Star Sums: Give (one word answer) and Take (one mark in future): 1. If is a rational number, what is the condition on q so that the decimal representation of is terminating. 2. Find the (H.C.F X L.C.M) for the numbers 100 and 190 3. State whether the number ( - )( + ) is rational or irrational justify. 4. Write one rational and one irrational number lying between 0.25 and 0.32. 5. Express 107 in the form of 4q+3 for some positive integer. 6. Write whether the rational number will have a terminating decimal expansion or a non terminating repeating decimal expansion. 7. State the fundamental theorem of arithmetic. 8. Express 2658 as a product of its prime factors. 9. The decimal expansion of will terminate after how many places of decimals 10. If p,q are two consecutive natural numbers, then HCF (p,q) is 11. How many prime factors are there in prime factorization of 5005? 12. If p,q are two prime numbers, then LCM (p,q) is --------- 13. A rational number can be expressed as a terminating decimal if the denominator has factors 14. Any one of the number a(a+2) and (a+4) is a multiple of 15. If HCF and LCM of two numbers are 4 and 9696, then the product of two numbers is 16. The largest which divides 70 and 125, leaving reminders 5 and 8, respectively,is 17. The product of non-zero rational and irrational number is ------------- 18. Which of the following numbers has terminating decimal expansion? 19. The [HCF LCM] for the numbers 50 and 20 is ------- 20. Euclid s division states that if a and b are two positive integers, then there exist unique integers q and r such that ------------- I. Do with care ( Each one catches Two marks): 1. Use Euclid s division algorithm to find the HCF of 1288 and 575. 2. Check whether 5 x 3 x 11 + 11 and 5 x 7 + 7 x 3 + 3 are composite number and justify. 3. Given that LCM (26,169) = 338, Write HCF (26,169). 4. Find the HCF and LCM of 6, 72 and 120 using the prime factorization method. 5. Using Euclid s algorithm, find the HCF of 960 and 1575. 6. By using prime factorization, Find the HCF and LCM of 24,36,40 7. By using prime factorization, find the HCF and LCM of 21, 28, 36 and 45.
8. The HCF of two numbers is 23 and their LCM is 1449. If one of the numbers is 161, find the other. 9. Find the largest number which divides 378 and 510 leaving remainder 6 in each case. 10. Find the simplest form of II. Solve with care ( Each one catches Three marks): 1. Show that 5 + 3 is an irrational number. 2. Show that square of an odd positive integer is of the form 8m+1, for some integer m. 3. Find the largest positive integer that will divide 122,150 and 115 leaving remainder 5, 7 and 11 respectively. 4. Find the LCM and HCF of 336 and 54 and verify that HCF LCM = Product of two numbers. 5. Show that one and only one out of n, (n+2) and (n+4) is divisible by 3, where n is any positive integer. VI. Enrich your knowledge by solving this: 1. Three pieces of timber 42m, 49m and 63m long have to be divided into planks of the same length. What is the greatest possible length of each plank? 2. Find the greatest possible length which can be used to measure exactly the lengths 7m, 3m85cm and 12m95cm. 3. Find the least number of square tiles required to pave the ceiling of the room 15m17cm long and 9m2cm broad. 4. Find the maximum number of students among whom 1001 pens and 910 pencils can be distributed in such a way that each student gets the same number of pens and the same number of pencils. 5. Three measuring rods are 64cm, 80cm and 96cm in length. Find the least length of cloth that can be measured an exact number of times, using any of the rods. 6. The traffic lights at three different road crossing change after every 48seconds and 108seconds respectively. If they all change simultaneously at 8 hours, then at what time will they again change simultaneously? 7. An electronic device makes a beep after every 60 seconds. Another device makes a beep after every 62seconds. They beeped together at 10a.m. At what time will they beep together at the earliest? 8. Six bells commence tolling together and toll at intervals of 2, 4, 6,8,10 and 12 minutes respectively. In 30 hours, how many times do they toll together? 9. Nitsha and Atsha drive around a circular sports field. Nitsha takes 16min to take one round, while Atsha completes the round in 20 min. If both start at the same time and go in the same direction, after how much time will they meet at the starting point?
Pair of Linear Equations in two variables I. Word problems on simultaneous linear equations. (from past papers) 1. The sum of the digits of a two-digit number is 12. The number obtained by interchanging its digits exceeds the given number by 18. Find the number. (CBSE 2006) 2. The sum of a two-digit number and the number obtained by reversing the order of its digits is 99. If the digits differ by3, find the number. (CBSE 2002) 3. A two-digit number is such that the product of its digit is 14. If 45 is added to the number, the digit interchange their places. Find the number. (CBSE 2005) 4. The sum of two number is 8 and the sum of their reciprocal is. Find the numbers. (CBSE 2009)
5. The difference of two numbers is 4 and the difference of their reciprocal is. Find the numbers. (CBSE 2008) 6. Five years ago, Ajith was thrice as old as Vijay and ten years later Ajith shall be twice as old as Vijay. What are the present ages? (CBSE 2002) 7. A fraction becomes, if 2 is added to both of its numerator and denominator. If 3 is added to bith of its numerator and denominator, then it becomes. Find the fraction. (CBSE 2009) 8. Ten years hence, a man s age will be twice the age of his son. Ten years ago, the man was four times as old as his son. Find their present ages. (CBSE 2003) 9. The monthly incomes of A and B are in the ratio of 5:4 and their monthly expenditures are in the ratio of 7:5. If each saves Rs.3000 per month, find the monthly income of each. (CBSE 2006) 10. Find the four angles of a cyclic quadrilateral ABCD in which = (x+y+10), = (y+20), = (x+y-30) and = (x+y) (CBSE 2005) 11. Draw the graph of the equations 4x-5y+16=0 and 2x+y-6=0. Determine the vertices of the triangle formed by these lines and the x-axis. 12. Solve the following system of linear equations graphically; 4x-5y-20=0 and 3x+5y-15=0. Determine the vertices of the triangle formed by the lines representing the above equations and the y axis. 13. Solve the following system of equations graphically: 3x+2y-11 = 0 and 2x-3y+10=0. Shade the region bounded by these lines and the x-axis. 14. Solve the following system of linear equations graphically: 3x+y-11=0, x-y-1 = 0. Shade the region bounded by these lines and the y-axis. Find the coordinates of the points where the graph lines cut the y-axis. 15. Check graphically whether the pair of equations 3x+5y=15 and x-y=5 is consistent. Also, find the coordinates of the points, where the graphs of the equations meet the y-axis.
Test (Real number) Section A (Each one carries one mark) 5 x 1 = 5 1. 3.24636363... is: (a) a terminating decimal number (b) a non-terminating repeating decimal number (c) a rational number (d) both (b) and (c) 2. For some integer q, every odd integer is of the form : (a) 2q (b) 2q + 1 (c) q (d) q + 1 3.If HCF of two numbers is 1, the two number are called relatively --------- or ------- (a) prime, co-prime (b) composite, prime (c) both a and b (d) none of these 4.Express 5005 as a product of its prime factors. 5. Find the LCM and HCF of 24,36 and 72 by prime factorization method. Section B (Each one carries one and half marks) 6 x 1 ½ = 9 6. Use Euclid s division algorithm to find the HCF of 196 and 38220 7. Find the H.C.F of k,2k,3k,4k,5k, where k is any positive integer. 8. Explain why 7 11 13 + 13 and 7 6 5 4 3 2 1 + 5 are composite numbers. 9. Prove that product of three consecutive positive integers is divisible by 24,15 and 36. 10. Show that 5-3 is irrational. 11. Check whether 6 n can end with the digit 0, for any natural number n. Section C (Each one carries two marks) 3 x 2 = 6 12. Prove that one of every three consecutive positive integers is divisible by 3. 13. Prove that is not a rational number, if n is not perfect square. 14. Use Euclid division lemma to show that cube of any positive integer is either of the form 9m, 9m + 1, or 9m + 8 OR, If d is the HCF of 45 and 27, find x & y satisfying d=27x +45y
Worksheet 1 (Real number) I. Complete the chart: REAL NUMBERS Rational Non terminating and repeating with examples ---------------------------- ---------------------------- -- Integers Whole Numbers Natural numbers Examples Examples Examples Examples 2/5, 27/30
II. Puzzle it out: Across 1. Number which has only two factors, 1 and number itself 3 4. Prince of mathematician 1 5. Gave division algorithm 6. A number which is neither prime nor composite. 2 4 7 Down 5 2. Any number which corresponds to a length on a number 6 line 3. Sum of rational number and irrational numbers 7. Short form of the greatest common divisor.
I. Use all your Muscles to solve these Puzzles: Across Worksheet 1 (Real number) 4. Fundamental theorem of ----- states that every composite number can be uniquely expressed as a product of primes, apart from the order of factors. 7. The ------------ factorization of composite numbers is unique. 10. --------------- numbers have either terminating or non-terminating repeating decimal expansion. Down: 1.------------------- is a sequence of well defined steps to solve any problem. 2.Numbers having non-terminating, non-repeating decimal expansion are known as ----------- 3.A proven statement used as a stepping stone towards the proof of another statement is known as --------- 5. Decimal expansion of 3/35 is ----------- 6. The ------ expansion of rational numbers is terminating if the denominator has 2 and 5 as its only factors. 8. --------- division algorithm is used to find the HCF of two positive numbers. 9. For any two numbers, HCF X LCM = ---------- of numbers. 3 5 4 6 9 7 8 10 Department of Mathematics (SSV, Karur) Page 1
Worksheet (Real number) I. First 20:20 Match, Only u can win this cup: 1. Using Euclid s division algorithm, find which of the following pairs of numbers are co-prime: (i) 231, 396 (ii) 847, 2160 2. Using Euclid s division algorithm to find HCF of 441,567,693 3. Show that the square of any positive integer cannot be of the form 5m+2 or 5m+3 for some integer m. 4. For any positive integer n, prove that n 3 n is divisible by 6. 5. Prove that if both x and y are positive odd integers, then x 2 + y 2 is an even integer but not divisible by 4. 6. Prove that one and only one out of n, n+2 and n+4 is divisible by 3, when n is any positive integer. 7. Prove that + is an irrational number. 8. Prove that + is an irrational, where p,q are primes. 9. There is a circular path around a sports field. Naveena takes 18 minutes to run one round of the field, while Dhanya takes 12 minutes for the same. Suppose they start at the same time and run in the same direction. After how many minutes will they meet again at the starting point? What values are depicted in the question? 10. Show that the cube of any positive integer is of the form 4m,4m+1 or 4m+3 for some integer m. Department of Mathematics (SSV, Karur) Page 1
Worksheet 4 (Real number) I. Treat this, as ur special guest: (V.I.Problems) 1. Using Euclid's division algorithm prove that : 847,2160 are co-primes or relatively prime. 2. Prove that one of any three consecutive positive integers must be divisible by 3. 3. Prove that product of any three consecutive positive integers is divisible by 6. (@) Prove that for any positive integer n, n 3 -n is divisible by 6. 4. In a group there are 21 children, 35 ladies and 49 gents. They all want to stay in same hotel. Find minimum number of rooms required for their stay so that in each room equal number of members stay also with the condition that children, female and male stay in separate rooms. 5. Check whether expansion of (252) n, n N can end with zero for any n as natural number. 6. Check whether expansion of (75) n, n N can end with zero for any n as natural number. 7. Prove that is an irrational number by contradiction method. 8. Show that is an irrational number. (Without using contradiction method) 9. Without actually performing the long division, state whether the has a terminating decimal expansion or non-terminating recurring decimal expansion. 10. Prove that is not a rational number, if n is not a perfect square.
Worksheet (Real number) I. Play with this Clay: 1. A number when divided by 53 gives 34 as quotient and 21 as remainder, Find the number. 2. If n is an odd positive integer, show that (n 2-1) is divisible by 8. 3. Show that every positive odd integers is of the form (6q+1) or (6q+3) or (6q+5) for some integer q. 4. Show that one and only one out of n, n+2, n+4 is divisible by 3, where n is any positive integer. 5. Find the largest number which divides 248 and 1032 leaving reminder 8 in each case. 6. Find the largest number which divides 546 and 764 leaving reminder 6 and 8 respectively. 7. Two tanks contain 504 and 735 liters of milk respectively. Find the maximum capacity of a container which can measure the milk of either tank an exact number of times. 8. Without actual division show that each of the following rational numbers is a non-terminating repeating decimal. (i) (ii) (iii) (iv) 9. The decimal expansion of the rational number, will terminate after how many places of decimals? 10. Express each of the following as a rational number in simplest form. (i) 0.8 (ii) 0.16 (iii) 0.365 11. If p is a prime number, then prove that is irrational. 12. Write a rational number between and. (13).Prove that 3 is an irrational number. 13. Give an example of two irrationals whose sum is rational. 14. State whether the given statement is true or false: The sum of two rationals is always rational. The product of two rationals is always rational. The sum of two irrationals is an irrational. The product of two irrationals is an irrational. The sum of a rational and an irrational is irrational. The product of a rational and an irrational is irrational.