Tenth Bit Bank Mathematics Real Numbers

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1 Tenth Bit Bank Mathematics Real Numbers 1. The rational number among the following is... i) 4.28 ii) iii) A) i) & ii) B) ii) & iii) C) i) & iii) D) All the above 2. A rational number can be in the form of... A) Terminating decimal B) Non - terminating recurring decimal C) Either A or B D) Neither A nor B 3. Which of the following is not correct? i) N Z Q ii) N W Q iii) N W Q iv) N Q Q A) i) & ii) B) ii) & iii) C) ii) & iv) D) i) & iii) 4. The sum of two Irrational numbers is... A) always Irrational number B) need not be an Irrational number C) always Rational number D) None of the above 5. The correct statement among the following is... i) The sum of two irrational numbers need not be Irrational. ii) The sum of two irrational numbers is always Irrational. iii) The product of two irrational numbers need not be Irrational. iv) The product of two irrational numbers is always Irrational. A) i) & iii) B) ii) & iv) C) i) & iv) D) ii) & iii) 6. The logarithmic form of 5 3 = 125 is... A) log 125 = 5 3 B) log 125 = 3 5 C) log 3 = D) log 5 = Exponential form of log 81 = 4 is... 3 A) 3 81 = 4 B) 3 4 = 81 C) 4 3 = 81 D) 4 81 = 3 8. log a 2 + log a a a 3 = A) 8 B) 9 C) 5 D) 6 9. log mn = a A) log a m log a n B) log a m + log a n C) log a m log a n D) log a m n 10. The rational number among the following is... i) log 1 a ii) log a a iii) log a 2 A) i) & ii) B) ii) & iii) C) i) & iii) D) All the above log log 3 log 7 = 288 A) log 7 B) log 288 log 7 C) log 32 + log 9 log 7 D) All the above 12. log a 3 a = A) 1 B) 2 C) 1 D) For any natural number 'n' 6 n always ends with... A) 0 B) 6 C) 36 D) 1 R

2 14. The relation between two positive integers a, b and their LCM and HCF is... A) a + b = LCM (a, b) + HCF (a, b) B) a b = LCM (a, b) HCF (a, b) C) a b = LCM (a, b) HCF (a, b) a LCM (a, b) D) = b HCF (a, b) 15. The rational number whose decimal expansion terminates can be expressed in p/q form where p and q are coprimes and the prime factorization of q is of the form... where n, m are non negative integers. A) 2 n 3 m B) 3 n 5 m C) 2 n 5 m D) 2 n 7 m 16. If p be a prime number. If p divides a 2 (where a is a positive integers) then... A) p divides a B) a divides p C) Neither A nor B D) Both A and B 17. p + q is Irrational if... i) p is prime and q is a perfect square number. ii) q is prime and p is a perfect square number. iii) p and q are primes. iv) p and q are perfect square numbers. A) i) B) i) & ii C) i), ii) & iii) D) i), ii), iii) & iv) 18. a + (b + c) = (a + b) + c when a, b, c are real numbers. The property that holds here under addition is... A) Closure B) Commutative C) Associative D) Identity Sets 19. If A = {a, e, i, o, u} which of the following are not correct? i) e A ii) c A iii) i A iv) t A A) i) & ii) B) ii) & iii) C) iii) & iv) D) i) & iv) 20. The set-builder form of the set A = {2, 4, 8, 16}... i) {x / x = 2y, y N, y < 5} ii) {x / x is a factor of 16 other than 1} iii) {x / x = 2 y, y N, y < 5} iv) {x / x is a factor of 16} A) Only i) B) Only iii) C) i) & iii) D) ii) & iii) 21. The set from the following collections. i) P = {x / x is a good book in the school library} ii) Q = {x / x is a prime number less than 100} iii) R = {x / x is a whole number less than 0} iv) S = {x / x is an intelligent student of class X} A) i) & ii) B) ii) & iii) C) iii) & iv) D) i) & iv) 22. If A = {a, b, c}, B = {c, d, e}, C = {d, e}, then the correct statement from the following is... i) A B ii) B C iii) C B A) i) & ii) B) ii) & iii) C) Only ii) D) Only iii) 23. Sneha has written symbol for null set as 'φ'. Shriya has written it as { } and Ridhi has written it as {φ}. Who is correct in representing the null set? A) Sneha & Shriya B) Shriya & Ridhi C) Sneha & Ridhi D) Sneha, Shriya & Ridhi

3 24. Which of the following statements are true? A) The cardinal number of the set of even prime numbers is 1. B) The cardinal number of a null set is 1. C) The cardinal number of an infinite set is a natural number. D) All the above 25. The equal sets among the following. i) A = {2, 3, 5, 7} ii) B = {x / x is a prime number less than 8} iii) C = {x / x is a prime number less than 10} A) i) & ii) B) i), ii) & iii) C) ii) & iii) D) i) & iii) 26. If A and B are not equal sets then which of the following are always disjoint sets? i) A B & B A ii) A B & B iii) A B & A A) i) & ii) B) ii) & iii) C) i) & iii) D) All the above 27. Which of the following is not true? i) A B = B A ii) A B = B A iii) A B = B A A) i) B) ii) C) iii) D) i), ii) & iii) 28. If A = {x / x is a player in Indian hockey team for olympics}, B = {x / x is a player in Indian cricket team selected to play World Cup 2015 matches} The way of representing the sets A, B through Venn diagram is... A B µ B µ A µ µ A A) B) A C) B D) B 29. If A B = A, then the sets A and B are represented using the Venn diagram... B µ A µ A A) A B) B C) 30. If P and Q are two sets. Which of the following are not correct? A) P Q = {x / x P or x Q} B) P Q = {x / x P and x Q} C) P Q = {x / x P and x Q} D) P Q = {x / x P or x Q} 31. The number of subsets of the set A = {p, q, r}... A) 2 B) 3 C) 6 D) If n(a) = 12, n(b) = 8, n( A B) = 5 then n(a B) =? A) 20 B) 15 C) 10 D) 5 Polynomials A B µ 33. The quadratic polynomial among the following is... i) 2x ii) 0x 2 + 5x 8 iii) 2x 2 + 3x + 5 iv) 3x 2 5x 7 A) i) & ii) B) ii) & iii) C) iii) & i) D) iv) & i) The degree of the polynomial x 5 + x 4 x 3 + is A) B) 5 C) D) µ B D)

4 35. If the quadratic polynomial ax 2 + bx + c (a 0) intersects X - axis at two distinct points then the number of its zeroes is... A) one B) two C) three D) None 36. The sum of the zeroes of the polynomial px 2 + qx + r (p 0) is... p q r r A) q B) p C) q D) p 37. The value of k if the product of zeroes of the polynomial kx 2 + 5x + 6 is 3... A) 1 B) 2 C) 3 D) The sum of the zeroes of the cubic polynomial ax 3 + bx 2 + cx + d (a 0) is... b c d a A) a B) a C) a D) b 39. When a polynomial 2x 2 + 3x + 1 is divided by x + 2 then the quotient q(x) and remainder r(x) are respectively... A) 2x 1, 3 B) 2x + 1, 3 C) 2x 1, 3 D) 2x + 1, The graph of y = cx +d where c, d are real numbers and c 0 intersects X - axis at... d d d d c c c c A) (0, 0) B) ( 0, ) C) (, 0 ) D) (, ) ANSWERS 1-D; 2-C; 3-C; 4-B; 5-A; 6-B; 7-B; 8-C; 9-B; 10-A; 11-D; 12-D; 13-B; 14-C; 15-C; 16-A; 17-C; 18-C; 19-B; 20-D; 21-B; 22-D; 23-A; 24-A; 25-B; 26-A; 27-C; 28-D; 29-C; 30-D; 31-D; 32-B; 33-C; 34-B; 35-B; 36-B; 37-B; 38-A; 39-A; 40-C. Writer: V. Padma Priya

5 Tenth Maths Real Numbers 2 p 1. Sugun said, 2 can be written as which is in the form of where q 0 1 q and hence 2 is a rational number". Do you agree? Why / Why not? Hint: A number which can be written in the form of p/q where q 0 and both 'p' and 'q' are integers is defined as a rational number. Check whether 'p' in the above case is an integer and write the solution giving your reasons. 2. Observe the following diagram. What are the mistakes you have observed in the diagram? Draw the diagram correctly. Hint: Recall the subsets of Real Numbers in the correct order. N W Z Q R and Q' R Q and Q' are disjoint sets. Draw the diagram correctly considering the above order of subsets. 3. Nazera has considered the following three cases. R (i) 2 and 3 are two irrational numbers, their sum is also an Irrational number. Irrational Whole numbers numbers Integers Natural numbers (N) (z) (W) (Q') Rational Numbers (Q) µ

6 (ii) 2 + 3, and 5 are Irrational numbers, their sum is also an Irrational number. (iii) 11 and 5 6 are irrational numbers, their sum is also an Irrational number. Based on the above statements Nazera has concluded that "The sum of two Irrational numbers is always an Irrational number". Do you agree with Nazera's conclusion? Why/ Why not? Hint: In all the above three cases it has been observed that the sum of two Irrational numbers is an Irrational number. To write your conclusion think of a possibility where two irrational numbers addition turns to be a rational number. Try to find the sum of and 2 3, and 5 6 etc., 4. The L.C.M. and H.C.F. of two positive numbers a, b is the sum of the elements in the sets A B and A B respectively. Where A = {1, 2, 3, 4, 5, 6, 7, 8} and B = {1, 2, 3}. Find the product of the numbers. Hint: In this problem the L.C.M. and H.C.F. of 'a' and 'b' are connected with the sum of the elements in A B and A B respectively. Find A B and the sum of its elements, similarly find A B and the sum of its elements. a b = L.C.M. (a, b) H.C.F. (a, b) You can observe that L.C.M. (a, b) = 36 and H.C.F. (a, b) = 6 using the above relation find the product of the numbers. 5. "The product of two Irrational numbers need not be Irrational". Justify your answer. Hint: We have discussed in the third question that the sum of two Irrational numbers need not be Irrational by writing a counter example for Nazera's statement. Now to justify the given statement, try to find two Irrational numbers whose product is rational. e.g.: Product of 3 and 27, and 5 3 etc.

7 6. Write 1000 in exponential form, express it in the logarithm form and expand log 1000 using the laws of logarithm. Hint: Write 1000 in exponential form. "If a n = x then log a x = n where 'a' and 'x' are positive numbers and a 1". After writing in logarithmic form expand using the following laws of logarithm. (1) log a xy = log a x + log a y (2) log a x m = m log a x In a Maths class, teacher has given four rational numbers,, and The task is to pick the rational numbers whose decimal expansion 16 1 terminates. Without actually dividing Samreen could find that and 25 8 are the rational numbers whose decimal expansion terminates. How could Samreen find the numbers without actual division? Explain. Hint: Samreen has used the statement of the theorem "Let x = p/q be a rational number, such that the prime factorization of q is of the form 2 n 5 m where n, m are non-negative integers. Then x has a decimal expansion which terminates" Try to show that the denominators in and can be written in 2 n 5 m 25 8 form. Multiple Choice Questions 1. The rational number among the following... A) B) C) D) Ans: C 2. The logarithmic form of 125 = 5 3 is... A) log = 3 B) log = 5 C) log 5 3 = 125 D) log 3 5 = 125 Ans: A

8 3. 2 log3 + 3 log 5 5 log 2 = 1125 (i) log (ii) log 1125 log (iii) log 9 + log 125 log 32 A) (i) Only B) (i) & (ii) Only C) (ii) & (iii) Only D) (i), (ii) & (iii) Ans: D Writer: V. Padma Priya

9 Tenth Maths Download Part of the Chapter Synopsis Real Numbers The set of rational and irrational numbers together are called Real Numbers. A rational number can be a terminating decimal or non-terminating recurring decimal. An irrational number's decimal expansion is non-terminating, non-recurring. All the real numbers (rational and irrational) can be represented on the number line. Fundamental Theorem of Arithmetic states that "Every composite number can be expressed as the product of primes which is unique, apart from the order in which the prime factors occur". Least Common Multiple (L.C.M.) Product of the greatest power of each prime factors, in the numbers represented as the product of primes. Highest Common Factor (H.C.F.) Product of the smallest power of each cammon prime factors in the numbers represented as the product of primes. The product of two positive numbers 'a' and 'b' is equal to the product of their L.C.M. and H.C.F. i.e., H.C.F. (a, b) L.C.M. (a, b) = a b. 4 n never ends with zero for any natural number 'n'. 12 n never ends with '0' or '5' for any natural number 'n'. 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 factorization of q is of the form 2 n 5 m where n, m are non-negative integers. Converse of the above statement also holds good. i.e., let x = p q be a rational

10 number, such that the prime factorization of q is of the form 2 n 5 m where n, m are non-negative integers. Then x has a decimal expansion that terminates. Let x = p q be a rational number, such that the prime factorization of 'q' is not of the form 2 n 5 m, where n, m are non-negative integers. then x as a decimal expansion which is non-terminating reapeating (recurring). An irrational number cannot be written in the form p/q where p and q are integers and q 0. p is irrational, where p is a prime. Let p be prime number. If p divides a 2 (where 'a' is a positive integer) then p divides 'a'. To prove a number as an irrational number we are using the method of contradiction. The sum, difference, product and quotient of two rational numbers is always a rational number. The sum of two irrational numbers need not be irrational. Similarly, the product of two irrational numbers need not be irrational. If a n = x where a 1 and 'a' and 'x' are positive numbers. then its the logarithmic form is written as log x a = n and is read as "log of 'x' to the base a is equal to n". Laws of Logarithms I Law log a xy = log a x + log a y Where x, y and a are positive numbers greater than zero and a 1. II Law x log a y = log a x - log a y Where x, y and a are positive numbers > 0 and a 1. III Law log a x m = m log a x Where a and x are positive numbers greater than zero and a 1.

11 Expansion of Logarithms using the Laws of Logarithms. Writing the Expansion as a single Logarithms by applying the Laws of logarithms. log a a = 1, a > 0 log a 1 = 0 Logarithms are used for all sorts of calculations in Engineering, Science, Business and Economics. Questions for Practice 1. Which of the following numbers are terminating and which are not? Are these Rational Numbers? Why/Why not? 2 1 i) ii) Find a rational number between and Represent - on the number line State Fundamental Theorem of Arithmetic. 5. Express 5005 as the product of its prime factors. 6. Find the L.C.M. and H.C.F. of 306 and 657 by Prime Factorization Method. 7. How do you relate the product of two positive numbers with their L.C.M. and H.C.F.? 8. Check whether 8 n can end with digit zero for any natural number 'n'. 9. Without actually performing division, How can you check whether the given rational number in p/q form is terminating or non-terminating repeating decimal? 10. Which of the following decimal form of real numbers are irrational? i) ii) How do you explain the following notation N W Z Q. 12. Is it correct to say that Q Q'? Why/Why not. 13. 'Rational and Irrational numbers sets are disjoint'. Justify your answer. 14. Show that 3 2 is Irrational. 15. How do you disprove the follownig statements?

12 (i) The sum of two Irrational numbers is always Irrational. (ii) The product of two Irrational numbers is always Irrational. 16. Write the following in logarithmic form (i) 2 10 = 1024 (ii) (0.1) 3 = Express the following in Exponential form (i) log = 2 (ii) log = Determine the value of log and log Write 3 log log 7-5 log 2 as a single logarithm. 20. x 5 y 2 z 3 Expand log where x, y, z, p, r are positive numbers. p 2 r Write the Laws of Logarithms. Formative Assessment A Student has to frame questions and has to find solution for it. The students will be awarded 5 marks for doing so. 1 For framing questions - 2 marks 2 1 For solving them - 2 marks 2 It is suggested that there is a scope of framing questions in the following topics from the chapter Real Numbers. Take any four rational numbers and try to find out whether they are terminating or non-terminating by actual division. Try to insert four rational numbers between any two rational numbers by mean method which you have learnt in IX class. Take any four rational numbers and try to represent them on the number line. Find the prime factors of a number in as many ways as possible and try to prove the uniqueness of Fundamental Theorem of Arithmetic. Find the L.C.M. and H.C.F. of any two positive integers and try to establish the relation between the numbers with their L.C.M. & H.C.F. Take any two rational numbers and try to write the denominator is 2 n 5 m form.

13 Take any four rational numbers and try to decide whether they are terminating or non-terminating recurring or non-terminating non-recurring by writing 'q' is the form of 2 n 5 m. Take any two irrational numbers of the form a ± b and try to prove them as irrational by the method of contradiction. Try to check the properties of real numbers by considering different sets of numbers. Try to convert Exponential form to logarithm form and vice-versa. Try to verify the Laws of Logarithms for different values of a, x and y. Convert the expanded form of the logarithm to a single logarithm. Try to write the single logarithm in the expanded form. Writer: V. Padmapriya

Tenth Maths Polynomials

Tenth Maths Polynomials Polynomials are algebraic expressions constructed using constants and variables. Coefficients operate on variables, which can be raised to various powers of non-negative integer