State whether the function f is bijective. Justify your answer.
Question: Let $f: \mathbf{N} \rightarrow \mathbf{N}$ be defined by $f(n)=\left\{\begin{array}{ll}\frac{n+1}{2}, \text { if } n \text { is odd } \\ \frac{n}{2}, \text { if } n \text { is even }\end{array}\right.$ for all $n \in \mathbf{N}$. State whether the function f is bijective. Justify your answer. Solution: $f: \mathbf{N} \rightarrow \mathbf{N}$ is defined as $f(n)=\left\{\begin{array}{ll}\frac{n+1}{2}, \text { if } n \text { is odd } \\ \frac{n}{2}, \text { if } n \text { is even }\end{arr...
Read More →A drinking glass is in the shape of a frustum of a cone of height 14 cm.
Question: A drinking glass is in the shape of a frustum of a cone of height 14 cm. The diameters of its two circular ends are 4 cm and 2 cm. Find the capacity of the glass. Solution: R = 2 cm, r = 1 cm, h = 14 cm Capacity of the glass = volume of the frustum with radii of ends as 2 cm and 1 cm and height 14 cm $=\frac{\mathbf{1}}{\mathbf{3}} \pi \mathrm{h}\left\{\mathrm{R}^{2}+\mathrm{r}^{2}+\mathrm{Rr}\right\}$ $=\frac{\mathbf{1}}{\mathbf{3}} \pi \times 14 \times\left\{(2)^{2}\right.$$\left.+(1...
Read More →Let A and B be sets. Show that f:
Question: LetAandBbe sets. Show thatf:ABBAsuch that (a,b) = (b,a) is bijective function. Solution: $f: A \times B \rightarrow B \times A$ is defined as $f(a, b)=(b, a)$. Let $\left(a_{1}, b_{1}\right),\left(a_{2}, b_{2}\right) \in \mathrm{A} \times \mathrm{B}$ such that $f\left(a_{1}, b_{1}\right)=f\left(a_{2}, b_{2}\right)$. $\Rightarrow\left(b_{1}, a_{1}\right)=\left(b_{2}, a_{2}\right)$ $\Rightarrow b_{1}=b_{2}$ and $a_{1}=a_{2}$ $\Rightarrow\left(a_{1}, b_{1}\right)=\left(a_{2}, b_{2}\right)...
Read More →A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank in her field,
Question: A farmer connects a pipe of internal diameter 20 cm from a canal into a cylindrical tank in her field, which is 10 m in diameter and 2 m deep. If water flows through the pipe at the rate of 3 km/h, in how much time will the tank be filled? Solution: Water speed $=3 \mathrm{~km} / \mathrm{hr}=\frac{\mathbf{3 0 0 0}}{\mathbf{6 0}} \mathbf{m} / \mathbf{m i n}$ $=50 \mathrm{~m} / \mathrm{min}$ Diameter of the pipe = 20 cm i.e., radius $=10 \mathrm{~cm}=\frac{\mathbf{1}}{\mathbf{1 0}} \math...
Read More →At 700 K, equilibrium constant for the reaction
Question: At 700 K, equilibrium constant for the reaction $\mathrm{H}_{2(g)}+\mathrm{I}_{2(g)} \longleftrightarrow 2 \mathrm{HI}_{(g)}$ is $54.8$. If $0.5 \mathrm{molL}^{-1}$ of $\mathrm{HI}_{(g)}$ is present at equilibrium at $700 \mathrm{~K}$, what are the concentration of $\mathrm{H}_{2(g)}$ and $\mathrm{I}_{2(g)}$ assuming that we initially started with $\mathrm{HI}_{(g)}$ and allowed it to reach equilibrium at $700 \mathrm{~K}$ ? Solution: It is given that equilibrium constant $K_{\mathrm{c...
Read More →Water in a canal 6 m wide and 1.5 m deep is flowing with a speed of 10 km/h.
Question: Water in a canal 6 m wide and 1.5 m deep is flowing with a speed of 10 km/h. How much area will it irrigate in 30 minutes, if 8 cm of standing water is needed? Solution: Depth of water in the canal = 1.5 m Width of canal = 6 m Volume of water flowing through the canal in 60 minutes $=(10 \times 1000) \times 6 \times 1.5 \mathrm{~cm}^{3}$ $(\because$ Speed $=10 \mathrm{~km} / \mathrm{hr}=10 \times 1000 \mathrm{~m}$ per $\mathrm{hr})$ Volume of water flowing through canal in 30 minutes $...
Read More →A cylindrical bucket, 32 cm high and with radius of base 18 cm, is filled with sand.
Question: A cylindrical bucket, 32 cm high and with radius of base 18 cm, is filled with sand. This bucket is emptied on the ground and a conical heap of sand is formed. If the height of the conical heap is 24 cm, find the radius and slant height of the heap. Solution: For the cylindrical bucket : Radius (r) = 18 cm and height (h) = 32 cm Volume $=\pi \mathrm{r}^{2} \mathrm{~h}=\frac{\mathbf{2 2}}{\mathbf{7}}(18)^{2} \times 32 \mathrm{~cm}^{3}$ $\Rightarrow$ Volume of the sand $=\left(\frac{22}{...
Read More →One mole of H2O and one mole of CO are taken in 10 L vessel and heated to
Question: One mole of $\mathrm{H}_{2} \mathrm{O}$ and one mole of $\mathrm{CO}$ are taken in $10 \mathrm{~L}$ vessel and heated to 725 K. At equilibrium 40% of water (by mass) reacts with CO according to the equation, $\mathrm{H}_{2} \mathrm{O}(\mathrm{g})+\mathrm{CO}(\mathrm{g}) \longleftrightarrow \mathrm{H}_{2}(\mathrm{~g})+\mathrm{CO}_{2}(\mathrm{~g})$ Calculate the equilibrium constant for the reaction. Solution: The given reaction is: Therefore, the equilibrium constant for the reaction, $...
Read More →$sin ^{2} rac{pi}{6}+cos ^{2} rac{pi}{3}- an ^{2} rac{pi}{4}=- rac{1}{2}$
Question: $\sin ^{2} \frac{\pi}{6}+\cos ^{2} \frac{\pi}{3}-\tan ^{2} \frac{\pi}{4}=-\frac{1}{2}$ Solution: L.H.S. $=\sin ^{2} \frac{\pi}{6}+\cos ^{2} \frac{\pi}{3}-\tan ^{2} \frac{\pi}{4}$ $=\left(\frac{1}{2}\right)^{2}+\left(\frac{1}{2}\right)^{2}-(1)^{2}$ $=\frac{1}{4}+\frac{1}{4}-1=-\frac{1}{2}$ $=$ R.H.S....
Read More →Find the value of the trigonometric function cot(-15π/4)
Question: Find the value of the trigonometric function $\cot \left(-\frac{15 \pi}{4}\right)$ Solution: It is known that the values of $\cot x$ repeat after an interval of $\pi$ or $180^{\circ}$. $\therefore \cot \left(-\frac{15 \pi}{4}\right)=\cot \left(-\frac{15 \pi}{4}+4 \pi\right)=\cot \frac{\pi}{4}=1$...
Read More →The equilibrium constant expression for a gas reaction is,
Question: The equilibrium constant expression for a gas reaction is, $K_{\mathrm{C}}=\frac{\left[\mathrm{NH}_{3}\right]^{4}\left[\mathrm{O}_{2}\right]^{5}}{[\mathrm{NO}]^{4}\left[\mathrm{H}_{2} \mathrm{O}\right]^{6}}$ Write the balanced chemical equation corresponding to this expression. Solution: The balanced chemical equation corresponding to the given expression can be written as: $4 \mathrm{NO}_{(g)}+6 \mathrm{H}_{2} \mathrm{O}_{(g)} \longleftrightarrow 4 \mathrm{NH}_{3(g)}+5 \mathrm{O}_{2(g...
Read More →How many silver coins 1.75 cm in diameter and of thickness 2 mm,
Question: How many silver coins 1.75 cm in diameter and of thickness 2 mm, must be melted to form a cuboid of dimensions 5.5 cm 10 cm 3.5 cm? Solution: For a circular coin : Diameter = 1.75 cm $\Rightarrow$ Radius $(r)=\frac{\mathbf{1 7 5}}{\mathbf{2 0 0}} \mathrm{cm}$ Thickness $(\mathrm{h})=2 \mathrm{~mm}=\frac{2}{10}$ $\therefore$ Volume $=\pi r^{2} h=\frac{22}{7} \times\left(\frac{175}{200}\right)^{2} \times \frac{2}{10} \mathrm{~cm}^{3}$ For a cuboid : Length $(\ell)=10 \mathrm{~cm}$, Bread...
Read More →Find the value of the trigonometric function sin(-11π/3)
Question: Find the value of the trigonometric function $\sin \left(-\frac{11 \pi}{3}\right)$ Solution: It is known that the values of $\sin x$ repeat after an interval of $2 \pi$ or $360^{\circ}$. $\therefore \sin \left(-\frac{11 \pi}{3}\right)=\sin \left(-\frac{11 \pi}{3}+2 \times 2 \pi\right)=\sin \left(\frac{\pi}{3}\right)=\frac{\sqrt{3}}{2}$...
Read More →Find the value of the trigonometric function tan19π/3
Question: Find the value of the trigonometric function $\tan \frac{19 \pi}{3}$ Solution: It is known that the values of $\tan x$ repeat after an interval of $\pi$ or $180^{\circ}$. $\therefore \tan \frac{19 \pi}{3}=\tan 6 \frac{1}{3} \pi=\tan \left(6 \pi+\frac{\pi}{3}\right)=\tan \frac{\pi}{3}=\tan 60^{\circ}=\sqrt{3}$...
Read More →A container shaped like a right circular cylinder having diameter 12 cm and height 15 cm is full of ice-cream.
Question: A container shaped like a right circular cylinder having diameter 12 cm and height 15 cm is full of ice-cream. The ice-cream is to be filled into cones of height 12 cm and diameter 6 cm, having a hemispherical shape on the top. Find the number of such cones which can be filled with ice-cream. Solution: Volume of one ice-cream cone as shown in figure. $=\frac{\mathbf{1}}{\mathbf{3}} \pi \times(3)^{2} \times 9+\frac{\mathbf{2}}{\mathbf{3}} \pi \times(3)^{3} \mathrm{~cm}^{3}$ $=27 \pi+18 ...
Read More →A mixture of 1.57 mol of N2, 1.92 mol of H2 and 8.13 mol of NH3 is introduced into a 20 L reaction vessel at 500
Question: A mixture of $1.57 \mathrm{~mol}$ of $\mathrm{N}_{2}, 1.92 \mathrm{~mol}$ of $\mathrm{H}_{2}$ and $8.13 \mathrm{~mol}$ of $\mathrm{NH}_{3}$ is introduced into a $20 \mathrm{~L}$ reaction vessel at $500 \mathrm{~K}$. At this temperature, the equilibrium constant, $K_{c}$ for the reaction $\mathrm{N}_{2}(\mathrm{~g})+3 \mathrm{H}_{2}(\mathrm{~g}) \longleftrightarrow 2 \mathrm{NH}_{3}(\mathrm{~g})$ is $1.7 \times 10^{2}$ Is the reaction mixture at equilibrium? If not, what is the directio...
Read More →In each of the following cases,
Question: In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer. (i) $f: \mathbf{R} \rightarrow \mathbf{R}$ defined by $f(x)=3-4 x$ (ii) $f: \mathbf{R} \rightarrow \mathbf{R}$ defined by $f(x)=1+x^{2}$ Solution: (i) $f: \mathbf{R} \rightarrow \mathbf{R}$ is defined as $f(x)=3-4 x$.\ Let $x_{1}, x_{2} \in \mathbf{R}$ such that $f\left(x_{1}\right)=f\left(x_{2}\right)$. $\Rightarrow 3-4 x_{1}=3-4 x_{2}$ $\Rightarrow-4 x_{1}=-4 x_{2}$ $\Righta...
Read More →Find the value of the trigonometric function cosec (–1410°)
Question: Find the value of the trigonometric function $\operatorname{cosec}\left(-1410^{\circ}\right)$ Solution: It is known that the values of $\operatorname{cosec} x$ repeat after an interval of $2 \pi$ or $360^{\circ}$. $\therefore \operatorname{cosec}\left(-1410^{\circ}\right)=\operatorname{cosec}\left(-1410^{\circ}+4 \times 360^{\circ}\right)$ $=\operatorname{cosec}\left(-1410^{\circ}+1440^{\circ}\right)$ $=\operatorname{cosec} 30^{\circ}=2$...
Read More →A well of diameter 3 m is dug 14 m deep.
Question: A well of diameter 3 m is dug 14 m deep. The earth taken out of it has been spread evenly all around it in the shape of a circular ring of width 4 m to form an embankment. Find the height of the embankment. Solution: Diameter of cylindrical well (d) = 3 m $\Rightarrow$ Radius of the cylindrical well $=\frac{\mathbf{3}}{\mathbf{2}} \mathrm{m}=1.5 \mathrm{~m}$ Depth of the well $(\mathrm{h})=14 \mathrm{~m}$ $\therefore$ Volume $=\pi \mathrm{r}^{2} \mathrm{~h}=\frac{\mathbf{2 2}}{\mathbf{...
Read More →Find the value of the trigonometric function sin 765°
Question: Find the value of the trigonometric function $\sin 765^{\circ}$ Solution: It is known that the values of $\sin x$ repeat after an interval of $2 \pi$ or $360^{\circ}$ $\therefore \sin 765^{\circ}=\sin \left(2 \times 360^{\circ}+45^{\circ}\right)=\sin 45^{\circ}=\frac{1}{\sqrt{2}}$...
Read More →Find the values of other five trigonometric functions if tan x = -5/12
Question: Find the values of other five trigonometric functions if $\tan x=-\frac{5}{12}, x$ lies in second quadrant. Solution: $\tan x=-\frac{5}{12}$ $\cot x=\frac{1}{\tan x}=\frac{1}{\left(-\frac{5}{12}\right)}=-\frac{12}{5}$ $1+\tan ^{2} x=\sec ^{2} x$ $\Rightarrow 1+\left(-\frac{5}{12}\right)^{2}=\sec ^{2} x$ $\Rightarrow 1+\frac{25}{144}=\sec ^{2} x$ $\Rightarrow \frac{169}{144}=\sec ^{2} x$ $\Rightarrow \sec x=\pm \frac{13}{12}$ Since $x$ lies in the $2^{\text {nd }}$ quadrant, the value o...
Read More →A 20 m deep well with diameter 7 m is dug and the earth from digging is evenly spread out to form a platform 22 m by 14 m.
Question: A 20 m deep well with diameter 7 m is dug and the earth from digging is evenly spread out to form a platform 22 m by 14 m. Find the height of the platform. Solution: Diameter of the cylindrical well = 7 m $\Rightarrow$ Radius of the cylindrical $(\mathrm{r})=\frac{\mathbf{7}}{\mathbf{2}} \mathrm{m}$ Depth of the well (h) = 20 m $\therefore$ Volume $=\pi \mathrm{r}^{2} \mathrm{~h}=\frac{22}{7} \times \frac{7}{2} \times \frac{7}{2} \times 20 \mathrm{~m}^{2}$ $=22 \times 7 \times 5 \mathr...
Read More →Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)}
Question: Let $A=\{1,2,3\}, B=\{4,5,6,7\}$ and let $f=\{(1,4),(2,5),(3,6)\}$ be a function from $A$ to $B$. Show that $f$ is one-one. Solution: It is given that $A=\{1,2,3\}, B=\{4,5,6,7\}$. $f: A \rightarrow B$ is defined as $f=\{(1,4),(2,5),(3,6)\}$. $\therefore f(1)=4, f(2)=5, f(3)=6$ It is seen that the images of distinct elements ofAunderfare distinct. Hence, functionfis one-one....
Read More →A sample of HI(g) is placed in flask at a pressure of 0.2 atm.
Question: A sample of $\mathrm{HI}_{(\mathrm{g})}$ is placed in flask at a pressure of $0.2$ atm. At equilibrium the partial pressure of $\mathrm{HI}_{(\mathrm{g})}$ is $0.04$ atm. What is $K_{\mathrm{p}}$ for the given equilibrium? $2 \mathrm{HI}(\mathrm{g}) \longleftrightarrow \mathrm{H}_{2}(\mathrm{~g})+\mathrm{I}_{2}(\mathrm{~g})$ Solution: The initial concentration of $\mathrm{HI}$ is $0.2 \mathrm{~atm}$. At equilibrium, it has a partial pressure of $0.04 \mathrm{~atm}$. Therefore, a decrea...
Read More →Show that the Signum Function
Question: Show that the Signum Function $f: \mathbf{R} \rightarrow \mathbf{R}$, given by $f(x)=\left\{\begin{array}{l}1, \text { if } x0 \\ 0, \text { if } x=0 \\ -1, \text { if } x0\end{array}\right.$ is neither one-one nor onto. Solution: $f: \mathbf{R} \rightarrow \mathbf{R}$ is given by, $f(x)=\left\{\begin{array}{l}1, \text { if } x0 \\ 0, \text { if } x=0 \\ -1, \text { if } x0\end{array}\right.$ It is seen that $f(1)=f(2)=1$, but $1 \neq 2$. fis not one-one. Now, as $f(x)$ takes only 3 va...
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