Evaluate the following integral:
Question: Evaluate the following integral: $\int \frac{1}{x\left\{6(\log x)^{2}+7 \log x+2\right\}} d x$ Solution: Let substitute $u=\log x \Rightarrow d u=\frac{1}{x} d x$, so the given equation becomes $\int \frac{1}{x\left\{6(\log x)^{2}+7 \log x+2\right\}} d x=\int \frac{1}{\left\{6 u^{2}+7 u+2\right\}} d u \ldots$ (i) Factorizing the denominator, we get $\int \frac{1}{(2 u+1)(3 u+2)} d u$ The denominator is factorized, so let separate the fraction through partial fraction, hence let $\frac{...
Read More →Assertion: Polytetrafluoroethene is used
Question: Assertion: Polytetrafluoroethene is used in making non-stick cookware. Reason: Fluorine has the highest electronegativity. (i) Assertion and reason both are correct statements but reason does not explain the assertion. (ii) Assertion and reason both are correct statements and reason explain the assertion. (iii) Both assertion and reason are the wrong statements. (iv) The assertion is correct statement and reason is the wrong statement. (v) The assertion is the wrong statement and reaso...
Read More →Assertion: Network polymers are thermosetting.
Question: Assertion: Network polymers are thermosetting. Reason: Network polymers have high molecular mass. (i) Assertion and reason both are correct statements but reason does not explain the assertion. (ii) Assertion and reason both are correct statements and reason explain the assertion. (iii) Both assertion and reason are the wrong statements. (iv) The assertion is correct statement and reason is the wrong statement. (v) The assertion is the wrong statement and reason is the correct statemen...
Read More →Assertion: For making rubber synthetically,
Question: Assertion: For making rubber synthetically, isoprene molecules are polymerised. Reason: Neoprene (a polymer of chloroprene) is a synthetic rubber. (i) Assertion and reason both are correct statements but reason does not explain the assertion. (ii) Assertion and reason both are correct statements and reason explain the assertion. (iii) Both assertion and reason are the wrong statements. (iv) The assertion is correct statement and reason is the wrong statement. (v) The assertion is the w...
Read More →Find the principal value of each of the following :
Question: Find the principal value of each of the following : $\tan ^{-1} \sqrt{3}-\cot ^{-1}(-\sqrt{3})^{3}$ Solution: $\tan ^{-1} \sqrt{3}-\cot ^{-1}(-\sqrt{3})$ Putting the value of $\tan ^{-1} \sqrt{3}$ and using the formula $\cot ^{-1}(-x)=\pi-\cot ^{-1} x$ $=\frac{\pi}{3}-\left(\pi-\cot ^{-1}(\sqrt{3})\right)$ Putting the value of $\cot ^{-1}(\sqrt{3})$ $=\frac{\pi}{3}-\left(\pi-\frac{\pi}{6}\right)$ $=\frac{\pi}{3}-\frac{5 \pi}{6}$ $=-\frac{3 \pi}{6}$ $=-\frac{\pi}{2}$...
Read More →Assertion: Polyamides are best used as fibres
Question: Assertion: Polyamides are best used as fibres because of high tensile strength. Reason: Strong intermolecular forces (like hydrogen bonding within polyamides) lead to close packing of chains and increase the crystalline character, hence, provide high tensile strength to polymers. (i) Assertion and reason both are correct statements but reason does not explain the assertion. (ii) Assertion and reason both are correct statements and reason explain the assertion. (iii) Both assertion and ...
Read More →Evaluate the following integral:
Question: Evaluate the following integral: $\int \frac{x^{2}+1}{(2 x+1)\left(x^{2}-1\right)} d x$ Solution: Denominator is factorized, so let separate the fraction through partial fraction, hence let $\frac{x^{2}+1}{(2 x+1)\left(x^{2}-1\right)}$ $=\frac{x^{2}+1}{(2 x+1)(x-1)(x+1)}$ $\frac{\mathrm{x}^{2}+1}{(2 \mathrm{x}+1)(\mathrm{x}-1)(\mathrm{x}+1)}=\frac{\mathrm{A}}{2 \mathrm{x}+1}+\frac{\mathrm{B}}{\mathrm{x}-1}+\frac{\mathrm{C}}{\mathrm{x}+1} \ldots \ldots$ (i) $\Rightarrow \frac{x^{2}+1}{(...
Read More →Assertion: Olefinic monomers undergo addition
Question: Assertion: Olefinic monomers undergo addition polymerisation. Reason: Polymerisation of vinyl chloride is initiated by peroxides/ persulphates. (i) Assertion and reason both are correct statements but reason does not explain the assertion. (ii) Assertion and reason both are correct statements and reason explain the assertion. (iii) Both assertion and reason are the wrong statements. (iv) The assertion is correct statement and reason is the wrong statement. (v) The assertion is the wron...
Read More →Find the principal value of each of the following :
Question: Find the principal value of each of the following : $\tan ^{-1}\left(\tan \frac{7 \pi}{6}\right)$ Solution: $\tan ^{-1}\left(\tan \frac{7 \pi}{6}\right)=\tan ^{-1}\left(\tan \left(\pi+\frac{\pi}{6}\right)\right)$ [ Formula: $\tan (\pi+x)=\tan x$, as tan is positive in the third quadrant.] $=\tan ^{-1}\left(\tan \frac{\pi}{6}\right)\left[\right.$ Formula: $\left.\tan ^{-1}(\tan x)=x\right]$ $=\frac{\pi}{6}$...
Read More →Assertion: Most of the Synthetic polymers are not biodegradable.
Question: Assertion: Most of the Synthetic polymers are not biodegradable. Reason: Polymerisation process induces toxic character in organic molecules. (i) Assertion and reason both are correct statements but reason does not explain the assertion. (ii) Assertion and reason both are correct statements and reason explain the assertion. (iii) Both assertion and reason are the wrong statements. (iv) The assertion is correct statement and reason is the wrong statement. (v) The assertion is the wrong ...
Read More →Assertion: Rayon is a semi-synthetic polymer
Question: Assertion: Rayon is a semi-synthetic polymer and is taken as a better choice than cotton fabric. Reason: Mechanical and aesthetic properties of cellulose can be improved by acetylation. (i) Assertion and reason both are correct statements but reason does not explain the assertion. (ii) Assertion and reason both are correct statements and reason explain the assertion. (iii) Both assertion and reason are the wrong statements. (iv) The assertion is correct statement and reason is the wron...
Read More →Find the principal value of each of the following :
Question: Find the principal value of each of the following : $\cos ^{-1}\left(\cos \frac{13 \pi}{6}\right)$ Solution: $\cos ^{-1}\left(\cos \frac{13 \pi}{6}\right)=\cos ^{-1}\left(\cos \left(2 \pi+\frac{\pi}{6}\right)\right)$ [ Formula: $\cos (2 \pi+x)=\cos x, \cos$ is positive in the first quadrant. ] $=\cos ^{-1}\left(\cos \frac{\pi}{6}\right)$ [Formula: $\left.\cos ^{-1}(\cos x)=x\right]$ $=\frac{\pi}{6}$...
Read More →Match the polymers given in Column
Question: Match the polymers given in Column I with their repeating units given in Column II. Solution: (i) is d (ii) is a (iii) is b (iv) is e (v) is c...
Read More →Find the principal value of each of the following :
Question: Find the principal value of each of the following : $\cos ^{-1}\left(\cos \frac{7 \pi}{6}\right)$ Solution: $\cos ^{-1}\left(\cos \frac{7 \pi}{6}\right)=\cos ^{-1}\left(\cos \left(2 \pi-\frac{5 \pi}{6}\right)\right)$ [Formula: $\cos (2 \pi-x)=\cos (x)$, as $\cos$ has a positive vaule in the fourth quadrant. ] $=\cos ^{-1}\left(\cos \frac{5 \pi}{6}\right)\left[\right.$ Formula: $\cos ^{-1}(\cos x)=x$ $=\frac{5 \pi}{6}$...
Read More →Match materials are given in Column I
Question: Match materials are given in Column I with the polymers given in Column II. Solution: (i) is f (ii) is e (iii) is a (iv) is c (v) is b (vi) is d...
Read More →Match the polymers given in Column
Question: Match the polymers given in Column I with the type of linkage present in they have given in Column II. Solution: (i) is b (ii) is d (iii) is a (iv) is d (v) is c...
Read More →Find the principal value of each of the following :
Question: Find the principal value of each of the following : $\tan ^{-1}\left(\tan \frac{3 \pi}{4}\right)$ Solution: $\tan ^{-1}\left(\tan \frac{3 \pi}{4}\right)=\tan ^{-1}\left(\tan \left(\pi-\frac{\pi}{4}\right)\right)$ [Formula: $\tan (\pi-x)=-\tan (x)$, as tan is negative in the second quadrant.] $=\tan ^{-1}\left(-\tan \frac{\pi}{4}\right)$ [Formula: $\tan ^{-1}(\tan x)=x$ ] $=-\frac{\pi}{4}$...
Read More →Match the polymers given in Column
Question: Match the polymers given in Column I with the preferred mode of polymerisation followed by their monomers. Solution: (i) is d (ii) is a (iii) is b...
Read More →Evaluate the following integral:
Question: Evaluate the following integral: $\int \frac{x^{2}+6 x-8}{x^{3}-4 x} d x$ Solution: Denominator is factorized, so let separate the fraction through partial fraction, hence let $\frac{x^{2}+6 x-8}{x^{3}-4 x}$ $=\frac{x^{2}+6 x-8}{x\left(x^{2}-4\right)}$ $\frac{\mathrm{x}^{2}+6 \mathrm{x}-8}{\mathrm{x}(\mathrm{x}-2)(\mathrm{x}+2)}=\frac{\mathrm{A}}{\mathrm{x}}+\frac{\mathrm{B}}{\mathrm{x}-2}+\frac{\mathrm{C}}{\mathrm{x}+2} \ldots \ldots$ (i) $\Rightarrow \frac{x^{2}+6 x-8}{x(x-2)(x+2)}=\...
Read More →Find the principal value of each of the following :
Question: Find the principal value of each of the following : $\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)$ Solution: $\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)=\sin ^{-1}\left(\sin \left(\pi-\frac{\pi}{3}\right)\right)$ [ Formula: $\sin (\pi-x)=\sin x$ ) $=\sin ^{-1}\left(\sin \frac{\pi}{3}\right)$ [ Formula: $\sin ^{-1}(\sin x)=x$ ] $=\frac{\pi}{3}$...
Read More →Match the polymers given in Column I
Question: Match the polymers given in Column I with their main applications given in Column II. Solution: (i) is d (ii) is e (iii) is a (iv) is f (v) is b (vi) is c...
Read More →Find the principal value of each of the following :
Question: Find the principal value of each of the following : $\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)$ Solution: $\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)=\sin ^{-1}\left(\sin \left(\pi-\frac{\pi}{3}\right)\right)$ [ Formula: $\sin (\pi-x)=\sin x$ ) $=\sin ^{-1}\left(\sin \frac{\pi}{3}\right)$ [ Formula: $\sin ^{-1}(\sin x)=x$ ] $=\frac{\pi}{3}$...
Read More →Match the polymers given in Column
Question: Match the polymers given in Column I with their commercial names given inColumn II. Solution: (i) is b (ii) is c (iii) is a (iv) is e (v) is d...
Read More →Find the principal value of each of the following :
Question: Find the principal value of each of the following : $\operatorname{cosec}^{-1}(2)$ Solution: $\operatorname{cosec}^{-1}(2)$ Putting the value directly $=\frac{\pi}{6}$...
Read More →Match the polymers given in Column
Question: Match the polymers given in Column I with their chemical names given in Column II. Solution: (i) is c (ii) is a (iii) is b (iv) is e (v) is d...
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