Counterexamples, "not", "and", "or", and Venn diagrams
Grade 10 Math Worksheet - Challenge Me
Circle the best answer.
Proof by contradiction starts by assuming the opposite of what is to be shown, then reaches a contradiction. Esme proves 'a triangle cannot have two right angles' by contradiction. Which assumption should the proof start from?
- Assume a triangle has no right angle, then show the angle sum is not 180°, a contradiction
- Assume the angle sum is 180°, then show a triangle has two right angles
- Assume a triangle has three right angles, then look for a contradiction
- Assume a triangle has two right angles, then show the sum exceeds 180°
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 4 by 8 rectangle, with area 32
- A 6 by 7 rectangle, with area 42
- A 6 by 6 square, with area 36
- A 5 by 7 rectangle, with area 35
Of Bo's 60 classmates, 35 take geometry, 28 take art and 12 take neither. How many take both?
- 20
- 48
- 3
- 15
S is true. Two statements are equivalent when they are true or false together. S: if a quadrilateral has four equal sides, then its diagonals are perpendicular. Milo wants a true statement equivalent to S. Which one?
- If a quadrilateral's diagonals are not perpendicular, then its four sides are not all equal
- If a quadrilateral's diagonals are perpendicular, then it has four equal sides
- If a quadrilateral does not have four equal sides, then its diagonals are not perpendicular
- A quadrilateral has four equal sides and its diagonals are not perpendicular
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 = 0 and y = 5
- x = 5 and y = 0
- x = 2 and y = 3
- x = 0 and y = 0
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The contrapositive of 'if P then Q' is 'if not Q then not P'. P: n² is even. Q: n is even.
Noah proves 'if n² is even, then n is even' by contrapositive. Which should the proof start from?
- Assume n² is odd, then show n is odd
- Assume n is odd, then show n² is odd
- Assume n is even, then show n² is even
- Assume n² is even, then show n is even
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 = 85 is more than the 80 students, so the data is impossible
- Yes: 45 + 40 + 5 − 12 = 78, which is close to 80
- No: 45 + 40 − 12 = 73 like at least one, but 80 − 5 = 75 should
- Yes: each count is less than 80
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
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), (8, 3), (2, 3): the diagonals are √73 and 5
- Corners (0, 0), (4, 0), (4, 4), (0, 4): equal diagonals √32, a square
- 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
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?
- True: the Chess-only region is empty, so every Chess member is in Art or Band
- 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: Chess has fewer members than Art, so they must sit inside Art's circle