Study HubHubPage 181/182Scan verified MCQs with the answer highlighted, then open explanations when you want the reasoning.
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Polygons (diagonals)
How many diagonals can be drawn in a figure of 5 sides?
The number of diagonals in a polygon with n sides is given by the formula n(n-3)/2. For a polygon with 5 sides (a pentagon), the number of diagonals is 5(5-3)/2 = 5(2)/2 = 5. Therefore, the correct answer is option C, which is 5 diagonals.
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Series
The sum of the first seventy positive odd integers is:
The sum of the first n odd positive integers is n². For n = 70, the sum is 70² = 4900. However, the options given are 4910, 4920, 4900, and None of these. Since 4900 is option C, the correct answer should be C, not B. Therefore, the original correct_answer 'C' is correct. The explanation was missing, so it has been added.
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Triangles (angles and sides)
In a triangle with sides a = 60, b = 40, c = 38, the largest angle is opposite which side?
In a triangle, the largest angle is opposite the longest side. Here, side a = 60 is the longest side, so the largest angle is opposite side a.
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Infinite geometric series
The infinite sum 2 + \frac{2}{5} + \frac{2}{25} + \frac{2}{125} + \dots to infinity is:
The series 2 + 2/5 + 2/25 + 2/125 + ... is a geometric series with first term a = 2 and common ratio r = 1/5. Since |r| < 1, the series converges and the sum to infinity is given by S = a / (1 - r) = 2 / (1 - 1/5) = 2 / (4/5) = 2 × 5/4 = 5/2. Therefore, the correct answer is B (5/2).
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Polynomials
If (z - a) is a factor of the polynomial P(z), then P(a) is always:
If (z - a) is a factor of the polynomial P(z), then by the Factor Theorem, P(a) = 0. This means the polynomial evaluates to zero at z = a.
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Centroid
The centroid of a triangle LMN with vertices L(1,1), M(3,3), and N(5,5) is the point
The centroid of a triangle is the average of the x-coordinates and y-coordinates of its vertices. For vertices L(1,1), M(3,3), and N(5,5), the centroid is ((1+3+5)/3, (1+3+5)/3) = (3,3). Options B and C include three coordinates, which is incorrect for a 2D point, and option D is incorrect since the centroid is (3,3).
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Matrices
Let A = [aij] be an n × n square matrix. If the second row (i = 2) is zero for all columns j = 1, 2, ..., n, then the matrix A is:
If the second row of the square matrix A consists entirely of zeros, then the matrix has a row of zeros, making its determinant zero. Hence, the matrix is singular (non-invertible).
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Permutations/Counting
A man has 6 coats, 3 shirts, and 5 trousers. In how many ways can he dress himself with a coat, a shirt, and trousers?
The total number of ways to dress is the product of the number of choices for each clothing item: 6 coats × 3 shirts × 5 trousers = 90 ways.
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Series
The sum of the first seventy positive even integers is:
The sum of the first n positive even integers is given by the formula n(n+1). For n = 70, the sum is 70 × 71 = 4970. Therefore, the correct answer is 4970, which corresponds to option B.
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Probability
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 a Red ball is 1, as these are the only possible outcomes.