Differentiate the following functions:
Question: Differentiate the following functions: (i) $(2 x+3)(3 x-5)$ (ii) $x(1+x)^{3}$ (iii) $\left(\sqrt{\mathrm{x}}+\frac{1}{\mathrm{x}}\right)\left(\mathrm{x}-\frac{1}{\sqrt{\mathrm{x}}}\right)$ (iv) $\left(x-\frac{1}{x}\right)^{2}$ (v) $\left(x^{2}-\frac{1}{x^{2}}\right)^{3}$ (vi) $\left(2 x^{2}+5 x-1\right)(x-3)$ Solution: Formula: $\frac{\mathrm{d}}{\mathrm{dx}} \mathrm{f}(\mathrm{g}(\mathrm{x}))=\frac{\mathrm{d}}{\mathrm{dg}} \mathrm{f}(\mathrm{g}) \frac{\mathrm{d}}{\mathrm{dx}} \mathrm{...
Read More →Different cells have different sizes.
Question: Different cells have different sizes. Arrange the following cells in an ascending order of their size. Choose the correct option among the followings i. Mycoplasma ii. Ostrich eggs iii. Human RBC iv. Bacteria Options: a. i, iv, iii ii b. i, iii, iv ii c. ii, i, iii iv d. iii, ii, i iv Solution: Option (a)i, iv, iii iiis the answer....
Read More →Evaluate:
Question: Evaluate: $\int \frac{1}{\sqrt{\mathrm{x}+\mathrm{a}}+\sqrt{\mathrm{x}+\mathrm{b}}} \mathrm{dx}$ Solution: Let $I=\int \frac{1}{\sqrt{x+a}+\sqrt{x+b}} d x$ $=\int \frac{1}{\sqrt{x+a}+\sqrt{x+b}} d x$ Now, Multiply with conjugate, we get $=\int \frac{1}{\sqrt{x+a}+\sqrt{x+b}} \times \frac{(\sqrt{x+a}-\sqrt{x+b})}{\sqrt{x+a}-\sqrt{(x+b)}} d x$ $=\int \frac{(\sqrt{x+a}-\sqrt{x+b})}{(\sqrt{x+a})^{2}-\sqrt{(x+b)}} d x$ $=\int \frac{(\sqrt{x+a}-\sqrt{x+b})}{a-b} d x$ $=\frac{1}{a-b}\left[\fr...
Read More →Which of the following stains is not used
Question: Which of the following stains is not used for staining chromosomes? a. Basic Fuchsin b. Safranin c. Methylene green d. Carmine Solution: Option (b)Safranin is the answer....
Read More →Which one of these is not a eukaryote?
Question: Which one of these is not a eukaryote? a. Euglena b. Anabaena c. Spirogyra d. Agaricus Solution: Option (b)Anabaenais the answer....
Read More →Evaluate:
Question: Evaluate: $\int \frac{2 \mathrm{x}}{(2 \mathrm{x}+1)^{2}} \mathrm{dx}$ Solution: Let $I=\int \frac{2 x}{(2 x+1)^{2}} d x$ $=\int \frac{2 x+1-1}{(2 x+1)^{2}} d x$ $=\int \frac{2 \mathrm{x}+1}{(2 \mathrm{x}+1)^{2}}-\frac{1}{(2 \mathrm{x}+1)^{2}} \mathrm{dx}$ $=\int \frac{1}{(2 \mathrm{x}+1)}-(2 \mathrm{x}+1)^{-2} \mathrm{dx}$ $=\frac{1}{2} \log |2 \mathrm{x}+1|-\frac{(2 \mathrm{x}+1)^{-2+1}}{-2+1(2)}$ $=\frac{1}{2} \log |2 \mathrm{x}+1|-\frac{(2 \mathrm{x}+1)^{-1}}{-2}$ Hence, $I=\frac{1...
Read More →Select one which is not true for ribosome
Question: Select one which is not true for ribosome a. Made of two sub-units b. Form polysome c. May attach to m+RNA d. Have no role in protein synthesis Solution: Option (d)Have no role in protein synthesisis the answer....
Read More →A common characteristic feature of plant sieve
Question: A common characteristic feature of plant sieve tube cells and most of the mammalian erythrocytes is a. Absence of mitochondria b. Presence of cell wall c. Presence of haemoglobin d. Absence of nucleus Solution: Option (d)Absence of nucleus is the answer....
Read More →Evaluate:
Question: Evaluate: $\int \frac{1}{\sqrt{2 x+3}+\sqrt{2 x-3}} \mathrm{dx}$ Solution: Let I $=\int \frac{1}{\sqrt{2 x+3}+\sqrt{2 x-3}} d x$ $I=\int \frac{1}{\sqrt{2 x+3}+\sqrt{2 x-3}} d x$ Now, Multiply with the conjugate, we get $=\int \frac{1}{\sqrt{2 x+3}+\sqrt{2 x-3}} \times \frac{(\sqrt{2 x+3}-\sqrt{2 x-3})}{\sqrt{2 x+3}-\sqrt{2 x-3}} d x$ $=\int \frac{(\sqrt{2 x+3}-\sqrt{2 x-3})}{(\sqrt{2 x+3})^{2}-(\sqrt{2 x-3})^{2}} d x$ $=\int \frac{(\sqrt{2 x+3}-\sqrt{2 x-3})}{2 x+3-2 x+3} d x$ $=\int \...
Read More →Evaluate:
Question: Evaluate: $\int \frac{1}{\sqrt{\mathrm{x}+1}+\sqrt{\mathrm{x}}} \mathrm{dx}$ Solution: Let $I=\int \frac{1}{\sqrt{x+1}+\sqrt{x}} d x$ $=\int \frac{1}{\sqrt{x+1}+\sqrt{x}} d x$ Now Multiply with the conjugate, we get $=\int \frac{1}{\sqrt{x+1}+\sqrt{x}} \cdot \frac{\sqrt{x+1}-\sqrt{x}}{\sqrt{x+1}-\sqrt{x}} d x$ $=\int \frac{\sqrt{x+1}-\sqrt{x}}{x+1-x} d x$ $=\int \sqrt{x+1}-\sqrt{x} d x$ $=\int(x+1)^{\frac{1}{2}}-x^{\frac{1}{2}}$ $=\frac{(x+1)^{\frac{3}{2}}}{\frac{3}{2}}-\frac{x^{\frac{...
Read More →Evaluate:
Question: Evaluate: $\int \frac{\mathrm{x}+3}{(\mathrm{x}+1)^{4}} \mathrm{dx}$ Solution: Let I $=\int \frac{x+3}{(x+1)^{4}} d x$ $I=\int \frac{x+3}{(x+1)^{4}} d x$ $=\int \frac{x+1}{x+1^{4}} d x+\int \frac{2}{(x+1)^{4}} d x$ $=\int \frac{1}{(x+1)^{3}} d x+\int \frac{2}{(x+1)^{4}} d x$ $=\int(x+1)^{-3} d x+\int 2(x+1)^{-4} d x$ $=\frac{[x+1]^{-2+1}}{-3+1}+\frac{2(x+1)^{-4+1}}{-4+1}$ $=\frac{[x+1]^{-2}}{-2}+\frac{2(x+1)^{-3}}{-3}$ Hence, $I=-\frac{1}{2(x+1)^{2}}-\frac{2}{3(x+1)^{3}}+C$...
Read More →Differentiate the following functions:
Question: Differentiate the following functions: (i) $4 \cot x-\frac{1}{2} \cos x+\frac{2}{\cos x}-\frac{3}{\sin x}+\frac{6 \cot x}{\operatorname{cosec} x}+9$ (ii) $-5 \tan x+4 \tan x \cos x-3 \cot x \sec x+2 \sec x-13$ Solution: Formulae: - $\frac{d}{d x} \cot x=-\operatorname{cosec}^{2} x$ $\frac{d}{d x} \cos x=-\sin x$ $\frac{d}{d x} \sec x=\sec x \tan x$ $\frac{d}{d x} \operatorname{cosecx}=-\operatorname{cosec} x \cot x$ $\frac{d}{d x} \tan x=\sec ^{2} x$ $\frac{d}{d x} \sin x=\cos x$ $\fra...
Read More →Evaluate:
Question: Evaluate: $\int \frac{1}{2-3 x}+\frac{1}{\sqrt{3 x-2}} d x$ Solution: Let I $=\int \frac{1}{2-3 x}+\frac{1}{\sqrt{3 x-2}} d x$ $\mathrm{I}=\int \frac{1}{2-3 \mathrm{x}}+\frac{1}{\sqrt{3 \mathrm{x}-2}} \mathrm{dx}$ We know $\int \frac{1}{x} d x=\log |x|+C$ $=\frac{\log |2-3 x|}{-3}+\frac{2}{3}(3 x-2)^{\frac{1}{2}}$ $=-\frac{1}{3} \mathrm{x} \cdot \log |2 \mathrm{x}-3|+\frac{2}{3} \sqrt{3 \mathrm{x}-3}+\mathrm{C}$...
Read More →Evaluate the problem
Question: Evaluate: $\int \frac{1}{(7 x-5)^{3}}+\frac{1}{\sqrt{5 x-4}} d x$ Solution: Let $I=\int \frac{1}{(7 x-5)^{2}}+\frac{1}{\sqrt{5 x-4}} \mathrm{dx}$ then, $I=\int(7 x-5)^{-3}+(5 x-4)^{-\frac{1}{2}}$ $=\frac{(7 x-5)^{-3+1}}{7(-3+1)}+\frac{(5 x-4)^{\frac{-1}{2}+1}}{5\left(-\frac{1}{2}+1\right)}$ $=\frac{(7 x-5)^{-2}}{-14}+\frac{(5 x-4)^{\frac{1}{2}}}{5\left(\frac{1}{2}\right)}$ Hence, $I=-\frac{1}{14}(7 x-5)^{-2}+\frac{2}{5} \sqrt{5 x-4}+C$...
Read More →Differentiate the following functions:
Question: Differentiate the following functions: (i) $4 \mathrm{x}^{3}+3 \cdot 2^{\mathrm{x}}+6 \cdot \sqrt[8]{\mathrm{x}^{-4}}+5 \cot \mathrm{x}$ (ii) $\frac{x}{3}-\frac{3}{x}+\sqrt{x}-\frac{1}{\sqrt{x}}+x^{2}-2^{x}+6 x^{-2 / 3}-\frac{2}{3} x^{6}$ Solution: (i) $4 x^{3}+3 \cdot 2^{x}+6 \cdot \sqrt[8]{x^{-4}}+5 \cot x$ $=4 x^{3}+3 \cdot 2^{x}+6 x^{-\frac{1}{2}}+5 \cot x$ Formulae: $\frac{d}{d x} x^{n}=n x^{n-1}$ $\frac{d}{d x} \cot x=-\operatorname{cosec}^{2} x$ $\frac{\mathrm{d}}{\mathrm{dx}} \...
Read More →Differentiate the following functions:
Question: Differentiate the following functions: (i) $4 \mathrm{x}^{3}+3 \cdot 2^{\mathrm{x}}+6 \cdot \sqrt[8]{\mathrm{x}^{-4}}+5 \cot \mathrm{x}$ (ii) $\frac{x}{3}-\frac{3}{x}+\sqrt{x}-\frac{1}{\sqrt{x}}+x^{2}-2^{x}+6 x^{-2 / 3}-\frac{2}{3} x^{6}$ Solution: (i) $4 x^{3}+3 \cdot 2^{x}+6 \cdot \sqrt[8]{x^{-4}}+5 \cot x$ $=4 x^{3}+3 \cdot 2^{x}+6 x^{-\frac{1}{2}}+5 \cot x$ Formulae: $\frac{d}{d x} x^{n}=n x^{n-1}$ $\frac{d}{d x} \cot x=-\operatorname{cosec}^{2} x$ $\frac{\mathrm{d}}{\mathrm{dx}} \...
Read More →If the solve the problem
Question: Evaluate: $\int(2 x-3)^{5}+\sqrt{3 x+2} d x$ Solution: Let $I=\int(2 x-3)^{5}+\sqrt{3 x+2}$ then, $I=\int(2 x-3)^{5}+(3 x+2)^{\frac{1}{2}}$ $=\frac{(2 x-3)^{5+1}}{2(5+1)}+\frac{(3 x+2)^{\frac{1}{2}+1}}{3\left(\frac{1}{2}+1\right)}$ $=\frac{(2 x-3)^{6}}{2(6)}+\frac{(3 x+2)^{\frac{3}{2}}}{3\left(\frac{2}{2}\right)}$ $=\frac{(2 x-3)^{6}}{12}+\frac{2(3 x+2)^{\frac{3}{2}}}{9}$ Hence, $I=\frac{(2 x-3)^{6}}{12}+\frac{2(3 x+2)^{\frac{3}{2}}}{9}+C$...
Read More →Differentiate the following functions:
Question: Differentiate the following functions: (i) $6 \times 5+4 \times 3-3 \times 2+2 \times-7$ (ii) $5 x^{-3 / 2}+\frac{4}{\sqrt{x}}+\sqrt{x}-\frac{7}{x}$ (iii) $a \times 3+b \times 2+c x+d$, where $a, b, c, d$ are constants Solution: (i) $6 x^{5}+4 x^{3}-3 x^{2}+2 x-7$ Formula:- $\frac{d}{d x} x^{n}=n x^{n-1}$ Differentiating with respect to $\mathrm{x}$, $\frac{d}{d x}\left(6 x^{5}+4 x^{3}-3 x^{2}+2 x-7\right)=30 x^{5-1}+12 x^{3-1}-6 x^{2-1}+2 x^{1-1}+0$ $=30 x^{4}+12 x^{2}-6 x^{1}+2 x$ (i...
Read More →Write the primitive or anti-derivative
Question: Write the primitive or anti-derivative of $f(x)=\sqrt{x}+\frac{1}{\sqrt{x}}$. Solution: Given $\mathrm{f}(\mathrm{x})=\sqrt{\mathrm{x}}+\frac{1}{\sqrt{\mathrm{x}}}$ Let $I=\int f(x) d x$ $\Rightarrow \mathrm{I}=\int\left(\sqrt{\mathrm{x}}+\frac{1}{\sqrt{\mathrm{x}}}\right) \mathrm{dx}$ $\Rightarrow \mathrm{I}=\int\left(\mathrm{x}^{\frac{1}{2}}+\frac{1}{\mathrm{x}^{\frac{1}{2}}}\right) \mathrm{dx}$ $\Rightarrow \mathrm{I}=\int\left(\mathrm{x}^{\frac{1}{2}}+\mathrm{x}^{-\frac{1}{2}}\righ...
Read More →If the solve the problem
Question: If $f^{\prime}(x)=a \sin x+b \cos x$ and $f^{\prime}(0)=4, f(0)=3, f\left(\frac{\pi}{2}\right)=5$, find $f(x)$ Solution: Given $f^{\prime}(x)=a \sin x+b \cos x$ and $f^{\prime}(0)=4$ On substituting $x=0$ in $f^{\prime}(x)$, we get $f^{\prime}(0)=a \sin 0+b \cos 0$ $\Rightarrow 4=a \times 0+b \times 1$ $\Rightarrow 4=0+b$ $\therefore b=4$ Hence, $f^{\prime}(x)=a \sin x+4 \cos x$ On integrating this equation, we have $\int \mathrm{f}^{\prime}(\mathrm{x}) \mathrm{dx}=\int(a \sin \mathrm{...
Read More →Differentiate the following functions:
Question: Differentiate the following functions: (i) $3 x^{-5}$ (ii) $\frac{1}{5 \mathrm{x}}$ (iii) $6 . \sqrt[3]{x^{2}}$ Solution: (i) $3 x^{-5}$ Formula:- $\frac{d}{d x} x^{n}=n x^{n-1}$ Differentiating with respect to $x$, $\frac{d}{d x} 3 x^{-5}=3(-5) x^{-5-1}$ $=-15 x^{-6}$ (ii) $1 / 5 x=\frac{1}{5} x^{-1}$ Formula:- $\frac{d}{d x} x^{n}=n x^{n-1}$ Differentiating with respect to $\mathrm{X}$, $\frac{1}{5} \frac{d}{d x} x^{-1}=\frac{-1}{5} x^{-1-1}$ $=-\frac{1}{5} x^{-2}$ (iii) 6. $\sqrt[3]...
Read More →Comment upon the gametic exchange
Question: Comment upon the gametic exchange in earthworm during mating. Solution: (i) Gamete exchange of earthworms occurs through cross-fertilization and external fertilization. Two earthworms come closer to each other and they get ventrally attached in the opposite direction. (ii) The copulation process begins and the male genital pore (i.e., papilla) of one earthworm is inserted into the spermathecal pore of the other earthworm. (iii) The earthworm remains mutually close together by the penet...
Read More →Differentiate the following functions:
Question: Differentiate the following functions: (i) $\frac{1}{x}$ (ii) $\frac{1}{\sqrt{\mathrm{x}}}$ (iii) $\frac{1}{\sqrt[3]{x}}$ Solution: (i) $\frac{1}{x}=x^{-1}$ Formula:- $\frac{d}{d x} x^{n}=n x^{n-1}$ Differentiating w.r.t $\mathrm{x}$, $\frac{d}{d x} x^{-1}=-1 x^{-1-1}$ $=-\mathrm{X}^{-2}$ (ii) $\frac{1}{\sqrt{x}}=x^{-\frac{1}{2}}$ Formula:- $\frac{d}{d x} x^{n}=n x^{n-1}$ Differentiating w.r.t $x$, $\frac{d}{d x} x^{\frac{-1}{2}}=\frac{-1}{2} x^{-\frac{1}{2}-1}$ $=\frac{-1}{2} x^{-\fra...
Read More →If the solve the problem
Question: If $f^{\prime}(x)=8 x^{3}-2 x, f(2)=8$, find $f(x)$ Solution: Given $f^{\prime}(x)=8 x^{3}-2 x$ and $f(2)=8$ On integrating the given equation, we have $\int f^{\prime}(x) d x=\int\left(8 x^{3}-2 x\right) d x$ We know $\int \mathrm{f}^{\prime}(\mathrm{x}) \mathrm{d} \mathrm{x}=\mathrm{f}(\mathrm{x})$ $\Rightarrow \mathrm{f}(\mathrm{x})=\int\left(8 \mathrm{x}^{3}-2 \mathrm{x}\right) \mathrm{dx}$ $\Rightarrow \mathrm{f}(\mathrm{x})=\int 8 \mathrm{x}^{3} \mathrm{dx}-\int 2 \mathrm{xdx}$ $...
Read More →Write down the common features of the connective tissue.
Question: Write down the common features of the connective tissue. Based on structure and function, differentiate between bones and cartilages. Solution: Connective tissues have an important role in the binding and connection of different tissues and organs. They provide structural rigidity support and protection of the body. They can regenerate Fibres, cells and matrix are present in connective tissues....
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