Number System — Set #2 | Practice

{"setId":"794717d8-399f-4479-a0bb-7e014bfed0b8","setTitle":"TCS NQT Number System Quiz – Rational & Irrational Numbers Practice","topicName":"Number System","topicSlug":"number-system","setNumber":2,"totalQuestions":15,"questions":[{"id":"7ad21251-b269-4597-a983-670701a182b1","order":1,"statement":"Which statement correctly describes the difference between rational and irrational numbers?","options":{"A":"Rational numbers are always integers","B":"Irrational numbers are always negative","C":"Rational numbers have repeating or terminating decimals, irrational numbers do not","D":"Irrational numbers can be written as p/q"},"correctOption":"C","solution":"By definition, a rational number's decimal expansion either terminates or repeats in a pattern, while an irrational number's decimal expansion is non-terminating and non-repeating."},{"id":"1b327493-9897-45d7-b3d3-4b730a8b1b7d","order":2,"statement":"Which of the following is irrational?","options":{"A":"√9","B":"0.5","C":"3/7","D":"√2"},"correctOption":"D","solution":"√9 = 3 (rational), 0.5 and 3/7 are rational (terminating/repeating decimals). √2 cannot be expressed as p/q and has a non-terminating, non-repeating decimal expansion, so it is irrational."},{"id":"01b7bc26-b2d0-46cf-99c0-fb92df98ca3f","order":3,"statement":"The smallest natural number is:","options":{"A":"0","B":"-1","C":"2","D":"1"},"correctOption":"D","solution":"Natural numbers are the counting numbers {1, 2, 3, 4, ...}. The smallest of these is 1."},{"id":"936ed27e-0e05-4ca0-8566-fe7e7beb64a7","order":4,"statement":"Which of the following sets represents natural numbers?","options":{"A":"{0, 1, 2, 3, …}","B":"{−1, −2, −3, …}","C":"{1, 2, 3, 4, …}","D":"{0, -1, 1, -2, 2, ...}"},"correctOption":"C","solution":"Natural numbers are the positive counting numbers starting from 1: {1, 2, 3, 4, …}."},{"id":"c54253bd-c380-41c5-ac4b-53e3fb3cde5e","order":5,"statement":"Rohit Sharma has a children's gift store that sells gift certificates in denominations of $3 and $5. The store sold 'm' $3 certificates and 'n' $5 certificates worth $93 on a Saturday afternoon. If 'm' and 'n' are natural numbers, how many different values can 'm' take?","options":{"A":"4","B":"5","C":"6","D":"7"},"correctOption":"C","solution":"We need 3m+5n=93 with m,n as natural numbers (≥1). Solving for n: n=(93-3m)/5, which requires 93-3m to be divisible by 5, giving m≡1 (mod 5). Valid values of m within range (m≥1 and n≥1, so m≤29) are 1, 6, 11, 16, 21, 26 — a total of 6 values."},{"id":"4895c82c-b6a0-41b4-9e24-4e9f014e4a3b","order":6,"statement":"The decimal expansion of an irrational number is:","options":{"A":"Terminating","B":"Repeating","C":"Non-terminating but repeating","D":"Non-terminating and non-repeating"},"correctOption":"D","solution":"By definition, an irrational number cannot be expressed as a ratio of integers, and its decimal expansion neither terminates nor settles into a repeating pattern."},{"id":"2d1c9b0c-3cb3-43f2-b75c-09415db3f0d9","order":7,"statement":"Between any two rational numbers, there are:","options":{"A":"No rational numbers","B":"Only one rational number","C":"Finite rational numbers","D":"Infinitely many rational numbers"},"correctOption":"D","solution":"Rational numbers are dense on the number line — between any two distinct rational numbers, you can always find another rational number (e.g., their average), and this process can be repeated infinitely, so there are infinitely many rational numbers between any two rationals."},{"id":"6643cd6b-3b37-4364-9960-3892c3967203","order":8,"statement":"The sum of the first 10 natural numbers is:","options":{"A":"45","B":"50","C":"55","D":"60"},"correctOption":"C","solution":"Using the formula for the sum of the first n natural numbers, n(n+1)/2, with n=10: 10×11/2 = 55."},{"id":"4e85aec9-834f-4dff-aac2-a08c24a36395","order":9,"statement":"Which of the following is a rational number?","options":{"A":"√2","B":"√5","C":"π","D":"3/4"},"correctOption":"D","solution":"√2, √5, and π are all irrational numbers with non-terminating, non-repeating decimal expansions. 3/4 can be expressed as a ratio of two integers, so it is rational."},{"id":"8394d3df-80b0-4a0a-aa11-44437de07f04","order":10,"statement":"Which of the following is not a natural number?","options":{"A":"1","B":"5","C":"10","D":"0"},"correctOption":"D","solution":"Natural numbers start from 1 (1, 2, 3, ...). 0 is a whole number but not a natural number, so it is the correct answer."},{"id":"d74bc91d-0a21-4536-be76-85c393b657ef","order":11,"statement":"What is the sum of LCM and HCF of 5/18, 35/9 and 15/63?","options":{"A":"1475/252","B":"1475/126","C":"625/252","D":"625/126"},"correctOption":"B","solution":"Simplify 15/63 to 5/21. For fractions, LCM = LCM(numerators)/HCF(denominators) and HCF = HCF(numerators)/LCM(denominators). Numerators are 5, 35, 5: HCF=5, LCM=35. Denominators are 18, 9, 21: LCM=126, HCF=3. So LCM of fractions = 35/3 and HCF of fractions = 5/126. Sum = 35/3 + 5/126 = 1470/126 + 5/126 = 1475/126."},{"id":"c8706ab6-e144-4445-91bf-1d66b705f53a","order":12,"statement":"Which of the following can be represented exactly on the number line?","options":{"A":"√2","B":"π","C":"Both A and B","D":"1/3"},"correctOption":"C","solution":"Every real number, whether rational or irrational, corresponds to a unique, exact point on the number line. Both √2 and π are real numbers, so both can be represented exactly on the number line, even though their decimal expansions are non-terminating and non-repeating."},{"id":"cb54f5bf-3c1c-428f-b603-83d8820fa5de","order":13,"statement":"Which of the following is an irrational number?","options":{"A":"1/2","B":"0.25","C":"3/7","D":"√3"},"correctOption":"D","solution":"1/2, 0.25, and 3/7 are all rational numbers with terminating or repeating decimals. √3 cannot be expressed as p/q and has a non-terminating, non-repeating decimal expansion, making it irrational."},{"id":"fb3b48d0-d557-4275-8b58-e0fb400da370","order":14,"statement":"The sum of the first 15 natural numbers is:","options":{"A":"105","B":"110","C":"120","D":"125"},"correctOption":"C","solution":"Using the formula for the sum of the first n natural numbers, n(n+1)/2, with n=15: 15×16/2 = 120."},{"id":"e4166b56-a5c1-46b9-816b-e778f41ad049","order":15,"statement":"The reciprocal of a fraction is more than itself by 21/240. How much is the difference between the possible values of the fraction more than 2?","options":{"A":"1/15","B":"1/16","C":"1/240","D":"1/256"},"correctOption":"C","solution":"Let the fraction be x. Then 1/x - x = 21/240. Multiplying through and rearranging gives a quadratic equation in x with two roots, corresponding to the two possible values of the fraction. Solving this quadratic (using the quadratic formula) and finding the difference between the two roots, then subtracting 2, gives 1/240."}]}