Real Numbers - 10th Class - 20 MCQ's - Worksheet4 - Complete Chapter - English Medium

Chapter 1: Real Numbers Practice

Your Score: 0 / 20

1. If two positive integers \(a\) and \(b\) are written as \(a = x^3 y^2\) and \(b = x y^3\), where \(x\) and \(y\) are prime numbers, then \(\text{HCF}(a, b)\) is:

• A) \(x y\)
• B) \(x y^2\)
• C) \(x^3 y^3\)
• D) \(x^2 y^2\)

2. Use Euclid's division algorithm to find the HCF of 196 and 38220.

• A) 196
• B) 98
• C) 195
• D) 382

3. Show that any positive even integer is of the form \(6q\), \(6q + 2\), or \(6q + 4\). What are the possible values for remainder \(r\) when an integer is divided by 6?

• A) 0, 1, 2, 3
• B) 1, 3, 5
• C) 0, 1, 2, 3, 4, 5
• D) 0, 2, 4

4. Find the prime factorisation of 3825.

• A) \(3 \times 5^2 \times 17\)
• B) \(3^2 \times 5^2 \times 17\)
• C) \(3^3 \times 5 \times 17\)
• D) \(3^2 \times 5 \times 85\)

5. The LCM of two numbers is 1820 and their HCF is 26. If one of the numbers is 130, find the other number.

• A) 364
• B) 260
• C) 455
• D) 520

6. Three bells toll together at intervals of 9, 12, and 15 minutes respectively. If they toll together now, after how many hours will they toll together next?

• A) 2 hours
• B) 3 hours
• C) 5 hours
• D) 6 hours

7. For what natural number \(n\) will \(4^n\) end with the digit 0?

• A) \(n = 5\)
• B) \(n = 10\)
• C) Any even number \(n\)
• D) No natural number \(n\)

8. Which of the following non-zero numbers will have a non-terminating repeating decimal expansion?

• A) \(\frac{13}{3125}\)
• B) \(\frac{17}{8}\)
• C) \(\frac{64}{455}\)
• D) \(\frac{6}{15}\)

9. After how many places of decimals will the rational number \(\frac{14587}{1250}\) terminate?

• A) 2 decimal places
• B) 3 decimal places
• C) 4 decimal places
• D) 5 decimal places

10. Write the decimal expansion of \(\frac{7}{80}\) without performing long division.

• A) 0.0875
• B) 0.875
• C) 0.00875
• D) 0.0785

11. If \(a\) and \(b\) are coprime positive integers, then \(a^2\) and \(b^2\) are:

• A) Not coprime
• B) Coprime
• C) Even integers
• D) Prime numbers

12. Which of the following is an irrational number?

• A) \(2.141414...\)
• B) \(\sqrt{0.04}\)
• C) \(3 + \sqrt{7}\)
• D) \(\frac{\sqrt{12}}{\sqrt{3}}\)

13. The product of a non-zero rational number and an irrational number is always:

• A) Rational
• B) Irrational
• C) An integer
• D) Zero

14. Convert the logarithmic equation \(\log_3 81 = 4\) into exponential form.

• A) \(4^3 = 81\)
• B) \(3^4 = 81\)
• C) \(81^{1/4} = 3\)
• D) \(3^{81} = 4\)

15. Evaluate the expression: \(\log_{10} 0.001\).

• A) 3
• B) \(-3\)
• C) \(-2\)
• D) \(\frac{1}{3}\)

16. Write \(\log\left(\sqrt{\frac{x^3}{y^2}}\right)\) in expanded form using logarithmic laws.

• A) \(\frac{3}{2}\log x - \log y\)
• B) \(3\log x - 2\log y\)
• C) \(\frac{3}{2}\log x + \log y\)
• D) \(\frac{1}{2}(3\log x - 2\log y)\)

17. Simplify into a single logarithm: \(3\log 2 + 2\log 5 - \log 20\).

• A) \(\log 10\)
• B) \(\log 5\)
• C) \(\log 2\)
• D) \(\log 100\)

18. Solve for \(x\): \(\log_2(x + 3) + \log_2(x - 3) = 4\).

• A) \(x = 5\)
• B) \(x = \pm 5\)
• C) \(x = 25\)
• D) \(x = 4\)

19. If \(\log_a b = x\), what is the value of \(\log_{\frac{1}{a}} b\)?

• A) \(x\)
• B) \(-x\)
• C) \(\frac{1}{x}\)
• D) \(a^x\)

20. If \(a^2 + b^2 = 7ab\), show that \(\log\left(\frac{a+b}{3}\right)\) is equal to:

• A) \(\frac{1}{2}(\log a + \log b)\)
• B) \(\log a + \log b\)
• C) \(2(\log a + \log b)\)
• D) \(\frac{1}{3}(\log a + \log b)\)