**NCERT Exemplar Problems Class 8 Mathematics Chapter 12 Introduction to Graphs**

**Question. 1 Comparison of parts of a whole may be done by a**

** (a) bar graph (b) pie chart**

** (c) linear graph (c) line graph**

** Solution.**

(b) There are various ways to represent and compare the data. One of them is pie chart.

Pie chart is a pictorial representation of the data in which the whole is represented by a circle and the parts, by non-intersecting adjacent sectors.

Hence, comparison of parts of a whole may be done by a pie chart.

**Question. 2 A graph that displays data that changes continuously over periods of time is 451**

** (a) bar graph (b) pie chart**

** (c) histogram (d) line graph**

** Solution.**

(d) Line graph is an important way to represent and compare the data which varies continuously. •

A line graph displays the relation between two varying quantities.

In a line graph, we connect all the points by a line segment while in bar graph and histogram, we use rectangles of uniform width.

**Question. 3 In the given graph the coordinates of point P are**

**Solution.**

(c) The foot of the perpendicular drawn from the indicated point on X-axis is at a distance of 3 units from the origin.

The x-coordinate of the point is 3.

Again, the perpendicular drawn from the given point on Y-axis meets the Y-axis at a point which is at a distance of 2 units from origin.

The y-coordinate of the point is 2.

Hence, the coordinates of the point are (3,2).

**Question. 4 In the given graph the letter that indicates the point (0,3) is**

**Solution.**

(c) The letter that indicates the point (0,3) is R as it lies on the Y-axis at a distance of 3 units from the origin.

The coordinates of the point P and Sare (3,0) and (3,3), respectively.

**Question. 5 The point (3,4) is at a distance of**

** (a) 3 from both the axes**

** (b) 4 from both the axes**

** (c) 4 from the X-axis and 3 from Y-axis**

** (d) 3 from X-axis and from Y-axis**

** Solution.**

(c) We know that, the x-coordinate is the distance of the point from Y-axis and that of y-coordinate is the distance from X-axis. .

Hence, the point (3,4) is at a distance of 4 from the X-axis and 3 from Y-axis.

**Question. 6 A point which lies on both the axes is**

** (a) (0,0) (b) (0,1) (0(1,0) (d) (1,1)**

** Solution.**

(a) We know that, the axes are two mutually perpendicular lines intersecting each other at the point (0,0) also known as the origin.

Hence, the point which lies on both the axes is (0,0).

**Question. 7 The coordinates of a point at a distance of 3 units from the X-axis and 6 units from the y-axis are**

** (a) (0,3) (b) (6,0) (c) (3,6) (d) (6,3)**

** Solution.**

(d) We know that, the x-coordinate is the distance of the point from /-axis and the y-coordinate is the distance of the point from X-axis.

Hence, the coordinates of the required point are (6,3).

**Question. 8 In the given figure, the position of the book on the table may be given by**

**Solution.**

(b) The book is at a distance of 3 units from Y-axis and 7 units from X-axis.

Hence, the position of the book on the table is given by (3,7).

**Question. 9 Data was collected on a student’s typing rate and graph was drawn as shown below. Approximately how many words had this student typed in 30 s?**

** Solution.**

(c) Observing the graph, we see that the graph intersects the line x = 30 just below the point (30,30), which is the intersection of the lines* = 30 and y- 30.

Since, the X-axis represents the time (in seconds) and the Y-axis represents the number of words typed, therefore we conclude that the students had typed approximately 28 words In 30 s.

**Question. 10 Which graph of the following represents the table below?**

** **

**Solution.**

(d) Here, the lengths of sides of squares are represented on X-axis and their perimeters are represented on Y-axis.

The points are (1,4),(2,8),(3,12),(4,16) and(5,20).

Now, observing all the 4 graphs carefully we see that these points lie on the graph(d).

**Fill in the Blanks**

**In questions 11 to 25, fill in the blanks to make the statements true.**

**Question. 11________.displays data that changes continuously over periods of time.**

** Solution.** Line graph

We have already discussed in the previous question that a line graph displays data that changes continuously over periods of time.

**Question. 12 The relation between dependent and independent variables is shown**

** through a________.**

** Solution.** graph .

Various types of graph depicts the relation between two variables, one of them is independent and the other is dependent.

A graph shows how a change in the independent variable changes the dependent variable.

**Question. 13 We need________coordinates for representing a point on the graph**

** sheet.**

** Solution.** pair of (or two)

To represent the position of a point on the graph sheet we require two measures namely x-coordinate and y-coordinate.

**Question. 14 A point in which the x-coordinate is zero and y-coordinate is non- zero**

** will lie on the________.**

** Solution.** Y-axis

Since, the x-coordinate is zero, i.e. the distance of the point from Y-axis is 0, the point lies on the Y-axis at a certain distance from the origin, which is given by y-coordinate.

**Question. 15 The horizontal and vertical lines in a graph are usually called________and**

** Solution.** X-axis, Y-axis

To draw a graph we need pairs of points known as coordinates and to plot a point we require two mutually perpendicular axes, also known as the X-axis (horizontal line) and the Y-axis (vertical line).

**Question. 16 The process of fixing a point with the help of the coordinates is known**

** as________of the point.**

** Solution.** plotting

To locate the exact position of a point we need two coordinates viz, the x-coordinate and the y-coordinate and this process of finding the position or representing the numbers on a graph sheet with the help of the coordinates is known as plotting of the point.

**Question. 17 The distance of any point from the y-axis is the________coordinate.**

** Solution.** x

x-coordinate of a point is the distance of any point from the Y-axis.

**Question. 18 All points with y-coordinate as zero lie on the________.**

** Solution.** X-axis

Since, y-coordinate is zero, i.e. the distance of the point from X-axis is zero.

Hence, the points lies on the X-axis.

**Question. 19 For the point (5,2), the distance from the X-axis is________units.**

** Solution.** 2

We know that, the y-coordinate represents the distance of the point from the X-axis. ;Hence, the point (5,2) is at a distance of 2 units from the X-axis.

**Question. 20 The x-coordinate of any point lying on the y-axis will be________.**

** Solution.** zero

Since, the x-coordinate represents the distance of the point from Y-axis is zero, therefore the points lying on the Y-axis have x-coordinate as zero.

**Question.21 The y-coordinate of the point (2,4) is________.**

** Solution.** 4

In the ordered pair (2,4), i.e. coordinates of a point, the second number is called as the y-coordinate of the point.

Hence, the y-coordinate of (2,4) is 4

**Question. 22 In the point (4,7), 4 denotes the________.**

** Solution.** x-coordinate (abscissa)

First number (coordinate) of the ordered pair is called as the x-coordinate or abscissa. Hence, in the point (4,7),4 denotes the x-coordinate.

**Question. 23 A point has 5 as its x-coordinate and 4 as its y-coordinate. Then, the**

** coordinates of the point are given by________.**

** Solution.** (5,4)

To denote a point in 2-D, we use two numbers viz. the x-coordinate, the y-coordinate.

In the ordered pair, the x-coordinate is written in the first slot and the y-coordinate in the second slot separated by comma. Hence, the required point is (5,4).

**Question. 24 In the coordinates of a point, the second number denotes the________.**

** Solution.** y-coordinate y ordinate

As we have discussed in the above question, that the second number denotes the . y-coordinate, also known as the ordinate.

**Question. 25 The point where the two axes intersect is called the________.**

** Solution.** origin

The X-axis and y-axis intersect each other at the point representing the origin. Coordinates of origin are (0,0).

**True/False**

**In questions 26 to 34, state whether the statements are True or False.**

** Question. 26 For fixing a point on the graph sheet we need two coordinates.**

** Solution.** True

To plot a point on the graph sheet we require two numbers, known as coordinates viz. the x-coordinate and the y-coordinate.

**Question. 27 A line graph can also be a whole unbroken line.**

** Solution.** True

A fine graph, which represents the variation of a quantity with respect to the other, may be an unbroken line.

**Question. 28 The distance of any point from the X-axis is called the x-coordinate.**

** Solution.** False

The distance of any point from the X-axis is called the y-coordinate.

**Question. 29 The distance of the point (3,5) from the Y-axis is 5.**

** Solution.** False

We know that the x-coordinate of a point represents the distance of the point from Y-axis. Here x-coordinate is 3, so the distance of the point (3,5) from the Y-axis is 3.

**Question. 30 The ordinate of a point is its distance from the Y-axis.**

** Solution.** False

The ordinate of a point is nothing but ^coordinate of the point and the y-coordinate denotes the distance of a point from X-axis.

**Question. 31 In the point (2,3), 3 denotes the y-coordinate.**

** Solution.** True

In the ordered pair (2,3), the second number is called the y-coordinate or ordinate of the number.

Hence, 3 denotes the y-coordinate of the point (2,3).

**Question. 32 The coordinates of the origin are (0,0).**

** Solution.** True

Origin is the point, where two axes meet and its coordinates are (0,0).

**Question. 33 The points (3,5) and (5,3) represent the same point.**

** Solution.** False

Two ordered pairs are equal, if they have same numbers at corresponding slot, i.e. x-coordinates are equal and Y-coordinates are equal. Hence, (3,5) and (5,3) are different points.

**Question. 34 The y-coordinate of any point lying on the X-axis will be zero.**

** Solution.** True

The distance of the points which lie on X-axis, will be zero from the X-axis, i.e. y-coordinate is zero for the points lying on X-axis.

**Question. 35 Match the coordinates given in Column A with the items mentioned in Column B.**

** **

** Solution.**

(a) In the pair (0,5), the second number also known as ordinate represents the distance from X-axis, i.e. 5.

(a) In the pair (2,3), 2 the first number, also known as abscissa represents the distance from Y-axis that is 2.

(b) We have the coordinates (4,8). Clearly, ^coordinate is double of x-coordinate.

(c) We have the coordinate (3,7), where x-coordinate = 3 and y-coordinate = 7

Evidently, y-coordinate = 2 x *x*-coordinate + 1

(d) (0,0) are the coordinates of origin.

(e) In the point (5,0), the y-coordinate is zero.

Hence, (a) —> iv, (b) —> vi, (c) —> v, (d) —> i, (e) —> ii, (f) —> iii.

**Question. 36 Match the ordinates of the points given in Column A with the items mentioned in Column B.**

** **

** Solution.**

(a) Clearly, the ordinate of the point (7,0) is zero.

(b) In the point (11,11), the ordinate is equal to the abscissa.

(c) In the point (4,8), the ordinate is double of the abscissa.

(d) In the point (6,2), the abscissa, i.e. x-coordinate is triple of the ordinate, i.e. y-coordinate.

(e) The abscissa of the point (0,9) is zero.

(f) Clearly,the abscissa is double of the ordinate.

Hence, (a)—> ii, (b) —> iii, (c) —> i, (d) -> v, (e) —> vi, (f) —> iv.

**Question. 37 From the given graph, choose the letters that indicate the location of the points given below.**

** **

** Solution.**

On observing the graph, we see that the point F is on X-axis, so its Y-coordinate will be zero.

Also, it is at a distance of 2 units from origin.

.^{.}. The coordinates of F are (2,0), similarly the coordinates of G are (4,0).

H is at a distance of 5 units from Y-axis and 1 unit from X-axis.

.^{.}. The coordinates of Hare (5,1).

/ is at a distance of 6 units from Y-axis and 2 units from X-axis.

.^{.}. The coordinates of / are (6,2).

The point D and A are on Y-axis at distances of 2 units and 4 units respectively from the origin.

Hence, the coordinates of D and A are (0,2) and (0,4), respectively.

B is at a distance of 1 unit from Y-axis and 5 units from X-axis.

.^{.}. The coordinates of Bare (1,5).

C is at a distance of 2 units from Y-axis and 6 units from X-axis.

.^{.}. The coordinates of C are (2,6).

E is at a distance of 3 units from Y-axis and X-axis both.

.^{.}. The coordinates of E are (3,3).

**Question. 38 Find the coordinates of all letters in the graph given below.**

**Solution.**

The point A is on the Y-axis at a distance of 7.5 units from the origin.

.^{.}. The coordinates are (0,7.5).

The point B is at a distance of 4 units from Y-axis and 5 units from X-axis.

.^{.}. The coordinates of B are (4,5).

The point C is at a distance of 7.5 units from Y-axis and 2.5 units from X-axis .

.^{.}. The coordinates of C denotes (7.5, 2.5).

The point D lies on X-axis at a distance of 11 units from the origin.

.^{.}. The coordinates of D are (11,0).

The point £ is at a distance of 14.5 units from Y-axis and 6.5 units from X-axis.

.^{.}. The coordinates of £ are (14.5, 6.5).

The point F is at a distance of 18 units from Y-axis and 9.5 units from X-axis.

.^{.}. The coordinates of £ are (18,9.5).

**Question. 39 Plot the given points on a graph sheet**

** (a) (5,4) (b) (2,0) (c) (3,1) (d)(0,4) (e) (4,5)**

** Solution.**

**Question. 40 Study the given map of a zoo and answer the following questions.**

**(a) Give the location of lions in the zoo.**

** (b) (D,f) and (C,d) represent locations of which animals in the zoo?**

** (c) Where are the toilets located?**

** (d) Give the location of canteen?**

** Solution. **

(a) Lions are at a distance of A units from road Y and f units from road X.

.^{.}. The location of lions is represented by the point (Af)

(b) Observing the map carefully, we see that (D, f) denotes the location of monkeys and (C,d) denotes the location of elephants.

(c) Toilets are located on the road Y at a distance of e units from the origin. Hence, the location of toilets is (0, e).

(d) Canteen is located at a distance of C units from road Y and c units from road X. Hence, the location of the canteen is (C,c).

**Question. 41 Write the x-coordinate (abscissa) of each of the given points.**

** (a) (7,3) (b) (5,7) (c) (0,5)**

** Solution.**

(a) The x-coordinate of the point (7,3) is 7

(b) The x-coordinate of the point (5,7) is 5.

(c) The x-ooordinate of the point (0,5) is 0,

**Question. 42 Write the y-coordinate (ordinate) of each of the given points.**

** (a) (3,5) (b) (4,0) (c) (2,7)**

** Solution.**

(a) The y-coordinate of the point (3,5) is 5.

(b) y-coordinate of the point (4,0) is 0.

(c) The ycoordinate of the point (2,7) is 7.

**Question. 43 Plot the given points on a graph sheet and check if the points lie on a straight line. If not, name the shape they form when joined in the given order.**

** (a) (1,2), (2,4), (3,6), (4,8)**

** (b) (14), (1,2), (2,1), (2,2)**

** (c) (4,2), (2,4), (3,3), (5,4)**

** Solution.**

Yes, the given points lie on a straight line.

No, the points do not form a straight line, they form a square.

**Question. 44 If y-coordinate is 3 times x coordinate, form a table for it and draw a graph.**

** Solution.**

Since, the ordinate is 3 times the abscissa, we get the following values.

**Question. 45 Make a line graph for the area of a square as per the given table.**

**Solution.**

**Question. 46 The cost of a notebook is Rs 10. Draw a graph after making a table showing cost of 2,3,4… notebooks. Use it to find**

** (a) the cost of 7 notebooks.**

** (b) the number of notebooks that can be purchased with Rs 50.**

** Solution.**

Let x : number of notebooks

y: cost of a notebook

(a) The cost of 7 notebooks is aqua) to the ordinate of the point (7,70), i.e. cost of 7 notebooks = Rs 70

(b) The number of notebooks that can be purchased with Rs 50 is equal to the abscissa of the point (5,50).

Hence, 5 notebooks can be purchased with Rs 50.

**Question. 47 Explain the situations represented by the following distance-time graphs.**

** **

** Solution.** Here X-axis represents time and y – axis represents distance.

(a) In the first graph, we observe that when time changes, distance also varies at the same rate.

When we move along time axis away from the origin, then the graph is strictly increasing.

Hence, the object is moving at a uniform speed.

(b) In the graph (b), we observe that initially graph increases steadily i.e. at uniform speed and after a certain period of time, it comes to rest position i.e. constant.

(c) In the graph (c), we see that the graph increases strictly with non-uniform speed and then slowly comes to the rest position.

**Question. 48 Complete the given tables and draw a graph for each.**

**Solution.**

**Question. 49 Study the given graphs (a) and (b) and complete the corresponding tables below**

**Solution.**

**Question. 50 Draw a graph for the radius and circumference of circle using a suitable scale.**

** [Hint Take radius = 7,14,21 units and so on.]**

** Form the graph,**

** (a) find the circumference of the circle when radius is 42 units.**

** (b) at what radius will the circumference of the circle be 220 units?**

** Solution.**

**Question. 51 The graph shows the maximum temperatures recorded for two consecutive**

** weeks of a town. Study the graph and answer the questions that follow.**

** (a) What information is given by the two axes?**

** (b) In which week was the temperature higher on most of the days?**

** (c) On which day was the temperature same in both the weeks?**

** (d) On which day was the difference in temperatures the maximum for both the weeks?**

** (e) What were the temperatures for both the weeks on Thursday?**

** (f) On which day was the temperature 35°C for the first week?**

** (g) On which day was the temperature highest for the second week?**

** Solution.**

(a) the X-axis represents days of a particular week and the X-axis represents the maximum temperature (in °C) recorded.

(b) Observing the graph, we see that in the first week temperature was higher on most of

the days.

(c) The temperature was same on Wednesday in both the weeks.

(d) The difference in temperatures was the maximum on Friday for both the weeks.

(e) The temperature for the first week on Thursday was 37°C and the temperature for the second week on the same day was 34°C.

(f) On Sunday, the temperature was 35° for the first week.

(g) On Wednesday, the temperature was highest for the second week.

**Question. 52 The graph given below gives the actual and expected sales of cars of a company for 6 months. Study the graph and answer the questions that follow.**

** **

**(a) In which month was the actual sales same as the expected sales?**

** (b) For which month(s) was (were) the difference in actual and expected sales the maximum?**

** (c) For which month(s) was (were) the difference in actual and expected**

** sales the least?**

** (d) What was the total sales of carc in the months-January, February and March?**

** (e) What is the average sales of cars in the last three months?**

** (f) Find the ratio of sales in the first three months to the last three months.**

** Solution.**Observing the graph carefully, we conclude that

(a) In April, the actual sales was same as the expected sales.

(b) In March, the difference in actual and expected sales was the maximum.

(c) In April, the difference in actual and expected sales was the least.

(d) The total sales of cars in the months January, February and March was (75 + 100+75) i.e. 250.

(e) The average sales of cars in the last three months is 125 i.e.125 + 100+ 150/3 = 125.

(f) The number of sales of car in the first three months = 250 and the number of sales of car in the last three months = 375

The required ratio is 250 : 375 i.e. 2:3,

**Question. 53 The graph given below shows the marks obtained out of 10 by Sonia in two different tests. Study the graph and answer the questions that follow.**

**(a) What information is represented by the axes?**

** (b) In which subject did she score the highest in Test I?**

** (c) In which subject did she score the least in Test II?**

** (d) In which subject did she score the same marks in both the Tests?**

** (e) What are the marks scored by her in English in Test II?**

** (f) In which test was the performance better?**

** (g) In which subject and which test did she score full marks?**

** Solution.** Observing the graph carefully, we conclude that

(a) The X-axis represents subjects and the Y-axis represents the marks obtained by Sonia.

(b) In Maths, she scored the highest in Test I.

(c) In English and Hindi, she scored the least in Test II.

(d) In Hindi and Maths, she scored the same marks in both tests.

(e) She scored 6 marks in English in Test II.

(f) Same performance in both tests.

(g) Test I in Maths, she scored full marks i.e. 10 marks.

**Question. 54 Find the coordinates of the vertices of the given figures.**

**Solution.**

**Question. 55 Study the graph given below of a person who started from his home and returned at the end of the day. Answer the questions that follow.**

**(a) At what time did the person start from his home?**

** (b) How much distance did he travel in the first four hours of his journey?**

** (c) What was he doing from 3 PM to 5 PM?**

** (d) What was the total distance travelled by him throughout the day?**

** (e) Calculate the distance covered by him in the first 8 h of his journey.**

** (f) At what did he cover 16 km of his journey?**

** (g) Calculate the average speed of the man from A to B and B to C.**

** (h) At what time did he return home?**

** Solution.** Observing the graph carefully, we conclude that

(a) At 10 AM, the person start from his home.

(b) In first 4 h (i.e. till 2PM), he travelled 16 km.

(c) He was taking rest from 3 PM to 5 PM.

(d) The total distance covered by the person throughout the day was 40 km, i.e. 20 km from A to Sand then 20 km from C to D.

(e) The distance covered by him in the first 8 h i.e. from 10 AM to 6 PM was 24 km,

(f) He covered 16 km of his journey at 2 PM,

(g) The total distance covered from A to S=20 km

and the time taken to travel from A to B = 5 h

.^{.}.Average speed of the man from A to B =20/5 = 4 km/h

and average speed from Sto C = 0/2 = 0 km/h

(h) He returned home at 10 PM.

**Question. 56 Plot a line graph for the variables p and q, where p is two times q i.e. the equation is p = 2q. Then, find**

** (a) the value of p when q = 3.**

** (b) the value of q when p = 8.**

** Solution.**

**Question. 57 Study the graph and answer the questions that follow.**

**(a) What information does the graph give?**

** (b) On which day was the temperature the least?**

** (c) On which day was the temperature 31 °C?**

** (d) Which was the hottest day?**

** Solution.**

(a) The information obtained from the given graph is that the maximum temperature is 34°C

and minimum temperature is 25°C in a week.

(b) On Sunday, the temperature was 25°C. So, it is least temperature in the week.

(c) On Saturday, the temperature was 31 °C.

(d) On Friday, the temperature was maximum i.e. 34°C. Hence, it is the hottest day of the week.

**Question. 58 Study the distance-time graph given below for a car to travel to certain places and answer the questions that follow.**

**(a) How far does the car travel in 2′-h?**

** (b) How much time does the car take to reach /??**

** (c) How long does the car take to cover 80 km?**

** (d) How far is Q from the starting point?**

** (e) When does the car reach the place S after starting?**

** Solution.**

(a) From the given graph, the car travels 80 km in 2h.

(b) 5 h taken by car to reach ft.

(c) 2 h taken by car to cover 80 km. •

(d) G is 120 km far from the starting point.

(e) The car reaches the places after starting in 6 h.

**Question. 59 Locate the points A (1, 2), B (4, 2) and C (1, 4) on a graph sheet taking suitable axes. Write the coordinates of the fourth point D to complete the rectangle ABCD.**

** Solution.**

**Question. 60 Locate the points .A <1, 2), B (3, 4) and C (5, 2) on a graph sheet taking suitable axes. Write the coordinates of the fourth point D to complete the rhombus ABCD. Measure the diagonals of this rhombus and find whether they are equal or not.**

** Solution.**

**Question. 61 Locate the points P (3,4), Q (1,0), R (0,4), S (4,1) on a graph sheet and write the coordinates of the point of intersection of line segments PQ and RS.**

** Observing the graph, we see that the line segments PQ and RS intersect at the point!**

** Solution.**

**Question. 62 The graph given below compares the sales of ice-creams of two vendors for a week.**

**Observe the graph and answer the following questions.**

** (a) Which vezndor has sold more ice-creams on Friday?**

** (b) For which day was the sales same for both the vendors?**

** (c) On which day did the sale of vendor A increase the most as compared to the previous day?**

** (d) On which day was the difference in sales the maximum?**

** (e) On which two days was the sales same for vendor B?**

** Solution.** Observing the graph carefully, we conclude that

(a) Vendor A has sold more ice-creams on Friday.

(b) On Sunday , the sales was the same for both the vendors.

(c) On Sunday, the sale of vendor A increased the most as compared to Saturday.

(d) The difference in sales was the maximum on Thursday.

(e) On Tuesday and Wednesday, the sales was the same for vendor B.

**Question. 63 The table given below shows the temperatures recorded on a day at different times.**

**Observe the graph and answer the following questions.**

** (a) What is the temperature at 8 AM?**

** (b) At what time is the temperature 3°C?**

** (c) During which hour did the temperature fall?**

** (d) What is the change in temperature between 7 AM and 10 AM?**

** (e) During which hour was there a constant temperature?**

** Solution.** Observing the given graph carefully, we have

(a) At 8 AM, the temperature is 7°C.

(b) At 6 AM, the temperature is 3°C.

(c) The temperature fall in the hour 5 AM to 6 AM.

(d) The change in temperature is 3°C between 7 AM and 10 AM.

(e) Between 8 AM to 9 AM, there was a constant temperature.

**Question. 64 The following table gives the growth chart of a child.**

**Draw a line graph for the table and answer the questions that follow.**

** (a) What is the height at the age 4 yr? ,**

** (b) How much taller was the child at the age of 10 yr than at the age of**

** 6 yr?**

** (c) Between which two consecutive periods did the child grow more faster?**

** Solution.**

**Question. 65 The following is the time-distance graph of Sneha’s walking.**

**(a) When does Sneha make the least progress ? Explain your reasoning.**

** (b) Find her average speed in km/h.**

** Solution.**

**Question. 66 Draw a parallelogram ABCD on a graph paper with the coordinates given in Table I. Use this table to complete Tables II and III to get the coordinates of E, F, G, H and J, K, L, M.**

**Draw parallelograms EFGH and JKLM on the same graph paper. Plot the points (2, 4) and (4, 2) on a graph paper, then draw a line segment joining these two points.**

** Solution.** Complete table is shown below

**Question. 67 Extend the line segment on both sides to meet the coordinate axes. What are the coordinate of the points, where this line meets the X-axis and the y-axis?**

** Solution.**

Let PQ is a line segment which is extened from both ends to meet the axes.

The coordinates of the point on y-axis, where the line segment meet will be of form (0,y) whereas the

coordinates of the point of interaction on X-axis will be of type (x,0).

**Question. 68 The following graph shows the change in temperature of a block of ice when heated. Use the graph to answer the following questions.**

**(a) For how many seconds did the ice block have no change in temperature?**

** (b) For how long was there a change in temperature?**

** (c) After how many seconds of heating did the temperature become constant at 100°C?**

** (d) What was the temperature after 25 s?**

** (e) What will be the temperature after 1.5 min? Justify your answer.**

** Solution.**

(a) In the first 20 s, the ice block have no change in temperature.

(b) There was a change in temperature from 20 s to 50 s, i.e. 50-20 = 30s.

(c) Observing the graph, we see that after 50 s of heating the temperature became constant.

(d) 20°C was the temperature after 25 s.

(e) Since, the temperature became constant at 100°C after 50 s heating, so the temperature will be 100°C even after 1.5 min.

**Question. 69 The following graph shows the number of people present at a certain shop at different times. Observe the graph and answer the following questions.**

** **

**(a) What types of a graph is this?**

** (b) What information does the graph give?**

** (c) What is the busiest time of day at the shop?**

** (d) How many people enter the shop when it opens?**

** (e) About how many people are there in the shop at 1:30 PM?**

** Solution.**

(a) This is a line graph.

(b) It represents the number of people, who visited the store at a particular time.

(c) The busiest time of day is 1 PM at a shop, as at this time maximum number of people i.e. 25 visited the shop.

(d) When it opens less than 5 people enter the shop.

(e) There are 20 people in the shop at 1:30 PM.

**Question. 70 A man started his journey on his car from Location A and came back. The given**

** graph shows his position at different times during the whole journey.**

**(a) At what time did he start and end his journey?**

** (b) What was the total duration of journey?**

** (c) Which journey, forward or return, was of longer duration?**

** (d) For how many hours did he not move?**

** (e) At what time did he have the faster speed?**

** Solution.** Analysing the graph carefully, we observe that

(a) He started his journey at 5:30 AM and end at 6 PM.

(b) Total duration of journey was 12:30 h.

(c) His forward journey is of duration 8:30 h and return journey is of duration 4 h. Forward journey was of longer duration.

(d) He did not move from 6:30 AM to 9:30 AM and 10 AM to 1 PM.

So, he did not move for 6 h.

(e) He have the fastest speed at 1 PM. ”

**Question. 71 The following graph shows the journey made by two cyclists, one from town A to B and the other from town B to A.**

**(a) At what time did cyclist II rest? How long did the cyclist rest?**

** (b) Was cyclist II cycling faster or slower after the rest?**

** (c) At what time did the two cyclists meet?**

** (d) How far had cyclist II travelled when he met cyclist I?**

** (e) When cyclist II reached town A, how far was cyclist I from town 6?**

** Solution.**

(a) On the basis of given graph, the cyclist II rest at 8 : 45 AM for 15 min.

(b) Cyclist II is cycling faster after rest as he has covered a distance of 20 km in 1 h.

(c) Both cyclists meet at 9:00 AM.

(d) The cyclist II had travelled 20 km, when he met cyclist I.

(e) When cyclist II reached town A, the cyclist I was 10 km for from town B.

**Question. 72 Ajita starts off from home at 07.00 h with her father on a scooter that goes at a uniform speed of 30 km/h and drops her at her school after half an hour. She stays in the school till 13.30 h and takes an auto rickshaw to returrThome. The rickshaw has a uniform speed of 10 km/h. Draw the graph for the above situation and also determine the distance of Ajita’s school from her house.**

**Solution.**

**Question. 73 Draw the line graph using suitable scale to show the annual gross profit of a company for a period of five years.**

**Solution.**

We have taken years on X-axis and gross profit on Y-axis. The line graph of an annual gross profit of a company for a period of five years are given below.

**Question. 74 The following chart gives the growth in height in terms of percentage of full height of boys and girls with their respective ages.**

**Draw the line graph of above data on the same sheet and answer the following questions.**

** (a) In which year both the boys and the girls achieve their maximum height?**

** (b) Who grows faster at puberty (14 yr to 16 yr of age)?**

** Solution.**

(a) In 18 yr, the boys and in 17 yr, the girls achieve their maximum height. I

(b) Boys grows faster the girls during puberty.

**Question. 75 The table shows the data collected for Dhruv’s walking on a road.**

**(a) Plot a line graph for the given data using a suitable scale.**

** (b) In what time periods did Dhruv make the most progress?**

** Solution.(a)**

**Question. 76 Observe the given graph carefully and complete the table given below.**

**Solution.**

**Question. 77 This graph shows the per cent of students who dropped out of school after completing high school. The point labelled A shown that, in 1996, about 4.7% of students dropped out.**

**(a) In which year was the drop out the rate’highest? In which year was it the lowest?**

** (b) When did the per cent of students who dropped out of high school first fall below 5%?**

** (c) About what per cent of students dropped out of high school in 2007? About what per cent of students stayed in high school in 2008?**

** Solution.** Observing the graph carefully, we have

(a) The drop out rate was the highest in the year 1990 and the least in 2000.

(b) In the year 1996, the per cent of students dropped out of high school first fall below 5%.

(c) About 4.7% students dropped out of high school in 2007.

**Question. 78 Observe the toothpick pattern given below**

**Solution.**

**Question. 79 Consider this in put/output table.**

**Solution.**

**Question. 80 This graph shows a map of an island just off the coast of a continent. The point labelled B represents a major city on the coast. The distance between grid lines represents 1 km.**

**Point A represents a resorts that is located 5 km East and 3 km North of point B. The values 5 and 3 are the coordinates of point A. The coordinates can be given as the ordered pair (5,3), where 5 is the horizontal coordinate and 3 is the vertical coordinate.**

** (i) On a copy of the map, mark the point that is 3 km East and 5 km North of point and lebel it S. Is point S in the water or on the island? Is point 5 in,the same place as point A?**

** (ii)Mark the point that is 7 km east and 5 km north of point B and label it C. Then, mark the point that is 5 km east and 7 km north of point B and label it D. Are points C and D in the same place? Give the coordinates of points C and D.**

** (iii)Which points is in the water (2, 7) or (7, 2)? Mark the point which is in water otLyour map and Label it f.**

** (iv)Give the coordinates of two points on the island that are exactly 2 km apart from point A.**

** (v) Give the coordinates of the point that is halfway between points L and P.**

** (vi)List three points on the island with their x-coordinates greater than 8.**

** (vii)List three points on the island with a y-coordinate less than 4.**

** Solution.**

(i) The points is in the water.

No, it is not in the same place as point A

(ii)No, they are not in the same place. The coordinates of points C and D are (7, 5) and

(5, 7), respectively.

(iii)(2, 7) is in the water.

(iv)(7, 3), (5, 5)

(v) (8.5, 3)

(vi)(9,4), (10,4), (11,5)

(vii)(5, 3), (6, 2), (7, 2)

Note Answer for option (vi) and (vii) may vary from student to student

**Question. 81 As part of his science project, Prithvi was supposed to record the temperature every hour one Saturday from 6 AM to midnight. At noon, he was taking lunch and forgot to record the temperature. At 8:00 PM, his favourite show came on and so forgot again. He recorded the data. So, collected on a graph sheet as shown below.**

**(a) Why does it make sense to connect the points in the situation?**

** (b) Describe the overall trend, or pattern, in the way the temperature changes over the time period shown on the graph.**

** (c) Estimate the temperature at noon and 8 PM.**

** Solution.**

(a) By connecting the points, it is easier to understand a change in the temperature.

(b) Initially the temperature was 8°C at 6 AM and started increasing strictly till 1 PM and after that it decreased to 8°C till 12 PM;

(c) At 12 PM 19°Candat8PM 10°C.

**Question. 82 The graph given below compares the price (in ?) and weight of 6 bags (in kg) of sugar of different brands A, B, C, D, E, F.**

**(a) Which brand(s) cost/costs more than brand 0?**

** (b) Bag of which brand of sugar**

** (c) Which brands weigh the same?**

** (d) Which brands are heavier than brand B?**

** (e) Which bag is the lightest?**

** (f) Which bags are of the same price?**

** Solution.** On observing the graph carefully, we note that

(a) The brands E and F cost more than brand D.

(b) The bag of sugar of brand D is the heaviest.

(c) The weights of bag of brand S and F; brand E and C weighs same.

(d) Brands C, D, E are heavier than brand B.

(e) Bag of brand A is the lightest.

(f) Bags of brand A and C are of the same price.

**Question. 83 The points on the graph below represent the height and weight of the donkey, dog, crocodile and ostrich shown in the drawing.**

**(a) What are the two variables represented in the graph?**

** (b) Which point represents each animals? Explain.**

** Solution.**

(a) Height and weight are the two variables in the graph.

(b) In the graph,we observe that the points A represents a crocodile as it has least height and greatest weight among all animals.

A-Crocodile ‘ [least height, greatest weight]

B – Donkey [height and weight more than dog]

C – Dog

D – Ostrich [greatest height]

**Question. 84 The two graph below compare car A and car B. The left graph shows the relationship between age and value. The right graph shows the relationship between size and maximum speed.**

**Use the graphs to determine whether each statement is true or false and explain your answer.**

** (a) The older car is less valuable.**

** (b) The faster car is larger.**

** (c) The larger car is older.**

** (d) The faster car is older.**

** (e) The more valuable car is slower.**

** Solution.**

(a) False, the older car is 8 i.e. 8 valuable more than car A.

(b) True, in the second graph 8 is larger car having greater speed.

(c) True, larger car is 8 which is older than A

(d) True, as 8 is faster as well as older than A.

(e) False, as 8 is more valuable but not slower.

**Question. 85 Sonal and Anmol made a sequence of the designs from square white tiles surrounding one square purple tile. The purple tiles come in many sizes. Three of the designs are shown below.**

** (a) Copy and complete the table**

**(b) Draw a graph using the first five pairs of numbers in your table.**

** (c) Do the points lie on a line?**

** Solution.**

(a) In side length 1 the number of white tile surrounding purple tile Is 4.

Similarly, in side length 2 the number of white tiles surrounding purple tile is 8.

Thus, we can arrange the following table which shows side length of purple corresponding to the number of white tiles in border.

**Question. 86 Sonal and Anmol then made another sequence of the designs. Three of the designs are shown below.**

**(b) Draw a graph of rows and number of white tiles. Draw another graph of the number of rows and the number of purple tiles. Put the number of rows on the horizontal axis.**

** (c) Which graph is linear?**

** Solution.**