Key Concept: Assertion-Reason

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

[Solution Description]

The assertion states that the sum of two even numbers is always even, which is true. For example, 2 + 4 = 6, which is even. The reason states that the product of two even numbers is always even, which is also true. For example, 2 * 4 = 8, which is even. However, the reason does not explain why the sum of two even numbers is even.

Key Concept: Introduction to Graphs

c) 30 books

[Solution Description]

First, identify the maximum and minimum number of books sold. The maximum is 80 books on Thursday, and the minimum is 50 books on Monday. Then, calculate the difference.

Calculation:

Difference = Maximum - Minimum

Difference = $80 - 50$ books

Difference = $30$ books

Key Concept: Purpose of Graphs

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that line graphs are used to display continuously changing data over time, which is true as per the syllabus. The reason claims that line graphs provide a clear visual representation of trends, making them easier to understand compared to tabular data. This is also correct and directly explains why line graphs are used for such purposes. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Linear Graph

b) $9$ cm$^2$

[Solution Description]

The area of a square is calculated using the formula:

$\text{Area} = \text{side length}^2$

Substituting the given side length:

$\text{Area} = 3^2 = 9$

Thus, the area is $9$ cm$^2$.

Key Concept: Graph Interpretation

d) Sales are increasing

[Solution Description]

The graph shows the sales increasing steadily over the five months. Starting from \$20,000 in January, the sales increase by \$5,000 each month, reaching \$40,000 in May. This indicates a consistent upward trend.

Key Concept: Graphs, Data Representation

d) Assertion is false, but Reason is true.

[Solution Description] The assertion states that graphs are always more effective than tables in presenting data. While it is true that graphs can make trends and comparisons easier to understand due to their visual nature, this does not mean they are always more effective than tables. Tables can be more effective for precise data comparison or when detailed numerical values are required. Therefore, the assertion is false. However, the reason is true because graphs do provide a visual representation that aids in understanding trends and comparisons quickly and clearly.

Key Concept: Purpose of Graphs

b) To show numerical facts visually for quick understanding

[Solution Description]

The main purpose of using graphs is to show numerical facts in visual form so that they can be understood quickly, easily, and clearly. Graphs are visual representations of data collected.

Key Concept: Graph Interpretation

d) Seasonal variation in consumer spending

[Solution Description] The graph shows that sales are highest in December, which is typically associated with holiday shopping, and then drop in January, which is common as people tend to reduce spending after the holidays. This pattern is consistent with seasonal trends in consumer behavior.

Key Concept: Graphs vs Tables

c) When showing trends or comparisons.

[Solution Description]

A graphical presentation is particularly useful when there is a trend or comparison to be shown. It helps in visualizing the data and understanding the patterns or relationships more easily than a table.

Key Concept: Tables vs Graphs, Data Visualization

b) Graph because it visually represents the proportion of each category

[Solution Description]

A graph, specifically a pie chart, is better suited for presenting percentage distribution because it provides a clear visual of how each category contributes to the whole. Tables, while accurate, do not offer an immediate visual understanding of the proportions.

Key Concept: Line Graph

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that a line graph displays data that changes continuously over periods of time, which is true as per the syllabus. The reason states that a line graph is a visual representation of data that helps in understanding trends quickly, which is also true. The reason correctly explains the assertion because the continuous change of data over time makes it easier to understand trends visually.

Key Concept: Graphs make trends and relationships easier to understand

c) 1/Mass

[Solution Description] The slope of a graph is calculated as the change in y divided by the change in x. In this case, y represents acceleration and x represents force. According to Newton's second law, force equals mass times acceleration ($F = ma$). Therefore, the slope represents the reciprocal of mass ($1/m$).

Key Concept: Line Graphs

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

[Solution Description]

The assertion states that a line graph is used to display data that changes continuously over periods of time, which is true as per the syllabus. The reason states that line graphs are useful for showing trends or comparisons, which is also true. However, the reason does not directly explain why a line graph is used to display continuous data over time. Therefore, both the assertion and reason are true, but the reason is not the correct explanation of the assertion.

Key Concept: Graphical Presentation

b) It is easier to understand

[Solution Description]

Graphical presentation of data is easier to understand as it shows numerical facts visually, making trends and comparisons clearer.

Key Concept: Line graphs, Data interpretation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The Assertion states that the slope of a line graph represents the rate of change of the dependent variable with respect to the independent variable. This is true because the slope is calculated as the change in the dependent variable divided by the change in the independent variable. The Reason states that a steeper slope indicates a faster rate of change, which is also true. The Reason correctly explains the Assertion because a steeper slope means a greater change in the dependent variable for a given change in the independent variable, indicating a faster rate of change.

Key Concept: Time-temperature graph

b) 0.5°C/hour

[Solution Description]

The rate of change of temperature is calculated as $\frac{\Delta \text{Temperature}}{\Delta \text{Time}}$. The change in temperature is $39 - 38 = 1°C$. The time taken is $10 - 8 = 2$ hours. Therefore, the rate of change of temperature is $\frac{1}{2} = 0.5°C/\text{hour}$.

Key Concept: Bar Graphs

d) City W

[Solution Description]

To determine which city received the least amount of rainfall, compare the rainfall values for all cities.

City X = 40 mm, City Y = 30 mm, City Z = 50 mm, City W = 20 mm.

The smallest value is 20 mm, which corresponds to City W.

Therefore, City W received the least amount of rainfall.

Key Concept: Bar Graphs

b) Categorical data[Solution Description]A bar graph is best suited for representing categorical data, where different categories are compared based on their values.

Key Concept: Pie Chart Calculation

c) 90 degrees

[Solution Description]

To find the angle of a sector in a pie chart that represents a certain percentage of the data, multiply the percentage by 3.6 (since 360 degrees / 100 = 3.6). For 25\%, the calculation is: $25 \times 3.6 = 90$ degrees.

Key Concept: Pie Chart Data Interpretation

b) \$1000

[Solution Description] The angle for groceries is $120^\circ$, and the total angle of the pie chart is $360^\circ$. The fraction of the total expenses spent on groceries is $\frac{120}{360} = \frac{1}{3}$. Therefore, the amount spent on groceries is $\frac{1}{3} \times 3000 = 1000$.

Key Concept: Line graph basics

b) Temperature

[Solution Description]

In a line graph displaying temperature changes over time, the y-axis represents the temperature values recorded at different time intervals.

Key Concept: Line graph interpretation

b) The temperature is increasing

[Solution Description]

In a line graph showing a patient's temperature over time, a rising line indicates an increase in the patient's temperature. This means the temperature is getting higher as time progresses.

Key Concept: Line Graph

b) Temperature

[Solution Description]

The y-axis in a time-temperature graph represents the temperature recorded at different timings. This is explicitly stated in the syllabus.

Key Concept: Line Graph Interpretation

c) 40°C

[Solution Description]

To determine the highest temperature recorded during the day, compare all the given temperatures: 37°C at 6 a.m., 40°C at 10 a.m., 38°C at 2 p.m., and 35°C at 6 p.m. The highest temperature among these is 40°C at 10 a.m.

Key Concept: Time-Temperature Graph

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that the temperature at 8 a.m. can be determined by connecting the points on the time-temperature graph. This is true because the graph is created by connecting points with line segments, allowing us to estimate intermediate values. The reason states that the graph suggests the temperature at 8 a.m. was more than 37°C because the line segment between 6 a.m. and 10 a.m. shows an increasing trend. This is also true, as the temperature increased from 37°C at 6 a.m. to 40°C at 10 a.m., indicating that at 8 a.m., the temperature was indeed more than 37°C. Moreover, the reason correctly explains the assertion by linking the increasing trend to the temperature at 8 a.m.

Key Concept: Understanding a time-temperature graph, trend analysis

d) Assertion is false, but Reason is true.

[Solution Description]

According to the syllabus, the temperature increased from $37^\circ C$ at 6 a.m. to $40^\circ C$ at 10 a.m. This indicates a rising trend in temperature during this period. Since the temperature at 6 a.m. was $37^\circ C$ and it was increasing, the temperature at 8 a.m. must have been more than $37^\circ C$. Therefore, the Assertion is false, but the Reason is true.

Key Concept: Identifying Trends in Graphs

d) No change in values

[Solution Description]

A horizontal line on a graph typically represents that there is no change in the values over time or across the measured variable.

Key Concept: Line Graph

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description] The assertion states that a line graph is used to display data that changes continuously over time, which is correct according to the syllabus. The reason explains that a line graph connects individual data points with line segments, making it easier to observe trends. Both the assertion and reason are true, and the reason correctly explains why a line graph is used for displaying continuous data. Therefore, the correct answer is option a).

Key Concept: Understanding Line Graphs

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The Assertion states that the slope of a line graph represents the rate of change between two variables. This is true because the slope indicates how much one variable changes in relation to another. The Reason explains that the slope is calculated as the ratio of the change in the vertical axis to the change in the horizontal axis. This is also true and directly explains how the slope is determined, which supports the Assertion.

Key Concept: Interpreting Line Graphs

d) Day 4

[Solution Description]

The graph increases steadily from Day 1 to Day 3 and remains constant on Day 4. Therefore, the highest number of books sold is on Day 3 and Day 4. Since the question asks for the highest number of books sold, the correct answer is Day 4.

Key Concept: Peak Performance

b) 100

[Solution Description] The highest score by Batsman B is given as 100 runs in a single match, as mentioned in the graph description.

Key Concept: Match Comparison

c) 4th match

[Solution Description] The graph indicates that both batsmen scored 60 runs during the 4th match. This is explicitly stated in the description.

Key Concept: Analyzing cricket performance using a line graph

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that Batsman A scored a zero in two matches. This is true as per the given information in the example. The reason states that Batsman A has a lot of ups and downs in his performance. This is also true as per the given information. Moreover, the reason correctly explains why Batsman A scored a zero in two matches, as his inconsistent performance led to such low scores.

Key Concept: Line Graph Interpretation

a) Matches played[Solution Description]The horizontal axis in a line graph representing runs scored by batsmen typically represents the matches played.

Key Concept: Comparing Runs Scored

b) 200

[Solution Description]

To find the average runs scored per match, add the runs from all three matches and divide by the number of matches. The calculation is as follows:

$\text{Average runs} = \frac{180 + 220 + 200}{3} = \frac{600}{3} = 200$

Therefore, the average runs scored per match by Team A is 200.

Key Concept: Comparing runs, percentages

c) 40\%

[Solution Description]

First, calculate the total runs: $80 + 120 + 100 = 300$. The percentage of runs scored in the second match is $\frac{120}{300} \times 100 = 40\%$.

Key Concept: Identifying trends, Consistency in scores

b) 1.37

[Solution Description]

To calculate the standard deviation, first find the mean of the sales data. The mean is (12 + 15 + 14 + 16 + 13 + 15) / 6 = 14.5. Next, find the squared differences from the mean for each month: (12 - 14.5)^2 = 6.25, (15 - 14.5)^2 = 0.25, (14 - 14.5)^2 = 0.25, (16 - 14.5)^2 = 2.25, (13 - 14.5)^2 = 2.25, (15 - 14.5)^2 = 0.25. The variance is the average of these squared differences: (6.25 + 0.25 + 0.25 + 2.25 + 2.25 + 0.25) / 6 = 1.875. The standard deviation is the square root of the variance: √1.875 ≈ 1.37.

Key Concept: Identifying trends and consistency in scores

d) Assertion is false, but Reason is true.

[Solution Description] To evaluate consistency, we need to analyze the variability in scores rather than just the peak score. The solution states that Batsman A has a lot of ups and downs, with zero runs in two matches and less than 40 runs in five matches, while Batsman B has never scored below 40 runs. This indicates that Batsman B is more consistent. Therefore, the Assertion that Batsman A is more consistent due to a higher peak score is false, but the Reason about judging consistency through variability is true.

Key Concept: Distance-Time Graph Slope

c) 60 km/h

[Solution Description]

The slope of a distance-time graph represents speed. Speed is calculated by dividing the distance covered by the time taken. Here, the distance is 300 km and the time is 5 hours. So, the speed = Distance / Time = 300 km / 5 h = 60 km/h. Therefore, the slope of the distance-time graph is 60 km/h.

Key Concept: Distance-Time Graph Calculation

c) 60 km/h

[Solution Description]

Speed is calculated using the formula: $speed = \frac{distance}{time}$. Here, distance = 60 km and time = 1 hour. Therefore, $speed = \frac{60}{1} = 60$ km/h.

Key Concept: Distance-Time Graph

c) 75 km/h

[Solution Description]

The total distance travelled between 9 a.m. and 11 a.m. is 100 km + 50 km = 150 km. The time taken is 2 hours. Average speed = Total distance / Time = 150 km / 2 hours = 75 km/h.

Key Concept: Distance-Time Graph

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

From the graph, it is clear that the car covered 100 km between 9 a.m. and 10 a.m. This corresponds to the change in distance from 50 km to 150 km. Since the slope of the distance-time graph represents speed, and the slope is constant during this interval, the car was moving at a constant speed. Therefore, both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

Key Concept: Identifying changes in speed

d) Assertion is false, but Reason is true.

[Solution Description]

From the graph, we can see that the car travelled 50 km in the first hour, 100 km in the second hour, and 50 km in the third hour. Since the distance covered in each hour is different, the speed of the car was not constant during the first three hours. Therefore, the Assertion is false. However, the Reason correctly states that the distance covered in each hour was not the same, which is true.

Key Concept: Identifying Changes in Speed

b) 20 km/h

[Solution Description]

The initial speed of the car is 60 km/h, and the final speed is 80 km/h. To find the change in speed, subtract the initial speed from the final speed: $80 - 60 = 20$ km/h.

Key Concept: Analyzing Stopping Points

d) The car is at rest

[Solution Description] A horizontal segment on a distance vs. time graph indicates that the object is not moving during that time interval. Since the distance remains constant, the car is at rest.

Key Concept: Analyzing stopping points using a horizontal segment in the graph

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description] The assertion states that the car was stationary between 11 a.m. and 12 noon because the distance from city P remained constant at 200 km during this interval. This is true because a constant distance over time implies no movement. The reason states that a horizontal line segment in the distance-time graph indicates that the car did not travel during that time interval. This is also true since a horizontal line in a distance-time graph represents zero speed or no movement. Additionally, the reason correctly explains the assertion because the horizontal line segment directly supports the claim of the car being stationary.

Key Concept: Graph Traversal

c) 5

[Solution Description]

In a cycle graph with $n$ vertices, the number of edges is equal to $n$. Here, the graph has 5 vertices, so the number of edges is 5.

Key Concept: Graph interpretation, Time-Distance

c) 30 km/h

[Solution Description]

To calculate the average speed, we need to find the total distance traveled and divide it by the total time taken.

Total distance = 30 km (first hour) + 60 km (second hour) = 90 km

Total time = 1 hour (first hour) + 1 hour (second hour) + 1 hour (stop) = 3 hours

Average speed = $\frac{Total \: distance}{Total \: time} = \frac{90 \: km}{3 \: hours} = 30 \: km/h$

Key Concept: Independent and Dependent Variables, Graphs

b) $A = s^2$

[Solution Description]

The area (\$A\$) of a square is given by the formula:

$A = s^2$

Here, \$A\$ is the dependent variable and \$s\$ is the independent variable. For every 1 cm increase in side length, the area increases by 9 cm², which confirms that the relationship is quadratic.

Key Concept: Independent and Dependent Variables

d) Assertion is false, but Reason is true.

[Solution Description]

The assertion states that the dependent variable is always plotted on the x-axis, which is incorrect. In graphs, the independent variable is typically plotted on the x-axis, while the dependent variable is plotted on the y-axis. The reason correctly explains that the independent variable is manipulated to observe its effect on the dependent variable. Therefore, the assertion is false, but the reason is true.

Key Concept: Graph Plotting

b) \$1000

[Solution Description] The cost of petrol is directly proportional to the quantity of petrol. If 10 litres cost \$500, then 20 litres will cost twice as much. Therefore, the cost of 20 litres = 2 * \$500 = \$1000.

Key Concept: Independent variable, Dependent variable

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that the amount of electric bill is directly proportional to the quantity of electricity used. This is true because as more electricity is consumed, the bill increases proportionally. The reason states that the quantity of electricity used is the independent variable, and the electric bill is the dependent variable. This is also true because the bill depends on the amount of electricity used. Additionally, the reason correctly explains the assertion because the direct proportionality is a result of the dependency relationship between the variables.

Key Concept: Quantity and Cost, Linear Graph

c) Rs.1500[Solution Description]Let us find the cost per liter first. Given that 18 liters cost \Rs.900, cost per liter is $\frac{900}{18} = \Rs.50$. Now, for 12 liters, the cost is given as \Rs.600, which confirms the cost per liter (\Rs.50). Therefore, the cost for 30 liters would be $30 \times 50 = \Rs.1500$.

Key Concept: Principal and Simple Interest, Linear Graph

c) Rs.8000[Solution Description]First, calculate the principal required for an interest of \Rs.1: $\frac{500}{5000} = 0.10$. To earn an interest of \Rs.800, the required deposit is $\frac{800}{0.10} = \Rs.8000$.

Key Concept: Quantity and Cost, Graphs

c) 1000 units

[Solution Description]

To find the profit-maximizing quantity, first derive the profit function. The profit function is given by:

$P(q) = R(q) - C(q) = 25q - (2000 + 5q + 0.01q^2)$

Simplify the expression:

$P(q) = 25q - 2000 - 5q - 0.01q^2$

Combine like terms:

$P(q) = -0.01q^2 + 20q - 2000$

To find the profit-maximizing quantity, take the derivative of $P(q)$ with respect to $q$ and set it to zero:

$\frac{dP}{dq} = -0.02q + 20 = 0$

Solve for $q$:

$-0.02q + 20 = 0 \implies q = \frac{20}{0.02} = 1000$

The profit-maximizing quantity is 1000 units.

Key Concept: Graph Applications, Cost Analysis

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

To determine whether the assertion and reason are correct, we need to analyze the relationship between marginal cost and average cost. The total cost for 100 units is \$10,000, so the average cost is \$100 per unit. The marginal cost at this production level is \$50 per unit. When the marginal cost is less than the average cost, increasing production decreases the average cost. This is because the additional cost of producing one more unit (\$50) is less than the current average cost (\$100), leading to a reduction in the overall average cost. The reason correctly explains the assertion as it highlights the condition under which the average cost decreases.

Key Concept: Graph plotting, Estimation

d) 1500

[Solution Description]

The graph represents a linear relationship where $C = k \cdot Q$.

For 20 litres, the cost is \$1000. Therefore:

$1000 = k \cdot 20$

Solving for $k$:

$k = \frac{1000}{20} = 50$

Now, we estimate the cost for 30 litres:

$C = 50 \cdot 30 = 1500$

So, the cost for 30 litres of petrol is \$1500.

Key Concept: Graph Plotting

c) \$1000

[Solution Description]

The cost of petrol is directly proportional to the quantity of petrol. Since 10 litres cost \$500, the cost per litre is \$50. Therefore, the cost of 20 litres is $50 \times 20 = \$1000$.

Key Concept: Using the graph to estimate costs for intermediate values

c) 1500 rupees

[Solution Description]

First, calculate the slope (cost per litre) using the two points (5, 250) and (25, 1250):

$m = \frac{1250 - 250}{25 - 5} = \frac{1000}{20} = 50 \text{ rupees per litre}$

Now, using the slope, the cost for 30 litres is:

$\text{Cost} = 30 \times 50 = 1500 \text{ rupees}$

Therefore, the estimated cost for purchasing 30 litres of petrol is 1500 rupees.

Key Concept: Using the graph to estimate costs for intermediate values

b) 900 rupees

[Solution Description]

The graph is linear and represents a direct variation between the number of litres and the cost of petrol. To estimate the cost for 18 litres, we can use the slope of the line. The slope (cost per litre) can be calculated using two points from the graph, say (10, 500) and (20, 1000). The slope $m$ is given by:

$m = \frac{1000 - 500}{20 - 10} = \frac{500}{10} = 50 \text{ rupees per litre}$

Using the slope, the cost for 18 litres is:

$\text{Cost} = 18 \times 50 = 900 \text{ rupees}$

Therefore, the estimated cost for 18 litres of petrol is 900 rupees.

Key Concept: Principal and Simple Interest

c) \$120

[Solution Description]

To find the simple interest (SI), we use the formula:

$SI = \frac{P \times R \times T}{100}$

Here, $P = \$500$, $R = 8\%$, and $T = 3$ years.

Plugging in the values:

$SI = \frac{500 \times 8 \times 3}{100}$

$SI = \frac{12000}{100}$

$SI = \$120$

Therefore, the total interest earned after 3 years is \$120.

Key Concept: Principal and Simple Interest, Graph Interpretation

c) \$700

[Solution Description]

To find the deposit amount $P$ that yields an annual simple interest of \$70 at 10\% interest rate:

$70 = \frac{10}{100} \times P$

Solving for $P$:

$P = \frac{70 \times 100}{10} = \$700$

Key Concept: Simple Interest, Direct Variation

b) $y = 0.12x$

[Solution Description]

The formula for simple interest is given by:

$y = \frac{P \times R \times T}{100}$

Here, \$P\$ is the principal amount (\$\$x), \$R\$ is the rate of interest (12\%), and \$T\$ is the time period (1 year). Substituting these values into the formula, we get:

$y = \frac{x \times 12 \times 1}{100} = \frac{12x}{100} = 0.12x$

Therefore, the correct equation representing the relationship between \$x\$ and \$y\$ is \$y = 0.12x\$.

Key Concept: Principal and Simple Interest

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The graph of simple interest versus principal amount is a straight line because simple interest is directly proportional to the principal amount. This means that as the principal increases, the simple interest increases at a constant rate, which results in a linear relationship. Therefore, both the assertion and the reason are true, and the reason correctly explains the assertion.

Key Concept: Graph Scale

c) 30 km/h

[Solution Description]

To find the speed of the car, we use the formula: speed = distance / time. Here, the distance covered is 60 km and the time taken is 2 hours. Plugging in these values, we get: speed = 60 km / 2 hours = 30 km/h. Therefore, the speed of the car is 30 km/h.

Key Concept: Graph Interpretation, Speed Calculation

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

To determine the speed of the car during the first hour, we need to calculate the distance traveled by the car in the first hour. According to the graph, the car traveled 60 km in the first hour. Speed is calculated using the formula: $Speed = \frac{Distance}{Time}$. Here, the distance is 60 km and the time is 1 hour. Therefore, the speed is $\frac{60\,km}{1\,hour} = 60\,km/h$. Thus, both Assertion and Reason are true, and Reason correctly explains Assertion.

Key Concept: Time Calculation

c) 3 hours

[Solution Description]

The car is traveling a distance of 180 km at a speed of 60 km/h. To find the time taken, we use the formula:

Time = Distance / Speed

Here, Distance = 180 km, Speed = 60 km/h.

Plugging in the values:

Time = 180 km / 60 km/h = 3 hours.

Key Concept: Time and Distance

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

First, let us verify the assertion. Given the speed of the car is 60 km/h and it travels for 2 hours, the distance covered can be calculated using the formula distance = speed × time. Substituting the values, we get distance = 60 km/h × 2 h = 120 km. Thus, the assertion is true. Now, let us analyze the reason. When an object moves at a constant speed, the relationship between distance and time is given by distance = speed × time. This shows that distance is directly proportional to time when speed is constant. Therefore, the reason is also true, and it correctly explains the assertion.

Key Concept: Distance-Time Graph

d) Assertion is false, but Reason is true.

[Solution Description]

The assertion claims that the speed of the car during the first three hours was constant. To verify this, we analyze the distance-time graph. During the first hour, the car traveled 50 km, in the second hour it traveled 100 km, and in the third hour it traveled 50 km. Since the distances covered per hour vary, the speed was not constant. Therefore, the assertion is false. The reason states that the distance-time graph for the first three hours is a straight line with a constant slope. However, since the distances covered per hour are different (50 km, 100 km, 50 km), the slope of the graph changes, indicating varying speeds. Thus, the reason is also false.

Key Concept: Distance-Time Graph

b) 50 km/h

[Solution Description]

Total distance traveled = $120 \text{ km} + 180 \text{ km} = 300 \text{ km}$.

Total time taken = $2 \text{ hours} + 1 \text{ hour} + 3 \text{ hours} = 6 \text{ hours}$.

Average speed = $\frac{\text{Total distance}}{\text{Total time}} = \frac{300 \text{ km}}{6 \text{ hours}} = 50 \text{ km/h}$.

Key Concept: Identifying Journey Start

b) 50 km

[Solution Description]

According to the graph, the car travelled 50 km between 8 a.m. and 9 a.m.

Key Concept: Identifying periods of rest, calculating speed

c) 26.67 km/h

[Solution Description]

To calculate the average speed, we need to find the total distance traveled and divide it by the total time taken.

Total distance traveled = Distance in first 45 minutes + Distance in next 50 minutes = 15 km + 25 km = 40 km.

Total time taken = Time for first 45 minutes + Rest period + Time for next 50 minutes = 0.75 hours + (10/60) hours + (50/60) hours = 1.5 hours.

Average speed = Total distance / Total time = 40 km / 1.5 hours = 26.67 km/h.

Key Concept: Linear Graphs, Direct Variation

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.

[Solution Description]

The assertion states that the graph of a situation where two quantities are in direct variation will always be a straight line passing through the origin. This is true because in direct variation, as one quantity increases, the other increases proportionally, resulting in a linear relationship. The reason states that when two quantities are in direct variation, their ratio remains constant. This is also true because direct variation implies that the ratio of the two quantities is always the same. However, the reason does not directly explain why the graph passes through the origin. The origin point (0,0) is included in the graph because if one quantity is zero, the other must also be zero in a direct variation relationship. Thus, both the assertion and reason are true, but the reason is not the correct explanation of the assertion.

Key Concept: Drawing Graphs

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The Assertion states that the slope of a straight-line graph represents the rate of change between two variables. This is true because the slope directly indicates how one variable changes with respect to the other. The Reason explains that the slope is calculated as the ratio of the vertical change to the horizontal change between two points on the graph, which is also correct and provides the mathematical basis for understanding the slope. Since both the Assertion and Reason are true, and the Reason correctly explains the Assertion, the correct answer is (a).

Key Concept: Graph Plotting, Linear Graph

d) (25, 1250)

[Solution Description]

The cost of petrol per litre is \$50. For any number of litres, say $x$, the total cost will be $50x$. Let’s check each option:

For option a) (10, 500): Total cost = $50 \times 10 = \$500$. This matches.

For option b) (15, 700): Total cost = $50 \times 15 = \$750$. This does not match.

For option c) (20, 800): Total cost = $50 \times 20 = \$1000$. This does not match.

For option d) (25, 1250): Total cost = $50 \times 25 = \$1250$. This matches.

Therefore, options a) and d) lie on the graph. However, since only one option is to be selected, option d) (25, 1250) is more appropriate as it represents a higher quantity.

Key Concept: Graph Plotting, Direct Variation

c) (3, 150)

[Solution Description]

Given the speed of the car is 50 km/h, the distance covered in 3 hours can be calculated using the formula: Distance = Speed × Time. So, Distance = $50 \times 3 = 150$ km. Therefore, the coordinates representing 3 hours of travel will be (3, 150).

Key Concept: Choosing an appropriate scale, Interpreting graphs

c) 120 km

[Solution Description]

The horizontal scale is 1 unit = 2 hours, so 3 units represent $3 \times 2 = 6$ hours. The vertical scale is 1 unit = 20 km, so 6 units represent $6 \times 20 = 120$ km. Therefore, the car has traveled 120 km.

Key Concept: Choosing an appropriate scale

b) Horizontal: 1 unit = 1 hour; Vertical: 1 unit = 50 km

[Solution Description]

To plot the data, we need to determine a suitable scale for both axes. Since the car travels 50 km every hour, we can use increments of 1 hour on the horizontal axis and 50 km on the vertical axis. For example:

- Horizontal: 1 hour is represented by 1 unit.

- Vertical: 50 km is represented by 1 unit.

This ensures that the points can be plotted accurately without compressing or expanding the graph too much.

Key Concept: Labeling the axes

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The assertion states that the x-axis in a time-temperature graph represents the timings at which temperatures are recorded. This is true, as the horizontal axis typically shows the independent variable, which in this case is time. The reason states that the x-axis is always used to represent the independent variable in a graph. This is also true, as it is a standard convention to plot the independent variable on the x-axis and the dependent variable on the y-axis. Additionally, the reason correctly explains why the x-axis represents timings in this context. Therefore, both the assertion and reason are true, and the reason is the correct explanation of the assertion.

Key Concept: Labeling the axes

a) Pressure at zero temperature[Solution Description]The point where the curve intersects the y-axis in a graph with temperature on the x-axis and pressure on the y-axis represents the pressure when the temperature is zero. This is because, at the y-intercept, the value of the x-axis (temperature) is zero.

Key Concept: Plotting given data points

d) Assertion is false, but Reason is true.

[Solution Description]

The assertion states that the graph of a linear relationship between two variables will always be a straight line passing through the origin. This is not necessarily true because a linear relationship can have a y-intercept other than zero, meaning the line does not pass through the origin. The reason states that in a linear relationship, the graph is linear because the variables are in direct variation. This is true because a linear relationship implies direct variation between the variables. Therefore, the assertion is false, but the reason is true.

Key Concept: Plotting given data points

c) (3, 6)

[Solution Description]

To identify which point does not lie on the line $y = 3x - 2$, substitute the x-coordinate of each point into the equation and check if it matches the y-coordinate.

For option (b) $(2, 4)$:

$y = 3(2) - 2 = 4$, which matches the y-coordinate.

For option (d) $(1, 1)$:

$y = 3(1) - 2 = 1$, which matches the y-coordinate.

For option (a) $(0, -2)$:

$y = 3(0) - 2 = -2$, which matches the y-coordinate.

For option (c) $(3, 6)$:

$y = 3(3) - 2 = 7$, which does not match the y-coordinate.

Key Concept: Curve Fitting

c) Exponential

[Solution Description]

The data points show that as $x$ increases, $y$ increases at an accelerating rate. This suggests that an exponential curve might best fit the data.

Key Concept: Linear Graph

b) 40 km/h

[Solution Description]

Speed is calculated as distance divided by time. Given the car covers 120 km in 3 hours, the speed is:

$\text{Speed} = \frac{120}{3} = 40 \text{ km/h}$

So, the speed of the car is 40 km/h.

Key Concept: Drawing Graphs, Examples of Graphs

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description] To determine the symmetry of the graph of $y = x^3$, we check if it satisfies the condition for symmetry about the origin. For a function to be symmetric about the origin, $f(-x) = -f(x)$ must hold true for all $x$ in the domain. Let's verify this for $f(x) = x^3$. Substitute $-x$ into the function: $f(-x) = (-x)^3 = -x^3$. Now, $-f(x) = -x^3$. Since $f(-x) = -f(x)$, the function $y = x^3$ is indeed symmetric about the origin. Therefore, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Graph Interpretation

b) $(2, -1)$

[Solution Description]

To find the vertex of the parabola given by $y = x^2 - 4x + 3$, we can use the formula for the vertex of a quadratic function $y = ax^2 + bx + c$, where the x-coordinate of the vertex is given by $x = -\frac{b}{2a}$. In this case, $a = 1$ and $b = -4$. Substituting these values, we get $x = -\frac{-4}{2 \times 1} = 2$. Now, substitute $x = 2$ back into the original equation to find the y-coordinate: $y = (2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1$. Therefore, the vertex of the parabola is $(2, -1)$.

Key Concept: Cost of Apples vs. Number of Apples

b) 5

[Solution Description]

To find the number of apples that can be bought for \$50, divide the total amount by the cost per apple: $\frac{50}{10} = 5$. Therefore, 5 apples can be bought for \$50.

Key Concept: Drawing Graphs

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The cost of apples (\$5 for 1 apple, \$10 for 2 apples, \$15 for 3 apples, \$20 for 4 apples, \$25 for 5 apples) shows that the cost increases linearly with the number of apples. This linear relationship indicates direct proportionality, where the ratio $\frac{\text{Cost}}{\text{Number of apples}}$ is constant (\$5 per apple). Since the relationship is directly proportional, the graph will be a straight line passing through the origin (0, 0). Thus, both Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Graph Interpretation

c) 120 km

[Solution Description]

The distance-time graph indicates that the car travels at a constant speed of 30 km/h. Therefore, the distance covered is directly proportional to the time taken. For 4 hours, we calculate the distance as follows:

$\text{Distance} = 30 \, \text{km/h} \times 4 \, \text{hours} = 120 \, \text{km}$

Thus, the distance covered by the car in 4 hours is 120 km.

Key Concept: Distance Travelled by a Car vs. Time

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description] When a car is moving at a constant speed, the distance it travels increases uniformly with time. This results in a straight-line graph for distance versus time. The slope of this graph is calculated as the change in distance divided by the change in time, which is equal to the speed of the car. Therefore, both the Assertion and Reason are true, and the Reason correctly explains the Assertion.

Key Concept: Interpreting Graphs

b) 2.5 hours

[Solution Description] From the table provided in the syllabus, for every 1 hour, the car travels 30 km. To find the time corresponding to 75 km, we use the formula $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$. Here, speed is 30 km/h, so $\text{Time} = \frac{75}{30} = 2.5$ hours.

Key Concept: Simple Interest Calculation

d) \$50

[Solution Description]

The simple interest formula is $SI = P \times r \times t$, where $P$ is the principal amount, $r$ is the rate, and $t$ is the time. Here, $P = 100$, $SI = 10$, and $t = 1$. Solving for $r$, we get $r = \frac{SI}{P \times t} = \frac{10}{100 \times 1} = 0.1$ or 10\%. Now, using the same rate for a deposit of \$500, $SI = 500 \times 0.1 \times 1 = 50$.

Key Concept: Interest on Deposits

a) \$13,468.55

[Solution Description]

Using the compound interest formula:

$A = P \left(1 + \frac{r}{n}\right)^{nt}$

Here, $P = 10000$, $r = 6\% = 0.06$, $n = 4$ (since it's compounded quarterly), and $t = 5$ years.

Plugging in the values:

$A = 10000 \left(1 + \frac{0.06}{4}\right)^{4 \times 5} = 10000 \left(1.015\right)^{20}$

Calculating $(1.015)^{20}$:

$1.015^{20} = 1.346855007$

Then:

$A = 10000 \times 1.346855007 = 13468.55$

So, the amount after 5 years is \$13,468.55.

Key Concept: Principal and Simple Interest

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The formula for Simple Interest is given by $S.I. = P \times r \times t$, where $P$ is the principal amount, $r$ is the rate of interest, and $t$ is the time period. Here, the rate of interest and time period are constant, so Simple Interest is directly proportional to the principal amount $P$. Therefore, the graph of Simple Interest vs. Deposit will be a straight line, as it represents a direct variation. Thus, both Assertion and Reason are true, and Reason correctly explains Assertion.

Key Concept: Side of a Square vs. Perimeter, Linear Graph

c) 32 cm

[Solution Description]

The original perimeter of the square is 16 cm. Using the formula $P = 4 \times \text{side}$, we can find the original side length as $\frac{16}{4} = 4$ cm. If each side is doubled, the new side length becomes $4 \times 2 = 8$ cm. The new perimeter is then calculated as $4 \times 8 = 32$ cm.

Key Concept: Linear Graph

c) A straight line passing through the origin

[Solution Description]

The relationship between the side and perimeter of a square is linear because the perimeter increases proportionally with the side. This means the graph should be a straight line passing through the origin. The correct graph is a straight line that goes upwards as the side increases, indicating a direct proportional relationship.

Key Concept: Side of a Square vs. Perimeter

b) 5 cm

[Solution Description]

The perimeter of a square is given by $P = 4 \times \text{side}$. To find the side, rearrange the formula: $\text{side} = P / 4$. Substituting the given perimeter of 20 cm, we get $\text{side} = 20 / 4 = 5$ cm. Hence, the side length is 5 cm.

Key Concept: Side of Square vs. Area

c) 6 cm

[Solution Description]

To find the side length of a square given its area, we use the formula $s = \sqrt{A}$, where $A$ is the area. For an area of 36 cm$^2$, substituting into the formula gives $s = \sqrt{36}$. Calculating this gives $s = 6$. Hence, the side length is 6 cm.

Key Concept: Side of a Square vs. Area

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.

[Solution Description]

The area of a square is given by $A = s^2$, where $s$ is the length of its side. If the side length is doubled, the new side length is $2s$. The new area is $(2s)^2 = 4s^2$, which is four times the original area $s^2$. Therefore, the assertion is true. The reason is also true because the area of a square is indeed proportional to the square of its side length. Moreover, the reason correctly explains the assertion as it mathematically justifies why doubling the side length results in the area becoming four times the original area.

Key Concept: Side of Square vs. Area

c) 16 cm$^2$

[Solution Description]

The area of a square is calculated using the formula $A = s^2$, where $A$ is the area and $s$ is the side length. Given the side length is 4 cm, we substitute this value into the formula: $A = 4^2$. Calculating this gives $A = 16$. Therefore, the area of the square is 16 cm$^2$.