At 1127 K and 1 atm pressure, a gaseous mixture
Question: At $1127 \mathrm{~K}$ and $1 \mathrm{~atm}$ pressure, a gaseous mixture of $\mathrm{CO}$ and $\mathrm{CO}_{2}$ in equilibrium with solid carbon has $90.55 \% \mathrm{CO}$ by mass $\mathrm{C}(\mathrm{s})+\mathrm{CO}_{2}(\mathrm{~g}) \longleftrightarrow 2 \mathrm{CO}(\mathrm{g})$ Calculate $K_{c}$ for this reaction at the above temperature. Solution: Let the total mass of the gaseous mixture be 100 g. Mass of $\mathrm{CO}=90.55 \mathrm{~g}$ And, mass of $\mathrm{CO}_{2}=(100-90.55)=9.45 ...
Read More →The length of 40 leaves of a plant are measured correct to the nearest millimetre,
Question: The length of 40 leaves of a plant are measured correct to the nearest millimetre, and the data obtained is represented in the following table. Find the median length of the leaves. Solution: The given series is in inclusive form. We may prepare the table in exclusive form and prepare the cumulative frequency table as given below : Here, N = 40 $\therefore \quad \frac{\mathbf{N}}{\mathbf{2}}=20$ The cumulative frequency just greater than 20 is 29 and the corresponding class is 144.5- 1...
Read More →Prove that cot 4x (sin 5x + sin 3x) = cot x (sin 5x – sin 3x)
Question: Prove that $\cot 4 x(\sin 5 x+\sin 3 x)=\cot x(\sin 5 x-\sin 3 x)$ Solution: L.H.S $=\cot 4 x(\sin 5 x+\sin 3 x)$ $=\frac{\cos 4 x}{\sin 4 x}\left[2 \sin \left(\frac{5 x+3 x}{2}\right) \cos \left(\frac{5 x-3 x}{2}\right)\right]$ $\left[\because \sin A+\sin B=2 \sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)\right]$ $=\left(\frac{\cos 4 x}{\sin 4 x}\right)[2 \sin 4 x \cos x]$ $=2 \cos 4 x \cos x$ R.H.S. $=\cot x(\sin 5 x-\sin 3 x)$ $=\frac{\cos x}{\sin x}\left[2 \cos \le...
Read More →A life insurance agent found the following data for distribution of ages of 100 policy holders.
Question: A life insurance agent found the following data for distribution of ages of 100 policy holders. Calculate the median age, if policies are only given to persons having age 18 years onwards but less than 60 year. Solution: Here, $\ell=35, \mathrm{n}=100, \mathrm{f}=33, \mathrm{cf}=45, \mathrm{~h}=5$ Median $=\ell+\left\{\frac{\frac{\mathbf{n}}{\mathbf{2}}-\mathbf{c f}}{\mathbf{f}}\right\} \times \mathrm{h}$ $=35+\left\{\frac{\mathbf{5 0}-\mathbf{4 5}}{\mathbf{3 3}}\right\} \times 5$ $=35...
Read More →Show that f is invertible. Find the inverse of f.
Question: Consider $f: \mathbf{R} \rightarrow \mathbf{R}$ given by $f(x)=4 x+3$. Show that $f$ is invertible. Find the inverse of $f$. Solution: $f: \mathbf{R} \rightarrow \mathbf{R}$ is given by, $f(x)=4 x+3$ One-one: Let $f(x)=f(y)$ fis a one-one function. Onto: For $y \in \mathbf{R}$, let $y=4 x+3$ $\Rightarrow x=\frac{y-3}{4} \in \mathbf{R}$ Therefore, for any $y \in \mathbf{R}$, there exists $x=\frac{y-3}{4} \in \mathbf{R}$ such that $f(x)=f\left(\frac{y-3}{4}\right)=4\left(\frac{y-3}{4}\ri...
Read More →Prove that $sin 2 x+2 sin 4 x+sin 6 x=4 cos ^{2} x sin 4 x$
Question: Prove that $\sin 2 x+2 \sin 4 x+\sin 6 x=4 \cos ^{2} x \sin 4 x$ Solution: L.H.S. $=\sin 2 x+2 \sin 4 x+\sin 6 x$ $=[\sin 2 x+\sin 6 x]+2 \sin 4 x$ $=\left[2 \sin \left(\frac{2 x+6 x}{2}\right) \cos \left(\frac{2 x-6 x}{2}\right)\right]+2 \sin 4 x$ $\left[\because \sin A+\sin B=2 \sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right)\right]$ $=2 \sin 4 x \cos (-2 x)+2 \sin 4 x$ $=2 \sin 4 x \cos 2 x+2 \sin 4 x$ $=2 \sin 4 x(\cos 2 x+1)$ $=2 \sin 4 x\left(2 \cos ^{2} x-1+1\righ...
Read More →If the median of the distribution given below is 28.5,
Question: If the median of the distribution given below is 28.5, find the values of x and y. Solution: Median = 28.5 lies in the class-interval (20-30). Then median class is (20-30). So, we have $\ell=20, \mathrm{f}=20, \mathrm{cf}=5+\mathrm{x}, \mathrm{h}=10, \mathrm{n}=60$ Median $=\ell+\left\{\frac{\frac{\mathbf{n}}{\mathbf{2}}-\mathbf{c f}}{\mathbf{f}}\right\} \times \mathrm{h}=28.528 .5=20+\left\{\frac{\mathbf{3 0}-\mathbf{( 5}+\mathbf{x})}{\mathbf{2 0}}\right\} \times 10$ $\Rightarrow 8.5=...
Read More →Prove that $cos ^{2} 2 x-cos ^{2} 6 x=sin 4 x sin 8 x$
Question: Prove that $\cos ^{2} 2 x-\cos ^{2} 6 x=\sin 4 x \sin 8 x$ Solution: It is known that $\cos A+\cos B=2 \cos \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right), \cos A-\cos B=-2 \sin \left(\frac{A+B}{2}\right) \sin \left(\frac{A-B}{2}\right)$ $\therefore$ L.H.S. $=\cos ^{2} 2 x-\cos ^{2} 6 x$ $=(\cos 2 x+\cos 6 x)(\cos 2 x-6 x)$ $=\left[2 \cos \left(\frac{2 x+6 x}{2}\right) \cos \left(\frac{2 x-6 x}{2}\right)\right]\left[-2 \sin \left(\frac{2 x+6 x}{2}\right) \sin \frac{(2 x-6 x...
Read More →If a and b are two odd positive integers such that a > b
Question: If $a$ and $b$ are two odd positive integers such that $ab$, then prove that one of the two numbers $\frac{a+b}{2}$ and $\frac{a-b}{2}$ is odd and the other is even. Solution: Given: Ifaandbare two odd positive integers such thatab. To Prove: That one of the two numbers $\frac{a+b}{2}$ and $\frac{a-b}{2}$ is odd and the other is even. Proof: Letaandbbe any odd odd positive integer such thatab. Since any positive integer is of the formq, 2q+ 1 Let $a=2 q+1$ and $b=2 m+1$, where, $q$ and...
Read More →The following frequency distribution gives the monthly consumption of electricity of 68 con sumers of a locality.
Question: The following frequency distribution gives the monthly consumption of electricity of 68 con sumers of a locality. Find the median, mean and mode of the data and compare them. Solution: (i) $\mathrm{n}=68$ gives $\frac{\mathbf{n}}{\mathbf{2}}=34$ So, we have the median class $(125-145)$ $\ell=125, \mathrm{n}=68, \mathrm{f}=20, \mathrm{cf}=22, \mathrm{~h}=20$ $\operatorname{Median}=\ell+\left\{\frac{\frac{\mathbf{n}}{\mathbf{z}}-\mathbf{c f}}{\mathbf{f}}\right\} \times \mathrm{h}$ $=125+...
Read More →Prove that $sin ^{2} 6 x-sin ^{2} 4 x=sin 2 x sin 10 x$
Question: Prove that $\sin ^{2} 6 x-\sin ^{2} 4 x=\sin 2 x \sin 10 x$ Solution: It is known that $\sin A+\sin B=2 \sin \left(\frac{A+B}{2}\right) \cos \left(\frac{A-B}{2}\right), \sin A-\sin B=2 \cos \left(\frac{A+B}{2}\right) \sin \left(\frac{A-B}{2}\right)$ $\therefore$ L.H.S. $=\sin ^{2} 6 x-\sin ^{2} 4 x$ $=(\sin 6 x+\sin 4 x)(\sin 6 x-\sin 4 x)=\left[2 \sin \left(\frac{6 x+4 x}{2}\right) \cos \left(\frac{6 x-4 x}{2}\right)\right]\left[2 \cos \left(\frac{6 x+4 x}{2}\right) \cdot \sin \left(\...
Read More →Show that $f:[-1,1] ightarrow mathbf{R}$, given by
Question: Show that $f:[-1,1] \rightarrow \mathbf{R}$, given by $f(x)=\frac{x}{(x+2)}$ is one-one. Find the inverse of the function $f:[-1,1] \rightarrow$ Range $f$. (Hint: For $y \in$ Range $f, y=f(x)=\frac{x}{x+2}$, for some $x$ in $[-1,1]$, i.e., $x=\frac{2 y}{(1-y)}$ ) Solution: $f:[-1,1] \rightarrow \mathrm{R}$ is given as $f(x)=\frac{x}{(x+2)}$. Let $f(x)=f(y)$. $\Rightarrow \frac{x}{x+2}=\frac{y}{y+2}$ $\Rightarrow x y+2 x=x y+2 y$ $\Rightarrow 2 x=2 y$ $\Rightarrow x=y$ fis a one-one fun...
Read More →Bromine monochloride, BrCl decomposes into bromine and chlorine and reaches the equilibrium:
Question: Bromine monochloride, BrCl decomposes into bromine and chlorine and reaches the equilibrium: $2 \mathrm{BrCl}(\mathrm{g}) \longleftrightarrow \mathrm{Br}_{2}(\mathrm{~g})+\mathrm{Cl}_{2}(\mathrm{~g})$ for which $K_{c}=32$ at $500 \mathrm{~K}$. If initially pure $\mathrm{BrCl}$ is present at a concentration of $3.3 \times 10^{-3} \mathrm{molL}^{-1}$, what is its molar concentration in the mixture at equilibrium? Solution: Let the amount of bromine and chlorine formed at equilibrium bex....
Read More →A student noted the number of cars passing through a spot on a road for 100 periods each of 3 minutes and summarised it in the table given below.
Question: A student noted the number of cars passing through a spot on a road for 100 periods each of 3 minutes and summarised it in the table given below. Find the mode of the data. Solution: Modal class = 40 50 Mode $=40+\left\{\frac{\mathbf{2 0}-\mathbf{1 2}}{\mathbf{2} \times \mathbf{2 0}-\mathbf{1 2}-\mathbf{1 1}}\right\} \times 10=40+\left\{\frac{\mathbf{8}}{\mathbf{4 0}-\mathbf{2 3}}\right\} \times 10$ $=40+4.706=44.706$...
Read More →Equilibrium constant, Kc for the reaction
Question: Equilibrium constant,Kcfor the reaction $\mathrm{N}_{2}(\mathrm{~g})+3 \mathrm{H}_{2}(\mathrm{~g}) \longleftrightarrow 2 \mathrm{NH}_{3}(\mathrm{~g})$ at $500 \mathrm{~K}$ is $0.061$ At a particular time, the analysis shows that composition of the reaction mixture is $3.0 \mathrm{~mol} \mathrm{~L}^{-1} \mathrm{~N}_{2}, 2.0 \mathrm{~mol} \mathrm{~L}^{-1} \mathrm{H}_{2}$ and $0.5 \mathrm{~mol} \mathrm{~L}^{-1} \mathrm{NH}_{3}$. Is the reaction at equilibrium? If not in which direction do...
Read More →The given distribution shows the number of runs scored by some top batsmen of the world in one day international cricket matches :
Question: The given distribution shows the number of runs scored by some top batsmen of the world in one day international cricket matches : Find the mode of the data. Solution: Modal class = 4000 5000 Mode $=\ell+\left\{\frac{\mathbf{f}-\mathbf{f}}{\boldsymbol{2} \mathbf{f}-\mathbf{f}_{\mathbf{0}}-\mathbf{f}_{2}}\right\} \times \mathrm{h}$ $=4000+\left\{\frac{18-4}{2 \times 18-4-9}\right\} \times 1000$ $=4000+\left\{\frac{14}{23}\right\} \times 1000$ = 4608.69...
Read More →State with reason whether following functions have inverse
Question: State with reason whether following functions have inverse (i) $f:\{1,2,3,4\} \rightarrow\{10\}$ with $f=\{(1,10),(2,10),(3,10),(4,10)\}$ (ii) $g:\{5,6,7,8\} \rightarrow\{1,2,3,4\}$ with $g=\{(5,4),(6,3),(7,4),(8,2)\}$ (iii) $h:\{2,3,4,5\} \rightarrow\{7,9,11,13\}$ with $h=\{(2,7),(3,9),(4,11),(5,13)\}$ Solution: (i) $f:\{1,2,3,4\} \rightarrow\{10\}$ defined as: $f=\{(1,10),(2,10),(3,10),(4,10)\}$ From the given definition off, we can see thatfis a many one function as:f(1) =f(2) =f(3)...
Read More →Prove that $sin (n+1) x sin (n+2) x+cos (n+1) x cos (n+2) x=cos x$
Question: Prove that $\sin (n+1) x \sin (n+2) x+\cos (n+1) x \cos (n+2) x=\cos x$ Solution: L.H.S. $=\sin (n+1) x \sin (n+2) x+\cos (n+1) x \cos (n+2) x$ $=\frac{1}{2}[2 \sin (n+1) x \sin (n+2) x+2 \cos (n+1) x \cos (n+2) x]$ $=\frac{1}{2}\left[\begin{array}{l}\cos \{(n+1) x-(n+2) x\}-\cos \{(n+1) x+(n+2) x\} \\ +\cos \{(n+1) x+(n+2) x\}+\cos \{(n+1) x-(n+2) x\}\end{array}\right]$ $\left[\begin{array}{l}\because-2 \sin A \sin B=\cos (A+B)-\cos (A-B) \\ 2 \cos A \cos B=\cos (A+B)+\cos (A-B)\end{a...
Read More →The following distribution gives the state-wise teacher-student ratio in higher secondary schools of India.
Question: The following distribution gives the state-wise teacher-student ratio in higher secondary schools of India. Find the mode and mean of this data. Interpret, the two measures. Solution: Modal class is (30-35) and its frequency is 10. So, $\ell=30, \mathrm{f}_{\mathrm{m}}=10, \mathrm{f}_{1}=9, \mathrm{f}_{2}=3, \mathrm{~h}=5$. Mode $=\ell+\left\{\frac{\mathbf{f}_{\mathbf{m}}-\mathbf{f}}{\mathbf{2 f}_{\mathbf{m}}-\mathbf{f}-\mathbf{f}_{2}}\right\} \times \mathbf{h}$ $=30+\left\{\frac{\math...
Read More →What is the inverse of f?
Question: If $f(x)=\frac{(4 x+3)}{(6 x-4)}, x \neq \frac{2}{3}$, show that $f \circ f(x)=x$, for all $x \neq \frac{2}{3} .$ What is the inverse of $f ?$ Solution: It is given that $f(x)=\frac{(4 x+3)}{(6 x-4)}, x \neq \frac{2}{3}$. Therefore, $f \circ f(x)=x$, for all $x \neq \frac{2}{3}$. Hence, the given functionfis invertible and the inverse offisfitself....
Read More →One of the reactions that takes place in producing steel from iron ore is the reduction of iron
Question: One of the reactions that takes place in producing steel from iron ore is the reduction of iron (II) oxide by carbon monoxide to give iron metal and CO2. $\mathrm{FeO}(\mathrm{s})+\mathrm{CO}(\mathrm{g}) \longleftrightarrow \mathrm{Fe}(\mathrm{s})+\mathrm{CO}_{2}(\mathrm{~g}) ; \mathrm{K}_{p}=0.265$ at $1050 \mathrm{~K}$ What are the equilibrium partial pressures of $\mathrm{CO}$ and $\mathrm{CO}_{2}$ at $1050 \mathrm{~K}$ if the initial partial pressures are: $p_{\mathrm{CO}}=1.4 \mat...
Read More →The following data gives the distribution of total monthly household expenditure of 200 families of a village.
Question: The following data gives the distribution of total monthly household expenditure of 200 families of a village. Find the modal monthly expenditure of the families. Also, find the mean monthly expenditure: Solution: Modal class $=1500-2000$ Mode $=\ell+\left\{\frac{\mathbf{f}-\mathbf{f}_{\mathbf{0}}}{\boldsymbol{2} \mathbf{f}-\mathbf{f}_{\mathbf{0}}-\mathbf{f}_{\mathbf{2}}}\right\} \times \mathrm{h}$ $=1500+\left\{\frac{\mathbf{4 0}-\mathbf{2 4}}{\mathbf{2} \times \mathbf{4 0}-\mathbf{2 ...
Read More →Find gof and fog, if
Question: Findgofandfog, if (i) $f(x)=|x|$ and $g(x)=|5 x-2|$ (ii) $f(x)=8 x^{3}$ and $g(x)=x^{\frac{1}{3}}$ Solution: (i) $f(x)=|x|$ and $g(x)=|5 x-2|$ (ii) $f(x)=8 x^{3}$ and $g(x)=x^{\frac{1}{3}}$...
Read More →The following data gives the information on the observed lifetimes (in hours) of 225 electrical components :
Question: The following data gives the information on the observed lifetimes (in hours) of 225 electrical components : Determine the modal lifetimes of the components. Solution: Modal class of the given data is 60-80. Here, $\ell=60, \mathrm{f}_{\mathrm{m}}=61, \mathrm{f}_{1}=52, \mathrm{f}_{2}=38$ and $\mathrm{h}=20$. Mode $=\ell+\left\{\frac{\mathbf{f}_{\mathbf{m}}-\mathbf{f}}{\mathbf{2 f}_{\mathbf{m}}-\mathbf{f}_{\mathbf{1}}-\mathbf{f}_{\mathbf{2}}}\right\} \times \mathbf{h}$ $=60+\left\{\fra...
Read More →The following table shows the ages of the patients admitted in a hospital during a year :
Question: The following table shows the ages of the patients admitted in a hospital during a year : Find the mode and the mean of the data given above. Compare and interpret the two measures of central tendency. Solution: From the given data, we have the modal class 35-45. $\{\because$ It has largest frequency among the given classes of the data $\}$ So, $\ell=35, \mathrm{f}_{\mathrm{m}}=23, \mathrm{f}_{1}=21, \mathrm{f}_{2}=14$ and $\mathrm{h}=10$. Mode $=\ell+\left\{\frac{\mathbf{f}_{\mathbf{m...
Read More →