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. $f(x)=2 x^{3}-15 x^{2}+36 x+1$ Solution: Given:- Function $f(x)=2 x^{3}-15 x^{2}+36 x+1$ 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...
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. $f(x)=x^{3}-6 x^{2}-36 x+2$ Solution: Given:- Function $f(x)=x^{3}-6 x^{2}-36 x+2$ 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 $f(x)$ (...
Read More →Find the equation of the line which passes through the point
Question: Find the equation of the line which passes through the point $(3,-5)$ and cuts off intercepts on the axes which are equal in magnitude but opposite in sign. Solution: To Find: The equation of the line passing through (3, -5) and cuts off intercepts on the axes which are equal in magnitude but opposite in sign. Given : Let a and b be two intercepts of $x$-axis and $y$-axis respectively. According to the question $a=-b$ or $b=-a$ And $(3,-5)$ passes through the point $(x, y)$, thus satis...
Read More →Name the compounds whose line formulae
Question: Name the compounds whose line formulae are given below: Solution: (i) The name of the compound is 3-ethyl-4-methyl-5-heptane-2-one. (ii) The name of the compound is; 1-nitro-cyclohexane-2-ene....
Read More →Three students, Manish, Ramesh and
Question: Three students, Manish, Ramesh and Rajni were determining the extra elements present in an organic compound given by their teacher. They prepared the Lassaignes extract (L.E.) independently by the fusion of the compound with sodium metal. Then they added solid FeSO4 and dilute sulphuric acid to a part of Lassaignes extract. Manish and Rajni obtained Prussian blue colour but Ramesh got a red colour. Ramesh repeated the test with the same Lassaignes extract but again got red colour only....
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. $f(x)=5 x^{3}-15 x^{2}-120 x+3$ Solution: Given:- Function $f(x)=5 x^{3}-15 x^{2}-120 x+3$ Theorem:- Let $f$ be a differentiable real function defined on an open interval $(a, b)$. (i) If $\mathrm{f}^{\prime}(\mathrm{x})0$ for all $x \in(a, b)$, then $\mathrm{f}(\mathrm{x})$ is increasing on $(\mathrm{a}, \mathrm{b})$ (ii) If $\mathrm{f}^{\prime}(\mathrm{x})0$ for all $x \in(a, b)$, then $\mathrm{f}(\math...
Read More →Write structures of various carbocation
Question: Write structures of various carbocation that can be obtained from 2-methyl butane. Arrange these carbocations in order of increasing stability. Solution: the stability order is found to be: (III) (II) (I) (IV) This is because (III) is tertiary carbocation, (II) is secondary carbocation, (I) and (IV) are primary,...
Read More →Find the equation of the line and cuts off equal intercepts
Question: Find the equation of the line and cuts off equal intercepts on the coordinate axes and passes through the point (4,7). Solution: To Find: The equation of the line with equal intercepts on the coordinate axes and that passes through the point (4,7). Given : Let a and b be two intercepts of x-axis and y-axis respectively. Also, given that two intercepts are equal, i.e., $a=b$ And $(4,7)$ passes through the point $(x, y)$ Formula used: Now since intercept form of a line is given: $\frac{x...
Read More →The structure of triphenylmethylcation is given below.
Question: The structure of triphenylmethylcation is given below. This is very stable and some of its salts can be stored for months. Explain the cause of the high stability of this cation. Solution: Triphenylcarbocation is tertiary carbocation and the positive charge is on the carbon atom which is stabilized by the three phenyl group by resonance. Due to the resonance the stability increases....
Read More →Which of the following ions is more stable?
Question: Which of the following ions is more stable? Use resonance to explain your answer. Solution: The stability of carbocation is decided by inductive effect, hyperconjugation, resonance etc. As structure A is having a carbocation, which is primary and allylic carbocation, whereas the carbocation in structure B has secondary and allylic carbocation. Hence, structure B is more stable....
Read More →Find the equation of the line which cuts off intercepts 4 and -6
Question: Find the equation of the line which cuts off intercepts 4 and -6 on the x-axis and y-axis respectively. Solution: To Find:The equation of the line with intercepts 4 and -6 on the x-axis and yaxis respectively. Given : Let a and b be the intercepts on $x$-axis and $y$-axis respectively. Then, $x$-intercept be $a=4$ $y$-intercept be $b=-6$ Formula used: we know that intercept form of a line is given by $\frac{x}{a}+\frac{y}{b}=1$ $\frac{x}{4}+\frac{y}{-6}=1$ $-3 x+2 y=-12$ $3 x-2 y-12=0$...
Read More →Draw the possible resonance structures for
Question: Draw the possible resonance structures for and predict which of the structures is more stable. Give the reason for your answer. Solution: The structure C is more stable than that of the structure A because, in structure C, the octet of all the atoms is complete, whereas, in structure A, C-atom having a positive charge is not having 8 electrons in its valence shell....
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. $f(x)=8+36 x+3 x^{2}-2 x^{3}$ Solution: Given:- Function $f(x)=8+36 x+3 x^{2}-2 x^{3}$ 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...
Read More →If a liquid compound decomposes
Question: If a liquid compound decomposes at its boiling point, which method(s) can you choose for its purification. It is known that the compound is stable at low pressure, steam volatile and insoluble in water. Solution: As the liquid compound decomposes at its boiling point which means that the compound is heat-sensitive, hence we use Steam distillation for its purification. It is done for temperature-sensitive material....
Read More →Find the equation of the line which cuts off intercepts -3 and 5 on the x-axis
Question: Find the equation of the line which cuts off intercepts -3 and 5 on the x-axis and y-axis respectively. Solution: To Find: The equation of a line with intercepts -3 and 5 on the x-axis and yaxis respectively. Given :Let $\mathrm{a}$ and $\mathrm{b}$ be the intercepts on $\mathrm{x}$-axis and $\mathrm{y}$-axis respectively. Then, the $x$-intercept is $a=-3$ $y$-intercept is $b=5$ Formula used: we know that intercept form of a line is given by: $\frac{x}{a}+\frac{y}{b}=1$ $\frac{x}{-3}+\...
Read More →In DNA and RNA, the nitrogen atom is present
Question: In DNA and RNA, the nitrogen atom is present in the ring system. Can the Kjeldahl method be used for the estimation of nitrogen present in these? Give reasons. Solution: The Kjeldahl method cannot be used for the estimation of nitrogen present in DNA and RNA because nitrogen cannot be converted into ammonium sulphate in the conditions provided in this method. The difficulty in decomposing ring structures limits this test for DNA and RNA....
Read More →A line perpendicular to the line segment joining the points
Question: A line perpendicular to the line segment joining the points (1, 0) and (2, 3) divides it in the ratio 1 : 2. Find the equation of the line. Solution: Given: A line perpendicular to the line segment joining the points $(1,0)$ and $(2,3)$ divides it in the ratio $1: 2$. Formula to be used: If $(a, b)$ and $(c, d)$ are two points then the equation of the line passing through them is $\frac{\mathrm{y}-\mathrm{b}}{\mathrm{x}-\mathrm{a}}=\frac{\mathrm{b}-\mathrm{d}}{\mathrm{a}-\mathrm{c}}$ I...
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. $f(x)=5+36 x+3 x^{2}-2 x^{3}$ Solution: Given:- Function $f(x)=5+36 x+3 x^{2}-2 x^{3}$ 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...
Read More →Compounds with same molecular formula
Question: Compounds with same molecular formula but differing in their structures are said to be structural isomers. What type of structural isomerism is shown by Solution: In the above structures, both have the same functional group (thioether/sulphide), but there is a difference in the arrangement of atoms in the main chain. So they can exhibit chain isomerism....
Read More →Show the polarisation of carbon-magnesium
Question: Show the polarisation of carbon-magnesium bond in the following structure. CH3CH2CH2CH2MgX Solution: CH3CH2CH2-CH2+Mg+ X- Carbon is more electronegative than Mg because of the electronegativity difference in the bond between C-Mg is highly polarised. C being more electronegative the electron density is towards C hence Mg readily loses its electron to form C-( carbanion)....
Read More →Explain, how is the electronegativity
Question: Explain, how is the electronegativity of carbon atoms related to their state of hybridisation in an organic compound? Solution: Hybridization for carbon in organic compounds is of three types, sp, sp2 and sp3. More the s character more is the electronegativity of the carbon as s orbitals are closer to the nucleus, hence electrons held more closely....
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. $f(x)=2 x^{3}-12 x^{2}+18 x+15$ Solution: Given:- Function $f(x)=2 x^{3}-12 x^{2}+18 x+15$ 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 ...
Read More →What is the hybridisation of each
Question: What is the hybridisation of each carbon in H2C = C = CH2? Solution: The terminal carbons are sp2 hybridised as they form 3 (2 with H and 1 with C) sigma bonds and 1 pi bond(between carbons) while the centre carbon is sp hybridised (as it forms 2 sigma bonds, 1 with each carbon and 2 pi bonds, one with each carbon)....
Read More →For testing halogens in an organic compound
Question: For testing halogens in an organic compound with AgNO3 solution, sodium extract (Lassaignes test) is acidified with dilute HNO3. What will happen if a student acidifies the extract with dilute H2 SO4 in place of dilute HNO3? Solution: If dilute H2 SO4is added then sodium sulphide and sodium cyanine are not decomposed thereby we will not get the correct result if halides are also present....
Read More →Find the equation of the line passing through
Question: Find the equation of the line passing through ( - 3, 5) and perpendicular to the line through the points (2, 5) and ( - 3, 6). Solution: Given: The line perpendicular to the line passing through $(2,5)$ and $(-3,6)$ passes through $(-3,5)$. Formula to be used: If $(a, b)$ and $(c, d)$ are two points then the equation of the line passing through them is $\frac{y-b}{x-a}=\frac{b-d}{a-c}$ Product of slopes of two perpendicular lines $=-1$ The equation of the line joining points $(2,5)$ an...
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