Official NCERT 2026-27 Solutions
NCERT Solutions for Class 8 Maths Chapter 12: Tales by Dots and Lines (Ganita Prakash Part II)
Authentic handwritten solutions in Topper Notebook style. Features statistical mean as a physical balance point, median shifts, seasonal daylight line graphs across India, lunar cycle phase tracking, and topological Game of Hex analysis from Pages 131 to 134 of Ganita Prakash Part II.
Quick Chapter Navigation & Blueprint
- Part A: Statistical Measures & Activity Strips (Questions 11 to 13) — Mean as equal share and center of mass, activity strip time budgeting, and comparative sleep duration across demographics (children, adults, seniors).
- Part B: Astronomical Line Graphs across India (Question 14) — Sunrise and sunset line graphs across 4 diverse Indian latitudes/longitudes, determining earliest sunrise (Arunachal Pradesh), and annual daylight variation.
- Part C: The Lunar Cycle & Moon Phase Curves (Question 15) — Moonrise-moonset line graphs, calculating exact dates of Amavasya (New Moon) and Purnima (Full Moon), and the daily 50-minute lunar lag.
- Part D: Strategic Game Theory (Game of Hex) — Hexagonal board connectivity, John Nash’s no-tie theorem, and topological path proofs.
Official Textbook PDF Reference: Download NCERT Class 8 Maths Chapter 12 (hegp205.pdf)
Part A: Mean, Median & Activity Time Strips (Pages 131–133)
The “Fair Share” & Balance Concept of Mean (Page 133):
Explain how the arithmetic mean acts as a physical balance point (fulcrum). What happens to the mean and median when a new value greater than the mean is added to a dataset?
Explain how the arithmetic mean acts as a physical balance point (fulcrum). What happens to the mean and median when a new value greater than the mean is added to a dataset?
Topper’s Statistical Proof:
- The Sum of Deviations is Zero:
Let the dataset be x1, x2, …, xn with mean M.
By definition, M = (∑ xi) / n → n × M = ∑ xi.
Now consider the deviation of each data point from the mean: (xi − M).
Sum of all deviations = ∑ (xi − M) = (∑ xi) − n × M = 0!
This means the total distance of all values to the left of the mean exactly equals the total distance of all values to the right of the mean — exactly like weights balancing on a seesaw! - Adding a Value Greater than the Mean (xnew > M):
• Impact on Mean: Since the new deviation (xnew − M) is strictly positive, the total sum increases faster than the denominator count, causing the mean to increase.
• Impact on Median: If the new value is inserted on the right, the median either shifts by half a step to the right or remains unchanged if repeated middle values exist. Unlike the mean, the median is highly resistant to extreme outliers!
Conclusion: The arithmetic mean is the unique center of gravity where ∑ (xi − Mean) = 0. Inserting values above the mean pulls the mean upward.
Question 13 (Page 131 – Demographic Sleep Time Study):
A student group tracks the daily sleep duration of family members across a week. Formulate average and median sleep expectations for children, adults, and the elderly.
A student group tracks the daily sleep duration of family members across a week. Formulate average and median sleep expectations for children, adults, and the elderly.
Topper’s Structured Data Analysis:
- 1. Children (Ages 5–14): Require significant sleep for physical and neurological growth. Typical sleep duration = 9 to 10.5 hours/day. Mean ≈ 9.5 hours.
- 2. Working Adults (Ages 20–55): Balances professional and household work. Typical sleep duration = 6.5 to 8 hours/day. Mean ≈ 7.2 hours.
- 3. Elderly (Ages 65+): Often experience fragmented night sleep complemented by afternoon naps. Total 24-hour sleep duration = 6 to 7.5 hours. Mean ≈ 6.8 hours.
Observation: Sleep requirement follows a declining gradient from childhood through adulthood, with children exhibiting the highest median sleep duration.
Part B: Sunrise, Sunset & Lunar Cycles (Pages 131–132)
Question 14 (Pages 131–132 – Annual Sunrise & Sunset Curves):
Graphs show sunrise and sunset times over 12 months at 4 locations in India:
(i) At which place does the Sun rise earliest in January? What is the day length?
(ii) Which place has the longest day length in June?
(iii) What geographic principle causes the curves to change throughout the year?
Graphs show sunrise and sunset times over 12 months at 4 locations in India:
(i) At which place does the Sun rise earliest in January? What is the day length?
(ii) Which place has the longest day length in June?
(iii) What geographic principle causes the curves to change throughout the year?
Topper’s Geographic Line Graph Interpretation:
- (i) Earliest Sunrise in January:
• The Earth rotates from west to east. Therefore, sunrise occurs earliest at the easternmost longitude.
• In India, the easternmost regions are in Arunachal Pradesh (e.g., Dong / Tezu at ≈ 97° E). The Sun rises as early as 5:45 AM even in winter.
• In January, day length in Northeast India is approximately 10 hours 15 minutes (sunrise ~6:00 AM, sunset ~4:15 PM). - (ii) Longest Summer Day Length:
• Around the Summer Solstice (June 21), places at higher northern latitudes experience significantly longer daylight hours.
• In northern India (e.g., Srinagar / Leh at ≈ 34° N), the day length reaches over 14 hours 25 minutes (sunrise ~5:15 AM, sunset ~7:40 PM).
• Conversely, places near the equator (e.g., Kanyakumari at 8° N) have nearly constant 12-hour days year-round with minimal variation.
Summary:
(i) Earliest Sunrise: Easternmost location (Arunachal Pradesh); Day length ≈ 10.25 hours.
(ii) Longest Summer Day: Northernmost location (Srinagar/Leh) with daylight exceeding 14.4 hours.
(iii) Governed by the Earth’s 23.5° axial tilt and revolution around the Sun.
(i) Earliest Sunrise: Easternmost location (Arunachal Pradesh); Day length ≈ 10.25 hours.
(ii) Longest Summer Day: Northernmost location (Srinagar/Leh) with daylight exceeding 14.4 hours.
(iii) Governed by the Earth’s 23.5° axial tilt and revolution around the Sun.
Question 15 (Page 132 – Moonrise & Moonset Monthly Plots):
The graph plots daily moonrise and moonset times over a lunar month:
(i) On which dates were Amavasya (New Moon) and Purnima (Full Moon)?
(ii) What mathematical pattern governs the daily shift in moonrise time?
The graph plots daily moonrise and moonset times over a lunar month:
(i) On which dates were Amavasya (New Moon) and Purnima (Full Moon)?
(ii) What mathematical pattern governs the daily shift in moonrise time?
Topper’s Astronomical Data Analysis:
- Amavasya (New Moon):
The Moon and Sun are on the same side of the Earth. Therefore, the Moon rises at roughly the same time as the Sun (sunrise around 6:00 AM) and sets at sunset (around 6:00 PM).
On the graph, Amavasya is the date where the moonrise line overlaps the 6:00 AM line. - Purnima (Full Moon):
The Moon is directly opposite the Sun. The Moon rises precisely as the Sun sets (around 6:00 PM) and sets as the Sun rises (around 6:00 AM), illuminating the sky all night long.
On the graph, Purnima is the date where moonrise occurs at ~6:00 PM (18:00 hrs). - The Daily 50-Minute Shift:
As Earth rotates on its axis every 24 hours, the Moon simultaneously orbits eastward around the Earth, covering ≈ 360° / 29.5 days ≈ 12.2° each day.
To face the Moon again, the Earth must rotate an extra 12.2°, which takes:
(12.2° / 360°) × 1440 minutes ≈ 48.8 minutes ≈ 50 minutes.
Thus, moonrise occurs approximately 50 minutes later every single day!
Conclusions:
• Amavasya: Moonrise occurs at sunrise (~6 AM).
• Purnima: Moonrise occurs at sunset (~6 PM).
• Daily Lag: Moonrise shifts later by approximately 50 minutes each day due to the Moon’s prograde orbit.
• Amavasya: Moonrise occurs at sunrise (~6 AM).
• Purnima: Moonrise occurs at sunset (~6 PM).
• Daily Lag: Moonrise shifts later by approximately 50 minutes each day due to the Moon’s prograde orbit.
Part C: Strategy & Topology — The Game of Hex (Page 134)
The Game of Hex (Page 134):
Hex is played on an 11 × 11 rhombus of hexagonal cells between two players (Red and Blue), each aiming to connect their opposite sides with an unbroken chain.
Prove mathematically: Can a game of Hex ever end in a draw/tie?
Hex is played on an 11 × 11 rhombus of hexagonal cells between two players (Red and Blue), each aiming to connect their opposite sides with an unbroken chain.
Prove mathematically: Can a game of Hex ever end in a draw/tie?
Topper’s Topological Proof (Nash-Gale Theorem):
- Planar Graph Connectivity: The game of Hex is played on a finite 2D hexagonal grid where each internal cell has exactly 6 neighbours.
- Blocking Means Connecting: Suppose Player 1 (Red) fails to connect top-to-bottom. In topology (Jordan Curve Theorem for finite planar lattices), the only way to block an unbroken path between two sides of a rectangle/rhombus is for the opponent to create an unbroken crossing path between the other two sides!
- A path of Blue hexagons that prevents Red from connecting top and bottom necessarily forms an unbroken chain connecting left and right.
- Therefore, once every single cell on the board is filled, exactly one player is guaranteed to have a connected path.
Theorem: A game of Hex can NEVER end in a tie. Exactly one player must win! Furthermore, by the Strategy-Stealing argument, the first player always has a theoretical winning strategy.