Simplify: \(1 \div [1 \div \{1 \div (1 \div 1)\}]\).
Answer: A
Working from the innermost bracket outwards:
Step 1: \((1 \div 1) = 1\).
Step 2: \(\{1 \div 1\} = 1\).
Step 3: \([1 \div 1] = 1\).
Step 4: \(1 \div 1 = 1\).
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Find the sum of first 30 natural numbers.
Answer: B
The sum of the first n natural numbers is given by the formula \(S_n = \frac{n(n+1)}{2}\).
Here, n = 30.
Sum = \(\frac{30(30+1)}{2} = \frac{30 \times 31}{2} = 15 \times 31 = 465\).
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Which of the following numbers is divisible by 9?
Answer: B
A number is divisible by 9 if the sum of its digits is a multiple of 9.
A) 7+5+3+2+0 = 17 (Not divisible by 9)
B) 7+5+3+2+1 = 18 (Divisible by 9, since 18 = 9 × 2)
C) 7+5+3+2+2 = 19 (Not divisible by 9)
D) 7+5+3+2+3 = 20 (Not divisible by 9)
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What is the value of the expression 101 + 102 + 103 + ... + 200?
Answer: B
The sum of an arithmetic progression is given by the formula \(S_n = \frac{n}{2}(a_1 + a_n)\), where \(n\) is the number of terms, \(a_1\) is the first term, and \(a_n\) is the last term.
Number of terms (n) = (Last Term - First Term) + 1 = (200 - 101) + 1 = 100.
First term (\(a_1\)) = 101.
Last term (\(a_n\)) = 200.
Sum = \(\frac{100}{2} \times (101 + 200) = 50 \times 301 = 15050\).
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If a number is decreased by 4 and divided by 6, the result is 8. What would be the result if 2 is subtracted from the number and then it is divided by 5?
Answer: B
Let the number be x.
According to the first condition: \(\frac{x-4}{6} = 8\).
\(x-4 = 48\), so \(x = 52\).
Now, according to the second condition, subtract 2 from the number: \(52 - 2 = 50\).
Then divide by 5: \(50 \div 5 = 10\).
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What is the result of \(3.5 \times (60 \div 2.5)\)?
Answer: A
Using BODMAS, we solve the bracket first.
\(60 \div 2.5 = 60 \div (\frac{5}{2}) = 60 \times \frac{2}{5} = 12 \times 2 = 24\).
Now, multiply by 3.5: \(3.5 \times 24 = (\frac{7}{2}) \times 24 = 7 \times 12 = 84\).
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What is the next number in the sequence: 1, 4, 9, 16, 25, ...?
Answer: B
The sequence consists of the squares of natural numbers.
\(1 = 1^2\)
\(4 = 2^2\)
\(9 = 3^2\)
\(16 = 4^2\)
\(25 = 5^2\)
The next term will be \(6^2 = 36\).
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What is the sum of all prime numbers between 60 and 80?
Answer: A
We need to identify the prime numbers between 60 and 80.
Prime numbers are: 61, 67, 71, 73, 79.
Sum = \(61 + 67 + 71 + 73 + 79\).
Sum = \((60+1) + (60+7) + (70+1) + (70+3) + (70+9) = 120 + 210 + (1+7+1+3+9) = 330 + 21 = 351\).
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The number 481481 is exactly divisible by which of the following?
Answer: D
Any number of the form 'abcabc' is a multiple of 1001.
\(481481 = 481 \times 1001\).
The prime factorization of 1001 is \(7 \times 11 \times 13\).
Therefore, any number of the form 'abcabc' will be divisible by 7, 11, and 13.
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Simplify: \(18 - [6 - \{4 - (8 - 6 + 3)\}]\).
Answer: B
Using BODMAS rule:
Step 1: Innermost bracket \((8 - 6 + 3) = (2 + 3) = 5\).
Step 2: Curly bracket \(\{4 - 5\} = -1\).
Step 3: Square bracket \([6 - (-1)] = [6 + 1] = 7\).
Step 4: Final calculation \(18 - 7 = 11\).
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