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كويز تفاعلي: Conjectures and Counterexamples - Reveal
Exploring Conjectures and Counterexamples in Mathematics. A conjecture is a mathematical statement that appears to be true based on patterns and observations but has not been formally proven. Inductive reasoning is the process of arriving at these conjectures by noticing recurring trends in data or sequences. However, to disprove a conjecture, one only needs to find a single counterexample—a specific case where the statement does not hold true. This worksheet covers identifying patterns, forming conjectures, and finding counterexamples across various mathematical scenarios, including number sequences and geometric properties.
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يرجى الانتباه إلى أن المعلم قام بإعداد الأسئلة فقط، ولم يقم بإعداد الإجابات أو الشروحات المرفقة. وقد تم توليد الإجابات باستخدام تقنيات الذكاء الاصطناعي، لذلك قد تتضمن بعض الأخطاء أو عدم الدقة.
للحصول على الإجابات الصحيحة والمضمونة، يُرجى الرجوع إلى المعلم أو المصدر الدراسي المعتمد.
Question 1
DB question no.: 1
Points: 1
To fully disprove a conjecture, one needs to find only ONE counterexample.
A counterexample must make the inequality false. If a = 6, then 2(6) + 7 = 12 + 7 = 19. Since 19 is not less than or equal to 17, a = 6 is a counterexample.
The sequence alternates between positive and negative integers: 1, -1, 2, -2, 3, so the next number is -3.
Question 10
DB question no.: 11
Points: 1
Determine if this conjecture is true. If not, give a counterexample. The difference of two negative numbers is a negative number.
Explanation
Subtracting a negative number is the same as adding its absolute value. In the case of -11 - (-13), the result is 2, which is positive, disproving the conjecture.
Question 11
DB question no.: 12
Points: 1
If it is an angle, then it is acute. What is an appropriate counterexample?
A counterexample must be an obtuse angle (greater than \(90^\circ\) and less than \(180^\circ\)) that is not \(125^\circ\). \(160^\circ\) is obtuse and not \(125^\circ\).
Question 15
DB question no.: 16
Points: 1
Which of the following provide a counterexample to the conjecture, 'If two angles are supplementary, then they are not congruent.'
Supplementary angles add up to \(180^\circ\). Two \(90^\circ\) angles are supplementary and they are congruent (equal), which disproves the conjecture.
Question 16
DB question no.: 17
Points: 1
What is a counterexample?
Explanation
This is the definition of a counterexample in the context of logic and mathematical conjectures.
Question 17
DB question no.: 18
Points: 1
Provide a counterexample to the following claim: 'If a number is divisible by 2, then it is divisible by 4.'
Explanation
14 is divisible by 2 (\(14 \div 2 = 7\)) but not divisible by 4 (\(14 \div 4 = 3.5\)), making it a counterexample.
Question 18
DB question no.: 19
Points: 1
Find the pattern to solve the sequence 2, 4, 7, 11...
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