Q1: How many numbers greater than 1,000,000 can be formed using the digits 1, 3, 0, 3, 4, 3, 4, without repeating any digit more than its frequency?
- 1) 240
- 2) 360
- 3) 480
- 4) 720
Correct Answer: 2) 360
Total valid numbers: \(7! / (3! \times 2!) = 420\). Subtracting those starting with 0 leaves 360 valid numbers.
Q2: There are 17 Biology books and 15 Mathematics books. How many ways can they be arranged such that no two Mathematics books are together?
- 1) 732
- 2) 768
- 3) 816
- 4) 854
Correct Answer: 3) 816
Total ways: \( \binom{18}{15} = 816 \).
Q3: If \( \tan A \) and \( \tan B \) are roots of \(x^2 – mx + n = 0\), find \( \sin^2(A + B) \).
- 1) \(1 – \frac{m^2}{4n}\)
- 2) \( \frac{4n – m^2}{4n} \)
- 3) \( \frac{m^2 – 4n}{4n} \)
- 4) \( 1 + \frac{m^2 – 4n}{4n} \)
Correct Answer: 2) \( \frac{4n – m^2}{4n} \)
Using the formula for \( \sin^2(A + B) \), we get the answer as \( \frac{4n – m^2}{4n} \).
Q4: Evaluate: \( \log 45600 + \log 32400 \).
- 1) 8.610
- 2) 9.860
- 3) 10.190
- 4) 11.370
Correct Answer: 2) 9.860
Using properties of logarithms: \( \log(45600 \times 32400) = \log(1.478 \times 10^9) = 9.860 \).
Q: A book has 512 pages, with an average of 3 mistakes per page. However, the first 204 pages contain a total of 156 mistakes. What is the average number of mistakes per page in the remaining pages?
- 1) 3 mistakes per page
- 2) 2 mistakes per page
- 3) 2.5 mistakes per page
- 4) None of the above
Correct Answer: 4) None of the above
Solution:
- Total pages: 512
- Expected total mistakes (3 mistakes per page): \(512 \times 3 = 1536\)
- Mistakes in the first 204 pages: 156
- Remaining pages: \(512 – 204 = 308\)
- Remaining mistakes: \(1536 – 156 = 1380\)
- Average mistakes per page in the remaining pages: \[ \frac{1380}{308} = 4.48 \]
Since 4.48 is not one of the given options, the correct answer is “None of the above.”
Q5: If the incomes of A, B, and C are in the ratio \(4:6:5\) and their expenditures are in the ratio \(4:6:7\), what is the ratio of their savings if A saves half of his income?
- 1) 3:6:4
- 2) 4:6:3
- 3) 6:4:3
- 4) 4:5:6
Correct Answer: 1) 3:6:4
The ratio of savings is calculated as \(3:6:4\) after simplifying the income and expenditure differences.
Q6: Complete the sequence: 1, 5, -4, 12, -13, ___, -26, 38.
- 1) 23
- 2) 22
- 3) 21
- 4) 20
Correct Answer: 1) 23
The pattern is alternating positive and negative steps. Adding 36 to -13 gives 23.
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