Chapter 1.5 Word Problems The product of two consecutive even integers. 1. Find two consecutive even integers whose product is 168 Sides of a Square 2. The length of each side of a square is 3 in. more than the length of each side of a smaller square. The sum of the areas of the squares is 149 in2. Find the lengths of the sides of the two squares. Uniform Strip 3. Cynthia Besch wants to buy a rug for a room that is 12 ft wide and 15 ft long. She wants to leave a uniform strip
Premium Mathematics Natural number Integer
ANSWERS/HINTS 345 APPENDIX 1 ANSWERS/ HINTS EXERCISE 1.1 1. (i) 45 3. 8 columns 4. An integer can be of the form 3q‚ 3q + 1 or 3q + 2. Square all of these integers. 5. An integer can be of the form 9q‚ 9q + 1‚ 9q + 2‚ 9q + 3‚ . . .‚ or 9q + 8. (ii) 196 (iii) 51 2. An integer can be of the form 6q‚ 6q + 1‚ 6q + 2‚ 6q + 3‚ 6q + 4 or 6q + 5. EXERCISE 1.2 1. 2. 3. (i) 2 × 5 × 7 (iv) 5 × 7 × 11 × 13 (i) LCM = 182; HCF = 13 (i) LCM = 420; HCF = 3 2 (ii) 22 × 3 × 13 (v) 17 × 19 × 23 (ii) LCM = 23460;
Premium Orders of magnitude Integer
in the above equations -12+13=23 02+13=13 S = {2/3‚ 1/3‚ 5/3‚ 17/3‚ 50/3} 22+13=53 42+13=173 72+13=503 “Q8: Construct a proof that a) If m is odd‚ then m^2 is odd b) for all integers m and n‚ if m is even and n is odd‚ the m+n is odd” a) If m is odd‚ then m=2k + 1 We have to prove that m^2
Premium Integer Function Reasoning
Diminishing returns From Wikipedia‚ the free encyclopedia Jump to: navigation‚ search In economics‚ diminishing returns (also called diminishing marginal returns) refers to how the marginal production of a factor of production starts to progressively decrease as the factor is increased‚ in contrast to the increase that would otherwise be normally expected. According to this relationship‚ in a production system with fixed and variable inputs (say factory size and labor)‚ each additional unit of
Premium Real number Addition Economics
of Euclid’s Lemma 13.3. The Lindemann-Zermelo Inductive Proof of FTA References 1 4 5 6 7 10 11 13 14 16 20 21 23 23 24 25 25 1. Introduction Principle of Mathematical Induction for sets Let S be a subset of the positive integers. Suppose that: (i) 1 ∈ S‚ and (ii) ∀ n ∈ Z+ ‚ n ∈ S =⇒ n + 1 ∈ S. Then S = Z+ . The intuitive justification is as follows: by (i)‚ we know that 1 ∈ S. Now apply (ii) with n = 1: since 1 ∈ S‚ we deduce 1 + 1 = 2 ∈ S. Now apply (ii) with n = 2:
Premium Natural number Prime number Integer
CLASS VII CBSE-i Introduction to Rational Numbers nt’s Section Stude Shiksha Kendra‚ 2‚ Community Centre‚ Preet Vihar‚Delhi-110 092 India UNIT-3 CLASS VII UNIT-3 CBSE-i Mathematics Introduction to Rational Numbers Shiksha Kendra‚ 2‚ Community Centre‚ Preet Vihar‚Delhi-110 092 India The CBSE-International is grateful for permission to reproduce and/or translate copyright material used in this publication. The acknowledgements have been included
Premium Number Elementary arithmetic Integer
Quadratic Equation For the equation ax2 + bx + c = 0‚ –b ± √ b2 – 4ac x = –––––––––––––– . 2a Binomial Theorem (a + b)n = an + (n) a 1 n – 1b + (n) a 2 n – 2 b2 +…+ ( nr ) a n – r br + … + b n‚ where n is a positive integer and –––––––– ( nr ) = (n –n! r! . r)! 2. TRIGONOMETRY Identities sin2 A + cos2 A = 1. sec2 A = 1 + tan2 A. cosec2 A = 1 + cot2 A. Formulae for ∆ ABC c b a –––– = –––– = –––– . sin A sin B sin C a2 = b 2 + c2 – 2bc cos A. 1 ∆ = – bc sin A
Premium Mathematics Real number Integer
Chapter 2 1. When data cannot be changed after a class is compiled‚ the data is A. constant 2. Which of the following is not a primitive data type in Java? D. sector 3. Which of the following element is not required in variable declaration? C. an assigned vale 4. The assignment operator in Java is A. = 5. Assuming you have declared shoeSize to be a variable of type int‚ which of the following is a valid assignment statement in Java? A. shoeSize = 9; 6. Which of the following data
Premium Equals sign Data type
Let’s Discuss (p. 1.30) The solution obtained by using the factor method is the exact value of the root. However‚ the solution obtained by using the graphical method is an approximation only. Classwork Classwork (p. 1.8) (a) Integer (b) Natural number (c) Negative integer (d) Terminating decimal (e) Recurring decimal (f) Fraction (g) Irrational number Classwork (p. 1.11) 1. (a) When x = 3‚ L.H.S. == 0 R.H.S. = 0 Since L.H.S. = R.H.S.‚ 3 is a root of the equation. (b) When x = 6
Premium Real number Quadratic equation Integer
OLYMPIC TIN HỌC HUFLIT 2014 BÀI 11 (NC) UNION-FIND DISJOINT SETS THEORY In computing‚ a disjoint-set data structure‚ also called a union–find data structure or merge–find set‚ is a data structure that keeps track of a set of elements partitioned into a number of disjoint (nonoverlapping) subsets. It supports two useful operations: o Find: Determine which subset a particular element is in. Find typically returns an item from this set that serves as its "representative"; by comparing the
Premium Natural number Integer Output