Connecting the Dots — Class 7 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 7 Mathematics chapter "Connecting the Dots" — 7 important questions with detailed answers for CBSE board exam preparation.
TL;DR: Free step-by-step NCERT solutions for Class 7 Mathematics chapter "Connecting the Dots" — 7 important questions with detailed answers for CBSE board e…
Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated
NCERT Solutions for Class 7 Mathematics — Connecting the Dots. Step-by-step answers to all 7 textbook questions from this chapter, written for CBSE board preparation and free to use.
Connecting the Dots — All 7 Questions Solved
Q1. Define mean, median, and mode. Find all three for the data set: 3, 7, 5, 3, 9, 3, 5.
Definitions:
1. Mean (Average):
The mean is the sum of all values divided by the number of values.
Formula: Mean = (Sum of all values) / (Number of values)
2. Median:
The median is the middle value when data is arranged in order.
If the number of values is even, median is the average of two middle values.
3. Mode:
The mode is the value that appears most frequently (most often) in the data set.
A data set can have no mode, one mode, or multiple modes.
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For the data set: 3, 7, 5, 3, 9, 3, 5
Finding the Mean:
Step 1: Add all values
Sum = 3 + 7 + 5 + 3 + 9 + 3 + 5 = 35
Step 2: Count the number of values
Number of values = 7
Step 3: Divide sum by count
Mean = 35 ÷ 7 = 5
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Finding the Median:
Step 1: Arrange values in ascending order
3, 3, 3, 5, 5, 7, 9
Step 2: Find the middle value
Number of values = 7 (odd), so the middle is the 4th value
Median = 5
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Finding the Mode:
Step 1: Count frequency of each value
3 appears 3 times
5 appears 2 times
7 appears 1 time
9 appears 1 time
Step 2: Identify the most frequent value
The value 3 appears most frequently (3 times)
Mode = 3
---
Answer:
Mean = 5
Median = 5
Mode = 3
Q2. A student scores: 80, 85, 78, 92, 88 in five tests. Calculate the mean and find how much above or below the mean each score is.
Given scores: 80, 85, 78, 92, 88
Step 1: Calculate the mean
Sum = 80 + 85 + 78 + 92 + 88 = 423
Number of scores = 5
Mean = 423 ÷ 5 = 84.6
Step 2: Find deviation from mean for each score
Deviation = Score - Mean
Score 1: 80 - 84.6 = -4.6 (below mean by 4.6)
Score 2: 85 - 84.6 = +0.4 (above mean by 0.4)
Score 3: 78 - 84.6 = -6.6 (below mean by 6.6)
Score 4: 92 - 84.6 = +7.4 (above mean by 7.4)
Score 5: 88 - 84.6 = +3.4 (above mean by 3.4)
Step 3: Summary table
| Score | Deviation from Mean | Status |
|-------|---------------------|--------|
| 80 | -4.6 | Below mean |
| 85 | +0.4 | Above mean |
| 78 | -6.6 | Below mean |
| 92 | +7.4 | Above mean |
| 88 | +3.4 | Above mean |
Step 4: Verify (sum of deviations should be 0)
-4.6 + 0.4 - 6.6 + 7.4 + 3.4 = 0 ✓
Answer: Mean = 84.6
Scores 80 and 78 are below the mean by 4.6 and 6.6 respectively.
Scores 85, 92, and 88 are above the mean by 0.4, 7.4, and 3.4 respectively.
Q3. Interpret a bar graph showing monthly rainfall: Jan-2, Feb-3, Mar-5, Apr-8, May-10, Jun-12 cm. Find the total and average rainfall.
Monthly Rainfall Data (in cm):
January: 2 cm
February: 3 cm
March: 5 cm
April: 8 cm
May: 10 cm
June: 12 cm
---
Bar Graph Description:
The bar graph would show:
• X-axis: Months (Jan, Feb, Mar, Apr, May, Jun)
• Y-axis: Rainfall in cm (scale: 0 to 12 cm)
• Bar heights increasing from Jan (2) to Jun (12)
Visual pattern:
June has the highest rainfall (12 cm)
January has the lowest rainfall (2 cm)
Rainfall increases continuously from January to June
---
Calculating Total Rainfall:
Total = 2 + 3 + 5 + 8 + 10 + 12
Total = 40 cm
---
Calculating Average Rainfall:
Number of months = 6
Average = Total ÷ Number of months
Average = 40 ÷ 6
Average = 6.67 cm (approximately)
Average = 6⅔ cm (exact value)
---
Interpretation:
1. Trend: Rainfall increases every month from January to June.
2. Highest: June received the maximum rainfall (12 cm).
3. Lowest: January received the minimum rainfall (2 cm).
4. Average: The average monthly rainfall was 6.67 cm.
5. Total: The total rainfall for these 6 months was 40 cm.
Conclusion: This indicates the monsoon or rainy season starts building up from January and peaks in June.
Q4. Define probability. A die is rolled. What is the probability of getting an even number?
Probability:
Probability is the chance or likelihood of an event occurring.
It is expressed as a ratio between 0 and 1, where:
0 = impossible event (will not happen)
1 = certain event (will definitely happen)
Formula: Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
---
Problem: A die is rolled. What is the probability of getting an even number?
Step 1: Identify all possible outcomes
When a die is rolled, the possible outcomes are: 1, 2, 3, 4, 5, 6
Total number of possible outcomes = 6
Step 2: Identify favorable outcomes (even numbers)
Even numbers on a die: 2, 4, 6
Number of favorable outcomes = 3
Step 3: Calculate probability
P(even number) = (Number of favorable outcomes) / (Total outcomes)
P(even number) = 3/6 = 1/2 = 0.5
Step 4: Express as percentage
P(even number) = 0.5 × 100% = 50%
---
Answer:
Probability of getting an even number = 1/2 or 0.5 or 50%
This means if you roll a die many times, you would expect to get an even number roughly half the time.
---
Other Probabilities for a Die Roll:
P(odd number) = 3/6 = 1/2 (numbers 1, 3, 5)
P(number less than 3) = 2/6 = 1/3 (numbers 1, 2)
P(number greater than 4) = 2/6 = 1/3 (numbers 5, 6)
P(getting 3) = 1/6
P(getting 7) = 0 (impossible, as 7 is not on the die)
Q5. A bag contains 5 red balls, 3 blue balls, and 2 green balls. If one ball is drawn at random, find the probability of drawing a red ball and a non-red ball.
Given:
Red balls: 5
Blue balls: 3
Green balls: 2
Step 1: Find total number of balls
Total = 5 + 3 + 2 = 10 balls
---
Part 1: Probability of drawing a red ball
Step 2a: Identify favorable outcomes
Favorable outcomes (red balls) = 5
Step 3a: Calculate probability
P(red) = (Number of red balls) / (Total balls)
P(red) = 5/10 = 1/2 = 0.5
P(red) = 50%
---
Part 2: Probability of drawing a non-red ball
Step 2b: Identify favorable outcomes
Non-red balls = Blue balls + Green balls
Non-red balls = 3 + 2 = 5
Step 3b: Calculate probability
P(non-red) = (Number of non-red balls) / (Total balls)
P(non-red) = 5/10 = 1/2 = 0.5
P(non-red) = 50%
---
Verification:
P(red) + P(non-red) = 1/2 + 1/2 = 1 ✓
(The sum must equal 1 because these are complementary events)
---
Answer:
Probability of drawing a red ball = 1/2 or 0.5 or 50%
Probability of drawing a non-red ball = 1/2 or 0.5 or 50%
Both probabilities are equal, meaning it is equally likely to draw a red ball or a non-red ball.
Q6. Find the median and mode of the following marks: 45, 50, 55, 50, 60, 45, 55, 50, 65.
Given marks: 45, 50, 55, 50, 60, 45, 55, 50, 65
Step 1: Arrange marks in ascending order
45, 45, 50, 50, 50, 55, 55, 60, 65
Number of values = 9 (odd number)
---
Finding the Median:
Step 2: Find the middle position
For odd number of values: Median position = (n + 1)/2
Median position = (9 + 1)/2 = 10/2 = 5th position
Step 3: Identify the median
Counting from the ordered list:
1st: 45, 2nd: 45, 3rd: 50, 4th: 50, 5th: 50, ...
Median = 50
---
Finding the Mode:
Step 4: Count frequency of each value
45 appears 2 times
50 appears 3 times (most frequent)
55 appears 2 times
60 appears 1 time
65 appears 1 time
Step 5: Identify the mode
The value 50 appears most frequently (3 times)
Mode = 50
---
Frequency Table:
| Mark | Frequency |
|------|----------|
| 45 | 2 |
| 50 | 3 |
| 55 | 2 |
| 60 | 1 |
| 65 | 1 |
---
Answer:
Median = 50
Mode = 50
Q7. A coin is tossed and a die is rolled together. How many total outcomes are possible? What is the probability of getting a head and an even number?
Problem: A coin is tossed and a die is rolled together.
Step 1: Identify outcomes for each event
Coin outcomes: Head (H) or Tail (T) = 2 outcomes
Die outcomes: 1, 2, 3, 4, 5, 6 = 6 outcomes
Step 2: Find total number of possible outcomes
Total outcomes = (Coin outcomes) × (Die outcomes)
Total outcomes = 2 × 6 = 12
---
All Possible Outcomes:
(H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6)
(T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6)
Total = 12 outcomes ✓
---
Part 2: Probability of getting Head and Even Number
Step 3: Identify favorable outcomes
Favorable outcomes = Head AND Even number
Even numbers on die: 2, 4, 6
Favorable outcomes: (H, 2), (H, 4), (H, 6)
Number of favorable outcomes = 3
Step 4: Calculate probability
P(Head and Even) = (Favorable outcomes) / (Total outcomes)
P(Head and Even) = 3/12 = 1/4 = 0.25
P(Head and Even) = 25%
---
Alternative Method (using multiplication rule):
P(Head) = 1/2
P(Even number) = 3/6 = 1/2
P(Head AND Even) = P(Head) × P(Even)
P(Head AND Even) = 1/2 × 1/2 = 1/4 = 25% ✓
---
Answer:
Total number of possible outcomes = 12
Probability of getting Head and Even number = 1/4 or 0.25 or 25%
Connecting the Dots — Practice MCQs with Answers
Attempt these 20 multiple-choice questions after working through the solutions above. Each carries the correct option and the reasoning behind it, so a wrong answer tells you which idea to revisit.
MCQ 1. What is formed when two distinct points are joined by the shortest path?
- A. A line segment
- B. A ray
- C. A line
- D. A curve
Answer: A. A line segment — A line segment has two distinct endpoints and a definite length, formed by connecting two points directly with the shortest path.
MCQ 2. How many endpoints does a line segment have?
- A. One
- B. Two
- C. Zero
- D. Infinite
Answer: B. Two — A line segment is defined by two distinct points, which act as its endpoints.
MCQ 3. Which of the following points lies on the x-axis?
- A. (0, 5)
- B. (5, 0)
- C. (5, 5)
- D. (0, 0)
Answer: B. (5, 0) — A point lies on the x-axis if its y-coordinate is 0. In (5, 0), the y-coordinate is 0.
MCQ 4. What is the shape formed by connecting four non-collinear points in order?
- A. Triangle
- B. Square
- C. Quadrilateral
- D. Pentagon
Answer: C. Quadrilateral — A quadrilateral is a polygon with four vertices (points) and four sides, formed by connecting four points in sequence.
MCQ 5. If the pattern is (1,2), (3,4), (5,6), what would be the next pair of dots in this sequence?
- A. (6,7)
- B. (7,8)
- C. (8,9)
- D. (9,10)
Answer: B. (7,8) — Each number in the pairs is consecutive, and the starting number of each new pair increases by 2 from the previous pair's starting number (1, 3, 5...). So, after (5,6), the next pair starts with 7, making it (7,8).
MCQ 6. When two distinct lines intersect, they form how many pairs of vertically opposite angles?
- A. One
- B. Two
- C. Three
- D. Four
Answer: B. Two — When two lines intersect, they form four angles. The angles opposite to each other are called vertically opposite angles, and there are two such pairs.
MCQ 7. On a number line, connecting the dot at -3 to the dot at 2 means covering how many units?
- A. 3
- B. 2
- C. 5
- D. 1
Answer: C. 5 — To find the distance between -3 and 2, you count the units: from -3 to 0 is 3 units, and from 0 to 2 is 2 units. Total units = 3 + 2 = 5.
MCQ 8. What are the coordinates of the origin on a Cartesian plane?
- A. (1, 0)
- B. (0, 1)
- C. (1, 1)
- D. (0, 0)
Answer: D. (0, 0) — The origin is the point where the x-axis and y-axis intersect, and its coordinates are (0, 0).
MCQ 9. Three points A(0,0), B(3,0), and C(0,4) are connected to form a triangle. What is the perimeter of this triangle?
- A. 7 units
- B. 12 units
- C. 10 units
- D. 8 units
Answer: B. 12 units — The length of AB is 3 units (along x-axis). The length of AC is 4 units (along y-axis). Using the Pythagorean theorem, BC = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5 units. Perimeter = AB + AC + BC = 3 + 4 + 5 = 12 units.
MCQ 10. Which of these geometric figures has no endpoints?
- A. Line segment
- B. Ray
- C. Line
- D. Angle
Answer: C. Line — A line extends infinitely in both directions and therefore has no endpoints. A line segment has two, and a ray has one.
MCQ 11. Consider the sequence of shapes: a dot, then two connected dots (line segment), then three connected dots (triangle). What would be the next logical shape if the pattern continues by adding one more dot and connecting all adjacent dots?
- A. Square
- B. Pentagon
- C. Quadrilateral
- D. Rectangle
Answer: C. Quadrilateral — The pattern shows polygons formed by sequentially increasing the number of vertices by one. After a triangle (3 dots/vertices), the next shape would be a quadrilateral (4 dots/vertices).
MCQ 12. Points P(1,1), Q(1,5), R(5,5), and S(5,1) are plotted and connected in order. What is the area of the shape formed?
- A. 4 square units
- B. 16 square units
- C. 8 square units
- D. 20 square units
Answer: B. 16 square units — The points form a square. The side length can be found by calculating the distance between P(1,1) and Q(1,5), which is |5-1| = 4 units. Area of square = side * side = 4 * 4 = 16 square units.
MCQ 13. How many dimensions does a point have?
- A. One
- B. Two
- C. Three
- D. Zero
Answer: D. Zero — A point represents a specific location in space and has no size or dimensions.
MCQ 14. If you plot the population of a city over 5 years, connecting the points would create which type of graph?
- A. Bar graph
- B. Pie chart
- C. Line graph
- D. Histogram
Answer: C. Line graph — A line graph is typically used to show trends over time, where points representing data at different times are connected by lines.
MCQ 15. A rectangular park has vertices at (1,1), (1,7), (5,7), and (5,1). If a path is built along all sides, what is the total length of the path?
- A. 10 units
- B. 12 units
- C. 20 units
- D. 24 units
Answer: C. 20 units — The length of the park is |5-1| = 4 units. The width of the park is |7-1| = 6 units. The total length of the path is the perimeter of the rectangle = 2 * (length + width) = 2 * (4 + 6) = 2 * 10 = 20 units.
MCQ 16. The point (3, -2) is located in which quadrant of the Cartesian plane?
- A. First Quadrant
- B. Second Quadrant
- C. Third Quadrant
- D. Fourth Quadrant
Answer: D. Fourth Quadrant — In the Cartesian plane, points with a positive x-coordinate and a negative y-coordinate lie in the Fourth Quadrant.
MCQ 17. Consider a sequence of points (1,1), (2,4), (3,9), (4,16). If this pattern continues, what would be the 6th point in the sequence?
- A. (5,25)
- B. (6,30)
- C. (6,36)
- D. (7,49)
Answer: C. (6,36) — The pattern shows that the y-coordinate is the square of the x-coordinate (y = x^2). For the 6th point, x=6, so y=6^2=36. The point is (6,36).
MCQ 18. A common point where two line segments meet in a polygon is called a:
- A. Side
- B. Vertex
- C. Edge
- D. Face
Answer: B. Vertex — In a polygon, the corners or points where two sides meet are called vertices.
MCQ 19. If two lines intersect to form a right angle (90 degrees), they are said to be:
- A. Parallel
- B. Skew
- C. Perpendicular
- D. Concurrent
Answer: C. Perpendicular — Perpendicular lines are two lines that intersect to form a right angle (90 degrees).
MCQ 20. A square ABCD has vertices A(2,2) and C(6,6). What are the coordinates of the other two vertices, B and D?
- A. B(2,6), D(6,2)
- B. B(6,2), D(2,6)
- C. B(4,4), D(4,4)
- D. B(2,4), D(6,4)
Answer: A. B(2,6), D(6,2) — Given A(2,2) and C(6,6) as opposite vertices, the other two vertices B and D must share x and y coordinates with A and C. For a square with sides parallel to the axes, B would be (2,6) and D would be (6,2) or vice-versa.
How to Use These Solutions
Attempt each question yourself first and only then compare with the worked answer. Marks in the CBSE board exam are awarded for the METHOD as much as the final result, so reproduce the steps rather than memorising the last line. Where a solution states a law, a formula or a definition, learn that wording — examiners look for it.
Related practice for this chapter: chapter MCQs, previous-year questions, revision notes and sample papers.