Find the intervals in which the following functions are increasing or decreasing.
Question: Find the intervals in which the following functions are increasing or decreasing. $f(x)=6-9 x-x^{2}$ Solution: Given:- Function $f(x)=6-9 x-x^{2}$ Theorem:- Let $f$ be a differentiable real function defined on an open interval $(a, b)$. (i) If $f^{\prime}(x)0$ for all $x \in(a, b)$, then $f(x)$ is increasing on $(a, b)$ (ii) If $f^{\prime}(x)0$ for all $x \in(a, b)$, then $f(x)$ is decreasing on $(a, b)$ Algorithm:- (i) Obtain the function and put it equal to $f(x)$ (ii) Find $f^{\prim...
Read More →Identify the pairs of compounds
Question: Identify the pairs of compounds that represent chain isomerism. Solution: I and III, I and IV, II and III, II and IV, II and IV, V and VII, VI and VII are the compounds which represent chain isomerism. Compounds having same molecular formula but vary in the length of the parent chain(as they are branched) is called chain isomerism....
Read More →Identify the pairs of compounds
Question: Identify the pairs of compounds that represent position isomerism. Solution: I and II, III and IV, V and VI are the compounds which represent the position isomerism. Position isomerism means, where the length of the parent chain and functional group is the same but the position of the functional group is different, then it is called position isomer....
Read More →Find the equation of the line whose y
Question: Find the equation of the line whose y - intercept is - 3 and which is perpendicular to the line joining the points ( - 2, 3) and (4, - 5). Solution: Given: The line perpendicular to the line passing through $(-2,3)$ and $(4,-5)$ has the $y-$ intercept of $-3$. Formula to be used: If (a,b) and (c,d) are two points then the equation of the line passing through them is $\frac{\mathrm{y}-\mathrm{d}}{\mathrm{x}-\mathrm{c}}=\frac{\mathrm{d}-\mathrm{b}}{\mathrm{c}-\mathrm{a}}$ Product of slop...
Read More →Identify the pairs of compounds
Question: Identify the pairs of compounds which are functional group isomers. Solution: I and V, I and VI, I and VII, II and V, II and VI, II and VII, III and V, III and VI, III and VII, IV and V, IV and VI, IV and VII. Alcholol is the function group of each ether in the given seven compounds....
Read More →Which of the above compounds
Question: Which of the above compounds form pairs of metamers? Solution: V and VI form pairs of metamers. (metamers of ethers). Here in compounds V and VI, the chain length is the same but the distribution of carbons is different around oxygen atom....
Read More →Find the intervals in which the following functions are increasing or decreasing.
Question: Find the intervals in which the following functions are increasing or decreasing. $f(x)=x^{2}+2 x-5$ Solution: Given:- Function $f(x)=x^{2}+2 x-5$ Theorem:- Let $f$ be a differentiable real function defined on an open interval $(a, b)$. (i) If $f^{\prime}(x)0$ for all $x \in(a, b)$, then $f(x)$ is increasing on $(a, b)$ (ii) If $f^{\prime}(x)0$ for all $x \in(a, b)$, then $f(x)$ is decreasing on $(a, b)$ Algorithm:- (i) Obtain the function and put it equal to $f(x)$ (ii) Find $f^{\prim...
Read More →Hyperconjugation involves delocalisation of ___________.
Question: Hyperconjugation involves delocalisation of ___________. (i) electrons of carbon-hydrogen bond of an alkyl group directly attached to an atom of the unsaturated system. (ii) electrons of carbon-hydrogen bond of alkyl group directly attached to the positively charged carbon atom. (iii) -electrons of carbon-carbon bond (iv) lone pair of electrons Solution: Option (i) and (ii) are the answers....
Read More →A nucleophile is a species that should have
Question: A nucleophile is a species that should have (i) a pair of electrons to donate (ii) positive charge (iii) negative charge (iv) electron-deficient species Solution: Option (i) and (iii) are the answers....
Read More →Which of the following pairs are not functional
Question: Which of the following pairs are not functional group isomers? (i) II and III (ii) II and IV (iii) I and IV (iv) I and II Solution: Option (i) and (iii) are the answers....
Read More →Which of the following pairs are position isomers?
Question: Which of the following pairs are position isomers? (i) I and II (ii) II and III (iii) II and IV (iv) III and IV Solution: Option (ii) is the answer....
Read More →Find the intervals in which the following functions are increasing or decreasing.
Question: Find the intervals in which the following functions are increasing or decreasing. $f(x)=10-6 x-2 x^{2}$ Solution: Given:- Function $f(x)=10-6 x-2 x^{2}$ Theorem:- Let $f$ be a differentiable real function defined on an open interval $(a, b)$. (i) If $f^{\prime}(x)0$ for $a l l_{x \in}(a, b)$, then $f(x)$ is increasing on $(a, b)$ (ii) If $f^{\prime}(x)0$ for all $x \in(a, b)$, then $f(x)$ is decreasing on $(a, b)$ Algorithm:- (i) Obtain the function and put it equal to $f(x)$ (ii) Find...
Read More →Find the equation of the line which is perpendicular to the line
Question: Find the equation of the line which is perpendicular to the line 3x + 2y = 8 and passes through the midpoint of the line joining the points (6, 4) and (4, - 2). Solution: Given: The given line is $3 x+2 y=8$. The perpendicular line passes through the midpoint of $(6,4)$ and $(4,-2)$. Formulae to be used: The product of slopes of two perpendicular lines $=-1$. If (a,b) and (c,d) be two points, then their midpoint is given by $\left(\frac{a+c}{2}, \frac{b+d}{2}\right)$ The slope of this ...
Read More →Electrophiles are electron seeking species.
Question: Electrophiles are electron seeking species. Which of the following groups contain only electrophiles? (i) BF3, NH3, H2O (ii) AlCl3, SO3 , + NO2 (iii) NO2+ , CH3+ , CH3 C+ = O (iv) C2H-5, C2H5, CH5+ Solution: Option (ii) AlCl3, SO3 , + NO2 and (iii)NO2+ , CH3+ , CH3 C+ = O are the answers....
Read More →In which of the following representations
Question: In which of the following representations given below spatial arrangement of group/ atom different from that given in structure A? Solution: Option (i), (iii) and (iv) are the answers....
Read More →Which of the following compounds contain all the carbon
Question: Which of the following compounds contain all the carbon atoms in the same hybridisation state? (i) HC CC CH (ii) CH3C CCH3 (iii) CH2 = C = CH2 (iv) CH2 = CHCH = CH2 Solution: Option (i)HC CC CH and (iv)CH2 = CHCH = CH2 are the answers....
Read More →Without using the derivative show that the function
Question: Without using the derivative show that the function $f(x)=7 x-3$ is strictly increasing function on $R$. Solution: Given, $f(x)=7 x-3$ Lets $\mathrm{x}_{1}, \mathrm{x}_{2} \in \mathrm{R}$ and $\mathrm{x}_{1}\mathrm{x}_{2}$ $\Rightarrow 7 \mathrm{X}_{1}7 \mathrm{X}_{2}$ $\Rightarrow 7 \mathrm{X}_{1}-37 \mathrm{X}_{2}-3$ $\Rightarrow \mathrm{f}\left(\mathrm{x}_{1}\right)\mathrm{f}\left(\mathrm{x}_{2}\right)$ $\therefore f(x)$ is strictly increasing on $R$....
Read More →The addition of HCl to alkene proceeds in two steps.
Question: The addition of HCl to alkene proceeds in two steps. The first step is the attack of H+ ion C=C to a portion which can be shown as Solution: Option (ii) is the answer....
Read More →Find the equation of the line that has x - intercept - 3
Question: Find the equation of the line that has x - intercept - 3 and which is perpendicular to the line 3x + 5y = 4 Solution: Given: The given line is $3 x+5 y=4$. The perpendicular line has an $x-$ intercept of $-3$. Formula to be used: The product of slopes of two perpendicular lines $=-1$. The slope of this line is $-3 / 5$. $\therefore$ the slope of the perpendicular line $=$ $\frac{-1}{-3 / 5}=5 / 3$ The equation of the line can be written in the form $y=\left(\frac{5}{3}\right) x+c$ (c i...
Read More →A covalent bond can undergo fission in two different ways.
Question: A covalent bond can undergo fission in two different ways. The correct representation involving heterolytic fission of CH3Br is Solution: Option (ii) is the answer....
Read More →Electrophilic addition reactions proceed in two steps.
Question: Electrophilic addition reactions proceed in two steps. The first step involves the addition of an electrophile. Name the type of intermediate formed in the first step of the following addition reaction. H3 CHC = CH2 + H+ ? (i) 2 Carbanion (ii) 1 Carbocation (iii) 2 Carbocation (iv) 1 Carbanion Solution: Option (iii)2 Carbocation is the answer....
Read More →Ionic species are stabilised by the dispersal
Question: Ionic species are stabilised by the dispersal of charge. Which of the following carboxylate ion is the most stable? Solution: Option (iv) is the answer....
Read More →Find the equation of the line passing through the point (2, 4) and
Question: Find the equation of the line passing through the point (2, 4) and perpendicular to the x - axis. Solution: Given: The line is perpendicular to $x$-axis and passes through $(2,4)$ The equation of the line perpendicular to the $x$-axis $(y=0)$ can be represented as $x=$ c, where $c$ is a real constant. Now, this line passes through (2,4). c = 2 The required equation is x = 2...
Read More →Find the equation of the line passing through the point (2, 4) and
Question: Find the equation of the line passing through the point (2, 4) and perpendicular to the x - axis. Solution: Given: The line is perpendicular to $x$-axis and passes through $(2,4)$ The equation of the line perpendicular to the $x$-axis $(y=0)$ can be represented as $x=$ c, where $c$ is a real constant. Now, this line passes through (2,4). c = 2 The required equation is x = 2...
Read More →In which of the following compounds
Question: In which of the following compounds the carbon marked with an asterisk is expected to have the greatest positive charge? (i) *CH3CH2Cl (ii) *CH3CH2Mg+Cl (iii) *CH3CH2Br (iv) *CH3CH2CH3 Solution: Option (i) *CH3CH2Cl is the answer....
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