mathrm{f}(\mathrm{x})=\mathrm{ax}^{2}+\frac{\mathrm{b}}{\mathrm{x}}$ We need to find the derivative of $f(x)$ i.e. $f^{\prime}(x)$ We know that, $f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}$ (i) $f(x)=a x^{2}+\frac{b}{x}$ $f(x+h)=a(x+h)^{2}+\frac{b}{(x+h)}$ Putting values in (i), we get $\mathrm{f}^{\prime}(\mathrm{x})=\lim _{\mathrm{h} \rightarrow 0...
Read More →Many elements are found in living organisms
Question: Many elements are found in living organisms either free or the form of compounds. Which of the following is not found in living organisms? a. Silicon b. Magnesium c. Iron d. Sodium Solution: Option (a)Silicon is the answer....
Read More →It is said that the elemental composition of living
Question: It is said that the elemental composition of living organisms and that of inanimate objects (like earths crust) are similar in the sense that all the major elements are present in both. Then what would be the difference between these two groups? Choose the correct answer from among the following: a. Living organisms have more gold in them than inanimate objects b. Living organisms have more water in their body than inanimate objects c. Living organisms have more carbon, oxygen and hydr...
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: $(a x+b)$ Solution: Let f(x) = ax + b 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) $f(x)=a x+b$ $f(x+h)=a(x+h)+b$ = ax + ah + b Putting values in (i), we get $\mathrm{f}^{\prime}(\mathrm{x})=\lim _{\mathrm{h} \rightarrow 0} \frac{\mathrm{ax}+\mathrm{ah}+\mathrm{b}-(\mathrm{ax}+\mathrm{b})}{\mathrm{h}}$ $=\lim _{h \rightarrow 0} \frac{...
Read More →Find the value
Question: $\mathrm{y}=\frac{1-\tan ^{2}(\mathrm{x} / 2)}{1+\tan ^{2}(\mathrm{x} / 2)}$, find $\frac{\mathrm{dy}}{\mathrm{dx}}$ Solution: Formula: Using Half angle formula, $\cos x=\frac{1-\tan ^{2}(x / 2)}{1+\tan ^{2}(x / 2)}$ y = cos x Differentiating y with respect to x $\frac{d y}{d x}=\frac{d}{d x} \cos x$ $=-\sin x$...
Read More →solve this
Question: $y=\sqrt{\frac{1+\cos 2 x}{1-\cos 2 x}}$, find $\frac{d y}{d x}$ Solution: $y=\sqrt{\frac{1+\cos 2 x}{1-\cos 2 x}}$ Formula: Using double angle formula: $\cos 2 x=2 \cos ^{2} x-1$ $=1-2 \sin ^{2} x$ $\therefore 1+\cos 2 x=2 \cos ^{2} x$ $1-\cos 2 x=2 \sin ^{2} x$ $\therefore y=\sqrt{\frac{2 \cos ^{2} x}{2 \sin x^{2} x}}$ $=\sqrt{\cot ^{2} x}$ $=\cot x$ Differentiating $y$ with respect to $x$ $\frac{d y}{d x}=\frac{d}{d x}(\cot x)$ $=-\operatorname{cosec}^{2} x$...
Read More →Are the different types of plastids interchangeable?
Question: Are the different types of plastids interchangeable? If yes, give examples where they are getting converted from one type to another. Solution: Yes, plastids are interchangeable in their form. There are three types of plastids Chloroplasts (green colour), Chromoplasts (red, yellow, orange colour), Leucoplasts (Colourless). Depending upon different circumstances, these plastids interchange. For example: Due to the replacement of chloroplast with chromoplasts the colour of green tomatoes...
Read More →Write the functions of the following a.
Question: Write the functions of the following a. Centromere b. Cell wall c. Smooth ER d. Golgi Apparatus e. Centrioles Solution: a. Centromere Centromere holds two chromatids or sister chromatids of a chromosome. b. The cell wall is present in plant cells which gives shape and protection to the cell from mechanical damages. c. Smooth ER It is a major site for lipid synthesis. In animal cells, lipid-like steroidal hormones are synthesized at Smooth ER. d. Golgi apparatus It is an important site ...
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