If a square is inscribed in a circle,
Question: If a square is inscribed in a circle, what is the ratio of the areas of the circle and the square? Solution: We have the following situation LetBDbe the diameter and diagonal of the circle and the square respectively. We know that area of the circle $=\pi r^{2}$ Area of the square $=\operatorname{side}^{2}$ As we know that diagonal of the square is the diameter of the square. Diagonal $=2 r$ Side of the square $=\frac{\text { diagonal }}{\sqrt{2}}$......(1) Substituting diagonal $=2 r$...
Read More →D and E are points on the sides AB and AC respectively of a
Question: $D$ and $E$ are points on the sides $A B$ and $A C$ respectively of a $\triangle A B C$. In each of the following cases, determine whether $D E \| B C$ or not. (i) $A D=5.7 \mathrm{~cm}, D B=9.5 \mathrm{~cm}, B D=4.8 \mathrm{~cm}$ and $E C=8 \mathrm{~cm}$. (ii) $A B=11.7 \mathrm{~cm}, A C=11.2 \mathrm{~cm}, B D=6.5 \mathrm{~cm}$ and $A E=4.2 \mathrm{~cm}$. (iii) $A B=10.8 \mathrm{~cm}, A D=6.3 \mathrm{~cm}, A C=9.6 \mathrm{~cm}$ and $E C=4 \mathrm{~cm}$. (iv) $A D=7.2 \mathrm{~cm}, A E...
Read More →In a circle of radius 10 cm,
Question: In a circle of radius 10 cm, an arc subtends an angle of 108 at the centre. what is the area of the sector in terms of ? Solution: We have given the radius of the circle and angle subtended at the centre of the circle. $r=10 \mathrm{~cm}$ $\theta=108^{\circ}$ Now we will find the area of the sector. Area of the sector $=\frac{\theta}{360} \times \pi r^{2}$ Substituting the values we get, Area of the sector $=\frac{108}{360} \times \pi \times 10^{2}$....(1) Now we will simplify the equa...
Read More →The half-life for radioactive decay of
Question: The half-life for radioactive decay of14C is 5730 years. An archaeological artifact containing wood had only 80% of the14C found in a living tree. Estimate the age of the sample. Solution: Here, $k=\frac{0.693}{t_{1 / 2}}$ $=\frac{0.693}{5730}$ years $^{-1}$ It isknown that, $t=\frac{2.303}{k} \log \frac{[\mathrm{R}]_{0}}{[\mathrm{R}]}$ $=\frac{2.303}{\frac{0.693}{5730}} \log \frac{100}{80}$ = 1845 years (approximately) Hence, the age of the sample is 1845 years....
Read More →What is the area of a sector of a circle of radius 5 cm
Question: What is the area of a sector of a circle of radius 5 cm formed by an arc of length 3.5 cm? Solution: We have $r=5 \mathrm{~cm}$ length of the $\operatorname{arc}(l)=3.5 \mathrm{~cm}$ Now we will find the area of the sector. Area of the sector $=\frac{1}{2} \times l r$ Substituting the values we get, Area of the sector $=\frac{1}{2} \times 3.5 \times 5$...(1) Now we will simplify the equation (1) as below, Area of the sector $=\frac{1}{2} \times 17.5$ Area of the sector $=8.75$ Therefor...
Read More →Calculate the half-life of a first order reaction from their rate constants given below:
Question: Calculate the half-life of a first order reaction from their rate constants given below: (i)200s1(ii)2min1(iii)4years1 Solution: (i) Half life, $t_{1 / 2}=\frac{0.693}{k}$ $=\frac{0.693}{200 \mathrm{~s}^{-1}}$ $=3.47 \times 10^{-3} \mathrm{~s}$ (approximately) (ii) Half life, $t_{1 / 2}=\frac{0.693}{k}$ $=\frac{0.693}{2 \mathrm{~min}^{-1}}$ = 0.35 min (approximately) (iii) Half life, $t_{1 / 2}=\frac{0.693}{k}$ $=\frac{0.693}{4 \text { years }^{-1}}$ = 0.173 years (approximately)...
Read More →D and E are points on the sides AB and AC respectively of a
Question: $D$ and $E$ are points on the sides $A B$ and $A C$ respectively of a $\triangle A B C$ such that $D E \| B C$. Find the value of $x$, when (i) $A D=x \mathrm{~cm}, D B=(x-2) \mathrm{cm}$, $A E=(x+2) \mathrm{cm}$ and $E C=(x-1) \mathrm{cm}$. (ii) $A D=4 \mathrm{~cm}, D B=(x-4) \mathrm{cm}, A E=8 \mathrm{~cm}$ and $E C=(3 x-19) \mathrm{cm}$. (iii) $A D=(7 x-4) \mathrm{cm}, A E=(5 x-2) \mathrm{cm}$, $D B=(3 x+4) \mathrm{cm}$ and $E C=3 x \mathrm{~cm}$. Solution: (i) In $\triangle A B C$,...
Read More →What is the angle subtended at the centre of
Question: What is the angle subtended at the centre of a circle of radius 6 cm by an arc of length 3 cm? Solution: We have $r=6 \mathrm{~cm}$ length of the $\operatorname{arc}=3 \pi \mathrm{cm}$ We will find the angle subtended at the centre of a circle. Length of the $\operatorname{arc}=\frac{\theta}{360} \times 2 \pi r$ Substituting the values we get, $3 \pi=\frac{\theta}{360} \times 2 \pi \times 6$....(1) Now we will simplify the equation (1) as below, $3 \pi=\frac{\theta}{360} \times 12 \pi$...
Read More →The reaction between A and B is first order with respect to A and zero order with respect to B.
Question: The reaction between A and B is first order with respect to A and zero order with respect to B. Fill in the blanks in the following table: Solution: The given reaction is of the first order with respect to A and of zero order with respect to B. Therefore, the rate of the reaction is given by, Rate =k[A]1[B]0 ⇒Rate =k[A] From experiment I, we obtain 2.0 102mol L1min1= k (0.1 mol L1) ⇒k= 0.2 min1 From experiment II, we obtain 4.0 102mol L1min1= 0.2 min1[A] ⇒[A] = 0.2 mol L1 From experime...
Read More →The following results have been obtained during the kinetic studies of the reaction:
Question: The followingresults have been obtained during the kinetic studies of the reaction: 2A + B C + D Determine the rate law and the rate constant for the reaction. Solution: Let the order of the reaction with respect to A bexand with respect to B bey. Therefore, rate of the reaction is given by, Rate $=k[\mathrm{~A}]^{x}[\mathrm{~B}]^{y}$ According to the question, $6.0 \times 10^{-3}=k[0.1]^{x}[0.1]^{y}$ ...(i) $7.2 \times 10^{-2}=k[0.3]^{x}[0.2]^{y}$ ...(ii) $2.88 \times 10^{-1}=k[0.3]^{...
Read More →What is the length (in terms of π) of the arc
Question: What is the length (in terms of ) of the arc that subtends an angle of 36 at the centre of a circle of radius 5 cm? Solution: We have $r=5 \mathrm{~cm}$ $\theta=36^{\circ}$ We have to find the length of the arc. Length of the $\operatorname{arc}=\frac{\theta}{360} \times 2 \pi r$ Substituting the values we get, Length of the $\operatorname{arc}=\frac{36}{360} \times 2 \pi \times 5$.....(1) Now we will simplify the equation (1) as below, Length of the $\operatorname{arc}=\frac{1}{10} \t...
Read More →Write the area of the sector of a circle whose radius is r and length of the arc is l.
Question: Write the area of the sector of a circle whose radius isrand length of the arc is l. Solution: We know that area of the sector of the circle of radius $r=\frac{\theta}{360} \times \pi r^{2}$ Length of the arc $=\frac{\theta}{360} \times 2 \pi r$ But we have given that length of the arc So, $l=\frac{\theta}{360} \times 2 \pi r$...(1) Area of the sector $=\frac{\theta}{360} \times \pi r^{2}$ Now we will adjust 2 in the following way, Area of the sector $=\frac{\theta}{360} \times \frac{2...
Read More →D and E are points on the sides AB and AC, respectively, of a
Question: $D$ and $E$ are points on the sides $A B$ and $A C$, respectively, of a $\triangle A B C$, such that $D E \| B C$. (i) IfAD= 3.6 cm,AB= 10 cm andAE= 4.5 cm, findECandAC.(ii) IfAB= 13.3 cm,AC= 11.9 cm andEC= 5.1 cm, findAD. (iii) If $\frac{A D}{D B}=\frac{4}{7}$ and $A C=6.6 \mathrm{~cm}$, find $A E$. (iv) If $\frac{A D}{A B}=\frac{8}{15}$ and $E C=3.5 \mathrm{~cm}$, find $A E$. Solution: (i) In $\triangle A B C$, it is given that $D E \| B C$. Applying Thales' theorem, we get: $\frac{A...
Read More →If the circumference of two circles are in the ratio 2 : 3,
Question: If the circumference of two circles are in the ratio 2 : 3, what is the ratio of their areas? Solution: We are given ratio of circumferences of two circles. If $C=2 \pi r$ and $C^{\prime}=2 \pi r^{\prime}$ are circumferences of two circles such that $\frac{C}{C^{\prime}}=\frac{2}{3}$ $\Rightarrow \frac{2 \pi r}{2 \pi r^{\prime}}=\frac{2}{3} \ldots \ldots$(1) Simplifying equation (1) we get, $\frac{r}{r^{\prime}}=\frac{2}{3}$ Let $A=\pi r^{2}$ and $A^{\prime}=\pi r^{\prime 2}$ are the a...
Read More →In a reaction between A and B,
Question: In a reaction between A and B, the initial rate of reaction (r0) was measured for different initial concentrations of A and B as given below: What is the order of the reaction with respect to A and B? Solution: Let the order of the reaction with respect to A bexand with respect to B bey. Therefore, $\mathrm{r}_{0}=k[\mathrm{~A}]^{x}[\mathrm{~B}]^{y}$ $5.07 \times 10^{-5}=k[0.20]^{x}[0.30]^{y}$ ...(i) $5.07 \times 10^{-5}=k[0.20]^{x}[0.10]^{y}$ ...(ii) $1.43 \times 10^{-4}=k[0.40]^{x}[0...
Read More →What is the ratio of the areas of a circle and an
Question: What is the ratio of the areas of a circle and an equilateral triangle whose diameter and a side are respectively equal? Solution: We are given that diameter and side of an equilateral triangle are equal. Letdandaare the diameter and side of circle and equilateral triangle respectively. Therefored=a We know that area of the circle $=\pi r^{2}$ Area of the equilateral triangle $=\frac{\sqrt{3}}{4} a^{2}$ Now we will find the ratio of the areas of circle and equilateral triangle. So, $\f...
Read More →A reaction is first order in A and second order in B.
Question: A reaction is first order in A and second order in B. (i)Write the differential rate equation. (ii)How is the rate affected on increasing the concentration of B three times? (iii)How is the rate affected when the concentrations of both A and B are doubled? Solution: (i)The differential rate equation will be $-\frac{d[\mathrm{R}]}{d t}=k[\mathrm{~A}][\mathrm{B}]^{2}$ (ii)If the concentration of B is increased three times, then $-\frac{d[\mathrm{R}]}{d t}=k[\mathrm{~A}][3 \mathrm{~B}]^{2...
Read More →In the following figure, ABC is a right angled triangle in which
Question: In the following figure,ABCis a right angled triangle in which A= 90,AB= 21 cm andAC= 28 cm. Semi-circles are described onAB,BCandACas diameters. Find the area of the shaded region. Solution: We have given three semi-circles and one right angled triangle. $\therefore$ Area of shaded region $=$ Area of semi-circle with $\mathrm{AB}$ as a diameter + Area of semi-circle with $\mathrm{AC}$ as a diameter + Area of right angled $\mathrm{ABC}$ - Area of semi-circle with $\mathrm{BC}$ as a dia...
Read More →In a pseudo first order hydrolysis of ester in water, the following results were obtained:
Question: In a pseudo first order hydrolysis of ester in water, the following results were obtained: (i)Calculate the average rate of reaction between the time interval 30 to 60 seconds. (ii)Calculate the pseudo first order rate constant for the hydrolysis of ester. Solution: (i) Average rate of reaction between the time interval, 30 to 60 seconds, $=\frac{d[\text { Ester }]}{d t}$ $=\frac{0.31-0.17}{60-30}$ $=\frac{0.14}{30}$ = 4.67 103mol L1s1 (ii)For a pseudo first order reaction, $k=\frac{2....
Read More →What is the effect of temperature on the rate constant of a reaction?
Question: What is the effect of temperature on the rate constant of a reaction? How can this temperature effect on rate constant be represented quantitatively? Solution: The rate constant is nearly doubled witha rise in temperature by 10 for a chemical reaction. The temperature effect on the rate constant can be represented quantitatively by Arrhenius equation, $k=\mathrm{A} e^{-E_{\mathrm{a}} / \mathrm{R} T}$ where,kis the rate constant, A is the Arrhenius factor or the frequency factor, R is t...
Read More →A reaction is second order with respect to a reactant.
Question: A reaction is second order with respect to a reactant. How is the rate of reaction affected if the concentration of the reactant is (i)doubled(ii)reduced to half? Solution: Letthe concentration of the reactant be [A] =a Rate of reaction, R =k[A]2 =ka2 (i)If the concentration of the reactant is doubled, i.e. [A] = 2a, then the rate of the reaction would be $\mathrm{R}^{\prime}=k(2 a)^{2}$ = 4ka2 = 4 R Therefore, the rate of the reaction would increase by 4 times. (ii) If the concentrati...
Read More →In the following figure, AB = 36 cm and M is mid-point of AB.
Question: In the following figure,AB= 36 cm and M is mid-point ofAB. Semi-circles are drawn onAB,AMandMBas diameters.Acircle with centreCtouches all the three circles. Find the area of the shaded region. Solution: We have given two semi-circles and one circle. Area of the shaded region = area of semicircle with diameter AB area of two semicircles with diameters AM and MB - area of circle ..(1) Let us calculate the area of the semi-circle with AB as a diameter. Area of semi-circle with $\mathrm{A...
Read More →Mention the factors that affect the rate of a chemical reaction.
Question: Mention the factors that affect the rate of a chemical reaction. Solution: The factors that affect the rate of a reaction areas follows. (i)Concentration of reactants (pressure in case of gases) (ii)Temperature (iii)Presence of a catalyst...
Read More →The decomposition of dimethyl ether leads to the formation of
Question: The decomposition of dimethyl ether leads to the formation of CH4, H2and CO and the reaction rate is given by Rate =k[CH3OCH3]3/2 The rate of reaction is followed by increase in pressure in a closed vessel, so the rate can also be expressed in terms of the partial pressure of dimethyl ether, i.e., Rate $=k\left(p_{\mathrm{CH}_{3} \mathrm{OCH}_{3}}\right)^{3 / 2}$ If the pressure is measured in bar andtime in minutes, then what are the units of rate and rate constants? Solution: If pres...
Read More →The decomposition of
Question: The decomposition of NH3on platinum surface is zero order reaction. What are the rates of production of N2and H2ifk= 2.5 104mol1L s1? Solution: The decomposition of NH3on platinum surface is represented by the following equation. $2 \mathrm{NH}_{3(g)} \stackrel{\mathrm{P}_{1}}{\longrightarrow} \mathrm{N}_{2(g)}+3 \mathrm{H}_{2(g)}$ Therefore, Rate $=-\frac{1}{2} \frac{d\left[\mathrm{NH}_{3}\right]}{d t}=\frac{d\left[\mathrm{~N}_{2}\right]}{d t}=\frac{1}{3} \frac{d\left[\mathrm{H}_{2}\r...
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