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About Mathematical Investigation — Class 10 IB

Conduct extended mathematical investigations using inquiry-based approaches. This topic is part of the IB Class 10 mathematics syllabus (chapter: Unit 10). On this page you can practice 59 questions across three difficulty levels — 20 easy, 19 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 35-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

Mathematical Investigation — solved examples for Class 10 IB

Example 1easy

Which of the following best describes the primary goal of a mathematical investigation?
  1. A)To memorize formulas and theorems for exams.
  2. B)To find and prove new mathematical truths, patterns, or relationships.
  3. C)To solve given textbook problems quickly using known methods.
  4. D)To practice calculation skills without necessarily understanding the underlying concepts.

Step-by-step solution

  1. A mathematical investigation involves exploring a problem, discovering patterns, forming conjectures, and then trying to prove or justify these conjectures. Its primary goal is to extend mathematical understanding and discover new relationships, rather than simply applying existing knowledge or memorizing.

Answer: To find and prove new mathematical truths, patterns, or relationships.

Example 2medium

Consider a sequence of patterns made with dots. Pattern 1 has 3 dots. Pattern 2 has 8 dots. Pattern 3 has 15 dots. If the pattern continues, how many dots will Pattern 5 have?
  1. A)24
  2. B)30
  3. C)35
  4. D)42

Step-by-step solution

  1. Observe the pattern: Pattern 1 (n=1) has 3 dots, Pattern 2 (n=2) has 8 dots, Pattern 3 (n=3) has 15 dots.
  2. Find the differences: 8 - 3 = 5, 15 - 8 = 7. The second differences are constant (7 - 5 = 2), indicating a quadratic relationship of the form An² + Bn + C.
  3. Alternatively, notice that 3 = (1+1)² - 1 = 2² - 1, 8 = (2+1)² - 1 = 3² - 1, 15 = (3+1)² - 1 = 4² - 1.
  4. The rule for the nth pattern is T_n = (n+1)² - 1. For Pattern 5 (n=5), T_5 = (5+1)² - 1 = 6² - 1 = 36 - 1 = 35.

Answer: 35

Example 3hard

A student investigates a sequence of numbers: 1, 6, 15, 28, 45, ... They want to find a general formula for the n-th term, T_n. Which of the following conjectures correctly represents the n-th term of this sequence?
  1. A)T_n = n² + 2n - 2
  2. B)T_n = 2n² - n
  3. C)T_n = n² + 4n - 4
  4. D)T_n = 2n² + n - 2

Step-by-step solution

  1. Calculate the first differences: 6-1=5, 15-6=9, 28-15=13, 45-28=17.
  2. Calculate the second differences: 9-5=4, 13-9=4, 17-13=4. Since the second differences are constant (4), the general term is a quadratic of the form T_n = an² + bn + c.
  3. Using the relations for quadratic sequences: 2a = second difference => 2a = 4 => a = 2. Then, 3a + b = first term of first differences => 3(2) + b = 5 => 6 + b = 5 => b = -1. Finally, a + b + c = first term of sequence => 2 + (-1) + c = 1 => 1 + c = 1 => c = 0.
  4. Thus, the general formula is T_n = 2n² - n + 0, or T_n = 2n² - n.

Answer: T_n = 2n² - n

Practice questions on Mathematical Investigation

  1. Q1.easy

    Consider the sequence: 5, 9, 13, 17, ... What is the next term in this sequence, and what type of pattern is it?
    1. A)20, Arithmetic
    2. B)21, Geometric
    3. C)20, Geometric
    4. D)21, Arithmetic
    Show answer

    Answer: 21, Arithmetic

    Hint: Look at the difference between consecutive terms to identify the rule governing the sequence.

  2. Q2.easy

    A student is investigating the number of matchsticks required to build a line of squares.
    Term (n) | Number of Squares | Matchsticks (M)
    ---|---|---
    1 | 1 | 4
    2 | 2 | 7
    3 | 3 | 10
    4 | 4 | 13
    Which formula correctly represents the number of matchsticks (M) for 'n' squares?
    1. A)M = 3n + 1
    2. B)M = 4n
    3. C)M = n + 3
    4. D)M = 2n + 2
    Show answer

    Answer: M = 3n + 1

    Hint: Look for a constant difference in the 'Matchsticks' column, then use that to find the coefficient of 'n' and the constant term.

  3. Q3.easy

    In a mathematical investigation, what is meant by a 'conjecture'?
    1. A)A proven mathematical theorem that is universally accepted.
    2. B)A statement that is assumed to be false and is the aim of the investigation to disprove.
    3. C)A conclusion reached through logical deduction from established axioms and definitions.
    4. D)An unproven statement or idea that is believed to be true, based on observations or evidence.
    Show answer

    Answer: An unproven statement or idea that is believed to be true, based on observations or evidence.

    Hint: A conjecture is an educated guess or hypothesis based on observed patterns, but it still needs to be proven or justified.

  4. Q4.medium

    A mathematician proposes the conjecture that for any positive integer 'n', the expression (n² + n + 2)/2 always results in an even number. Which of the following values of 'n' serves as a counterexample to this conjecture?
    1. A)n = 1
    2. B)n = 2
    3. C)n = 3
    4. D)n = 4
    Show answer

    Answer: n = 3

    Hint: Substitute each value of 'n' into the expression and check if the result is even or odd. A counterexample will produce an odd number.

  5. Q5.medium

    A school canteen wants to optimize its food ordering to minimize waste. They decide to investigate the relationship between the number of students attending school on a given day and the quantity of a specific dish (e.g., 'Chole Bhature') consumed. Which pair of variables would be most appropriate to identify as independent and dependent variables for this investigation?
    1. A)Independent: Quantity of food consumed; Dependent: Number of students attending
    2. B)Independent: Number of students attending; Dependent: Quantity of food consumed
    3. C)Independent: Time of day; Dependent: Quantity of food consumed
    4. D)Independent: Type of food; Dependent: Number of students attending
    Show answer

    Answer: Independent: Number of students attending; Dependent: Quantity of food consumed

    Hint: The independent variable is the one you change or observe, and the dependent variable is the one that changes in response. Think about what influences what.

  6. Q6.medium

    Imagine constructing larger equilateral triangles from smaller unit equilateral triangles.
    * Figure 1 is a single unit triangle (1 triangle).
    * Figure 2 is an equilateral triangle with 2 unit triangles along its base (total 4 unit triangles).
    * Figure 3 is an equilateral triangle with 3 unit triangles along its base (total 9 unit triangles).
    If this pattern continues, how many unit triangles would be required to form Figure 8?
    1. A)36
    2. B)49
    3. C)64
    4. D)81
    Show answer

    Answer: 64

    Hint: Look for the relationship between the figure number and the total number of unit triangles. The number of unit triangles along the base is a key indicator.

  7. Q7.hard

    During a mathematical investigation, a student draws circles and divides them into the maximum possible number of regions by drawing chords. They record the number of regions for n=0, 1, 2, 3, 4 chords. The results are: 1, 2, 4, 7, 11. If they continue this pattern, what would be their correct conjecture for the maximum number of regions, R_n, created by 'n' chords?
    1. A)R_n = 2^n
    2. B)R_n = n² + n + 1
    3. C)R_n = (n² + n + 2)/2
    4. D)R_n = (n² - n + 2)/2
    Show answer

    Answer: R_n = (n² + n + 2)/2

    Hint: Generate the next term in the sequence (R_5) by following the pattern of differences. The sequence is R_0=1, R_1=2, R_2=4, R_3=7, R_4=11.

  8. Q8.hard

    A student investigates the expression P(n) = n² + n + 41, conjecturing that it always produces a prime number for any positive integer 'n'. Which is the smallest positive integer value of 'n' that disproves this conjecture?
    1. A)n = 39
    2. B)n = 40
    3. C)n = 41
    4. D)n = 42
    Show answer

    Answer: n = 40

    Hint: Test values of 'n' by substituting them into the expression. Remember that a prime number has exactly two distinct positive divisors: 1 and itself. A composite number has more than two. Consider values that might create common factors.

  9. Q9.hard

    A mathematician observes that all swans they have ever seen are white. They conclude, 'Therefore, all swans are white.' Another mathematician starts with the axioms of Euclidean geometry and proves that the sum of angles in a triangle is 180°. Which statement accurately describes the reasoning used by each mathematician?
    1. A)The first uses deductive reasoning, the second uses inductive reasoning.
    2. B)Both use inductive reasoning.
    3. C)The first uses inductive reasoning, the second uses deductive reasoning.
    4. D)Both use deductive reasoning.
    Show answer

    Answer: The first uses inductive reasoning, the second uses deductive reasoning.

    Hint: Inductive reasoning moves from specific observations to a general conclusion, which may or may not be true. Deductive reasoning starts from general principles or axioms to reach a specific, logically certain conclusion.

These are 9 of the 59 questions available for Mathematical Investigation. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.