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Arithmetic progression
The population of a city increases by 700 each year. If the population grows to 25,000 in five years, what was its initial population?
The population increases by 700 each year for 5 years, so total increase = 700 × 5 = 3500. Given the population after 5 years is 25000, the initial population = 25000 - 3500 = 21500.
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Sentence transformation
He made us work: (Change into passive voice)
The correct passive transformation of 'He made us work' is 'We are made to work.' Option A correctly reflects this transformation. Option C is a duplicate of A but the question format typically expects the first correct option to be chosen.
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Vocabulary (deliberated)
The jury members deliberated for almost an hour before singing the verdict.
The verb 'deliberated' means to carefully consider or discuss something, which fits the context of jury members discussing before giving a verdict. 'Deferred' means to postpone, and 'adjourned' means to suspend proceedings, neither of which fit the sentence context. Therefore, 'deliberated' is the correct choice.
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Polynomials (factor theorem)
If (z - a) is a factor of polynomial P(z), then P(a) is always:
According to the factor theorem, if (z - a) is a factor of polynomial P(z), then P(a) = 0. None of the options A (z), B (o), or C (a) represent zero, so the correct answer is D (None of these).
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Complex numbers
What is the imaginary part of the multiplicative inverse of the complex number 6 - 5i?
The multiplicative inverse of a complex number a + bi is given by (a - bi) / (a^2 + b^2). For 6 - 5i, the inverse is (6 + 5i) / (6^2 + (-5)^2) = (6 + 5i) / 61. The imaginary part is therefore 5/61.
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Triangle angles
In a triangle with sides a = 60, b = 40, and 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, so the largest angle is opposite side a.
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Probability
A ball is selected at random from a box. If the box has 9 white, 8 green, and 3 orange balls, what is the probability that the ball selected is not orange?
The total number of balls is 9 + 8 + 3 = 20. The number of balls that are not orange is 9 + 8 = 17. Therefore, the probability of selecting a ball that is not orange is 17/20.
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Olympics 2020 host city
The 2020 Summer Olympics were held in which city?
The 2020 Summer Olympics were held in Tokyo, Japan. Although originally scheduled for 2020, the event was postponed to 2021 due to the COVID-19 pandemic but retained the name 'Tokyo 2020.'
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Synonyms
Which of the following is not a synonym of "investigate"?
"Elucidate" means to make something clear or explain, which is not a synonym of "investigate." "Examine" and "Scrutinize" both mean to inspect or investigate closely, making them synonyms of "investigate."
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Fundamental counting principle
A man has 6 coats, 3 shirts, and 5 trousers. In how many ways can he dress himself with a coat, shirt, and trouser?
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.