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Chapter 2: Basic Mathematics and Number System Chapter 2: basic mathematics And NUMBER SYSTEM Level – I (QUESTIONS EXACTLY COPIED FROM NCERT) 1. Find the remainder when x3 – ax2 + 6x – a is divided by x – a ? (a) 2a (b) 3a (c) 4a (d) 5a 2. Find the value of k, if x – 1 is a factor of 4x + 3x2 – 4x + k ? 3 (a) 3 (b) –3 (c) 1 (d) 5 3. Factorise : 3x – x – 4 2 (a) ( x − 1)( 3 x − 4 ) (b) ( x + 1)( 3 x − 4 ) (c) ( x − 1)( 3 x + 4 ) (d) ( x + 1)( 3 x + 4 ) 4. Factorise : 2y3 + y2 – 2y – 1 (a) ( y + 1)( y − 2 )( 2 y + 1) (b) ( y − 1)( y + 1)( 2 y − 1) (c) ( y − 1)( y + 1)( 2 y − 3) (d) ( y − 1)( y + 1)( 2 y + 1) 5. Expand (4a – 2b – 3c)2 (a) 16a2 + 4b2 + 9c2 + 16ab −12bc − 24ac (b) 16a2 + 4b2 + 9c2 −16ab + 12bc − 24ac (c) 16a2 − 4b2 − 9c2 −16ab + 12bc − 24ac (d) 16a2 + 4b2 + 9c2 −16ab −12bc + 24ac 6. Factorise : 4x2 + y2 + z2 – 4xy – 2yz + 4xz (a) ( 2x − y + z ) (b) ( 2x − y + z ) (c) ( 2x − y − z ) (d) ( 2x − y − z ) 3 2 3 2 7. Evaluate (104)3 using suitable identities (a) 1124863 (b) 1124864 (c) 1124865 (d) 1124866 8. Factorise : 64a3 – 27b3 – 144a2b + 108ab2 (a) ( 4a − 3b ) (b) ( 4a + 3b ) (c) ( 4a − 6b ) (d) ( 4a + 6b ) 3 3 3 3 9. Factorise : 27x3 + y3 + z3 – 9xyz ( (a) ( 3x + y − z ) 9 x + y + z − 3xy − yz + 3xz 2 2 2 ) (b) ( 3x − y − z ) ( 9 x + y + z + 3xy − yz − 3xz ) 2 2 2 (c) ( 3x + y + z ) ( 9 x + y + z − 3xy − yz − 3xz ) 2 2 2 (d) ( 3x + y + z ) ( 9 x + y + z + 3xy + yz + 3xz ) 2 2 2 10. Without actually calculating the cubes, find the value of (28)3 + (–15)3 + (–13)3 ? (a) –16380 (b) 16380 (c) 15389 (d) –15389 11. Which is the possible expressions for the dimensions of the cuboids whose volumes is 12ky2 + 8ky – 20k ? (a) 4k ,( 3 y − 5) ,( y + 1) (b) 4k ,( 3 y + 5) ,( y − 1) (c) 4k ,( 3 y − 5) ,( y − 1) (d) 4k ,( 3 y + 5) ,( y + 5) Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458 Chapter 2: Basic Mathematics and Number System 3 4 12. The rational numbers between and is 5 5 3 4 8 7 (a) (b) (c) (d) 4 3 7 8 13. Which of the following statements are true. I. Every natural number is a whole number. II. Every integer is a whole number. III. Every rational number is a whole number. (a) Only I (b) I & II (c) II & III (d) I & III 1 3 14. Find 125 (a) 25 (b) 5 (c) 5 (d) 125 2 5 15. Find 32 2 (a) 5 (b) 2 (c) 4 (d) 5 3 1 16. Simplify 2 2 2 5 7 13 17 12 15 15 10 15 (a) 2 (b) 2 (c) 2 (d) 2 1 1 17. Simplify 7 8 2 2 (a) 561/ 4 (b) 56 (c) 56 (d) 1 18. By Euclid’s division algorithm the HCF of 867 and 255 is (a) 50 (b) 51 (c) 52 (d) 53 19. The LCM of 510 and 92 is (a) 2 (b) 37650 (c) 23460 (d) 7690 20. Given that HCF (306, 657) = 9, then LCM (306, 657) is (a) 22338 (b) 22337 (c) 22336 (d) 22335 21. There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. 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 again at the starting point? (a) 6 (b) 18 (c) 12 (d) 36 22. Without actually performing the long division, state among the following rational numbers will have a non-terminating repeating decimal expansion: 15 29 23 129 I. II. III. IV. 1600 343 2352 2 25775 (a) I, II (b) I, III (c) II, IV (d) III, IV 23. Which of the following are rational? I. 43.123456789 II. 0.120120012000120000…… III. 43.123456789 (a) Only I (b) Only II (c) Only III (d) I & III Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458 Chapter 2: Basic Mathematics and Number System LEVEL – II (QUESTIONS BASED ON NCERT) 1. If then is x + y + z = 9 & xy + yz + zx = 23, then ( x 3 + y 3 + z 3 − 3xyz ) is (a) 108 (b) 207 (c) 669 (d) 729 If  x + 1  = 3, then x 2 + 2 is equal to 1 2.  x x 10 82 (a) (b) (c) 7 (d) 11 3 9 3. If x + y = 7 and xy = 12, the value of x + y is 2 2 ( ) (a) 25 (b) 29 (c) 37 (d) 49 1 4. If x 4 + 4 = 322 , the then of  x − 1  is x  x (a) 4 (b) 6 (c) 8 (d) 3 2  3 1 If  x +  = 3 , then the value of 1 5.  x + 3  is  x  x  (a) 0 (b) 3 3 (c) 3 ( 3 −1 ) (d) 3 ( 3 +1) If  x 4 + 4  = 727 , the value of  x 3 − 13  is 1 6.  x   x  (a) 124 (b) 140 (c) 155 (d) 170 2 2 2 7. If a + b + c = 0, then the value of a + b + c bc ca ab (a) 1 (b) 0 (c) –1 (d) 3  ( 2.3)3 − 0.027  8. The value of   is  ( 2.3)2 + 0.69 + 0.09    (a) 2 (b) 3 (c) 2.327 (d) 2.273 9. The value of  343  343  343 + 257  257  257   343  343 − 343  257 + 257  257  (a) 100 (b) 300 (c) 600 (d) 1200 1 10. 2 ( The value of 17 − 8 ) 2 2 is (a) 5 (b) 10 (c) 15 (d) 20 11. ( The factor of x + 5 x + 6 x is equal to 4 3 2 ) (a) 𝑥 (𝑥 + 3)(𝑥 2 + 2) (b) 𝑥(𝑥 2 + 3)(𝑥 + 2) (c) x 2 ( x − 2 )( x − 3) (d) x 2 ( x + 3)( x + 2 ) 12. 3 ( The factor of x − 5 x + 8 x − 4 are 2 ) (a) ( x + 2 )( x − 2 )( x − 1) (b) ( x + 1)( x + 2 )( x − 2 ) (c) ( x − 2 ) ( x − 1) (d) ( x − 2 ) ( x + 1) 2 2 5 2 13. Find the solution of equation, 2 x −  4 8 3 3 3 3 (a) x  (b) x  (c) x  (d) x = 4 4 4 4 Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458 Chapter 2: Basic Mathematics and Number System 14. On dividing 16890 by a certain number, the quotient is 123 and the remainder is 39. Find the divisor (a) 173 (b) 137 (c) 237 (d) 371 15. Find the number which is nearest to 6444 and exactly divisible by 42 is (a) 18 (b) 6426 (c) 6462 (d) 6420 1 1 16. Simplify : 2+ 1 2+ 1 5− 130 5 153 53 53 (a) (b) (c) (d) 53 130 130 13 17. When simplified the product  1 − 1   1 − 1   1 − 1  ................ 1 − 1   3  4  5  n (c) ( 1 2 2 n − 1) 2 (a) (b) (d) n n n n ( n + 1) 18. If 2 x − 3 = 7 , then the value of x are (a) 5, –5 (b) 2, –5 (c) –2, 5 (d) 2, –2 19. On simplification  6 + 9  − 0.35  3 give  8 11   5 76 67 33 33 (a) (b) (c) (d) 33 33 76 67 20. The value of 9 + 9  9 − 9  9 is (a) 17 (b) 89 (c) 9 (d) None of these 21. When x = 1, then 2 x − 1 + x − 3 (a) 0 (b) 2 (c) 3 (d) 4 22. ( If ( x − a ) is a factor of x − 3x a + 2a x + b , then the value of b is 3 2 2 ) (a) 0 (b) 2 (c) 1 (d) 3 23. On dividing ( x − 6 x + 7 ) by ( x + 1) , the remainder is 3 (a) 2 (b) 12 (c) 0 (d) 7 24. Let f ( x ) = a0 x + a1 x + a2 x + ......an −1 x + an , where a0 ,a1 ,a2 ,.........an are constant. n n −1 n−2 If f ( x ) is divided by ( ax − b ) then the remainder is 𝑏 𝑏 𝑎 −𝑎 (a) 𝑓 (𝑎) (b) 𝑓 (− 𝑎) (c) 𝑓 (𝑏 ) (d) 𝑓 ( 𝑏 ) 25. Let f ( x ) = x 3 − 6 x 2 + 11x − 6, then which one of the following is not a factor of f ( x ) ? (a) ( x − 1) (b) ( x − 2 ) (c) ( x + 3) (d) ( x − 3) 26. 3 2 ( If ( x + 2 ) and ( x − 1) are the factors of x + 10 x + mx + n , the value of m and n are ) (a) m = 5,m = −3 (b) m = 17,n = −8 (c) m = 7,n = −18 (d) m = 23,n = −19 27. Which of the following is correct Euclid division algorithm, when a and b are two integer’s and b>0 (a) b = aq + r, and 0  r  b (b) b = rq + a, where 0  r  b (c) a = bq + r, where 0  r  b (d) None of these. 28. If x x x x.......  = 4 , then x is equal to (a) 2 (b) 2 (c) 4 (d) Infinity Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458 Chapter 2: Basic Mathematics and Number System 29. The HCF of 𝐴 = 22 × 33 × 55 and 𝐵 = 23 × 32 × 5 is: (a) 90 (b) 180 (c) 360 (d) 720 30. LCM of 3  5  2 , 2  5  3 and 2  3  5 11 3 2 3 2 2 2 (a) 600 (b) 5400 (c) 10800 (d) 59400 4 3 7 31. LCM of , and 5 10 15 84 5 10 5 (a) (b) (c) (d) 5 27 5 3 32. Greatest number which divides 15,45 and 75 and leaves no remainder in each case :- (a) 9 (b) 15 (c) 25 (d) 35 33. The greatest number which can divide 290,460 and 552 leaving respectively 4,5,6 as remainder, is (a) 11 (b) 13 (c) 12 (d) 14 34. Four bells begin to ring together and ring respectively at internals of 6,7,8 and 12 second. How many times they will ring together in one hour, excluding the one at the start ? (a) 13 (b) 15 (c) 21 (d) 19 35. The product of two numbers is 1600 and their HCF is 5. The LCM of the number is (a) 8000 (b) 1600 (c) 320 (d) 1605 36. The solution of ( ) x I. x x x = x x II. 33 x+5  23 x+3 = 9 III. 4x+2 + 22 x+1 − 36 = 0 are respectively 9 1 9 1 1 9 9 1 (a) ,−1 & (b) , & −1 (c) −1, & (d) −1, & 4 2 4 2 2 4 4 2 37. Arrange 2 , 3 4 and 4 6 in ascending order (a) 2,3 4,4 6 (b) 4 6, 3 4, 2 (c) 2,4 6, 3 4 (d) 4 6, 2, 3 4 38. 2 2 2 2 2 is equal to (a) 0 (b) 2 (c) 1 (d) 231/32 1 1 1 39. The expression (b−a ) ( c −a ) + ( a −b ) ( c −b ) + (b −c ) is + x( ) a −c 1+ x +x 1+ x +x 1+ x a −b − c (a) x (b) 1 (c) 0 (d) None of these 40. Consider the following statements I. If a x = b,b y = c,c z = a then xyz = 0 II. If p = a x , q = a y ,( p y q x ) = a 2 , then xyz = 1 z III. If x a = y b = z c &ab + bc + ca = 0, then xyz = 1 of these statements (a) 1 & 2 are correct (b) 2 & 3 are correct (c) only 1 is correct (d) 1 & 3 are correct 1 1 1 𝑥 𝑎 𝑎𝑏 𝑥 𝑏 𝑏𝑐 𝑥 𝑐 𝑐𝑎 41. The value of ( ) . (𝑥 𝑐 ) . ( ) is 𝑥𝑏 𝑥𝑎 1 1 ( ab +bc + ca ) (a) 1 (b) 0 (c) abc (d) x x  3 42.  0.756   is equivalent to:  4 (a) 18.9% (b) 37.8% (c) 56.7% (d) 75% Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458 Chapter 2: Basic Mathematics and Number System 43. What percent of 7.2 kg is 18 gms (a) 25% (b) 2.5% (c) 0.25% (d) 0.025% 44. If the cost of a book worth Rs 50 is increased by Rs 25. The rate of increase is (a) 50% (b) 25% (c) 20% (d) 10% 1 4 45. Which is greatest 33 %, and 0.35 ? 3 15 1 4 (a) 33 % (b) (c) 0.35 (d) all are equal 3 15 46. If 75% of the student in a school are boys and the number of girls is 420, the number of boys is (a) 1176 (b) 1350 (c) 1260 (d) 1125 47. The price of an item is increased by 20% and then decreased by 20%. The final price as compared to original price is (a) 20% less (b) 20% more (c) 4% more (d) 4% less 48. The value of machine depreciates at the rate of 16% per annum. If the price of a new machine is Rs 62500, its value after 3 years (in rupees) will be (a) 32500 (b) 36054 (c) 37044 (d) 38400 49. ( ) 2 : 1 + 3 :: 6 : x , then x is equal to (a) 3 − 3 (b) 1 + 3 (c) 1 − 3 (d) 3 + 3 50. The ratio of number of boys and girls in a school of 720 students is 7 : 5. How many more girls should be admitted to male the ratio 1 : 1 ? (a) 90 (b) 120 (c) 220 (d) 240 51. The ratio of zinc and copper in a brass piece is 13 : 7. How much zinc will be there is 100 kg of such a piece ? (a) 20 kg (b) 35 kg (c) 55 kg (d) 65 kg a b c a+b+c 52. If = = , then =? 3 4 7 c 1 1 (a) 7 (b) 2 (c) (d) 2 7 53. 94 is divided into two parts in such a way that the fifth part of the first and the eight part of the second are in the ratio 3 : 4. The first part is (a) 27 (b) 30 (c) 36 (d) 48 2 54. What is the ratio whose terms differ by 40 and the measure of which is ? 7 (a) 16 : 56 (b) 14 : 56 (c) 15 : 56 (d) 16 : 72 55. 20 liters of a mixture contain milk and water in the ratio 5 : 3. If 4 liters of this mixture are replaced by 4 liters of milk, the ratio of milk to water in the new mixture will become ? (a) 2 : 1 (b) 6 : 3 (c) 7 : 3 (d) 8 : 3 56. Which of the following statements are true? (a) on number line between two Q, there lies infinite Qc (b) on number line between two Qc, there lies infinite Q (c) on number line between two R, there lies infinite Q and Qc (d) All the above are correct Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458 Chapter 2: Basic Mathematics and Number System 57. Which of the following is terminating decimal 310 60 2 367 (a) (b) (c) (d) 15 7000 50 28 58. Which of the following statements is true :- I. terminating and repeating decimal are always rational II. Natural no, Even no, odd no, Whole no are always rational x III. , Where y  0 , is always rational y (a) Only I (b) Only I & II (c) All (d) II & III 59. Express 0.18 in P/Q form. 17 18 17 17 (a) (b) (c) (d) 98 90 99 90 60. 0.2313131…. is equal to 229 229 231 229 (a) (b) (c) (d) 999 990 990 90 61. Which of the following is true? 22 (a) π is a rational No (b) is an Qc 7 (c) ’e’ is an irrational No. (d) 1.34 is an Qc 62. Product of three consecutive natural numbers always divisible by (a) 2 (b) 3 (c) 6 (d) All of these 63. The number x,x + 2,x + 4 are all primes so x is :- (a) 3 (b) 6 (c) 11 (d) 17 64. Every Composite number has : (a) no prime divisor (b) at least one prime divisor (c) At least two prime divisor (d) one and only one prime divisor 65. If −1  x  3 and 1  y  3 , then maximum value of ( 3 y − 4 x ) is :- (a) 18 (b) 13 (c) 5 (d) –6 66. composite No can be (a) rational no (b) irrational no (c) both (d) None 67. ( ) The value of 0.6 + 0.8 + 0.7 is 1 1 1 (a) 2 (b) 2 (c) 2 (d) None of these 10 9 3 68. 235.235235235............. is a/an (a) integer (b) Whole Number (c) rational number (d) Qc 69. If n is an rational number, then n is (a) always a Q number (b) always a rational number c (c) always Q number (d) Sometimes a rational no and sometimes an irrational number 70. Which of the following cannot be represented on number line ? (a) 9.6 (b) 1.69 (c) 3 (d) 3 − 4i 71. Which of the following is a prime no ? (a) 165 (b) 247 (c) 572 (d) 101 Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458 Chapter 2: Basic Mathematics and Number System 72. Which of the statements is/are correct? A. Every point on the number line represents a R. B. Every point on the number line represents a Q. (a) A and B (b) A only (c) B only (d) neither A nor B 73. There are …....... prime Numbers between 0 to 100. (a) 20 (b) 25 (c) 30 (d) 35 74. Consider the following statements :- A. Every whole number is a natural number B. Every rational number is an integer (a) Only A is true (b) Both A and B are false (c) Both A and B are true (d) only B is true 75. Consider the following statements about irrational no. A. The decimal expansion of an irrational number is non-repeating and non-terminating B. An irrational number can’t be represented by a point on the number line. Which of these statements are/is correct? (a) A only (b) B only (c) Both A and B (d) Neither A nor B 76. The set of real number consists of (a) All positive and Q no. (b) All Q numbers. c (c) All positive, Q and Q (d) All Q and Qc numbers. 77. The set of all integers is not closed with respect to (a) Subtraction & division (b) Division alone (c) Division & multiplication (d) Multiplication, division & subtraction 78. Sum of two irrational numbers is (a) Always irrational (b) always rational no (c) may be rational and irrational (d) None 79. Which of the following statements is true ? (a) Products of two irrational numbers is always irrational. (b) Products of a rational and a irrational number is always Qc. (c) Sum of two irrational numbers can never be irrational (d) Sum of a irrational and a rational number is always irrational. 80. Which of the following is not a true statement ? (a) Even + Even = Even (b) odd + odd = Even (c) Even + odd = odd (d) odd + odd = odd 81. Match List I to List II List – I List – II (i) Q + Qc (A) always rational (ii) Qc × Qc (B) always irrational (iii) Qc ÷ Qc (C) sometime Q & sometimes Qc (iv) Qc – Qc (D) Can’t say (v) Q × Q (a) (i) A (ii) B (iii) C (iv) D (v) A (b) (i) C (ii) C (iii) C (iv) B (v) A (c) (i) B (ii) C (iii) C (iv) C (v) A (d) (i) D (ii) A (iii) B (iv) C (v) A Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458 Chapter 2: Basic Mathematics and Number System 82. Which one of the following is a prime number? (a) 165 (b) 171 (c) 173 (d) 221 1 1 83. If y + = 2 then the value of 16y3 + is: 4y 4 y3 (a) 102 (b) 104 (c) 105 (d) 106 1 1 1  xb   x c   x a  bc ca ab 84. The value of  c   a   b  on simplifying is: x  x  x  (a) x (b) 1/x (c) 1 (d) –1 85. (x + y) –(x – y) – 6y (x –y ) is equal to: 3 3 2 2 (a) x + y (b) x – y (c) 8x3 (d) 8y3 1 1 1 1 86. If x = + + +. . . . . . + , then the value of x is: √121+√122 √123+√124 √125+√126 √323+√324 (a) 5 (b) 6 (c) 7 (d) 8 87. 0. 001 is equal to: 1 1 1 1 (a) (b) (c) (d) 1000 999 990 99 88. Consider the following statements: A : Every whole number is a natural number B : Every rational number is an integer In Your opinion ? (a) Only A is true (b) Only B is true (c) Both A and B are true (d) Both A and B are false 89. 0.2313131 …….. is equal to: 229 229 231 231 (a) (b) (c) (d) 999 990 990 99 90. A bag contains Rs. 102.00 in the form of rupee, fifty paise and ten paise coins in the ratio 3:4:10. Find the number of 10 paise coins: (a) 17 (b) 60 (c) 120 (d) 170 91. A man starts from a given point. Each time he takes three steps forward, he must take two steps back. How many steps must he take in order to reach a point six steps ahead of the starting point: (a) 23 (b) 25 (c) 17 (d) 18 92. Solve 2 – 3. 2 + 32 = 0: 2x x+2 (a) {2,3} (b) {l,3} (c) {1,2} (d) {3,4} 93. If a3+b3+l 5 ab = 125 find a+b : (a) 3 (b) 5 (c) 7 (d) 9 94. If the length of a rectangular field be increased by 50% and the breadth be decreased by 25%, find the % change in area: (a) 10½ % (b) 12½ % (c) 15½% (d) 17½% bc ca ab 95. If bc + ca + ab = 0, find the value of 2 + 2 + 2 : a b c (a) – 2 (b) 2 (c) 3 (d) – 3 Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458 Chapter 2: Basic Mathematics and Number System ANSWERS LEVEL – I 1.d 2.b 3.b 4.d 5.b 6.b 7.b 8.a 9.c 10.b 11.b 12.a 13.a 14.c 15.c 16.c 17.c 18.b 19.c 20.a 21.d 22.c 23.d LEVEL – II 1.a 2.c 3.a 4.a 5.a 6.b 7.d 8.a 9.c 10.c 11.d 12.c 13.c 14.b 15.b 16.c 17.b 18.c 19.b 20.b 21.c 22.a 23.b 24.a 25.c 26.c 27.c 28.c 29.b 30.d 31.a 32.b 33.b 34.c 35.c 36.a 37.c 38.d 39.b 40.b 41.a 42.c 43.c 44.a 45.c 46.c 47.d 48.c 49.d 50.b 51.d 52.b 53.b 54.a 55.c 56.d 57.c 58. b 59. d 60. b 61. c 62. d 63. a 64. b 65. b 66. c 67. c 68. c 69. d 70. d 71. d 72. b 73. b 74. b 75. a 76. d 77. b 78. c 79. d 80. d 81. c 82. c 83. b 84. c 85.d 86. c 87. b 88. d 89. b 90. d 91. d 92. a 93. b 94. b 95. c Old ICC building, near Firoz Hospital, Sir Syed Nagar, Medical Road, Aligarh: 9997607607, 9997394458