1. Real Numbers
рдЗрд╕ рдкреЗрдЬ рдкрд░ рдЖрдкрдХреЛ 1. Real Numbers рдХреЗ рд╕рднреА рдорд╣рддреНрд╡рдкреВрд░реНрдг рдмрд╣реБрд╡рд┐рдХрд▓реНрдкреАрдп рдкреНрд░рд╢реНрди (MCQ), рдкрд░реАрдХреНрд╖рд╛ рдЙрдкрдпреЛрдЧреА рдиреЛрдЯреНрд╕ рдФрд░ рдСрдирд▓рд╛рдЗрди рдХреНрд╡рд┐рдЬрд╝ рдорд┐рд▓реЗрдВрдЧреЗред рдЕрдкрдиреЗ рдЬреНрдЮрд╛рди рдХрд╛ рдкрд░реАрдХреНрд╖рдг рдХрд░рдиреЗ рдФрд░ рдмреЛрд░реНрдб/рдкреНрд░рддрд┐рдпреЛрдЧреА рдкрд░реАрдХреНрд╖рд╛ рдореЗрдВ рд╢рдд-рдкреНрд░рддрд┐рд╢рдд рдЕрдВрдХ рдкреНрд░рд╛рдкреНрдд рдХрд░рдиреЗ рдХреЗ рд▓рд┐рдП рдиреАрдЪреЗ рджрд┐рдП рдЧрдП рдореЙрдХ рдЯреЗрд╕реНрдЯ рдореЗрдВ рдЕрд╡рд╢реНрдп рднрд╛рдЧ рд▓реЗрдВред
ЁЯСЙ рдЕрдзреНрдпрд╛рдп рдХрд╛ рд╕рд╛рд░рд╛рдВрд╢ (Summary)
Dear students, welcome to the very first and one of the most fundamental chapters of Class 10 Mathematics, Real Numbers. In our everyday lives, we constantly rely on numbers for counting, measuring, conducting financial transactions, and solving various practical problems. The real number system is essentially a combination of both rational numbers and irrational numbers. These numbers represent any continuous quantity that can be accurately located on an infinite number line. This chapter is a natural progression and a more advanced continuation of the number system concepts you have already explored in your previous classes. Now, our primary focus shifts towards understanding and applying two deeply significant properties of positive integers: Euclid's Division Algorithm and the Fundamental Theorem of Arithmetic.
Euclid's Division Lemma provides a formal mathematical statement for the long division process we have known for years. It states that given any two positive integers a and b, there must exist unique integers q and r satisfying the linear equation a = bq + r, under the strict condition that 0 тЙд r < b. We utilize this algorithm as a highly effective technique primarily to compute the Highest Common Factor (HCF) of two given positive integers. For example, if a store owner needs to find the HCF of 124 and 24 to organize items evenly, they can repeatedly apply this step-by-step division algorithm until the mathematical remainder inevitably reaches exactly zero.
The Fundamental Theorem of Arithmetic is another immensely crucial and powerful mathematical concept. It guarantees that every single composite number can be uniquely expressed, or factorized, strictly as a product of prime numbers. This distinct prime factorization remains completely unique for every composite number, apart from the specific order in which these prime factors happen to occur. This theorem serves multiple essential applications in higher mathematics. We extensively use it to efficiently find the Least Common Multiple (LCM) as well as the HCF of two or more positive integers. Furthermore, it forms the logical backbone when we mathematically prove the irrationality of specific numbers, such as the square root of 2, the square root of 3, the square root of 5, and so forth.
Throughout this chapter, we also carefully study and analyze the decimal expansions of various rational numbers. Any rational number expressed as p/q possesses a terminating decimal expansion strictly if the prime factorization of its denominator q is perfectly in the mathematical form of 2^n 5^m, where both n and m are non-negative integers. If the prime factorization of the denominator q contains any prime factors other than 2 or 5, then the rational number is guaranteed to have a non-terminating but repeating decimal expansion. Understanding these foundational concepts with absolute clarity is highly beneficial and strongly recommended for excelling in your NCERT preparations and upcoming board exams.
Let us consider a highly relatable real-life example to solidify this. If you are packing two completely different types of manufactured items into uniform boxes and you desperately want every single box to contain the exact same number of items without leaving any leftovers, you are practically finding the HCF. Similarly, if two independent events occur at different continuous intervals and you want to predict exactly when they will simultaneously happen together again, you are actively looking for the LCM. For instance, if two church bells ring at strict intervals of 10 and 15 minutes respectively, they will harmoniously ring together once again after exactly 30 minutes, which represents their LCM. These highly practical applications undeniably make the deep study of real numbers very engaging, highly interesting, and directly relevant to everyday logical problem-solving. Always ensure to vigorously practice the formal proofs of irrationality, as examiners frequently ask these foundational questions in the main board exams.
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рдорд╣рддреНрд╡рдкреВрд░реНрдг рдкреНрд░рд╢реНрди (Important Questions)
PDF рдбрд╛рдЙрдирд▓реЛрдб рдХрд░реЗрдВрдЕрдзреНрдпрд╛рдп рдХреЗ рд╕рднреА рдкреНрд░рд╢реНрди (Revision Notes)
*рдиреЛрдЯ: рдпрд╣ рд╕реЗрдХреНрд╢рди рд░рд┐рд╡реАрдЬрди рдХреЗ рд▓рд┐рдП рд╣реИред рдЕрдкрдирд╛ рдЬреНрдЮрд╛рди рдкрд░рдЦрдиреЗ рдХреЗ рд▓рд┐рдП рдКрдкрд░ рджрд┐рдП рдЧрдП рдХреНрд╡рд┐рдЬрд╝ рдореЗрдВ рднрд╛рдЧ рд▓реЗрдВред
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What is the HCF of 12 and 15?
A) 3 | B) 4 | C) 5 | D) 6 -
What is the prime factorization of 140?
A) 2 x 2 x 5 x 7 | B) 2 x 3 x 5 x 7 | C) 2 x 2 x 3 x 5 | D) 2 x 5 x 5 x 7 -
For any two positive integers a and b, what is HCF(a,b) x LCM(a,b) equal to?
A) a + b | B) a - b | C) a x b | D) a / b -
The decimal expansion of 17/8 will terminate after how many decimal places?
A) 1 | B) 2 | C) 3 | D) 4 -
Which of the following is an irrational number?
A) sqrt(4) | B) sqrt(9) | C) sqrt(16) | D) sqrt(3) -
What is the LCM of 6 and 20?
A) 30 | B) 60 | C) 120 | D) 15 -
If p is a prime number, then what type of number is sqrt(p)?
A) Rational | B) Irrational | C) Integer | D) Whole number -
What is the HCF of two co-prime numbers always equal to?
A) 0 | B) 1 | C) 2 | D) 3 -
Every composite number can be expressed as a product of which numbers?
A) Even numbers | B) Odd numbers | C) Prime numbers | D) Natural numbers -
What type of number is 5 - sqrt(3)?
A) Rational | B) Irrational | C) Integer | D) Natural -
If HCF(26, 91) = 13, then what is the LCM(26, 91)?
A) 182 | B) 2366 | C) 13 | D) 26 -
The decimal expansion of the rational number 14587/1250 will terminate after how many decimal places?
A) 1 decimal place | B) 2 decimal places | C) 3 decimal places | D) 4 decimal places -
What is the largest number that divides 70 and 125, leaving remainders 5 and 8 respectively?
A) 13 | B) 65 | C) 875 | D) 1750 -
If two positive integers p and q can be expressed as p = ab^2 and q = a^3b (where a and b are prime numbers), then what is LCM(p, q)?
A) ab | B) a^2b^2 | C) a^3b^2 | D) a^3b^3 -
What is the product of a non-zero rational and an irrational number?
A) Always irrational | B) Always rational | C) Rational or irrational | D) One -
What is the least number that is exactly divisible by all the numbers from 1 to 10 (both inclusive)?
A) 10 | B) 100 | C) 504 | D) 2520 -
If the HCF of 65 and 117 is expressible in the form 65m - 117, what is the value of m?
A) 4 | B) 2 | C) 1 | D) 3 -
The expression (7 x 11 x 13) + 13 represents what type of number?
A) Prime number | B) Composite number | C) Odd number | D) Irrational number -
Which of the following rational numbers has a terminating decimal expansion?
A) 11/700 | B) 91/2100 | C) 15/1600 | D) 29/343 -
A rational number can be expressed as a terminating decimal if its denominator has factors of:
A) 2 or 5 only | B) 2 or 3 only | C) 3 or 5 only | D) Any prime numbers -
If n is a natural number, then 6^n can never end with which digit?
A) 0 | B) 6 | C) 2 | D) 4 -
Let x = p/q be a rational number such that prime factorization of q is 2^n 5^m. If x terminates, what determines the number of decimal places?
A) Sum of n and m | B) Product of n and m | C) Maximum of n and m | D) Minimum of n and m -
Three bells ring at intervals of 4, 7 and 14 minutes. If all three rang at 6 AM, when will they ring together again?
A) 6:28 AM | B) 6:56 AM | C) 6:14 AM | D) 7:00 AM -
If n is an odd positive integer, then n^2 - 1 is always divisible by:
A) 8 | B) 16 | C) 24 | D) 32 -
What is the largest number which divides 320 and 457, leaving remainders 5 and 7 respectively?
A) 45 | B) 60 | C) 75 | D) 90 -
Determine the smallest number which when increased by 17 is exactly divisible by both 520 and 468.
A) 4663 | B) 4680 | C) 4697 | D) 4643 -
Given that the LCM of 91 and 26 is 182, find the HCF of 91 and 26.
A) 13 | B) 26 | C) 7 | D) 91 -
What is the HCF of the smallest composite number and the smallest prime number?
A) 1 | B) 2 | C) 3 | D) 4 -
If p and q are two distinct prime numbers, what is their Least Common Multiple (LCM)?
A) 1 | B) p + q | C) p - q | D) pq -
A sweet seller has 420 kaju barfis and 130 badam barfis. She wants to stack them equally. What is the maximum number of barfis in each stack?
A) 10 | B) 20 | C) 30 | D) 40
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