Let $f=left{left(x, rac{x^{2}}{1+x^{2}} ight): x in mathbf{R} ight}$ be a function from $mathbf{R}$ into $mathbf{R}$. Determine the range of $f$.
Question: Let $f=\left\{\left(x, \frac{x^{2}}{1+x^{2}}\right): x \in \mathbf{R}\right\}$ be a function from $\mathbf{R}$ into $\mathbf{R}$. Determine the range of $f$. Solution: $f=\left\{\left(x, \frac{x^{2}}{1+x^{2}}\right): x \in \mathbf{R}\right\}$ $=\left\{(0,0),\left(\pm 0.5, \frac{1}{5}\right),\left(\pm 1, \frac{1}{2}\right),\left(\pm 1.5, \frac{9}{13}\right),\left(\pm 2, \frac{4}{5}\right),\left(3, \frac{9}{10}\right),\left(4, \frac{16}{17}\right), \ldots\right\}$ The range offis the set...
Read More →Find the domain and the range of the real function f defined by f (x) = |x – 1|.
Question: Find the domain and the range of the real function $f$ defined by $f(x)=|x-1|$. Solution: The given real function is $f(x)=|x-1|$. It is clear that $|x-1|$ is defined for all real numbers $\therefore$ Domain of $f=\mathbf{R}$ Also, for $x \in \mathbf{R},|x-1|$ assumes all real numbers. Hence, the range of $f$ is the set of all non-negative real numbers....
Read More →In fig., AB and CD are two diameters of a circle (with centre O) perpendicular to each other and OD is the diameter of the smaller circle.
Question: In fig., AB and CD are two diameters of a circle (with centre O) perpendicular to each other and OD is the diameter of the smaller circle. If OA = 7 cm. find the area of the shaded region. Solution: O is the centre of the circle, OA = 7 cm AB = 2(OA) = 2 7 = 14 cm OC = OA = 7 cm $\because \quad \mathrm{AB}$ and $\mathrm{CD}$ are perpendicular to each other $\Rightarrow \mathrm{OC} \perp \mathrm{AB}$ $\therefore \quad$ Area of $\Delta \mathrm{ABC}$ $=\frac{1}{2} \times \mathrm{AB} \time...
Read More →Calculate the enthalpy change for the process
Question: Calculate the enthalpy change for the process $\mathrm{CCl}_{4(g)} \rightarrow \mathrm{C}_{(g)}+4 \mathrm{Cl}_{(g)}$ and calculate bond enthalpy of $\mathrm{C}-\mathrm{Cl}$ in $\mathrm{CCl}_{4(g)}$. $\Delta_{\text {vap }} H^{\theta}\left(\mathrm{CCl}_{4}\right)=30.5 \mathrm{~kJ} \mathrm{~mol}^{-1}$ $\Delta_{f} H^{\theta}\left(\mathrm{CCl}_{4}\right)=-135.5 \mathrm{~kJ} \mathrm{~mol}^{-1}$ $\Delta_{a} H^{\theta}(\mathrm{C})=715.0 \mathrm{~kJ} \mathrm{~mol}^{-1}$, where $\Delta_{a} H^{\t...
Read More →Find the domain and the range of the real function f defined by.
Question: Find the domain and the range of the real function $t$ defined by $f(x)=\sqrt{(x-1)}$ Solution: The given real function is $f(x)=\sqrt{x-1}$. It can be seen that $\sqrt{x-1}$ is defined for $(x-1) \geq 0$. i.e., $f(x)=\sqrt{(x-1)}$ is defined for $x \geq 1$. Therefore, the domain of $f$ is the set of all real numbers greater than or equal to 1 i.e., the domain of $f=[1, \infty)$. As $x \geq 1 \Rightarrow(x-1) \geq 0 \Rightarrow \sqrt{x-1} \geq 0$ Therefore, the range of $f$ is the set ...
Read More →Show that the relation R in the set A of all the books in a library of a college,
Question: Show that the relation R in the setAof all the books in a library of a college, given by R = {(x,y):xandyhave same number of pages} is an equivalence relation. Solution: SetAis the set of all books in the library of a college. R = {x,y):xandyhave the same number of pages} Now, $R$ is reflexive since $(x, x) \in R$ as $x$ and $x$ has the same number of pages. Let $(x, y) \in R \Rightarrow x$ and $y$ have the same number of pages. $\Rightarrow y$ and $x$ have the same number of pages. $\...
Read More →Find the domain of the function $f(x)= rac{x^{2}+2 x+1}{x^{2}-8 x+12}$
Question: Find the domain of the function $f(x)=\frac{x^{2}+2 x+1}{x^{2}-8 x+12}$ Solution: The given function is $f(x)=\frac{x^{2}+2 x+1}{x^{2}-8 x+12}$ $f(x)=\frac{x^{2}+2 x+1}{x^{2}-8 x+12}=\frac{x^{2}+2 x+1}{(x-6)(x-2)}$ It can be seen that functionfis defined for all real numbers except atx= 6 andx= 2. Hence, the domain of $f$ is $\mathbf{R}-\{2,6\}$....
Read More →If $f(x)=x^{2}$, find $ rac{f(1.1)-f(1)}{(1.1-1)}$.
Question: If $f(x)=x^{2}$, find $\frac{f(1.1)-f(1)}{(1.1-1)}$. Solution: $f(x)=x^{2}$ $\therefore \frac{f(1.1)-f(1)}{(1.1-1)}=\frac{(1.1)^{2}-(1)^{2}}{(1.1-1)}=\frac{1.21-1}{0.1}=\frac{0.21}{0.1}=2.1$...
Read More →The relation $f$ is defined by $f(x)=left{egin{array}{l}x^{2}, 0 leq x leq 3 \ 3 x, 3 leq x leq 10end{array} ight.$
Question: The relation $f$ is defined by $f(x)=\left\{\begin{array}{l}x^{2}, 0 \leq x \leq 3 \\ 3 x, 3 \leq x \leq 10\end{array}\right.$ The relation $g$ is defined by $g(x)=\left\{\begin{array}{l}x^{2}, 0 \leq x \leq 2 \\ 3 x, 2 \leq x \leq 10\end{array}\right.$ Show thatfis a function andgis not a function. Solution: The relation $f$ is defined as $f(x)=\left\{\begin{array}{l}x^{2}, 0 \leq x \leq 3 \\ 3 x, 3 \leq x \leq 10\end{array}\right.$ It is observed that for $0 \leq x3, f(x)=x^{2}$ $3x ...
Read More →Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive.
Question: Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is symmetric but neither reflexive nor transitive. Solution: LetA= {1, 2, 3}. A relation R onAis defined as R = {(1, 2), (2, 1)}. It is seen that $(1,1),(2,2),(3,3) \notin \mathrm{R}$. R is not reflexive. Now, as $(1,2) \in R$ and $(2,1) \in R$, then $R$ is symmetric. Now, $(1,2)$ and $(2,1) \in R$ However, $(1,1) \notin R$ R is not transitive. Hence, R is symmetric but neither reflexive nor transitive....
Read More →Calculate the standard enthalpy of formation of CH3OH(l) from the following data:
Question: Calculate the standard enthalpy of formation of $\mathrm{CH}_{3} \mathrm{OH}_{(n)}$ from the following data: $\mathrm{CH}_{3} \mathrm{OH}_{(f)}+\frac{3}{2} \mathrm{O}_{2(g)} \longrightarrow \mathrm{CO}_{2(g)}+2 \mathrm{H}_{2} \mathrm{O}_{(f)} ; \Delta_{r} H^{\theta}=-726 \mathrm{~kJ} \mathrm{~mol}^{-1}$ $\mathrm{C}_{(g)}+\mathrm{O}_{2(g)} \longrightarrow \mathrm{CO}_{2(g)} ; \Delta_{c} H^{\theta}=-393 \mathrm{~kJ} \mathrm{~mol}^{-1}$ $\mathrm{H}_{2(g)}+\frac{1}{2} \mathrm{O}_{2(g)} \lo...
Read More →Fig. depicts a racing track whose left and right ends are semicircular.
Question: Fig. depicts a racing track whose left and right ends are semicircular. The distance between the two inner parallel line segments is 60 m and they are each 106 m long. If the track is 10 m wide, find: (i) the distance around the track along its inneredge (ii) the area of the track. Solution: (i) The distance around the track along the inner edge (as seen from figure) = Perimeter of APB + BC + Perimeter CQD + AD $=\{\pi \times 30+106+\pi \times 30+106\} \mathrm{m}$ $=\{60 \pi+212\} \mat...
Read More →Check whether the relation R in R defined
Question: Check whether the relation $R$ in $R$ defined as $R=\left\{(a, b): a \leq b^{3}\right\}$ is reflexive, symmetric or transitive. Solution: $R=\left\{(a, b): a \leq b^{3}\right\}$ It is observed that $\left(\frac{1}{2}, \frac{1}{2}\right) \notin \mathrm{R}$ as $\frac{1}{2}\left(\frac{1}{2}\right)^{3}=\frac{1}{8}$. R is not reflexive. Now, $(1,2) \in R\left(\right.$ as $\left.12^{3}=8\right)$ But, $(2,1) \notin R\left(\operatorname{as} 21^{3}=1\right)$ R is not symmetric. We have $\left(3...
Read More →Find the range of each of the following functions.
Question: Find therange of each of the following functions. (i) $f(x)=2-3 x, x \in \mathbf{R}, x0$. (ii) $f(x)=x^{2}+2, x$, is a real number. (iii) $f(x)=x, x$ is a real number Solution: (i) $f(x)=2-3 x, x \in \mathbf{R}, x0$ The values of $f(x)$ for various values of real numbers $x0$ can be written in the tabular form as Thus, it can be clearly observed that the range offis the set of all real numbers less than 2. i.e., range of $f=(-\infty, 2)$ Alter: Let $x0$ $\Rightarrow 3 x0$ $\Rightarrow ...
Read More →Given
Question: Given $\mathrm{N}_{2(g)}+3 \mathrm{H}_{2(g)} \longrightarrow 2 \mathrm{NH}_{3(g)} ; \Delta_{r} H^{\theta}=-92.4 \mathrm{~kJ} \mathrm{~mol}^{-1}$ What is the standard enthalpy of formation of NH3gas? Solution: Standard enthalpy of formation of a compound is the change in enthalpy that takes place during the formation of 1 mole of a substance in its standard form from its constituent elements in their standard state. Re-writing the given equation for 1 mole of NH3(g), $\frac{1}{2} \mathr...
Read More →In fig., ABCD is a square of side 14 cm. With centres A, B, C and D,
Question: In fig., ABCD is a square of side 14 cm. With centres A, B, C and D, four circles are drawn such that each circle touch externally two of the remaining three circles. Find the area of the shaded region. Solution: Side of the square ABCD = 14 cm $\therefore \quad$ Area of the sqaure $\mathrm{ABCD}=14 \times 14 \mathrm{~cm}^{2}$ $=196 \mathrm{~cm}^{2}$ $\because \quad$ Circles touch each other Radius of the circle $=\frac{\mathbf{1 4}}{\mathbf{2}}=7 \mathrm{~cm}$ Now, area of a sector of...
Read More →Show that the relation R in R defined as R = {(a, b): a ≤ b}, is reflexive and transitive but not symmetric.
Question: Show that the relation R inRdefined as R = {(a,b):ab}, is reflexive and transitive but not symmetric. Solution: $R=\{(a, b) ; a \leq b\}$ Clearly $(a, a) \in R$ as $a=a$. R is reflexive. Now, $(2,4) \in R(\operatorname{as} 24)$ But, $(4,2) \notin R$ as 4 is greater than 2 . $\therefore R$ is not symmetric. Now, let $(a, b),(b, c) \in R$. Then, $a \leq b$ and $b \leq c$ $\Rightarrow a \leq c$ $\Rightarrow(a, c) \in R$ $\therefore R$ is transitive. Hence,R is reflexive and transitive but...
Read More →Enthalpies of formation of CO(g), CO2(g), N2O(g) and N2O4(g) are –110 kJ mol–1,
Question: Enthalpies of formation of $\mathrm{CO}_{(g)}, \mathrm{CO}_{2(g)}, \mathrm{N}_{2} \mathrm{O}_{(g)}$ and $\mathrm{N}_{2} \mathrm{O}_{4(g)}$ are $-110 \mathrm{~kJ} \mathrm{~mol}^{-1},-393 \mathrm{~kJ} \mathrm{~mol}^{-1}, 81 \mathrm{~kJ} \mathrm{~mol}^{-1}$ and $9.7 \mathrm{~kJ} \mathrm{~mol}^{-1}$ respectively. Find the value of $\Delta, \mathrm{H}$ for the reaction: $\mathrm{N}_{2} \mathrm{O}_{4(g)}+3 \mathrm{CO}_{(g)} \longrightarrow \mathrm{N}_{2} \mathrm{O}_{(g)}+3 \mathrm{CO}_{2(g)}...
Read More →The function ‘t’ which maps temperature in degree Celsius into temperature in degree
Question: The function ' $t$ ' which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by $t(\mathrm{C})=\frac{9 \mathrm{C}}{5}+32$. Find (i) $t$ ( 0 ) (ii) $t$ (28) (iii) $t(-10)$ (iv) The value of $\mathrm{C}$, when $t(\mathrm{C})=212$ Solution: The given function is $t(\mathrm{C})=\frac{9 \mathrm{C}}{5}+32$. Therefore, (i) $t(0)=\frac{9 \times 0}{5}+32=0+32=32$ (ii) $t(28)=\frac{9 \times 28}{5}+32=\frac{252+160}{5}=\frac{412}{5}$ (iii) $t(-10)=\frac{9 \times(...
Read More →In a circular table cover of radius 32 cm,
Question: In a circular table cover of radius 32 cm, a design is formed leaving an equilateral triangle ABC in the middle as shown in fig. Find the area of the design. Solution: $\mathrm{O}$ is the centre of the circular table cover and radius $=32 \mathrm{~cm} . \triangle \mathrm{ABC}$ is equilateral. Join $\mathrm{OA}, \mathrm{OB}, \mathrm{OC}$. Now, $\angle \mathrm{AOB}=\angle \mathrm{BOC}=\angle \mathrm{COA}=120^{\circ}$ In $\Delta \mathrm{OBC}$, we have $\mathrm{OB}=\mathrm{OC}$ Draw $\math...
Read More →Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as
Question: Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as $\mathrm{R}=\{(a, b): b=a+1\}$ is reflexive, symmetric or transitive. Solution: LetA= {1, 2, 3, 4, 5, 6}. A relation R is defined on setAas: R = {(a,b):b=a+ 1} R = {(1, 2), (2, 3), (3, 4), (4, 5), (5, 6)} We can find $(a, a) \notin R$, where $a \in A$. For instance, (1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6) R R is not reflexive. It can be observed that $(1,2) \in R$, but $(2,1) \notin R$. R is not symmetric.. No...
Read More →A function f is defined by f(x) = 2x – 5.
Question: A function $t$ is defined by $t(x)=2 x-5$. Write down the values of (i) $f(0)$, (ii) $f(7)$, (iii) $f(-3)$ Solution: The given function is $f(x)=2 x-5$. Therefore, (i) $f(0)=2 \times 0-5=0-5=-5$ (ii) $f(7)=2 \times 7-5=14-5=9$ (iii) $f(-3)=2 \times(-3)-5=-6-5=-11$...
Read More →Find the domain and range of the following real function:
Question: Find the domain and range of the following real function: (i) $f(x)=-|x|$ (ii) $f(x)=\sqrt{9-x^{2}}$ Solution: (i) $f(x)=-|x|, x \in \mathbf{R}$ We know that $|x|=\left\{\begin{array}{l}x, x \geq 0 \\ -x, x0\end{array}\right.$ $\therefore f(x)=-|x|=\left\{\begin{array}{l}-x, x \geq 0 \\ x, x0\end{array}\right.$ Since $f(x)$ is defined for $x \in \mathbf{R}$, the domain of $f$ is $\mathbf{R}$. It can be observed that the range of $f(x)=-|x|$ is all real numbers except positive real numb...
Read More →From each corner of a square of side 4 cm a quadrant of a circle of radius 1 cm is cut and also a circle of diameter 2 cm is cut as shown in fig.
Question: From each corner of a square of side 4 cm a quadrant of a circle of radius 1 cm is cut and also a circle of diameter 2 cm is cut as shown in fig. Find the area of the remaining portion of the square. Solution: Side of the square = 4 cm $\therefore$ Area of the square $\mathrm{ABCD}=4 \times 4 \mathrm{~cm}^{2}$ $=16 \mathrm{~cm}^{2}$ $\because$ Each corner has a quadrant circle of radius $1 \mathrm{~cm}$. $\therefore \quad$ Area of all the 4 quadrant squares $4 \times \frac{1}{4} \pi r^...
Read More →Find the area of the shaded region in fig.,
Question: Find the area of the shaded region in fig., where a circular arc of radius 6 cm has been drawn with vertex O of an equilateral triangle OAB of side 12 cm as centre. Solution: Area of the circle with radius 6 cm $=\pi \mathrm{r}^{2}=\frac{22}{7} \times 6 \times 6 \mathrm{~cm}^{2}=\frac{792}{7} \mathrm{~cm}^{2}$ Area of equilateral triangle, having side a = 12 cm, is given by $\frac{\sqrt{3}}{4} a^{2}=\frac{\sqrt{3}}{4} \times 12 \times 12 \mathrm{~cm}^{2}=36 \sqrt{3} \mathrm{~cm}^{2}$ $...
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