eSaral - JEE | NEET | Class 8-10 Exam Prep App

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...

Let f, g and h be functions from R to R. Show that

Question: Letf,gandhbe functions fromRtoR. Show that Solution: To prove:...

The following table gives the literacy rate (in percentage) of 35 cities.

Question: The following table gives the literacy rate (in percentage) of 35 cities. Find the mean literacy rate. Solution:...

Let f: {1, 3, 4} → {1, 2, 5} and g: {1, 2, 5} → {1, 3} be given

Question: Let $f:\{1,3,4\} \rightarrow\{1,2,5\}$ and $g:\{1,2,5\} \rightarrow\{1,3\}$ be given by $f=\{(1,2),(3,5),(4,1)\}$ and $g=\{(1,3),(2,3),(5,1)\}$. Write down gof. Solution: The functions $f:\{1,3,4\} \rightarrow\{1,2,5\}$ and $g:\{1,2,5\} \rightarrow\{1,3\}$ are defined as f= {(1, 2), (3, 5), (4, 1)} andg= {(1, 3), (2, 3), (5, 1)}....

A class teacher has the following absentee record of 40 students of a class for the whole term.

Question: A class teacher has the following absentee record of 40 students of a class for the whole term. Find the mean number of days a student was absent. Solution:...

Let f: {1, 3, 4} → {1, 2, 5} and g: {1, 2, 5} → {1, 3} be given by

Question: Letf: {1, 3, 4} {1, 2, 5} andg: {1, 2, 5} {1, 3} be given byf= {(1, 2), (3, 5), (4, 1)} andg= {(1, 3), (2, 3), (5, 1)}. Write downgof. Solution: The functions $f:\{1,3,4\} \rightarrow\{1,2,5\}$ and $g:\{1,2,5\} \rightarrow\{1,3\}$ are defined as $f=\{(1,2),(3,5),(4,1)\}$ and $g=\{(1,3),(2,3),(5,1)\}$ $\begin{array}{ll}g \circ f(1)=g(f(1))=g(2)=3 {[f(1)=2 \text { and } g(2)=3]} \\ g \circ f(3)=g(f(3))=g(5)=1 {[f(3)=5 \text { and } g(5)=1]} \\ g \circ f(4)=g(f(4))=g(1)=3 {[f(4)=1 \text {...

To find out the concentration of

Question: To find out the concentration of $\mathrm{SO}_{2}$ in the air (in parts per million, i.e., ppm), the data was collected for 30 localities in a certain city and is presented below : Find the mean concentration of $\mathrm{SO}_{2}$ in the air. Solution:...

Let f: R → R be defined as f(x)

Question: Letf:RRbe defined asf(x) (A) $f$ is one-one onto (B) $f$ is many-one onto (C) $f$ is one-one but not onto (D) $f$ is neither one-one nor onto Solution: $f: \mathbf{R} \rightarrow \mathbf{R}$ is defined as $f(x)=3 x$. Let $x, y \in \mathbf{R}$ such that $f(x)=f(y)$. $\Rightarrow 3 x=3 y$ $\Rightarrow x=y$ fis one-one. Also, for any real number $(y)$ in co-domain $\mathbf{R}$, there exists $\frac{y}{3}$ in $\mathbf{R}$ such that $f\left(\frac{y}{3}\right)=3\left(\frac{y}{3}\right)=y$. fi...

The table below shows the daily expenditure on food of 25 households in a locality.

Question: The table below shows the daily expenditure on food of 25 households in a locality. Find the mean daily expenditure on food by a suitable method. Solution:...

In a retail market, fruit vendors were selling mangoes kept in packing boxes.

Question: In a retail market, fruit vendors were selling mangoes kept in packing boxes. These boxes contained varying number of mangoes. The following was the distribution of mangoes according to the number of boxes. Find the mean number of mangoes kept in a packing box. Which method of finding the mean did you choose? Solution: $\mathrm{a}=57, \mathrm{~h}=2, \mathrm{n}=400$ and $\Sigma \mathrm{f}_{\mathrm{i}} \mathrm{u}_{\mathrm{i}}=25$. By step deviation method, Mean $=\mathrm{a}+\mathrm{h} \t...

A sample of pure PCl5 was introduced into an evacuated vessel at 473 K.

Question: A sample of pure $\mathrm{PCl}_{5}$ was introduced into an evacuated vessel at $473 \mathrm{~K}$. After equilibrium was attained, concentration of $\mathrm{PCl}_{5}$ was found to be $0.5 \times 10^{-1}$ mol $\mathrm{L}^{-1}$. If value of $K_{c}$ is $8.3 \times 10^{-3}$, what are the concentrations of $\mathrm{PCl}_{3}$ and $\mathrm{Cl}_{2}$ at equilibrium? $\mathrm{PCl}_{5}(\mathrm{~g}) \longleftrightarrow \mathrm{PCl}_{3}(\mathrm{~g})+\mathrm{Cl}_{2}(\mathrm{~g})$ Solution: Let the co...

Thirty women were examined in a hospital by a doctor and the number of heart beats per minute were recorded and summarised as follows.

Question: Thirty women were examined in a hospital by a doctor and the number of heart beats per minute were recorded and summarised as follows. Find the mean heart beats per minute for these women, choosing a suitable method. Solution:...

The following distribution shows the daily pocket allowance of children of a locality.

Question: The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is Rs. 18. Find the missing frequency f. Solution: We may prepare the table as given below : It is given that mean = 18. From the table, we have $\mathrm{a}=18, \mathrm{n}=44+\mathrm{f}$ and $\Sigma \mathrm{f}_{\mathrm{d}} \mathrm{d}=2 \mathrm{f}-40$ Now, mean $=\mathrm{a}+\frac{\mathbf{1}}{\mathbf{n}} \times \Sigma \mathrm{f}_{\mathrm{i}} \mathrm{d}_{\mathrm{i}}$ Then subst...

Consider the following distribution of daily wages of 50 workers of a factory.

Question: Consider the following distribution of daily wages of 50 workers of a factory. Find the mean daily wages of the workers of the factory by using an appropriate method. Solution: We have $\Sigma f_{i}=50$ and $\sum f_{i} x_{i}=7260$ Mean $=\frac{\sum \mathbf{f}_{\mathbf{x}} \mathbf{i}}{\sum \mathbf{f}}=\frac{\mathbf{7 2 6 0}}{\mathbf{5 0}}=145.2$...

Find the value of: (i) sin 75°

Question: Find the value of: (i) $\sin 75^{\circ}$ (ii) $\tan 15^{\circ}$ Solution: (i) $\sin 75^{\circ}=\sin \left(45^{\circ}+30^{\circ}\right)$ $=\sin 45^{\circ} \cos 30^{\circ}+\cos 45^{\circ} \sin 30^{\circ}$ $[\sin (x+y)=\sin x \cos y+\cos x \sin y]$ $=\left(\frac{1}{\sqrt{2}}\right)\left(\frac{\sqrt{3}}{2}\right)+\left(\frac{1}{\sqrt{2}}\right)\left(\frac{1}{2}\right)$ $=\frac{\sqrt{3}}{2 \sqrt{2}}+\frac{1}{2 \sqrt{2}}=\frac{\sqrt{3}+1}{2 \sqrt{2}}$ (ii) $\tan 15^{\circ}=\tan \left(45^{\ci...

A survery was conducted by a group of students as a part of their environment awareness programme,

Question: A survery was conducted by a group of students as a part of their environment awareness programme, in which they collected the following data regarding the number of plants in 20 houses in a locality. Find the mean number of plants per house. Which method did you use for finding the mean, and why? Solution: Let us find mean of the data by direct method because the figures are small. We have, $\mathrm{n}=\Sigma \mathrm{f}_{\mathrm{i}}=20$ and $\Sigma \mathrm{f}_{\mathrm{i}} \mathrm{x}_{...

Let f: R → R be defined as

Question: Let $f: \mathbf{R} \rightarrow \mathbf{R}$ be defined as $f(x)=x^{4}$. Choose the correct answer. (A) $f$ is one-one onto (B) $f$ is many-one onto (C) $f$ is one-one but not onto (D) $f$ is neither one-one nor onto Solution: $f: \mathbf{R} \rightarrow \mathbf{R}$ is defined as $f(x)=x^{4}$. Let $x, y \in \mathbf{R}$ such that $f(x)=f(y)$. $\Rightarrow x^{4}=y^{4}$ $\Rightarrow x=\pm y$ $\therefore f\left(x_{1}\right)=f\left(x_{2}\right)$ does not imply that $x_{1}=x_{2}$. For instance,...

Ethyl acetate is formed by the reaction between ethanol and acetic acid and the equilibrium is represented as:

Question: Ethyl acetate is formed by the reaction between ethanol and acetic acid and the equilibrium is represented as: $\mathrm{CH}_{3} \mathrm{COOH}(\mathrm{I})+\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OH}(\mathrm{I}) \longleftrightarrow \mathrm{CH}_{3} \mathrm{COOC}_{2} \mathrm{H}_{5}(\mathrm{I})+\mathrm{H}_{2} \mathrm{O}(\mathrm{l})$ (i) Write the concentration ratio (reaction quotient),Qc, for this reaction (note: water is not in excess and is not a solvent in this reaction) (ii) At 293 K, if...

A metallic right circular cone 20 cm high and whose vertical angle is 60° is cut into two parts at the middle of its height by a plane parallel to its base.

Question: A metallic right circular cone 20 cm high and whose vertical angle is 60 is cut into two parts at the middle of its height by a plane parallel to its base. If the frustum so obtained be drawn into a wire of diameter 1/16 cm, find the length of the wire. Solution: From the figure $\frac{\mathbf{R}}{\mathbf{2 0}}=\tan 30^{\circ}=\frac{\mathbf{1}}{\sqrt{3}}$, i.e., $\mathrm{R}=\frac{\mathbf{2 0}}{\sqrt{\mathbf{3}}} \mathrm{cm}$ $\frac{\mathbf{r}}{\mathbf{1 0}}=\tan 30^{\circ}=\frac{\mathb...

Let A = R − {3} and B = R − {1}.

Question: Let $A=\mathbf{R}-\{3\}$ and $B=\mathbf{R}-\{1\} .$ Consider the function $f: A \rightarrow B$ defined by $f(x)=\left(\frac{x-2}{x-3}\right) .$ Is $f$ one-one and onto? Justify your answer. Solution: $A=\mathbf{R}-\{3\}, B=\mathbf{R}-\{1\}$ $f: \mathrm{A} \rightarrow \mathrm{B}$ is defined as $f(x)=\left(\frac{x-2}{x-3}\right)$. Let $x, y \in$ A such that $f(x)=f(y)$. $\Rightarrow \frac{x-2}{x-3}=\frac{y-2}{y-3}$ $\Rightarrow(x-2)(y-3)=(y-2)(x-3)$ $\Rightarrow x y-3 x-2 y+6=x y-3 y-2 x...

A container, opened from the top and made up of a metal sheet,

Question: A container, opened from the top and made up of a metal sheet, is in the form of a frustum of a cone of height $16 \mathrm{~cm}$ with radii of its lower and upper ends as $8 \mathrm{~cm}$ and $20 \mathrm{~cm}$, respectively. Find the cost of the milk which can completely fill the container, at the rate of Rs. 20 per litre. Also find the cost of metal sheet used to make the container, if it costs Rs. 8 per $100 \mathrm{~cm}^{2}$. (Take $\pi=3.14$ ). Solution: We have $: r_{1}=20 \mathrm...

Kp = 0.04 atm at 899 K for the equilibrium shown below

Question: $\mathrm{K}_{\mathrm{p}}=0.04 \mathrm{~atm}$ at $899 \mathrm{~K}$ for the equilibrium shown below. What is the equilibrium concentration of $\mathrm{C}_{2} \mathrm{H}_{6}$ when it is placed in a flask at $4.0$ atm pressure and allowed to come to equilibrium? $\mathrm{C}_{2} \mathrm{H}_{6}(\mathrm{~g}) \longleftrightarrow \mathrm{C}_{2} \mathrm{H}_{4}(\mathrm{~g})+\mathrm{H}_{2}(\mathrm{~g})$ Solution: Letpbe the pressure exerted by ethene and hydrogen gas (each) at equilibrium. Now, ac...

A fez, the cap used by the Turks, is shaped like the frustum of a cone (see fig.).

Question: A fez, the cap used by the Turks, is shaped like the frustum of a cone (see fig.). If its radius on the open side is 10 cm, radius at the upper base is 4 cm and its slant height is 15 cm, find the area of material used for making it. Solution: $\mathrm{R}=10 \mathrm{~cm}, \mathrm{r}=4 \mathrm{~cm}, \ell=15 \mathrm{~cm}$ Curved surface area $=\pi \times \ell \times\{R+r\}$ $=\pi \times 15 \times\{10+4\} \mathrm{cm}^{2}$ $=\frac{22}{7} \times 15 \times 14 \mathrm{~cm}^{2}=660 \mathrm{~cm...

What is the equilibrium concentration of each of the substances in the equilibrium when the initial concentration of ICl was 0.78 M?

Question: What is the equilibrium concentration of each of the substances in the equilibrium when the initial concentration of ICl was 0.78 M? $2 \mathrm{ICl}(\mathrm{g}) \rightleftharpoons \mathrm{I}_{2}(\mathrm{~g})+\mathrm{Cl}_{2}(\mathrm{~g}) ; \mathrm{K}_{\mathrm{C}}=0.14$ Solution: The given reaction is: Now, we can write, $\frac{\left[\mathrm{I}_{2}\right]\left[\mathrm{Cl}_{2}\right]}{[\mathrm{ICl}]^{2}}=K_{\mathrm{C}}$ $\Rightarrow \frac{x \times x}{(0.78-2 x)^{2}}=0.14$ $\Rightarrow \fr...

The slant height of a frustum of a cone is 4 cm and the perimeters (circumference) of its circular ends are 18 cm and 6 cm.

Question: The slant height of a frustum of a cone is 4 cm and the perimeters (circumference) of its circular ends are 18 cm and 6 cm. Find the curved surface area of the frustum. Solution: We have Slant height $(\ell)=4 \mathrm{~cm}$ $2 \pi r_{1}=18 \mathrm{~cm}$ and $2 \pi r_{2}=6 \mathrm{~cm}$ $\Rightarrow \pi r_{1}=\frac{\mathbf{1 8}}{\mathbf{2}}=9 \mathrm{~cm}$ and $\pi r_{2}=\frac{\mathbf{6}}{\mathbf{2}}=3 \mathrm{~cm}$ $\therefore$ Curved surface area of the frustum of the cone $=\pi\left(...

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