Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
EXERCISE 1 Write the correct answer in each of the following: 1. Every rational number is (A) a natural number (B) (C) a real number (D) 2. Between two rational numbers (A) there is no rational number (B) there is exactly one rational number (C) there are infinitely many rational numbers an integer a whole number (D) 4. The product of any two irrational numbers is (A) always an irrational number (B) always a rational number (C) always an integer 3. Decimal representation of a rational number cannot be (A) terminating (B) non-terminating (C) non-terminating repeating (D) (D) non-terminating non-repeating 5. The decimal expansion of the number (A) a finite decimal (B) 1.41421 (C) non-terminating recurring (D) non-terminating non-recurring 2 is (B) 0.1416 8. A rational number between (A) (A) 2+ 3 2 (C) 0.1416 2 and 2⋅ 3 2 (B) (D) 0.4014001400014... 3 is (C) 1.5 (D) 1.8 p 9. The value of 1.999... in the form q , where p and q are integers and q ≠ 0 , is (A) 19 10 (B) 1999 1000 (C) 2 (D) 1 9 6 (C) 3 3 (D) 4 6 5 6 (C) 25 (D) 10 5 10. 2 3 + 3 is equal to (A) 11. 2 6 (B) 10 × 15 is equal to (A) 6 5 (B) sometimes rational, sometimes irrational 6. Which of the following is irrational? 7. Which of the following is irrational? (A) 0.14 there are only rational numbers and no irrational numbers 4 9 (B) 12 3 (C) 7 (D) 81 1 12. The number obtained on rationalising the denominator of 7+2 3 (A) 7–2 3 (B) 7+2 5 (B) 1 3+2 2 (D) 3+2 2 is 7+2 45 (D) 1 is equal to 9– 8 13. (A) 1 3– 2 2 2 (C) 3– 2 2 ( ) 14. After rationalising the denominator of (A) 13 (B) (A) 16. If 4 3 (A) (B) 2 2 = 1.4142, then (C) 5 (D) 35 2 (C) 4 (D) 8 2 –1 is equal to 2 +1 (A) 2.4142 (B) 5.8282 (C) (D) 0.1718 0.4142 2 2 equals − 2 1 6 (B) 18. The product (A) 2 19. Value of 4 (A) 19 7 , we get the denominator as 3 3 –2 2 32 + 48 is equal to 8 + 12 15. The value of 17. (C) 7–2 1 9 3 2– 6 (C) 1 26 (D) 26 (D) 12 (D) 1 81 2 ⋅ 4 2 ⋅12 32 equals ( 81) −2 (B) 2 (C) 12 (B) 1 3 (C) 9 2 32 is 20. Value of (256)0.16 × (256)0.09 is (A) 4 (B) 16 (C) 64 (D) 256.25 (D) x 7 × x 12 21. Which of the following is equal to x? 12 7 (A) x –x 5 7 12 (B) 1 4 3 (x ) 2 3 3 12 (x) (C) 7 EXERCISE 2 1. Let x and y be rational and irrational numbers, respectively. Is x + y necessarily an irrational number? Give an example in support of your answer. 2. Let x be rational and y be irrational. Is xy necessarily irrational? Justify your answer by an example. 3. State whether the following statements are true or false? Justify your answer. (i) 2 is a rational number. 3 There are infinitely many integers between any two integers. (ii) (iii) Number of rational numbers between 15 and 18 is finite. p (iv) There are numbers which cannot be written in the form q , q ≠ 0 , p, q both are integers. (v) The square of an irrational number is always rational. (vi) 12 is not a rational number as 12 and 3 (vii) 15 p is written in the form , q ≠ 0 and so it is a rational number. q 3 3 are not integers. 4. Classify the following numbers as rational or irrational with justification : (i) 196 (ii) 3 18 (iii) 12 75 9 27 (v) – 0.4 (vi) (viii) (1 + 5) – (4 + 5 ) (ix) 10.124124... (x) (iv) (vii) 0.5918 1.010010001... 28 343 EXERCISE 3 1. Find which of the variables x, y, z and u represent rational numbers and which irrational numbers: x2 = 5 (i) (ii) y2 = 9 (iii) z2 = .04 (ii) 0.1 and 0.11 (iv) 1 1 and 4 5 (iv) u2 = 17 4 2. Find three rational numbers between (i) –1 and –2 (iii) 5 6 and 7 7 3. Insert a rational number and an irrational number between the following : (i) 2 and 3 (ii) 0 and 0.1 (iii) (iv) –2 1 and 5 2 (v) 0.15 and 0.16 (vi) (vii) (x) 2.357 and 3.121 (viii) .0001 and .001 6.375289 and 6.375738 1 1 and 3 2 2 and 3 (ix) 3.623623 and 0.484848 p 4. Express the following in the form q , where p and q are integers and q ≠ 0 : (i) 0.2 (ii) 0.888... (iii) 5.2 (iv) (v) 0.2555... (vi) 0.134 (vii) .00323232... (viii) .404040... 0.001 5. Simplify the following: (i) (ii) 45 – 3 20 + 4 5 3 3 + 2 27 + (v) 7 (vi) 3 2 3 (ix) 3 – ( 3– 2 ) 4 (iii) 24 54 + 8 9 2 (vii) 4 4 28 ÷ 3 7 ÷ 3 7 (iv) 12 × 7 6 81 – 8 3 216 + 15 5 32 + 225 (viii) 3 1 + 8 2 3 6 6. Rationalise the denominator of the following: 2 (i) 3 3 3+ 2 3– 2 (vii) (ii) 40 3 (viii) 3 5+ 3 5– 3 (iii) (ix) 3+ 2 4 2 (iv) 16 41 – 5 2+ 3 2– 3 (v) 6 2+ 3 (vi) 4 3+5 2 48 + 18 7. Find the values of a and b in each of the following: (i) 5+ 2 3 =a−6 3 7 +4 3 3– 5 19 =a 5– 3+2 5 11 (ii) 8. If a = 2 + 3 , then find the value of a – (iii) 2+ 3 = 2– b 6 3 2 –2 3 (iv) 7+ 5 7 – 5 7 – =a+ 5b 7 – 5 7+ 5 11 1 . a 3 = 1.732 and 5 = 2.236 9. Rationalise the denominator in each of the following and hence evaluate by taking 2 = 1.414 , upto three places of decimal. 4 3 (i) 6 6 (ii) (iii) 10 – 5 2 (iv) 2 2+ 2 10. Simplify : (i) 1 13 + 2 3 + 33 2 ( ) − (vi) 64 1 3 1 64 3 (ii) – 2 64 3 3 5 4 8 5 1 83 −12 × (vii) 32 1 16 3 − 1 3 32 5 6 (iii) 1 27 −2 3 (iv) 1 3+ 2 (v) (625) 1 − 2 − 1 4 2 1 93 (v) 1 36 × 27 ×3 − − 1 2 2 3 EXERCISE 4 1. Express 0.6 + 0.7 + 0.47 in the form 2. Simplify : 3. If p , where p and q are integers and q ≠ 0 . q 7 3 2 5 3 2 – – . 10 + 3 6+ 5 15 + 3 2 2 = 1.414, 3 = 1.732 , then find the value of 4. If a = 5. If x = 3+ 5 2 2 , then find the value of a + 4 3 + . 3 3–2 2 3 3+2 2 1 . a2 3+ 2 3– 2 and y = , then find the value of x2 + y2. 3– 2 3+ 2 ( ) 6. Simplify : (256 ) − 4 7. Find the value of −3 2 4 (216 ) 2 − 3 + 1 ( 256) 3 − 4 + 2 1 (243 )− 5 ANSWERS EXERCISE 1 1. (C) 2. 9. (C) 10. 16. (C) 3. (D) 4. (D) (C) 11. (B) 12. (A) (C) 18. (C) 17. (B) 19. 5. 13. (A) (D) (D) 20. (A) EXERCISE 2 1. Yes. Let x = 21, y = 2 be a rational number. 2 = 21 + 1.4142 ... = 22.4142 ... Which is non-terminating and non-recurring. Hence x + y is irrational. Now x + y = 21 + 2. No. 0 × 2 = 0 which is not irrational . 3. (i) False. Although p 2 is of the form but here p, i.e., 2 is not an integer. q 3 (ii) (iii) False. Between 2 and 3, there is no integer. False, because between any two rational numbers we can find infinitely many rational numbers. (iv) True. (v) False, as (vi) False, because 2 3 is of the form (4 2) 2 p but p and q here are not integers. q = 2 which is not a rational number. 12 = 4 = 2 which is a rational number. 3 (vii) False, because 15 3 = 5= 5 which is p, i.e., 1 5 is not an integer. 6. (C) 14. (B) 21. (C) 7. 8. (D) 15. (B) (C) 4. (i) Rational, as (ii) 3 18 = 9 2 , which is the product of a rational and an irrational number and so an irrational number . 9 1 = , which is the quotient of a rational and an irrational number and so an irrational number 27 3 . (iii) 28 (iv) (v) 196 = 14 = 343 2 , which is a rational number. 7 Irrational, − 0.4 = − 12 (vi) = 75 2 10 , which is the quotient of a rational and an irrational. 2 , which is a rational number. 7 (vii) Rational, as decimal expansion is terminating. (viii) (1 + 5) − (4 + 5 ) = –3 , which is a rational number. (ix) Rational, as decimal expansion is non-terminating recurring. (x) Irrational, as decimal expansion is non-terminating non-recurring. EXERCISE 3 1. Rational numbers: (ii), (iii) Irrational numbers: (i), (iv) 2. 3. 4. (i) –1.1, –1.2, –1.3 (ii) 0.101, 0.102, 0.103 (i) 2.1, 2.040040004 ... (ii) 0.03, 0.007000700007, ... (v) 0.151, 0.151551555 ... (vi) 1.5, 1.585585558 ... (ix) 1, 1.909009000 ... (x) 6.3753, 6.375414114111 ... (i) (vi) 1 5 133 990 (ii) (vii) 8 9 8 2475 (iii) 47 9 (viii) 40 99 (iv) 1 999 (iii) 51 52 53 , , 70 70 70 (iii) 5 , 0.414114111 ... 12 (vii) (v) 23 90 3, 3.101101110 ... (iv) 9 17 19 , , 40 80 80 (iv) (viii) 0, 0.151151115 ... 0.00011, .0001131331333 ... 5. (i) 5 (ii) 7 6 12 (iii) 168 2 6 (i) 2 3 9 (ii) 2 30 3 (iii) (ix) (iv) 2+3 2 8 8 3 (iv) (v) 34 3 3 (vi) 5 − 2 6 (v) 7 + 4 3 41 + 5 (vii) 0 (vi) 3 2 − 2 3 9+4 6 15 (ii) a = 7. (i) a = 11 8. 2 3 9. (i) 2.309 9 11 (ii) 2.449 10. (i) 6 (ii) 2025 64 (iii) b = −5 6 (iv) a = 0, b = 1 (iii) 0.463 (iv) 0.414 (v) 0.318 (iii) 9 (iv) 5 (v) 3– 3 1 (vi) –3 (vii) 16 EXERCISE 4 1. 167 90 (vii) 5 + 2 6 2. 1 3. 2.063 4. 7 5. 98 6. 1 2 7. 214 (viii) 5 2 4 (viii) 9 + 2 15 (ix) 3 2