MATHS_HOLIDAY_HOMEWORK.pptx useeee fulll

MATHS_HOLIDAY_HOMEWORK.pptx useeee fulll

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  MATHS VACATION ASSIGNMENT  SUBMITTEDBY : ATHARV 3385/S
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  REAL NUMBERS
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  INTRODUCTON  Real numbersinclude rational numbers like positive and negative integers, fractions, and irrational numbers. In other words, any number that we can think of, except complex numbers, is a real number. For example, 3, 0, 1.5, 3/2, √5, and so on are real numbers. Mathematician Richard Dedekind asked these questions 159 years ago at ETH Zurich, and became the first person to define real numbers.16 Nov 2017 Mathematician Richard Dedekind asked these questions 159 years ago at ETH Zurich, and became the first person to define real numbers. Every real number can be represented by a different point on the number line since we know that real numbers can be either rational or irrational. A real number line, also known as a number line, represents real numbers on the line with distinct points assigned to each of the numbers.
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  EUCLID DIVISION LEMMA Euclid’s division lemma states that given two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b. This means that any positive integer a can be divided by another positive integer b, with a unique quotient q and remainder r. Dividend = (Divisor × Quotient) + Remainde An algorithm is a series of well defined steps which gives a procedure for solving a type of problem. The word algorithm comes from the name of the 9th century Persian mathematician al-Khwarizmi. In fact, even the word ‘algebra’ is derived from a book, he wrote, called Hisab al-jabr w’al-muqabala. A lemma is a proven statement used for proving another statement.
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  FUNDAMENTAL THEORM OFARTHEMETIC  In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors. Theorem (Fundamental Theorem of Arithmetic) : Every composite number can be expressed ( factorised ) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.
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  REVISITING IRRATIONAL NUMBERS Theorem: Let p be a prime number. If p divides a, then p divides a, where a is a positive integer. An irrational number is a real number that cannot be expressed as a ratio of integers; for example, √2 is an irrational number. We cannot express any irrational number in the form of a ratio, such as p/q, where p and q are integers, q≠0. EUCLIDS DIVISION ALGORITHM Euclid’s division algorithm : This is based on Euclid’s division lemma. According to this, the HCF of any two positive integers a and b, with a > b, is obtained as follows: Step 1 : Apply the division lemma to find q and r where a = bq + r, 0 ≤ r < b. Step 2 : If r = 0, the HCF is b. If r ≠ 0, apply Euclid’s lemma to b and r. Step 3 : Continue the process till the remainder is zero. The divisor at this stage will be HCF (a, b). Also, HCF(a, b) = HCF(b, r).
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  PROVING √2 ASIRRATIONAL  Let us assume that 2 is a √ rational number with p and q as co-prime integers and q ≠ 0  ⇒ √2 = p/q  On squaring both sides we get,  ⇒ 2q2 = p2  ⇒ Here, 2q2 is a multiple of 2 and hence it is even. Thus, p2 is an even number. Therefore, p is also even.  So we can assume that p = 2x where x is an integer.  By substituting this value of p in 2q2 = p2 , we get  ⇒ 2q2 = (2x)2  ⇒ 2q2 = 4x2  ⇒ q2 = 2x2  ⇒ q2 is an even number. Therefore, q is also even.  Since p and q both are even numbers, they have 2 as a common multiple which means that p and q are not co-prime numbers as their HCF is 2.  This leads to the contradiction that root 2 is a rational number in the form of p/q with "p and q both co-prime numbers" and q ≠ 0.
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  Revisiting Rational Numbersand Their Decimal Expansions  Theorem : Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form , p/q where p and q are coprime, and the prime factorisation of q is of the form 2n5mwhere n, m are non-negative integers.  Theorem : Let x = p/q be a rational number, such that the prime factorisation of q is of the form 2n 5m, where n, m are non-negative integers. Then x has decimal expansion which terminates.  Theorem: Let x = p/q be a rational number, such that the prime factorisation of q is not of the form 2n 5m, where n, m are non-negative integers. Then, x has a decimal expansion which is non-terminating repeating (recurring).
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  SUMMARY  Euclid’s divisionlemma : Given positive integers a and b, there exist whole numbers q and r satisfying a = bq + r, 0 ≤ r < b.  The Fundamental Theorem of Arithmetic : Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.  If p is a prime and p divides a2, then p divides a, where a is a positive integer.
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  MULTIPLE CHOICE QUESTIONS 1)What is the least number that must be added to 1056, so the number is divisible by 23 A)0 B)3 C)2 D)1 Answer: C.
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  2)Which of thefollowing is not irrational? (a) (2 – 3)2 √ (b) ( 2 + 3)2 √ √ (c) ( 2 - 3)( 2 + 3) √ √ √ √ (d)27 7/ 7 √ √ ANSWER :C
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  3) The productof a rational and irrational number is (a) rational (b) irrational (c) both of above (d) none of above  ANSWER:B 4)The sum of a rational and irrational number is (a) rational (b) irrational (c) both of above (d) none of above ANSWER:B
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  5) HCF of8, 9, 25 is (a) 8 (b) 9 (c) 25 (d) 1 ANSWER: (D) 6)The product of three consecutive positive integers is divisible by (a) 4 (b) 6 (c) no common factor (d) only 1 ANSWER: (B)
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  7)LCM of thegiven number ‘x’ and ‘y’ where y is a multiple of ‘x’ is given by (a) x (b) y (c) xy (d) x/y ANSWER: (B) 8)The largest number that will divide 398,436 and 542 leaving remainders 7,11 and 15 respectively is (a) 17 (b) 11 (c) 34 (d) 45 Answer: (a) Explanation:(a); [Hint. Algorithm 398 – 7 – 391; 436 – 11 = 425; 542 – 15 = 527; HCF of 391, 425, 527 = 17]
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  9)There are 312,260 and 156 students in class X, XI and XII respectively. Buses are to be hired to take these students to a picnic. Find the maximum number of students who can sit in a bus if each bus takes equal number of students (a) 52 (b) 56 (c) 48 (d) 63 Answer: (a) Explanation:(a); [Hint. HCF of 312, 260, 156 = 52]
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  10) There isa circular path around a sports field. Priya takes 18 minutes to drive one round of the field. Harish takes 12 minutes. Suppose they both start at the same point and at the same time and go in the same direction. After how many minutes will they meet ? (a) 36 minutes (b) 18 minutes (c) 6 minutes (d) They will not meet Answer: (a) Explanation:(a); [Hint. LCM of 18 and 12 = 36]