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. $f(x)=-2 x^{3}-9 x^{2}-12 x+1$ Solution: Given:- Function $f(x)=-2 x^{3}-9 x^{2}-12 x+1$ 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 ...
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. $f(x)=2 x^{3}-24 x+107$ Solution: Given:- Function $f(x)=2 x^{3}-24 x+107$ 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 $a l l x \in(a, b)$, then $f(x)$ is decreasing on $(a, b)$ Algorithm:- (i) Obtain the function and put it equal to $f(x)$ (ii) ...
Read More →A straight line passes through the point
Question: A straight line passes through the point (5, -2) and the portion of the line intercepted between the axes is divided at this point in the ratio 2 : 3. Find the equation of the line. Solution: Given : The ratio of the line intercepted between the axes is 2 :3 Let $\left(x_{1}, y_{1}\right)=A(a, 0)$ And $\left(x_{2}, y_{2}\right)=B(0, b)$ Where a and b are intercepts of the line. Formula used: The equation of the line is : $\frac{x}{a}+\frac{y}{b}=1$ And the co-ordinate axis is divided a...
Read More →Find the equation of the line whose portion intercepted between the
Question: Find the equation of the line whose portion intercepted between the coordinate axes is divided at the point (5, 6) in the ratio 3 : 1. Solution: To Find: The equation of the line whose portion intercepted between the coordinate axes is divided at the point (5, 6) in the ratio 3 : 1. Given : The coordinate axes is divided in the ratio 3 : 1 $\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)=\mathrm{A}(\mathrm{a}, 0)$ $\left(\mathrm{x}_{2}, \mathrm{y}_{2}\right)=\mathrm{B}(0, \mathrm{~b})$ Whe...
Read More →Match the ions given in Column I
Question: Match the ions given in Column I with their nature-given in Column II. Solution: (i) is a, b and d (ii) is b (iii) is b (iv) are c and d...
Read More →Find the equation of the line whose portion intercepted between the axes is
Question: Find the equation of the line whose portion intercepted between the axes is bisected at the point (3, -2). Solution: To Find: The equation of the line whose portion intercepted between the axes is bisected at the point (3, -2). Formula used: Let the equation of the line be $\frac{x}{a}+\frac{y}{b}=1$ Since it is given that this equation, whose portion is intercepted between the axes is bisected i.e.; is divided into ratio $1: 1$. Let $A(a, 0)$ and $B(0, b)$ be the points foring the coo...
Read More →Match the intermediates given in Column
Question: Match the intermediates given in Column I with their probable structure in Column II. Solution: (i) is a (ii) is a (iii) is b...
Read More →Match Column I with Column II.
Question: Match Column I with Column II. Solution: (i) is c (ii) is e (iii) is a (iv) is b (v) is d...
Read More →Match the terms mentioned in Column
Question: Match the terms mentioned in Column I with the terms in Column II. Solution: (i) is c (ii) is f (iii) is b (iv) is a (v) is d (vi) is e...
Read More →Match the type of mixture of compounds
Question: Match the type of mixture of compounds in Column I with the technique of separation/purification given in Column II. Solution: (i) is e (ii) is d (iii) is a (iv) is b (v) is c...
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 (22, -6) and whose intercept on the x-axis exceeds the intercept on the y-axis by 5. Solution: To Find:The equation of the line that passes through the point (22, -6) and intercepts on the x-axis exceeds the intercept on the y-axis by 5. Given : let x-intercept be a and $y$-intercept be $b$. According to the question : $a=b+5$ Formula used: And the given point satisfies the equation of the line, so $\frac{x}{a}+\frac{y}{b}=1$...
Read More →Which of the two structures (A) and (B)
Question: Which of the two structures (A) and (B) given below is more stabilised by resonance? Explain. (A) CH3COOH (B) CH3COO- Solution: The two resonating structure in compound A, are not equivalent, whereas compound B, having a negative charge on the oxygen atom is having the two equivalent structures, hence the compound B is more stable....
Read More →By mistake, an alcohol (boiling point 97°C)
Question: By mistake, an alcohol (boiling point 97C) was mixed with a hydrocarbon (boiling point 68C). Suggest a suitable method to separate the two compounds. Explain the reason for your choice. Solution: We can separate the mixture by steam distillation. Both alcohol and hydrocarbon dont have much difference in the boiling point. Steam distillation is particularly used for the purification of temperature-sensitive material....
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 (22, -6) and whose intercept on the x-axis exceeds the intercept on the y-axis by 5. Solution: To Find:The equation of the line that passes through the point (22, -6) and intercepts on the x-axis exceeds the intercept on the y-axis by 5. Given : let x-intercept be a and $y$-intercept be $b$. According to the question : $a=b+5$ Formula used: And the given point satisfies the equation of the line, so $\frac{x}{a}+\frac{y}{b}=1$...
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. $f(x)=6+12 x+3 x^{2}-2 x^{3}$ Solution: Given:- Function $f(x)=-2 x^{3}+3 x^{2}+12 x+6$ 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 →Resonance structures of propenal are given below.
Question: Resonance structures of propenal are given below. Which of these resonating structures is more stable? Give a reason for your answer. Solution: The structure I will be more stable than structure II because all the atoms are having complete octet in structure I and the carbon atom having positive charge dont have a complete octet in structure II....
Read More →Why does SO3 act as an electrophile?
Question: Why does SO3 act as an electrophile? Solution: Oxygen is more electronegative than sulphur and the electron density at S in SO3 is reduced and they need more electrons to suppress the partial positive charge and hence they act as an electrophile....
Read More →Which of the following compounds
Question: Which of the following compounds will not exist as a resonance hybrid. Give the reason for your answer: Solution: Conjugation is only possible when an atom has any of the charge or alternative position of the double bond. Here CH3OH will not exist as resonance hybrid because of the reason....
Read More →Give three points of differences between
Question: Give three points of differences between the inductive effect and resonance effect. Solution: 1. In the inductive effect, the transmission of the electron is only from sigma bond whereas in resonance effect the transmission occurs from both sigma and pi bond. 2. It is possible when the polarization of bond takes place whereas the resonance effect is possible only if the system is conjugated. 3. Electronegativity plays role in inductive effect whereas in resonance effect more number of ...
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. $f(x)=2 x^{3}-9 x^{2}+12 x-5$ Solution: Given:- Function $f(x)=2 x^{3}-9 x^{2}+12 x-5$ 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...
Read More →Identify the most stable species in the
Question: Identify the most stable species in the following set of ions giving reasons: Solution: (i) C+H3will be more stable as the bromine atom destabilizes the positive charge on a carbon atom. Bromine atom is an electron-withdrawing group and has a lone pair of electrons. (ii) CCl3will be most stable because chlorine is more electron-withdrawing atom and the negative charge on carbon will be stabilized by the chlorine atom. As the number of chlorine atom attached to carbocation increases the...
Read More →Find the equation of the line passing through the point
Question: Find the equation of the line passing through the point (2, 2) and cutting off intercepts on the axes, whose sum is 9. Solution: To Find: The equation of the line passing through the point (2, 2) and cutting off intercepts on the axes, whose sum is 9. Given : Let $\mathrm{a}$ and $\mathrm{b}$ be two intercepts of $\mathrm{x}$-axis and $\mathrm{y}$-axis respectively. sum of the intercepts is $9, i . e, a+b=9$ $\Rightarrow a=9-b$ or $b=9-a$ Formula used: The equation of a line is given b...
Read More →Draw the resonance structures
Question: Draw the resonance structures of the following compounds; Solution:...
Read More →Write structural formulae for compounds named as-
Question: Write structural formulae for compounds named as- (a) 1-Bromoheptane (b) 5-Bromoheptanoic acid Solution:...
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. $f(x)=2 x^{3}+9 x^{2}+12 x+20$ Solution: Given:- Function $f(x)=2 x^{3}+9 x^{2}+12 x+20$ 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...
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