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1 Number System CHAPTER We are Starting from a Point but want to Make it a Circle of Infinite Radius We use numbers in our daily life for counting, measuring, etc. Let us recall the types of numbers. Natural Numbers: Counting numbers are called Natural numbers, e.g., 1, 2, 3, 4, ... are Natural Numbers. It is denoted by N. There are infinite natural numbers and the smallest natural number is 1. Whole Numbers: The natural numbers along with zero form the system of whole numbers, e.g., 0, 1, 2, 3, 4, … . It is denoted W. There is no largest whole number and the smallest whole number is 0. Integers: The number system consisting of natural numbers, their negative and zero is called integers, e.g., , … - 2, - 1, 0, 1, 2, 3, … + . It is denoted by Z or I. The smallest and the largest integers cannot be determined. Even numbers: Natural numbers which are divisible by 2 are called even numbers, e.g., 2, 4, 6, 8 … . Smallest even number is 2. There is no largest even number. Odd Numbers: Natural numbers which are not divisible by 2 are called odd number. e.g, 1, 3, 5, 7, … . Smallest odd number is 1. There is no largest odd number Prime Numbers: Natural numbers which have exactly two factors, i.e., 1 and the number itself are called prime numbers, e.g., 2, 3, 5, 7, ... . Rational numbers: If a number can be expressed in the form of p , where q 0 , where p and q are integers, then the q number is called rational number. It is denoted by Q. For Example, 9 16 8 27 , , , etc. 25 7 1 51 Irrational Numbers: The numbers which cannot be expressed in the form of p , where p and q are integers and q 0 , is q called irrational number. It is denoted by I. For example, irrational numbers? 2, 3 , 5 ,2 3 , 3 5 , 3 3 are Important Point: Every positive irrational number has a negative irrational number corresponding to it. 011-26925013/14 NTSE, NSO +91-9811134008 Diploma, XI Entrance +91-9582231489 1 2 3 5, 5 3 2, 3 2 3 2 6, 6 2 6 3 2 Sometimes, product of two irrational numbers is a rational number For example: 2 2 (i) 2 2 2 (ii) 2 3 2 3 (2) 2 3 2 43 1 The Number Line: The number line is a straight line between negative infinity on the left to positive infinity on the right. On number line integers are placed at a equal distance. ... -4 -3 -2 -1 0 1 2 3 4... Real Numbers: A set of rational and irrational numbers is called the set of real numbers. It is denoted by R. Therefore, Real numbers = Rational numbers + Irrational numbers. In other words, all numbers that can be represented on the number line are called real numbers. R+: Positive real numbers and R–: Negative real numbers. TYPES OF NUMBERS Real numbers (R) Rational numbers (Q) Imaginary numbers Z (a ib), a, b R Irrational numbers (T) The number which cannot be Expressed as Natural numbers(N) Whole numbers (W) Integer(I) Negative Integers Positive Integers Even Odd number number Prime numbers Composite numbers {- 1, - 2, - 3, - 4…} {1, 2, 3, 4…} Important Points: 1 is neither prime nor composite. 1 is an odd integer. 0 is neither positive nor negative. 2 is smallest even prime number. All prime numbers (except 2) are odd. 011-26925013/14 +91-9811134008 +91-9582231489 form NTSE, NSO Diploma, XI Entrance 2 Non-negative Integers {0, 1, 2, 3,…} Non-positive Integers {0, - 1, - 2, - 3, …} Fraction: A fraction is a quantity which expresses a part of a whole. Numerator Fraction = Deno min ator Types of Fractions: (a) Proper Fraction: If numerator is less than its denominator, then it is called a proper fraction. 2 6 For example: , . 5 18 (b) Improper Fraction : If numerator is greater than or equal to its denominator, then it is called an improper fraction. For example: (c) 5 18 13 , , . 2 7 13 1 2 5 Mixed fraction: It consists of an integer a proper fraction. For example: 1 , 3 ,7 . 2 3 9 Mixed fraction can always be changed into improper fraction advice versa. For example: 7 (d) 19 9 2 1 1 1 5 7 9 5 63 5 68 9 9 and 2 2 2 2 9 9 9 9 Equivalent fractions / Equal fractions: Fractions with the same value. For example: 2 4 6 8 2 , , , 3 6 9 12 3 Important Points: Value of fraction is not change by multiplying or dividing the numerator or denominator by the same number. For example: (i) 2 2 5 10 5 5 5 25 So, 2 10 5 25 (ii) 36 36 4 9 16 16 4 4 So, 36 9 16 4 If in a fraction, its numerator and denominator are of equal value then fraction is equal to unity i.e. 1. (e) Like fractions: Fraction with same denominators. For example: (f) 2 3 9 11 , , , 7 7 7 7 Unlike Fractions: Fractions with different denominators. For example: 2 4 9 9 , , , 5 7 8 2 Important Point: Unlike fractions can be converted into like fractions. 011-26925013/14 +91-9811134008 +91-9582231489 NTSE, NSO Diploma, XI Entrance 3 For example: 3 4 and 5 7 3 7 21 4 5 20 and 5 7 35 7 5 35 (g) Decimal fraction: The fractions whose denominators are of the powers of 10. For example: (h) 2 9 0.2, 0.09 10 100 Vulgar fraction: Denominators are not the power of 10. For example: 3 9 5 , , 7 2 193 We can also study real numbers by studying classification of decimal expansion. Decimal Expansion of real numbers Terminating Non-Terminating Recurring Pure Recurring Non-recurring Mixed Recurring (a) Terminating (or finite decimal fractions) : For example: 7 21 4.2 0.875 , 5 8 (b) Non-terminating decimal fractions : There are two types of Non-terminating decimal fractions : (i) Non-terminating periodic fractions or non-terminating recurring (repeating) decimal factions : For example: 10 3.333... 3. 3 3 1 0.142857142857... 0.142857 7 (ii) Non-terminating non-periodic fraction or non-terminating non-recurring fractions : For example 15.2731259629 Important point: The decimal expansion of a rational number is either terminating or non-terminating recurring. Moreover, a number whose decimal expansion is terminating or non-terminating recurring is rational. The decimal expansion of an irrational number is non-terminating non recurring. Moreover, a number whose decimal expansion is non-terminating non recurring is irrational. 011-26925013/14 +91-9811134008 +91-9582231489 NTSE, NSO Diploma, XI Entrance 4 2 = 1.41421356237309504880… For example: = 3.1415926535897932384626433… We often take 22 22 as an approximate value of , but . 7 7 Rational Numbers To find out rational numbers between two Rational If r and s be two rational numbers then rs is between r and s. 2 rs Then find the rational number between r and , i.e. 2 r rs 2 . 2 Example 1: Find two rational numbers between 1 and 2. A rational number between 1 and 2 = A rational number between 1 and 1 2 3 2 2 1 3 1 5 5 3 = 1 . 2 2 2 2 2 4 So, two rational numbers between 1 and 2 is 3 5 and . 2 4 Example 2: Write five rational numbers between 3 and 4. 3 3 10 30 10 10 4 4 10 40 \ 10 10 Five rational numbers between 3 and 4 are 31 32 33 34 35 , , , , and . 10 10 10 10 10 Example 3: Find four rational numbers between 2 5 and . 3 3 2 10 20 5 10 50 and 3 10 30 3 10 30 Four rational number between 2 5 21 22 23 24 , , and . and are 30 30 30 30 3 3 011-26925013/14 +91-9811134008 +91-9582231489 NTSE, NSO Diploma, XI Entrance 5 IX ACADEMIC QUESTIONS Subjective Assignment-1 1. Is zero a rational number? If yes, write it in the form of p/q. 2. Find six rational numbers between (i) 3 and 4 (ii) 2 and 3 Find 5 rational numbers between 4. 5. Find five rational numbers between 1 and 2. Find six rational numbers between -1 and 3. 2 5 Find four rational numbers between and . 3 3 3 7 Find six numbers between and 4 4 7. 8. 12. Find nine rational numbers between 0 and 0.1. 1 1 Find three rational numbers lying between and 5 4 2 3 Find five rational numbers lying between and 5 4 1 Find a rational numbers lying between 1and 2 State true or false for the following. Justify your answer (i) Every whole number is a natural number. (ii) Every integer is a rational number. (iii) Every rational number is an integer. (iv) Every integer is a whole number. (v) Every natural number is a whole number. 13. State true or false. Justify by giving example. (i) Every real number is an irrational number. (ii) Every real no. is either rational or irrational (iii) is an irrational number (iv) Irrational numbers cannot be represented by points on the number line. (v) The sum of two rational numbers is rational. (vi) The sum of two irrational numbers is an irrational. 9. 10. 11. (iv) 15 and 16 1 2 and . 2 5 3. 6. (iii) 9 and 10 (vii) The product of two rational numbers is rational. (viii) The product of two irrational numbers is an irrational. (ix) The sum of a rational number and an irrational number is an irrational. (x) The product of a non-zero rational number and an irrational number is a rational number. 011-26925013/14 +91-9811134008 +91-9582231489 NTSE, NSO Diploma, XI Entrance 6 14. Represent 6 on the number line. 15. Represent 7 on the number line. 16. Represent 3 on the number line. CONVERSION OF A RATIONAL NUMBER INTO DECIMAL Terminating Decimals (Finite Decimals) 3 Let us convert into decimal from. 4 3 (Rational number) 0.75 (Terminating decimal) 4 This is a terminating decimal form. Hence, we conclude that terminating decimals are rational numbers, since the process of division terminates (comes to an end). Here, the division stops at a point, where there is no remainder. Non-Terminating and repeating decimals (Recurring Decimals) 2 Let us convert into decimal form 3 2 0.6666.... 0.6 3 0.6666…. is a non-terminating and repeating decimal and is represented as 0.6 . Non terminating and repeating decimals are also known as recurring decimals. 2 is a rational number whose decimal expansion is a non-terminating and repeating decimal (recurring 3 decimal). Q.1 Write the following in decimal form and say what kind of decimal expansion each has: 36 1 1 (i) (NCERT) (ii) (NCERT) (iii) 4 (NCERT) 11 100 8 3 145 2 (iv) (NCERT) (v) (NCERT) (vi) (NCERT) 11 8 13 3 874 (vii) (viii) 40 999 p Conversion of terminating decimals into form q Steps: 1. Assume the given recurring decimal as x. 2. In the given number, check the number of digits which have bar on their heads. 3. If only one digit has bar on its head, multiply the number by 10; if two digits have bar on its head, multiply the number by 100; similarly if three digits have bars on its head, multiply the number by 100; similarly if three digits have bars on its head, multiply the number by 1000 and so on. When 011-26925013/14 +91-9811134008 +91-9582231489 NTSE, NSO Diploma, XI Entrance 7 we multiple the number by 10, the decimal will shift one place to the right i.e., if x = 0.6 then 10x = 6.6 4. Subtract the numbers obtained in step 1 and that obtained in step 3. 5. Simplify the equation obtained in step 4 and calculate x. 6. Write the value of x in simplest form. p Example: 1.8: Express each of the following in form. q (i) 0.7 (ii) 0.32 (iii) 0.459 Solution: (i) Let x = 0.7 Multiplying both sides by 10, we get 10x = 7.7 Subtracting (i) and (ii), we get 10x x 7.7 0.7 9x = 7 7 x 9 7 0.7 9 (ii) Let x = 0.32 Multiplying both sides by 100, we get 100x = 32. 32 99x = 32 32 x= 99 32 0.32 = 99 …(i) …(ii) …(i) …(ii) (iii) Let x 0.459 Multiplying both sides by 1000, we get 1000x = 459. 459 Subtracting (i) and (ii), we get 1000x – x = 459.459 − 0.459 999x = 459 459 51 x= 999 111 51 0.459 111 011-26925013/14 +91-9811134008 +91-9582231489 …(i) …(ii) NTSE, NSO Diploma, XI Entrance 8 IX ACADEMIC QUESTIONS Subjective Assignment-2 Convert the following decimal numbers in the form p . q 1. (i) 0.15 (ii) 0.00026 (iii) 8.0025 2. (i) 0.6 (NCERT) (ii) 0.47 (NCERT) (iii) 0.001 (NCERT) 3. (i) 0.25 (ii) 0.1532 (iii) 2.2612 4. Express (2.23 0.32) in 5. If 10x 0.3 0.2, find the value of x. 6. If p form q x 0.6 0.4, find the value of x. 9 Irrational Number A number which cannot be expressed in the form of p , where p and q are integers and q 0 , is known as q irrational number. Example: 2, 3, 5, 6, 21 are irrational numbers. A non terminating and non-repeating decimal is known as irrational numbers. Examples: 1.4123567……………. 2.101001000…………. 0.121221222…………. All the above decimal numbers are non-terminating and non-repeating decimal and therefore these are irrational numbers. Properties of Irrational Numbers: (i) The sum of two irrational numbers need not be an irrational number. Example: Sum of 3 and 3 3 ( 3) 3 3 0 , which is a rational numbers Sum of (3 5) and (2 5) 3 5 2 5 3 2 5, which is a rational number. (ii) The difference of two irrational numbers need not be an irrational number. 011-26925013/14 +91-9811134008 +91-9582231489 NTSE, NSO Diploma, XI Entrance 9 Example: Difference of (2 6) and 6 2 6 6 2 , which is a rational number. (iii) The product of two irrational numbers need not be an irrational number. Examples: Product of 5 and 5 5 5 5 is a rational number. (iv) The division of two irrational numbers need not be an irrational number. Examples: Division of 3 2 and Division of (v) 8 and 2 2 3 2 2 8 2 = 3, is a rational number. 4 = 2, is a rational number. The sum of a rational and irrational number is irrational number. Examples: Sum of 2 and 3 = (2 3) , which is an irrational number. (vi) The difference of a rational and an irrational number is irrational number. Examples: Difference of 2 and 3 (2 3) , which is an irrational number. (vii) The product of a rational and an irrational number is irrational number. Examples: The product of 2 and 5 2 5 2 5, which is an irrational number. (viii) The division of a rational and an irrational number is irrational numbers. Examples: The division of 5 and 5 5 5 5 , which is an irrational number. Addition and Subtraction of Irrational Numbers: Example: Add (3 2 5 3) and ( 2 3) . Solution: (3 2 5 3) ( 2 3) = (3 2 2) (5 3 3) = (3 1) 2 (5 1) 3 4 2 6 3 011-26925013/14 +91-9811134008 +91-9582231489 NTSE, NSO Diploma, XI Entrance 10 IX ACADEMIC QUESTIONS Subjective Assignment-3 1. Add (4 5 3 7 6) and (2 5 4 6) 2. Add (7 7 4 3) and (4 7 3 3) 3. Add (2 3 5 2) and ( 3 2 2) 4. Add (2 2 5 3 7 5)and (3 3 2 5) 5. 1 3 2 1 7 2 6 11 and 7 2 11 6. 2 2 3 3 7. 27 3. 12 8. 9. 50 98 162 10. 3 a 4 b 3 ab4 12. 4 81 8. 3 216 15. 3 32 225 11. 13. 5 20 45 8 6. 1 2 147 108 3 16 3 54 3 192 3 375 Multiplication of Irrational Numbers To multiply irrational numbers, multiply the rational parts and irrational parts separately. For example: Multiply 2 2 and 5 3 , write (2 × 5) and ( 2 3) separately. 2 2 5 3 (2 5) ( 2 3) 10 6 Division of Irrational Numbers: To divide irrational numbers, divide the rational parts and irrational parts separately. For example: Divide 10 6 and 5 2 , write (10 5) and ( 6 2) 10 6 5 2 10 6 10 6 2 3 5 5 2 2 011-26925013/14 +91-9811134008 +91-9582231489 NTSE, NSO Diploma, XI Entrance 11 IX ACADEMIC QUESTIONS Subjective Assignment-4 1. Multiply (3 3) by (2 2) . 2. Simplify ( 5 2)2 3. Simplify ( 5 2)( 5 2) 4. 2. 3 4 3. 3 16 5. 2.4 3 5. 4 81 6. 7. 3. 3 32 3. 3 4 8. 10 4 6 3 9. 4 43 4 10. 11. 21. 384 8. 96 13. ( 6 7)( 6 7) 5 3 10 12. ( 7 5)( 7 5) 14. 4 28 3 7 Rationalisation of the Denominator of an Irrational number having two terms in the denominator, multiply the numerator and denominator of the number by the conjugate of its denominator. For example: To rationalize 1 , multiply the numerator and denominator by ( a b) . a b 1 1 a b a b a b 2 2 ab a b a b a b ( a ) ( b) Example: Rationalise the denominator of 1 . 3 2 Solution: Multiplying numerator and denominator by ( 3 2) 1 3 2 011-26925013/14 +91-9811134008 +91-9582231489 = 1 3 2 3 2 3 2 = 3 2 ( 3) 2 ( 2) 2 = 3 2 = 3 2 3 2. NTSE, NSO Diploma, XI Entrance 12 IX ACADEMIC QUESTIONS Subjective Assignment-5 1. 2. Rationalise the denominators of the following real numbers: (i) 7 2 (v) 1 1 (vi) 3 7 5 2 3 (ii) 2 3 (iii) (vii) 4 3 5 1 3 5 5 3 5 4 3 Simplify the following by rationalizing the denominator: (i) 5 6 3 5 (ii) 5 6 7 6 3 1.732 and (iii) 2 6 5 5 3 5 2 (iv) 3 5 2 6 48 18 5 2.236, find the value of 3. If 4. Rationalise the denominator of the following: (i) 7 2 9 2 14 Rationalise the denominator of 6. Rationalise the denominator of 7. Simplify each of the following: (i) 3 2 5 3 5 3 (iii) 1 4 3 3 5 y2 (ii) 5. 8. (iv) x 2 y2 x 4 . 2 3 7 1 5 7 2 (ii) 3 2 2 3 3 2 2 3 3 2 2 3 3 2 2 3 7 1 7 1 7 1 7 1 Find the values of ‘a’ and ‘b’ in each of the following: (i) (ii) (v) 3 7 ab 7 3 7 11 7 a b 77 11 7 5 27 ab 3 7 48 011-26925013/14 +91-9811134008 +91-9582231489 (ii) 5 3 a b 15 5 3 (iv) 2 3 a b 6 18 12 (vi) 73 5 a b 5 2 7 3 5 2 NTSE, NSO Diploma, XI Entrance 13 9. Find the values of a and b 5 2 52 a b 5 52 5 2 1 1 1 13 4 15 14 14 13 10. Show that 11. Prove that 12. If x = 2 3, find the value of x 13. If x = 3 5, find the value of x 2 4 15 1 1 2 1 2 3 1 3 4 1 4 5 5 1 0 1 . x 1 x2 . Laws of Exponents for Real Numbers If a, n and m are natural numbers, then (i) am × an = am × n (iii) am n a (ii) (am)n = amn a m n , where m > n. If a , b, m are natural numbers then Examples: (i) 23 × 24 = 23 + 4 = 27 (ii) 55 (ii) (32)3 = 32 × 3 = 36 55 2 53 52 (iv) 23 33 (2 3)3 63 Laws of Radicals If n is a positive integer and a and b are positive rational numbers, then n (i) ( a) (ii) n (iii) (iv) (v) n a 1 n n nn 1 1 1 n a a n b a n b n (ab)1/n n ab mn n a n b n m a n a a1/n b1/n 1 m 1 1 m an 1 1 m an 1 a mn mn a 1 a a n n b b (a p ) m 1 pm n m a a 011-26925013/14 +91-9811134008 +91-9582231489 pm 1 1 n m 1 (a p ) n n a p NTSE, NSO Diploma, XI Entrance 14 (iv) am × bm = (ab)m Examples: (i) 4 (iii) (v) 3 31/4 5 (ii) 2 5 3 21/5 31/5 (2 3)1/5 5 6 3 2 5 ( 2 5)1/3 51/6 6 5 011-26925013/14 +91-9811134008 +91-9582231489 4 3 4 (31/4 )4 34/4 31 3 3 (iv) (vi) 12 3 3 3 5 121/3 31/3 (a 2 )5 1/3 12 3 1 10 3 5 a NTSE, NSO Diploma, XI Entrance 15 41/3 3 4 1 10 3 a5 1 (a 2 ) 3 3 a 2 IX ACADEMIC QUESTIONS Subjective Assignment-6 1 1 1 1. Find: (i) 64 2 (ii) 32 5 (iii) 1253 (NCERT) 3 2 3 1 2. Find: (i) 9 2 (ii) 32 5 (iii) 16 4 (iv) 1253 3. Find: (i) 2 23 1 25 1 (ii) 3 3 7 (iii) 1 112 1 114 1 (NCERT) 1 (iv) 7 2 8 2 (NCERT) 4. Find the value of x in each of the following: (ii) 5x 2 32x 3 135 (i) 25x 2x 220 5 3 5 5. If 52x 1 25x 1 2500, find the value of x. 6. Prove that 7. If 528 527 526 5 5 5 29 28 27 31 . 145 1 274 272 , then find the value of x. x 272 270 8. Prove that 2 1 x 2a 2b xa 9. Prove that b x a b 2 1 x 2b 2a xb c x bc 10. If 3x = 5y = 15−z, then show that 011-26925013/14 +91-9811134008 +91-9582231489 2 xc a x x 5 3 (iii) ca 1 1 1 1 0. x y z NTSE, NSO Diploma, XI Entrance 16 2x 125 27 XI SCIENCE & DIP. ENTRANCE Multiple Choice Questions Assignment – 7 1. 1.272727 ………. can be expressed in rational form as (a) 14 99 (b) 14 11 (c) 11 14 (d) 2. The sum of two rational numbers is………. number (a) Rational (b) Irrational (c) Either a& or 3. 99 14 (d) Natural 0.123 can be expressed in rational form as (a) 900 111 (b) 2( 2 6) 4. The fraction (a) 3( 2 3) 2 2 3 111 900 (c) 123 10 (d) 121 900 (c) 2 3 3 (d) 4 3 (d) 3 8 4 is equal to (b) 1 5. Which of the following is a pure surd? (a) 6. If x 0, then (a) x x 7. 256x16 4 81y (a) 3y 4x 4 33 5 (c) 12 (b) x4 x (c) (b) 2y 6x 4 (c) (b) 3 x x x 8 x (d) 3y 8x 4 (d) 8 x7 1/ 4 8. Set of natural numbers is a subset of (a) Set of even numbers (c) Set of composite numbers 011-26925013/14 +91-9811134008 +91-9582231489 (b) Set of odd numbers (d) Set of real numbers NTSE, NSO Diploma, XI Entrance 17 4y 5x 4 9. For an integer n, a student states the following: I. If n is odd, (n + 1)2 is even. II. If n is even, (n – 1)2 is odd. (n 1) is irrational III. If n is even Which of the above statements would be true? (a) I and III (b) I and II (c) I, II and III (d) II and III 10. The irrational number between 2 and 3 is (a) 11. If (b) 2 7 3 ( 10 3) (b) 4.398 2 5 ( 6 5) (a) 1 (a) 15 ( 15 3 2) 5 a 3 b2 c (b) (c) (a) 2 , 6 1 4 (b) (d) 6.398 1 2 (d) 3 a 2 b3c4 is 4 (c) 3 a3b2c a 3 b2 c 14. Two irrational numbers between 1 2 is (b) 2 5 10 20 40 5 80 (c) 3.398 3 2 13. The rationalizing factor of (d) 11 5 5 = 2.236 and 10 = 3.162, then the value of (a) 5.398 12. (c) 3 1 4 3 ,3 2 and 1 6 (d) a3b2c 3 are 1 1 1 (c) 6 8 , 3 4 1 (d) 3 2 , 38 15. Which of the following sets of fraction is in ascending order? (a) 7 9 11 , , 9 11 13 (b) 11 9 7 , , 13 11 9 (c) 11 7 9 , , 13 9 11 (d) 9 7 11 , , 11 9 13 16. 1/ ( 3 2) is not equal to (a) 3 2 17. The value of (b) 2 / ( 6 2) (c) ( 3 2) / (5 2 6) (d) 3 / (9 6) 1 1 1 1 ..... is 1 2 2 3 3 4 99 100 (a) Less than 99 100 (c) Greater than 99 100 011-26925013/14 +91-9811134008 +91-9582231489 (b) Equal to 99 100 (d) Equal to 99 100 NTSE, NSO Diploma, XI Entrance 18 18. The numerator of a a 2 b2 a a 2 b2 (a) a2 a a 2 b2 (c) a2 – b2 (b) b2 19. The ascending order of the surds (a) 9 4, 6 3, 3 2 9 (b) 20. Rational number between 2 3 2 (a) is a a 2 b2 3 2, 6 3, 9 4 is 4, 3 2, 6 3 2 and (c) 3 2, 6 3, 9 4 (d) 6 3, 9 4, 3 2 3 is 2 3 2 (b) (d) 4a2 – 2b2 (c) 1.5 (d) 1.8 21. The greater between 17 12 and 11 6 is 17 12 (a) 11 6 (c) Both are equal (b) 22. Which of the expressions is the same as (a) 3 2 1 23. If m (a) 3 (b) 1 ( 2 1) 3 4 1 (c) (d) Cannot compare ? 3 4 3 2 1 (d) 3 4 23 2 1 cad , then b equals ab m(a b) ca 1 cab ma (c) m 1 c (b) 1 1 (d) ma m ca 1 x q qr x r rp x p pq 24. The value of r p q is equal to x x x 1 1 1 q r (a) x p (b) 0 3 4 (c) x pq qr rp (d) 1 2 25. The value of 6 27 6 equals 3 2 (a) (b) 26. The rationalizing factor of (a) 3 5 (b) 011-26925013/14 +91-9811134008 +91-9582231489 3 2 (c) 23 3 3 4 (d) 3 4 5 is 52 (c) 52 (d) 53 NTSE, NSO Diploma, XI Entrance 19 3 1 1 then x 3 3 2 x 27. If x = (a) 28 3 15 8 (b) 27 3 35 4 28. The rational number between (a) 2 5 28 3 15 8 (d) 27 3 35 4 1 1 and is 2 3 (b) 1 5 (c) 3 5 (d) 4 5 (b) 2n 1 (c) 1 – 2n (d) 7 8 (c) 18 (d) 7 7 2n 4 2(2n ) 2(2n 3 ) 29. Simplify : (a) 2n 1 (c) 1 8 2 1 1 30. If a 9 , then a 3 3 equals a a (a) 10 3 3 (b) 3 3 31. If both ‘a’ and ‘b’ are rational numbers, then ‘a’ and ‘b’ from (a) a 32. 9 19 ,b 11 11 (b) a 19 9 ,b 11 11 (c) a 3 5 3 2 5 2 8 ,b 11 11 a 5 b , are (d) a 10 21 ,b 11 11 a b c d where d, c, b, a are consecutive natural numbers. Then which of the following is true? 33. (a) a b c d (b) a b c d (c) a c b d (d) c d a b 2 6 2 3 5 equals (a) 2 3 5 (b) 4 2 3 (c) 2 3 6 5 (d) 34. The value of (a) 527 1 ( 2 5 3) 2 1 of 1527 is 3 (b) 159 011-26925013/14 +91-9811134008 +91-9582231489 (c) 5 1526 NTSE, NSO Diploma, XI Entrance 20 (d) 5 39 35. Which one of the following is correct? (a) (c) ( 7 7 3)( 7 3 7 3 2) 24 3 ( 3 7 7 3 4)( 3 7 7 3 6) ( 3 ( (b) 7 7 3)( 3 7 7 3 2) 24 7 7 3)( 7 3 3 7 2) 24 ( 3 7 7 3 4)( 3 7 7 3 6) ( (d) ( 3 7 7 3 4)( 3 7 7 3 6) 3 3 ( 3 7 7 3 4)( 3 7 7 3 6) 36. If 25x – 1 = 52x – 1 – 100, then the value of x is (a) 3 (b) 2 (c) 4 37. If x = 2 3 , then the value of x2 + (a) 14, 8 3 (b) 10 39. The 100th root of 10(10 10 (a) 108 ) (d) 1 1 1 and x 2 2 is 2 x x 14, 8 3 38. If 444 + 444 + 444 + 444 = 4x, then x is (a) 45 (b) 44 7 7 3)( 3 7 7 3 2) 24 (c) 14, 8 3 (d) 14, 8 3 (c) 176 (d) 11 is 8 (b) 1010 (c) ( 10)( 10 )10 (d) 10( (10)) 40. Which of the following numbers has the terminal decimal representation? (a) 1 7 (b) 1 3 41. 2.6 0.82 is equal to………… (a) 82/99 (b) 182/99 (c) 3 5 (d) (c) 24/99 17 3 (d) 3/99 42. If x 2 5 3 and y 2 5 3 , then x4 + y4 = (a) 1538 (b) 1200 (c) 1048 (d) 149 43. If x 3 3 2 and y 3 3 2, then x 4 y4 8x 2 y2 (a) 3914 (b) 3010 (c) –486 (d) –856 44. If a 1 2 3 and b 1 2 3, then a 2 b2 2a 2b (a) 11 (b) 8 011-26925013/14 +91-9811134008 +91-9582231489 (c) 152 NTSE, NSO Diploma, XI Entrance 21 (d) 15 10 45. If x 1 3 2 2 and y (a) 4 46. If x 3 2 2 , then value of xy2 + x2y is (b) 12 5 3 80 48 45 27 (a) 15 47. If x 1 (c) 6 , then value of 4x 2 3x 5 (b) 2 7 3 10 3 (a) 2 3 2 15 3 2 (d) 9 (c) 12 2 5 6 5 (b) 1 (d) 5 , then value of x 4 x 2 is (c) 0 (d) 12 Passage: Given that 2 = 1.414, 3 = 1.732, 5 = 2.236, 10 =3.162 and = 3.141 48. Evaluate 10 2 5 (a) 5.141 49. Evaluate (b) 0.823 (c) 3.084 (d) 0.915 (c) 12.413 (d) 0.674 (c) 3.146 (d) 6.813 10 2 5 3 5 8 (a) 1.529 (b) 8.236 50. Find the value of (a) 12.479 2 3 2 (b) 9.428 011-26925013/14 +91-9811134008 +91-9582231489 NTSE, NSO Diploma, XI Entrance 22 ANSWER Assignment – 1: 0 0 0 22 23 24 25 26 27 15 16 17 18 19 20 2.(i) , , , , , , , , , , 1. Yes, , , , etc. (ii) 1 2 3 7 7 7 7 7 7 7 7 7 7 7 7 64 65 66 67 68 69 106 107 108 109 110 111 25 26 27 28 29 , , , , , , , , , , , , , , (iii) (iv) 3. 7 7 7 7 7 7 7 7 7 7 7 7 60 60 60 60 60 7 8 9 10 11 6 5 4 3 2 11 12 13 14 , , , , , , , 4. , , , , 5. 6. 6 6 6 6 6 7 7 7 7 7 15 15 15 15 22 23 24 25 26 27 1 2 3 4 5 6 7 8 9 , , , , , , , , , , , , , 7. 8. 28 28 28 28 28 28 100 100 100 100 100 100 100 100 100 17 18 19 9 10 11 12 13 1 , , , , , , 9. 10. 11. 4 80 80 80 20 20 20 20 20 12. (i) (iii) (iv) are false and (ii) and (v) is True 13. (ii), (iii) (iv) (v) (vi) and (ix) are True and (i) (viii) (x) are false. Assignment – 2 1. (i) 3/20 2. (i) 2/3 3. (i) 25/99 4. 23/9 (ii) 13/5000 (ii) 43/90 (ii) 1517/9900 5. 1/90 (iii) 3201/400 (iii) 1/999 (iii) 5599/2475 6. 10 Assignment – 3 1. 6 5 3 7 5 6 5. 7 2 5 11 9. 7 2 13. 30 2. 11 7 3 3. 3 3 3 2 4. 6. 6 5 7. 9 3 8. 2 3 3 10. 2 11. ab a b 2 8 3 6 5 12. 5 3 2 3 3 2 Assignment – 4 1. 6 3 2 2 3 6 2. 7 2 10 5. 30 4 3 6. 9. 2 13. –1 12 10. 4 1 2 14. 8/3 011-26925013/14 +91-9811134008 +91-9582231489 3. 3 4. 24 7. 36 3 2 8. 21 11. 4 12. 2 NTSE, NSO Diploma, XI Entrance 23 5 2 Assignment – 5 7 2 2 (ii) 2 3 3 (iii) 4 5 15 (v) 2 3 (vi) 3 7 5 38 (vii) 5(3 5 4 3) 3 1. (i) 2. (i) 31 10 6 19 (ii) 3. 4.54 6. 4. (i) 2 5 70 5 2 20 8. a = 8, b = 3 39 8 30 21 21 3 2 42 6 (iii) 7. (i) 7 2 5 x 2 y2 x 5. (ii) 25 3 22 (ii) 10 3 5 4 (iv) 6 6 6 3 2 3 21 3 2 7 3 5 (iv) a = 2, b = 6 (iii) 9 1 (iii) a , b 2 2 (ii) a = 4, b = 1 (iv) (v) a = –1, b = 1 (vi) a = 47, b = 21 9. a = 0, b = 8 12. 4 13. 119 45 5 8 Assignment – 6 1. (i) 8 2. (i) 27 13 3. (i) 2 15 4. (i) x = 1 5. x = 3 7. x (ii) 2 (ii) 4 (iii) 5 (iii) 8 (iv) 5 1 4 1 (ii) 3–21 (iii) 11 (ii) x = 3 (iii) x = 3 (iv) 56 2 1 4 Assignment – 7 1.b 2.c 3.b 4.d 5.a 6.d 7.a 8.d 9.b 10.c 11.a 12.a 13.a 14.c 15.a 16.d 17.b 18.d 19.a 20.c 21.b 22.c 23.d 24.d 25.d 26.b 27.b 28.a 29.d 30.c 31.a 32.b 33.a 34.c 35.a 36.b 37.c 38.a 39. b 40.c 41.b 42.a 43.c 44.b 45.c 46.c 47.a 48.d 49.d 50.b 011-26925013/14 +91-9811134008 +91-9582231489 NTSE, NSO Diploma, XI Entrance 24