Counterexamples, "not", "and", "or", and Venn diagrams

Grade 10 Math Worksheet - Challenge Me

Circle the best answer.

1. 1.

   A counterexample has perimeter 24 and area greater than 35. Ari claims: if a rectangle has perimeter 24, its area is at most 35. Which rectangle is a counterexample?

   • A 5 by 7 rectangle, with area 35
   • A 6 by 7 rectangle, with area 42
   • A 6 by 6 square, with area 36
   • A 4 by 8 rectangle, with area 32
2. 2.

   Of Bo's 60 classmates, 35 take geometry, 28 take art and 12 take neither. How many take both?

   • 20
   • 48
   • 3
   • 15

3. Read this first

   Use this

   Art 20, Band 18, Chess 15. Art and Band 7, Art and Chess 6, Band and Chess 5, each including the students who like all three. All three: 3.

4. 3.

   Jun surveys students about Art, Band and Chess. Using the counts on the card, how many like at least one?

   • 38
   • 53
   • 32
   • 35

5. Use this

   Total 40. Art 17, Band 15, Chess 12. Art and Band 6, Art and Chess 4, Band and Chess 5, each including those in all three clubs. 9 are in no club. The number in all three clubs is not given.

6. 4.

   Kira's class has 40 students. Using the counts on the card, how many are in all three clubs?

   • 13
   • 9
   • 11
   • 2
7. 5.

   A counterexample makes xy = 0 true but x ≠ 0. Tala claims: if xy = 0, then x = 0. Which pair of values is a counterexample?

   • x = 5 and y = 0
   • x = 0 and y = 0
   • x = 2 and y = 3
   • x = 0 and y = 5

8. Use this

   The contrapositive of 'if P then Q' is 'if not Q then not P'. P: n² is even. Q: n is even.

9. 6.

   Noah proves 'if n² is even, then n is even' by contrapositive. Which should the proof start from?

   • Assume n is even, then show n² is even
   • Assume n² is even, then show n is even
   • Assume n² is odd, then show n is odd
   • Assume n is odd, then show n² is odd
10. 7.

    Total 80. Soccer 45, basketball 40, both 12, neither 5. Zee reads a survey of 80 students: 45 like soccer, 40 like basketball, 12 like both and 5 like neither. Does the data fit together?

    • No: 45 + 40 − 12 = 73 like at least one, but 80 − 5 = 75 should
    • Yes: 45 + 40 + 5 − 12 = 78, which is close to 80
    • Yes: each count is less than 80
    • No: 45 + 40 = 85 is more than the 80 students, so the data is impossible
11. 8.

    For every triangle, at least one side is shorter than 10. Ren says: for every triangle there is a side shorter than 10. Which statement is its negation?

    • For every triangle there is a side longer than 10
    • There is a triangle with a side longer than 10
    • For every triangle, all sides are at least 10
    • There is a triangle whose sides are all at least 10
12. 9.

    A counterexample has equal diagonals and is not a rectangle. Luz claims: if a quadrilateral's diagonals have equal length, it is a rectangle. Which quadrilateral is a counterexample?

    • Corners (0, 0), (6, 0), (6, 3), (0, 3): equal diagonals √45, a rectangle
    • Corners (0, 0), (6, 0), (5, 3), (1, 3): equal diagonals √34, no right angles
    • Corners (0, 0), (6, 0), (8, 3), (2, 3): the diagonals are √73 and 5
    • Corners (0, 0), (4, 0), (4, 4), (0, 4): equal diagonals √32, a square
13. 10.

    A new club survey has Chess only 0, Art+Chess only 3, Band+Chess only 2, all three 1. Pia claims everyone in Chess is also in Art or Band. Is it true?

    • False: Chess has 6 members, and only the 3 in Art and Chess count as covered
    • False: the 9 students in no club break the claim about Chess members
    • True: the Chess-only region is empty, so every Chess member is in Art or Band
    • True: Chess has fewer members than Art, so they must sit inside Art's circle