5 Marks Questions

Question 15 Marks

$100$ surnames were randomly picked up from a local telephone directory and a frequency distribution of the number of letters in the English alphabets in the surnames was found as follows :

Number of letters	Number of surnames
$1-4$	$6$
$4-6$	$30$
$6-8$	$44$
$8-12$	$16$
$12-20$	$4$

$i.$ Draw a histogram to depict the given information.
$ii.$ Write the class interval in which the maximum number of surnames lie.

Answer

$i.$

Number of letters	Number of surnames	Width of the class	Length of the rectangle
$1-4$	$6$	$3$	$\frac{6}{3} \times 2 = 4$
$4-6$	$30$	$2$	$\frac{30}{2} \times 2 = 30$
$6-8$	$44$	$2$	$\frac{44}{2} \times 2 = 44$
$8-12$	$16$	$4$	$\frac{16}{4} \times 2 = 8$
$12-20$	$4$	$8$	$\frac{4}{8} \times 2 = 1$

$ii.$The class interval in which the maximum number of surnames lie is $6-8.$

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Question 25 Marks

A random survey of the number of children of various age groups playing in a park was found as follows

Age (in years)	Number of children
$1-2$	$5$
$2-3$	$3$
$3-5$	$6$
$5-7$	$12$
$7-10$	$9$
$10-15$	$10$
$15-17$	$4$

Draw a histogram to represent the data above.

Answer

Here, the widths of the rectangles are varying. So, we need to make certain modifications in the lengths of the rectangles, so that the areas are became proportional to the frequencies.
The minimum class size is $1$ .
Length of rectangle (Adjusted frequency) $= \frac{\text { Minimum class size }}{\text { Class size of this class }} \times Frequency$
Then modified table of given data is shown below

Age (in years)	Number of children (Frequency)	Width of the class	Length of the rectangle
$1-2$	$5$	$1$	$\frac{1}{1} \times 5 = 5$
$2-3$	$3$	$1$	$\frac{1}{1} \times 3 = 3$
$3-5$	$6$	$2$	$\frac{1}{2} \times 6 = 3$
$5-7$	$12$	$2$	$\frac{1}{2} \times 12 = 6$
$7-10$	$9$	$3$	$\frac{1}{3} \times 9 = 3$
$10-15$	$10$	$5$	$\frac{1}{5} \times 10 = 2$
$15-17$	$4$	$2$	$\frac{1}{2} \times 4 = 2$

So, the histogram with varying width is given below

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Question 35 Marks

The runs scored by two teams $A$ and $B$ on the first $60$ balls in a cricket match are given below :

Number of balls	Team A	Team B
$1-6$	$2$	$5$
$7-12$	$1$	$6$
$13-18$	$8$	$2$
$19-24$	$9$	$10$
$25-30$	$4$	$5$
$31-36$	$5$	$6$
$37-42$	$6$	$3$
$43-48$	$10$	$4$
$49-54$	$6$	$8$
$55-60$	$2$	$10$

Represent the data of both the teams on the same graph by frequency polygons.
[Hint: First make the class intervals continuous.]

Answer

Number of balls	Class-Marks	Team A	Team B
$0.5-6.5$	$3.5$	$2$	$5$
$6.5-12.5$	$9.5$	$1$	$6$
$12.5-18.5$	$15.5$	$8$	$2$
$18.5-24.5$	$21.5$	$9$	$10$
$24.5-30.5$	$27.5$	$4$	$5$
$30.5-36.5$	$33.5$	$5$	$6$
$36.5-42.5$	$39.5$	$6$	$3$
$42.5-48.5$	$45.5$	$10$	$4$
$48.5-54.5$	$51.5$	$6$	$8$
$54.5-60.5$	$57.5$	$2$	$10$

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Question 45 Marks

The following table gives the distribution of students of two sections according to the marks obtained by them:

Section A		Section B
Marks	Frequency	Marks	Frequency
$0-10$	$3$	$0-10$	$5$
$10-20$	$9$	$10-20$	$19$
$20-30$	$17$	$20-30$	$15$
$30-40$	$12$	$30-40$	$10$
$40-50$	$9$	$40-50$	$1$

Represent the marks of the students of both the sections on the same graph by frequency polygons.

Answer

For section $A$

Classes	Class-Marks	Frequency
$0-10$	$5$	$3$
$10-20$	$15$	$9$
$20-30$	$25$	$17$
$30-40$	$35$	$12$
$40-50$	$45$	$9$

For section $B$

Classes	Class-Marks	Frequency
$0-10$	$5$	$5$
$10-20$	$15$	$19$
$20-30$	$25$	$15$
$30-40$	$35$	$10$
$40-50$	$45$	$1$

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Question 55 Marks

The length of $40$ leaves of a plant are measured correct to one millimetre, and the obtained data is represented in the following table :

Length (in mm)	Number of leaves
$118-126$	$3$
$127-135$	$5$
$136-144$	$9$
$145-153$	$12$
$154-162$	$5$
$163-171$	$4$
$172-180$	$2$

$i.$ Draw a histogram to represent the given data. $[$Hint: First make the class intervals continous$]$
$ii.$ Is there any other suitable graphical representation for the same data?
$iii.$ Is it correct to conclude that the maximum number of leaves are $153$ mm long$?$ Why$?$

Answer

$i.$ Modified continuous Distribution

Length $($in mm$)$	Number of leaves
$117.5-126.5$	$3$
$126.5-135.5$	$5$
$135.5-144.5$	$9$
$144.5-153.5$	$12$
$153.5-162.5$	$5$
$162.5-171.5$	$4$
$171.5-180.5$	$2$

$i.$ Frequency Polygon.
$ii.$ No because the maximum number of leaves have their lengths lying in the interval $145-153.$

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Question 65 Marks

Given below are the seats won by different political parties in the polling outcome of a state assembly elections:

Political party	$A$	$B$	$C$	$D$	$E$	$F$
Seats won	$75$	$55$	$37$	$29$	$10$	$37$

$i.$ Draw a bar graph to represent the polling results.
$ii.$ Which political party won the maximum number of seats$?$

Answer

$i.$ Political party $A$ won the maximum number of seats.

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Question 75 Marks

The following data on the number of girls $($to the nearest ten$)$ per thousand boys in different sections of the society is given below :

Section	Number of girls per thousand boys
Scheduled caste	$940$
scheduled tribe	$970$
Non $SC/ST$	$920$
Backward districts	$950$
Non$-$backward districts	$920$
Rural	$930$
Urban	$910$

$i.$ Represent the information above by a bar graph.
$ii.$ In the classroom discuss what conclusion can be arrived at from the graph.

Answer

$i.$ The two conclusions we can arrive at from the graph are as follows:
$ii.$ The numbers of girls to the nearest ten per thousand boys is maximum in Scheduled Tribe section of the society and minimum in Urban section of the society.
$iii.$ The number of girls to the nearest ten per thousand boys is the same for $'$Non $SC/ST'$ and $'$Non$-$backward Districts$'$ sections of the society.

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Question 85 Marks

A survey conducted by an organisation for the cause of illness and death among the women between the ages $15-44 ($in years$)$ worldwide, found the following figures $($in $\%)$ :

S.No.	Causes	Female fatality rate (%)
$1$	Reproductive health conditions	$31.8$
$2$	Neuropsychiatric conditions	$25.4$
$3$	Injuries	$12.4$
$4$	Cardiovascular conditions	$4.3$
$5$	Respiratory conditions	$4.1$
$6$	Other causes	$22.0$

$i.$ Represent the information given above graphically.
$ii.$ Which condition is the major cause of women's ill health and death worldwide?
$iii.$ Try to find out, with the help of your teacher, any two factors which play a major role in the cause in $(ii)$ above being the major cause.

Answer

$i.$ Reproductive health conditions is the major cause of women's ill health and death worldwide.
$ii.$ Lack of proper diet, lack of advised exercises.

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Question 95 Marks

In a city, the weekly observations made in a study on the cost of living index are given in the following table:

Cost of living index	Number of weeks
$140 - 150$	$5$
$150 - 160$	$10$
$160 - 170$	$20$
$170 - 180$	$9$
$180 - 190$	$6$
$190 - 200$	$2$
Total	$52$

Draw a frequency polygon for the data above (without constructing a histogram)

Answer

Since we want to draw a frequency polygon without a histogram, let us find the class-marks of the classes given above, that is of $140 - 150, 150 - 160,....$ For $140 - 150,$ the upper limit $= 150,$ and the lower limit $= 140$
So, the class-mark = $\begin{equation} \frac{150+140}{2}=\frac{290}{2}=145 \end{equation}$
Continuing in the same manner, we find the class-marks of the other classes as well.
So, the new table obtained is as shown in the following table:
Table

Cost of living index	Class-marks	Number of weeks
$140 - 150$	$145$	$5$
$150 - 160$	$155$	$10$
$160 - 170$	$165$	$20$
$170 - 180$	$175$	$9$
$180 - 190$	$185$	$6$
$190 - 200$	$195$	$2$
Total		$52$

We can now draw a frequency polygon by plotting the class-marks along the horizontal axis, the frequencies along the vertical-axis, and then plotting and joining the points $B(145, 5), C(155, 10), D(165, 20), E(175, 9), F(185, 6)$ and $G(195, 2)$ by line segments. We should not forget to plot the point corresponding to the class-mark of the class $130 - 140 ($just before the lowest class $140 - 150)$ with zero frequency, that is, $A(135, 0),$ and the point $H (205, 0)$ occurs immediately after $G(195, 2).$ So, the resultant frequency polygon will be $ABCDEFGH$ (see Fig.)

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Question 105 Marks

Consider the marks, out of $100,$ obtained by $51$ students of a class in a test, given in Table.

Marks	Number of students
$0 - 10$	$5$
$10 - 20$	$10$
$20 - 30$	$4$
$30 - 40$	$6$
$40 - 50$	$7$
$50 - 60$	$3$
$60 - 70$	$2$
$70 - 80$	$2$
$80 - 90$	$3$
$90 - 100$	$9$
Total	$51$

Draw a frequency polygon corresponding to this frequency distribution table.

Answer

The $OABCDEFGHIJKL$ is the frequency polygon, which is shown in Fig.

Frequency polygons can also be drawn independently without drawing histograms. For this, we require the mid-points of the class-intervals used in the data. These mid-points of the class-intervals are called class-marks.

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Question 115 Marks

A teacher wanted to analyse the performance of two sections of students in a mathematics test of $100$ marks. Looking at their performances, she found that a few students got under $20$ marks and a few got $70$ marks or above. So she decided to group them into intervals of varying sizes as follows: $0 - 20, 20 - 30, . . ., 60 - 70, 70 - 100.$ Then she formed the following table:

Marks	Number of students
$0 - 20$	$7$
$20 - 30$	$10$
$30 - 40$	$10$
$40 - 50$	$20$
$50 - 60$	$20$
$60 - 70$	$15$
$70 -$ above	$8$
Total	$90$

Draw a histogram for this table$?$

Answer

The histrogram is given below.

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Question 125 Marks

A family with a monthly income of $₹. 20,000$ had planned the following expenditures per month under various heads:

Heads	Expenditure $($in thousand rupees$)$
Grocery	$4$
Rent	$5$
Education of children	$5$
Medicine	$5$
Fuel	$2$
Entertainment	$1$
Miscellaneous	$1$

Draw a bar graph for the data above.

Answer

We draw the bar graph of this data in the following steps.
Note that the unit in the second column is thousand rupees.
So, $'4\ ’$ against $'$grocery$’$ means $₹.4000.$
$i.$ We represent the Heads $($variable$)$ on the horizontal axis choosing any scale, since the width of the bar is not important.
But for clarity, we take equal widths for all bars and maintain equal gaps in between.
Let one Head be represented by one unit.
$ii.$ We represent the expenditure $($value$)$ on the vertical axis.
Since the maximum expenditure is $₹.5000,$ we can choose the scale as $1$ unit $= ₹.1000.$
$iii.$ The bar graph is drawn in Fig.

$i.$ Here, you can easily visualise the relative characteristics of the data at a glance, e.g., the expenditure on education is more than double that of medical expenses.
Therefore, in some ways it serves as a better representation of data than the tabular form.

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Question 135 Marks

Consider the marks obtained by $10$ students in a mathematics test as given below:
$55, 36, 95, 73, 60, 42, 25, 78, 75, 62$
Find the maximum, minimum scores and range.

Answer

Let us arrange the marks in ascending order as $25, 36, 42, 55, 60, 62, 73, 75, 78, 95$
Now, the lowest marks are $25$ and the highest marks are $95.$
The range, in this case, is $95 – 25 = 70$

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