IB myp 4-5 MATHEMATICS – Practice Questions- All Topics
Topic :Trigonometry-Bearings
Topic :Trigonometry- Weightage : 21 %
All Questions for Topic : Triangle properties,Bearings,Pythagoras’ theorem,Trigonometric ratios in right angled triangles
Question (31 marks)
The following animation shows patterns in triangular numbers.
Question (a) : 2 marks
The table below is populated with some of the results.
Write down the missing values in the table up to row 5 .
▶️Answer/Explanation
Ans:
To fill in the missing values in the table up to row 5, we need to determine the total number of all triangles (T) for rows 3, 4, and 5.
To find the missing values, we can observe the pattern in the given data:
For row 1:
$\bullet $ Number of up triangles (U) $= 1$
$\bullet $ Number of down triangles (D) $= 0$
$\bullet $ Total number of all triangles (T) $= 1$
For row 2:
$\bullet $ Number of up triangles (U) $= 3$
$\bullet $ Number of down triangles (D) $= 1$
$\bullet $ Total number of all triangles (T) $= 4$
Now, let’s analyze the pattern:
Each row contributes to the total number of all triangles (T) by adding the number of up triangles (U) and the number of down triangles (D) from that row. So, the formula to calculate T would be $T = U + D.$
Using this formula, we can calculate the missing values in the table:
For row 3:
$\bullet $ Number of up triangles (U) $= 6$
$\bullet $ Number of down triangles (D) $= 3$
$\bullet $ Total number of all triangles (T) $= 6 + 3 = 9$
For row 4:
$\bullet $ Number of up triangles (U) $= 10 $
$\bullet $ Number of down triangles (D) $= 6$
$\bullet $ Total number of all triangles (T) $= 10 + 6 = 16$
For row 5:
$\bullet $ Number of up triangles (U) $= 15 $
$\bullet $ Number of down triangles (D) $= 10 $
$\bullet $ Total number of all triangles (T) $= 15 + 10 = 25$
Now, let’s update the table with the missing values:
The missing values have been filled in up to row 5.
Question (b) : 3 marks
Describe in words three patterns you see in the table.
▶️Answer/Explanation
Ans:
Based on the given table, three patterns can be observed:
1. Pattern of Triangle Numbers: The third column represents the total number of triangles (T) in each row. The values in this column follow a specific pattern. It can be observed that the number of triangles in each row (T) is equal to the sum of the number of up triangles (U) and the number of down triangles (D). For example, in row 1, there is 1 up triangle and 0 down triangles, so the total number of triangles is 1. In row 2, there are 3 up triangles and 1 down triangle, resulting in a total of 4 triangles. This pattern continues in subsequent rows.
2. Increasing Number of Triangles: As we move from row to row, the number of triangles (T) increases. In each row, the number of triangles is greater than or equal to the row number (n). For instance, in row 3, the number of triangles is 9, which is equal to the row number. In row 4, the number of triangles is 16, greater than the row number. This increasing pattern continues in subsequent rows.
3. Relationship between Up and Down Triangles: The second and third columns represent the number of up triangles (U) and down triangles (D), respectively. A pattern emerges in the relationship between these two quantities. The number of up triangles in each row is equal to the sum of the row number (n) and the number of down triangles in the previous row. For example, in row 3, there are 6 up triangles, which is the sum of 3 (the row number) and 3 (the number of down triangles in the previous row). This pattern holds true for subsequent rows.