Polynomials-10th Class-English Medium-20-MCQ's-Worksheet - Complete Chapter

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Project Report - similar triangles - 10th class - English Medium - Formative Assessment

Sets - 10th Class - 20 MCQ's - Worksheet1 - Complete Chapter - English Medium

Class 10 Mathematics: Sets

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Period Plans-Quadratic Equations-10th Class-English Medium-Complete Chapter-Eight Periods-Easy to download

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

Chapter 1: Real Numbers Practice

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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) \(\frac{1}{3}(\log a + \log b)\)

Real Numbers - 10th Class - 20 MCQ's - Worksheet3 - Complete Chapter - English Medium-Textbook

Chapter 1: Real Numbers

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1. According to Euclid's Division Algorithm, for positive integers \(a\) and \(b\), there exist unique integers \(q\) and \(r\) satisfying \(a = bq + r\). Which condition must the remainder \(r\) satisfy?

• A) \(0 < r \le b\)
• B) \(0 \le r < b\)
• C) \(0 \le r \le b\)
• D) \(r > b\)

2. Use Euclid's division algorithm to find the Highest Common Factor (HCF) of 135 and 225.

• A) 15
• B) 30
• C) 45
• D) 90

3. Any positive odd integer can be expressed in which of the following forms (where \(q\) is some integer)?

• A) \(4q\) or \(4q + 2\)
• B) \(4q + 1\) or \(4q + 3\)
• C) \(2q\) only
• D) \(6q\) or \(6q + 4\)

4. Expressing 5005 as a product of prime factors yields:

• A) \(5 \times 7 \times 11 \times 13\)
• B) \(5 \times 11 \times 13 \times 17\)
• C) \(3 \times 5 \times 7 \times 13\)
• D) \(5^2 \times 7 \times 11\)

5. For any two positive integers \(a\) and \(b\), which relation between their HCF and LCM is always true?

• A) \(\text{HCF}(a, b) + \text{LCM}(a, b) = a \times b\)
• B) \(\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b\)
• C) \(\frac{\text{HCF}(a, b)}{\text{LCM}(a, b)} = a \times b\)
• D) \(\text{HCF}(a, b) \times a = \text{LCM}(a, b) \times b\)

6. Given that \(\text{HCF}(306, 657) = 9\), what is the \(\text{LCM}(306, 657)\)?

• A) 22338
• B) 12338
• C) 22448
• D) 30600

7. Why can't the number \(6^n\) end with the digit zero for any natural number \(n\)?

• A) Because 6 is not a prime number.
• B) Because its prime factorisation \(2^n \times 3^n\) does not contain 5.
• C) Because \(6^n\) is always odd.
• D) Because prime factorisation of 6 contains only odd numbers.

8. A rational number \(\frac{p}{q}\) (in simplest form) has a terminating decimal expansion if and only if the prime factorization of \(q\) is of the form:

• A) \(2^n 3^m\)
• B) \(2^n 5^m\)
• C) \(3^n 5^m\)
• D) \(7^n 5^m\)

9. Which of the following rational numbers has a terminating decimal expansion?

• A) \(\frac{77}{210}\)
• B) \(\frac{129}{2^2 \cdot 5^7 \cdot 7^5}\)
• C) \(\frac{15}{1600}\)
• D) \(\frac{29}{343}\)

10. Convert the rational number \(\frac{23}{2^3 \cdot 5^2}\) into decimal form without actual division.

• A) 0.23
• B) 0.115
• C) 0.0115
• D) 1.15

11. Let \(p\) be a prime number. If \(p\) divides \(a^2\) (where \(a\) is a positive integer), then:

• A) \(p\) must be equal to \(a\)
• B) \(p\) divides \(a\)
• C) \(a\) divides \(p\)
• D) \(p^2\) divides \(a\)

12. If \(p\) and \(q\) are distinct prime numbers, then \(\sqrt{p} + \sqrt{q}\) is always:

• A) Rational
• B) An integer
• C) Irrational
• D) A whole number

13. Evaluating the expression \((2\sqrt{3} + \sqrt{5})(2\sqrt{3} - \sqrt{5})\) yields:

• A) 7 (which is rational)
• B) 7 (which is irrational)
• C) 13 (which is rational)
• D) \(\sqrt{7}\) (which is irrational)

14. Convert the exponential statement \(5^3 = 125\) into its logarithmic equivalent.

• A) \(\log_3 125 = 5\)
• B) \(\log_5 125 = 3\)
• C) \(\log_{125} 5 = 3\)
• D) \(\log_5 3 = 125\)

15. What is the value of \(\log_2 \left(\frac{1}{16}\right)\)?

• A) 4
• B) \(-4\)
• C) \(\frac{1}{4}\)
• D) \(-2\)

16. Express \(\log\left(\frac{343}{125}\right)\) in expanded logarithmic form.

• A) \(3\log 7 + 3\log 5\)
• B) \(3(\log 7 - \log 5)\)
• C) \(7\log 3 - 5\log 3\)
• D) \(\frac{3\log 7}{3\log 5}\)

17. Express \(2\log 3 + 3\log 5 - 5\log 2\) as a single logarithm.

• A) \(\log\left(\frac{1125}{32}\right)\)
• B) \(\log\left(\frac{225}{32}\right)\)
• C) \(\log\left(\frac{1125}{10}\right)\)
• D) \(\log(1125 - 32)\)

18. Solve for \(x\) if \(2\log 5 + \frac{1}{2}\log 9 - \log 3 = \log x\).

• A) \(x = 5\)
• B) \(x = 15\)
• C) \(x = 25\)
• D) \(x = 75\)

19. What is the value of \(2^{2 + \log_2 3}\)?

• A) 7
• B) 12
• C) 6
• D) 14

20. If \(x^2 + y^2 = 25xy\), which of the following logarithmic identities holds true?

• A) \(2\log(x+y) = 3\log 3 + \log x + \log y\)
• B) \(2\log(x+y) = \log x + \log y + 3\log 2\)
• C) \(\log(x+y) = \log 25 + \log x + \log y\)
• D) \(2\log(x-y) = 3\log 3 + \log x + \log y\)

Similar Triangles - 10th Class - Worksheet1- English Medium - 20MCQ's-Telangana

Chapter 8: Similar Triangles

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1. If \(\triangle ABC \sim \triangle PQR\), which of the following ratios is correct?

• A) \(\frac{AB}{PQ} = \frac{BC}{PR} = \frac{AC}{QR}\)
• B) \(\frac{AB}{PQ} = \frac{BC}{QR} = \frac{AC}{PR}\)
• C) \(\frac{AB}{QR} = \frac{BC}{PQ} = \frac{AC}{PR}\)
• D) \(\frac{AB}{PR} = \frac{BC}{QR} = \frac{AC}{PQ}\)

2. All congruent figures are similar, but similar figures are:

• A) Always congruent
• B) Never congruent
• C) Not necessarily congruent
• D) Equal in area

3. In \(\triangle XYZ\), \(ST \parallel YZ\) with \(S\) on \(XY\) and \(T\) on \(XZ\). If \(XS = 3\text{ cm}\), \(SY = 6\text{ cm}\), and \(XT = 4\text{ cm}\), find the length of \(XZ\).

• A) \(8\text{ cm}\)
• B) \(12\text{ cm}\)
• C) \(9\text{ cm}\)
• D) \(10\text{ cm}\)

4. In \(\triangle LMN\), a line \(PQ\) cuts \(LM\) at \(P\) and \(LN\) at \(Q\) such that \(\frac{LP}{PM} = \frac{LQ}{QN}\). Which theorem justifies that \(PQ \parallel MN\)?

• A) Pythagoras Theorem
• B) Basic Proportionality Theorem (Thales Theorem)
• C) Converse of Basic Proportionality Theorem
• D) AAA Similarity Criterion

5. Two triangles are similar if their corresponding angles are equal. This is known as which criterion?

• A) SSS
• B) SAS
• C) AAA
• D) RHS

6. If in \(\triangle DEF\) and \(\triangle PQR\), \(\frac{DE}{PQ} = \frac{EF}{QR}\) and \(\angle E = \angle Q\), then the triangles are similar by which criterion?

• A) SSS
• B) SAS
• C) AAA
• D) RHS

7. In two similar triangles \(\triangle ABC\) and \(\triangle DEF\), the ratio of their corresponding sides is \(3:5\). What is the ratio of their areas?

• A) \(3:5\)
• B) \(6:10\)
• C) \(9:25\)
• D) \(27:125\)

8. If the areas of two similar triangles are in the ratio \(81:49\), then the ratio of their corresponding altitudes is:

• A) \(9:7\)
• B) \(81:49\)
• C) \(3:7\)
• D) \(7:9\)

9. In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. This statement is known as:

• A) Thales Theorem
• B) Mid-point Theorem
• C) Pythagoras Theorem
• D) Apollonius Theorem

10. Which of the following sets of side lengths forms a right-angled triangle?

• A) \(5\text{ cm}, 7\text{ cm}, 9\text{ cm}\)
• B) \(6\text{ cm}, 8\text{ cm}, 10\text{ cm}\)
• C) \(4\text{ cm}, 5\text{ cm}, 6\text{ cm}\)
• D) \(3\text{ cm}, 4\text{ cm}, 6\text{ cm}\)

11. A man travels \(12\text{ m}\) due West and then \(5\text{ m}\) due North. How far is he from his starting point?

• A) \(17\text{ m}\)
• B) \(7\text{ m}\)
• C) \(13\text{ m}\)
• D) \(15\text{ m}\)

12. In \(\triangle ABC\), \(\angle B = 90^\circ\) and \(BD \perp AC\). If \(AD = 4\text{ cm}\) and \(DC = 9\text{ cm}\), what is the length of \(BD\)?

• A) \(6\text{ cm}\)
• B) \(5\text{ cm}\)
• C) \(13\text{ cm}\)
• D) \(36\text{ cm}\)

13. In an equilateral triangle of side \(2a\), the length of each altitude is:

• A) \(a\sqrt{2}\)
• B) \(a\sqrt{3}\)
• C) \(2a\sqrt{3}\)
• D) \(\frac{a\sqrt{3}}{2}\)

14. If a line divides any two sides of a triangle in the same ratio, then the line must be:

• A) Perpendicular to the third side
• B) Parallel to the third side
• C) Equal to half the third side
• D) Bisecting the third side

15. If \(\triangle ABC \sim \triangle DEF\), \(\text{Area}(\triangle ABC) = 64\text{ cm}^2\), \(\text{Area}(\triangle DEF) = 121\text{ cm}^2\), and \(EF = 11\text{ cm}\), find \(BC\).

• A) \(8\text{ cm}\)
• B) \(6.4\text{ cm}\)
• C) \(11\text{ cm}\)
• D) \(8.8\text{ cm}\)

16. Vertical poles of heights \(6\text{ m}\) and \(11\text{ m}\) stand vertically on a plane ground. If the distance between their feet is \(12\text{ m}\), what is the distance between their tops?

• A) \(13\text{ m}\)
• B) \(14\text{ m}\)
• C) \(15\text{ m}\)
• D) \(17\text{ m}\)

17. If \(\triangle ABC\) is an isosceles right triangle right-angled at \(C\), then \(AB^2 =\)

• A) \(AC^2\)
• B) \(2AC^2\)
• C) \(3AC^2\)
• D) \(4AC^2\)

18. A vertical stick \(1.8\text{ m}\) long casts a shadow \(1.2\text{ m}\) long on the ground. At the same time, a tower casts a shadow \(24\text{ m}\) long. What is the height of the tower?

• A) \(36\text{ m}\)
• B) \(16\text{ m}\)
• C) \(32\text{ m}\)
• D) \(40\text{ m}\)

19. In \(\triangle ABC\), \(D\) and \(E\) are points on sides \(AB\) and \(AC\) respectively such that \(DE \parallel BC\). If \(AD = x\), \(DB = x - 2\), \(AE = x + 2\), and \(EC = x - 1\), find the value of \(x\).

• A) \(3\)
• B) \(4\)
• C) \(5\)
• D) \(6\)

20. The perimeter of two similar triangles \(\triangle ABC\) and \(\triangle PQR\) are \(30\text{ cm}\) and \(20\text{ cm}\) respectively. If \(AB = 12\text{ cm}\), find the corresponding side \(PQ\).

• A) \(6\text{ cm}\)
• B) \(8\text{ cm}\)
• C) \(10\text{ cm}\)
• D) \(9\text{ cm}\)