answer:To answer this question, we need to understand the concept of proof by contradiction. Proof by contradiction, also known as reductio ad absurdum, involves negating the conclusion that we want to prove as false, not both the conclusion and conditions. Then, we deduce a contradiction or an absurdity from this assumption. When we reach an impossibility or contradiction, we conclude that the negation of the conclusion must be false; therefore, the original conclusion is true. We do not simultaneously negate both the conclusion and the conditions. So, the correct statement should be: Proof by contradiction involves negating the conclusion and deducing a contradiction from that negation. Therefore, the correct answer to the problem statement is: [boxed{B}] (False).

answer:1. **Analyze (5^a)**: Since (5) is an odd number, and (a) is an integer, (5^a) will also be odd. This is because the product or exponentiation of odd numbers always results in an odd number. 2. **Analyze ((b+1)^2c)**: Since (b) is an odd integer, (b+1) is an even integer (adding 1 to an odd number results in an even number). Squaring an even number, ((b+1)^2), results in an even number because the square of any even number is even. 3. **Multiplication by (c)**: Multiplying an even number ((b+1)^2) by any integer (c) (odd or even) still gives an even number, because the product of an even number and any integer still results in an even number due to one of the factors being even. 4. **Sum of (5^a + (b+1)^2c)**: Adding an odd number (5^a) to an even number ((b+1)^2c) consistently results in an odd number because the sum of an odd number and an even number is always odd. 5. **Conclusion:** The expression (5^a + (b+1)^2c) is odd for all choices of (c), thus the answer is text{odd for all choices of c}. The final answer is boxed{A) odd for all choices of c}

answer:Step 1: Understand the transformations about new point (1,1) and triangle vertices. - **Rotation by 60^circ, 120^circ, and 180^circ counterclockwise** about (1,1) rotates coordinates in specific ways around this pivot. - **Reflection across the x-axis** changes (x,y) to (x,-y+2) for vertices above y=1. - **Reflection across the y-axis** changes (x,y) to (-x+2,y) for vertices right of x=1. Step 2: Account for orientation changes and count valid sequences. Rotations preserve orientation; reflections change it. Hence, an even number of reflections, similar to the original approach, is required. Case 1: No reflections Using only rotations, the sum of the angles must be 360^circ modulo. Combinations include: - Three 120^circ rotations. - One 60^circ and one 300^circ (combination of 120^circ and 180^circ). - One 120^circ and one 240^circ (two 180^circ rotations). Valid combinations: - (120^circ, 120^circ, 120^circ) perhaps in `3!` ways. Case 2: Two reflections Using two reflections (one along x-axis, one along y-axis) and one rotation: - 180^circ rotation still counters both reflections effectively. - Sequence could be ordered in `3!` ways. Conclusion: Count the valid sequences from both cases: - Case 1: 6 - Case 2: 6 (as both reflections need to be included in the sequence) Overall, there are 6 + 6 = 12 sequence setups that keep T at its original position after the transformations. 12 The final answer is boxed{B) 12}

answer:Given that the random variable eta has all possible values of 1, 2, 3, 4, and P(eta=k)=ak (k=1,2,3,4), The sum of probabilities for all possible values of eta must be equal to 1, therefore P(eta=1)+P(eta=2)+P(eta=3)+P(eta=4)=a+2a+3a+4a=1, Solving for a, we get a=dfrac{1}{10}. therefore The value of a is boxed{dfrac{1}{10}}. Thus, the answer is B. This problem requires you to understand the properties of a probability distribution for a discrete random variable. By summing the probabilities for all possible values of the random variable, you can find the value of a.