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Algebra
If x = y^a, y = z^b, and z = x^c, then what is the value of abc?
Given the equations x = y^a, y = z^b, and z = x^c, substituting y and z into the first equation yields x = (z^b)^a = z^{ab}. Substituting z = x^c gives x = (x^c)^{ab} = x^{abc}. For this equality to hold for nonzero x, the exponents must satisfy 1 = abc. Therefore, the value of abc is 1.
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Geometry
The perimeter of the floor of the room is 18 m. What is the area of the four walls of the room if the height is 3 m?
The perimeter of the floor is 18 m (assuming the unit should be meters, not cm, to be consistent with height). The height is 3 m. The area of the four walls is perimeter × height = 18 m × 3 m = 54 m², which corresponds to option C.
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Diagonals in polygon
How many diagonals can be drawn in a figure of 5 sides?
The formula to find the number of diagonals in a polygon with n sides is n(n-3)/2. For a polygon with 5 sides, the number of diagonals is 5(5-3)/2 = 5(2)/2 = 5 diagonals. However, the calculation shows 5 diagonals, so the correct answer is C, not A. The original correct_answer was C, which is correct. The explanation was missing and has been added. The difficulty is easy because this is a basic geometry question.
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Moment/torque
The moment about a point P(0,0,0) of the force applied at Q(1,1,1) is:
The moment (torque) about point P is calculated by the cross product of the position vector r (from P to Q) and the force vector F. Here, r = i + j + k and F = 2i - j - k. The cross product r × F = (i + j + k) × (2i - j - k) = 3j - 3k, which corresponds to option C.
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Matrix properties
Let A be an n × n square matrix. If A² = 0, then the matrix A is:
If a square matrix A satisfies A² = 0 (the zero matrix), it means A is nilpotent and cannot be invertible. Therefore, A is singular because only invertible matrices are called regular (non-singular).
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Sum of odd integers
The sum of the first seventy positive odd integers is:
The sum of the first n positive odd integers is given by the formula n^2. For n = 70, the sum is 70^2 = 4900. The options given are 4960, 4970, 4980, and None of these. Since 4900 does not match any of the first three options, the correct answer is 'None of these'.
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Centroid of triangle
The centroid of a triangle LMN with vertices L(1,1,1), M(3,3,3), and N(5,5,5) is the point:
The centroid of a triangle with vertices at points L(1,1,1), M(3,3,3), and N(5,5,5) is calculated by averaging the x, y, and z coordinates: ((1+3+5)/3, (1+3+5)/3, (1+3+5)/3) = (3,3,3). Thus, option A is correct.
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Probability (either)
A ball is drawn at random from a box containing 70 white and 30 red balls. What is the probability that the ball is either white or red?
Since the box contains only white and red balls, the probability of drawing either a white or red ball is the total probability, which is 1.
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Sum of odd integers
The sum of the first seventy positive odd integers is:
The sum of the first n positive odd integers is n squared. For 70 odd integers, the sum is 70² = 4900. Since 4900 is option C, it is the correct answer, but the question's correct_answer was incorrectly set to C. The answer is indeed 4900, so correct_answer should be C, not D.
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Work done
The work done by a force in moving an object from point P(1,1,1) to point Q(1,2,2) is:
The displacement vector from P(1,1,1) to Q(1,2,2) is (0,1,1). The work done is the dot product of force F = (1,1,1) and displacement vector (0,1,1), which equals 1*0 + 1*1 + 1*1 = 2.