Fundamentals of Mathematics Exercises

Fundamentals of Mathematics-I  Marked questions are recommended for Revision. PART - I : SUBJECTIVE QUESTIONS Section (A) : Representation of sets, Types of sets, subset and power set A-1. State wether the following collections is a set or not ? (i) The collection of natural numbers between 2 and 20 (ii) The collection of numbers which satisfy the equation x 2 –5x + 6 = 0 (iii) The collection of prime numbers between 1 and 100. (iv) The collection of all intelligent women in Jalandhar. A-2. Write the following set in tabular form (i) A = {x : x is a positive prime < 10} (ii) B = {x : x = 3, x  , 1    3} Write the following set in builder form (i) set of all rational number (ii) {2, 5, 10, 17, 26, 37, .........} A-3. A-4. Identifiy type of set in terms of empty/singleton/finite/infinite (i) {x : x is a real number and x2 – 1 = 0} (ii) {x : x is a real number and x2 + 1 = 0} 2 (iii) {x : x is positive real number and x – 9 = 0} (iv) {x : x is a real number and x2 + 2x + 2  0} A-5. Write power set of the set A = {, {}}. Section (B) : Operations on sets, Law of Algebra of sets B-1. Given the sets A = {1, 2, 3}, B = {3, 4}, C = {4, 5, 6}, then find the following (i) A  (B  C) (ii) A – (B  C) (iii) (B  C) – A B-2. Find the smallest set A such that A  {1, 2} = {1, 2, 3, 5, 9} B-3. If aN = {ax : x  N} and bN  cN = dN, where b, c  N, b  2, c  2 are relatively prime, then write 'd' in terms of b and c. B-4. Sets A and B have 3 and 6 elements respectively. What can be the minimum and maximum number of elements in (i) AB (ii) AB Section (C) : Cardinal number Problems C-1. Let n(U) = 700, n(A) = 200, n(B) = 300 and n(A  B) = 100, then find n(A'  B') C-2. In a college of 300 students, every student reads 5 newspapers and every newspaper is read by 60 students. Find the number of newspaper. C-3. In a town of 10,000 families it was found that 40% families buy newspaper A, 20% families buy newspaper B and 10% families buy newspaper C, 5% families buy A and B, 3 % buy B and C and 4% buy A and C. If 2% families buy all the three news papers, then find number of families which buy newspaper A only. Fundamentals of Mathematics-I Section (D) : Rational Inequalities D-1. Solve the following Inequalities x2 + 4 x + 4 (i) 2x2 − x − 1 ( x + 2)( x2 − 2x + 1) (iv) D-2. 4 + 3x − x 2 x4 + x2 + 1 (ii) 0 x2 − 4 x − 5 0 x 4 − 3 x 3 + 2x 2 (v) (2 − x2 )( x − 3)3 (iii) ( x + 1)( x2 − 3 x − 4) 0 0 x2 − x − 30 Solve the following Inequalities 7x − 5 4 8x + 3 ( x − 2)( x − 4)( x − 7) 1 ( x + 2)( x + 4)( x + 7) (i) (iv) D-3. 0 2x2 − 3 x − 459 (ii) x2 + 1 1 x2 − 5 x + 12 (iii) x2 − 4 x + 5 3 Find the number of integers satisfying the following Inequalities (i) x4 – 5x2 + 4  0 (ii) (iv) 14x 9x − 30  x +1 x−4 (v) x4 – 2x2 – 63  0 x2 + 2 x2 − 1 x2 + 6 x − 7  2 (iii)  −2 Section (E) : Logarithm E-1. Find the value of E-2. E-3. (v) log7(2.11) (ix) log10 (log10 9) 5 2 + 9log3 7 − 8log2 5 log3 ( 7 − 2) (vi) (vii) log1/3(1/5) (iv) log3(4)  2 + 1 log4   2 − 1    (viii)  2· 3 3 log3   3  log 45 144 Compute the following 3 5 1 log7 5 1 + ( − log10 0.1) log3 5 log5 7 +3 (ii) log0.75 log2 log3 7 −5 1+log7 2  1  (iii)    49  + 5– log1/ 5 7 log5 3 −7 (iv) 7 (i) Let n = 75600, then find the value of (ii) If ( 1 0.125 ) log2 log3 ( log4 (x) ) = 0 4 3 2 1 + + + log2 n log3 n log5 n log7 n and log3 ( log4 ( log2 (y) ) ) = 0 find the sum of x, y and z is E-6.    Let log102 = a and log103 = b then determine the following logarithms in terms of a and b.   (i) log10  sin2  (ii) log1004 + 2 log10027 (iii) log29 + log38 3  (i) E-5. 5 Which of the following numbers are positive/negative (i) log 3 2 (ii) log1/7(2) (iii) (iv) E-4. log log105.log1020 + (log102)2 (ii) (i) Show that the number log 2 7 is an irrational number. and ( ) log4 log2 ( log3 (z) ) = 0 then Fundamentals of Mathematics-I 1 . Suppose b = a2 + a3 + a4 + a5 and logn 2002 c = a10 + a11 + a12 + a13 + a14. Then find the value of (b – c) E-7. Suppose n be an integer greater than 1. let an = E-8. If E-9. loga logb logc , show that a a . b b . c c = 1. = = b−c c −a a−b Solve the following equations : (i) logx(4x – 3) = 2 log2(log3(x2 – 1)) = 0 5 − log2 x = 3 – log2 x (iii) 4log2 x − 2x − 3 = 0 (iv) (v) log 4 (log 2 x) + log 2 (log 4 x) = 2. (vi) log 4 log 3 log 2 x = 0 (vii) 1   log 3  log9 x + + 9 x  = 2x. 2   (viii) 2log 4 (4 – x) = 4 – log 2 (– 2 – x). (ix) x (i) Find the real value of 'x' satisfying the equation x (ii) Solve for x : xlog10 x+2 = 10log10 x+2 (iii) Solve for x : x (iv) Find the product of roots of the equation (log3x)2 – 2(log3x) – 5 = 0 (v) Find sum of roots of the equation 4x – 7.2x + 6 = 0 (log E-10. (ii) x 2x) =4 log10 x + 5 3 0.5 log x (x 2 − x) = 3log9 4 . = 105+log10 x Section (F) : Logarithmic inequalities F-1. Solve the following inequalities 3  (i) log 5  2x 2 − x −   1 8 8  2x − 6 0 2x − 1 (iii) log7 (v) log1/3(2x+2 – 4x)  – 2 (ii) log 1 ( x2 − 5 x + 6)  −1 2 F G2 IJ (iv) x + 1K log1/4(2 – x) > log1/4 H (vi) logx(4x – 3)  2 4x + 6 0 x F-2. Find the number of integers satisfying log1/5 F-3. Find the number of positive integers not satisfying the inequality log2(4x – 2.2x + 17) > 5. F-4. Solve the following inequalities : 1 log(3x 2 +1) 2 < (i) 2 (ii) logx² (2 + x) < 1 Fundamentals of Mathematics-I PART - II : ONLY ONE OPTION CORRECT TYPE Section (A) : Representation of sets, Types of sets, subset and power set A-1. A-2. A-3. A-4. The set of intelligent students in a class is(A) a null set (C) a finite set (B) a singleton set (D) not a well defined collection The set A = {x : x  R, x2 = 16 and 2x = 6} is (A) Null set (C) Infinite set (B) Singleton set (D) not a well defined collection If A = {x : –3 < x < 3, x  Z} then the number of subsets of A is – (A) 120 (B) 30 (C) 31 (D) 32 Which of the following are true ? (A) [3, 7]  (2, 10) (B) (0, )  (4, ) (D) [2, 7]  (2.9, 8) (C) (5, 7]  [5, 7) A-5. The number of subsets of the power set of set A = {7, 10, 11} is (A) 32 (B) 16 (C) 64 A-6. (D) 256 Which of the folowing sets is an infinite set ? (A) Set of divisors of 24 (B) Set of all real number which lie between 1 and 2 (C) Set of all humman beings living in India. (D) Set of all three digit natural numbers Section (B) : Operations on sets, Law of Algebra of sets B-1. Let A = {x : x  R, –1 < x < 1} , B = {x : x  R, x  0 or x  2} and A  B = R – D, then the set D is (A) {x : 1 < x  2} (B) {x : 1  x < 2} (C) {x : 1  x  2} (D) {x : 1 < x < 2} B-2. If A = {2, 3, 4, 8, 10}, B = {3, 4, 5, 10, 12}, C = {4, 5, 6, 12, 14} then (A  B)  (A  C) is equal to (A) {3, 4, 10} (B) {2, 8, 10} (C) {4, 5, 6} (D) {3, 5, 14} B-3. The shaded region in the given figure is (A) A  (B  C) (B) A  (B  C) (C) A  (B – C) (D) A – (B  C) B-4. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2, 5}, B = {6, 7}, then A  B is (A) B (B) A (C) A (D) B B-5. If A = {x : x = 4n + 1, n  5, n  N} and B {3n : n  8, n  N}, then A – (A – B) is : (A) {9, 21} (B) {9, 12} (C) {6, 12} (D) {6, 21} B-6. A  B = A  B iff : (A) A  B (B) A = B (C) A  B (D) A  B B-7. Consider the following statements : 1. N  (B  Z) = (N  B)  Z for any subset B of R, where N is the set of positive integers, Z is the set of integers, R is the set of real numbers. 2. Let A = {n  N : 1  n  24, n is a multiple of 3}. There exists no subset B of N such that the number of elemets in A is equal to the number of elements in B. Which of the above statements is/are correct ? (A) 1 only (B) 2 only (C) Both 1 and 2 (D) Neither 1 nor 2 Fundamentals of Mathematics-I B-8.Which of the following venn-diagrams best represents the sets of females, mothers and doctors ? (A) (B) (C) (D) Section (C) : Cardinal number Problems C-1. Let A and B be two sets. Then (A) n(A  B)  n(A  B) (B) n(A  B)  n(A  B) (C) n(A  B) = n(A  B) (D) can't be say C-2. In a city 20 percent of the population travels by car, 50 percent travels by bus and 10 percent travels by both car and bus. Then persons travelling by car or bus is (A) 80 percent (B) 40 percent (C) 60 percent (D) 70 percent C-3. A class has 175 students. The following data shows the number of students obtaining one or more subjects : Mathematics 100, Physics 70, Chemistry 40, Mathematics and Physics 30, Mathematics and Chemistry 28, Physics and Chemistry 23, Mathematics & Physics & Chemistry 18. How many students have offered Mathematics alone ? (A) 35 (B) 48 (C) 60 (D) 22 C-4. 31 candidates appeared for an examination, 15 candidates passed in English, 15 candidates passed in Hindi, 20 candidates passed in Sanskrit. 3 candidates passed only in English. 4. candidates passed only in Hindi, 7 candidates passed only in Sanskrit. 2 candidates passed in all the three subjects How many candidates passed only in two subjects ? (A) 17 (B) 15 (C) 22 (D) 14 Section (D) : Rational Inequalities D-1. The complete solution set of the inequality (A) (− , − 5)  (1, 2)  (6, )  {0} (C) (− , − 5]  [1, 2]  [6, )  {0} x 4 − 3x 3 + 2x 2  0 is: x 2 − x − 30 (B) (− , − 5)  [1, 2]  (6, )  {0} (D) (− , − 5]  [1, 2]  [6, ) D-2. Number of positive integral values of x satisfying the inequality (x − 4)2017 . (x + 8)2016 (x + 1)  0 is x2016 (x − 2)3 . (x + 3)5 . (x − 6) (x + 9)2018 (A) 0 (B) 1 (C) 2 2 x −1 D-3. The complete solution of  0 & x2 – 5x + 2  0 is : x+3  5 − 17 5 + 17   5 + 17  , (A) x   (B) x  1,   2 2 2     (C) x  (–3, –1] D-4. D-5. D-6. (D) x  (–3, –1]  [1, ) The solution of the inequality 2x – 1  x2 + 3  x – 1 is (A) x  R (B) [–2, 2] (C) (–2, 2) The number of prime numbers satisfying the inequality (A) 1 (D) 3 (B) 2 (D) x   x2 − 1 < 3 is 2x + 5 (C) 3 The number of the integral solutions of x2 + 9 < (x + 3)2 < 8x + 25 is : (A) 1 (B) 3 (C) 4 (D) 4 (D) 5 Fundamentals of Mathematics-I D-7. Number of non-negative integral values of x satisfying the inequality (A) 0 (B) 1 (C) 2 2 x − x +1 (D) 3 2 − 1 2x − 1  0 is − x + 1 x3 + 1 Section (E) : Logarithm E-1. If a4 · b5 = 1 then the value of loga(a5b4) equals (A) 9/5 E-2. (B) 4 (D) 8/5 1 1 1 + + has the value equal to 1 + logb a + logb c 1 + logc a + logc b 1 + loga b + loga c (A) abc E-3. (C) 5 (B) 1 log abc + 1 log bc abc + 1 abc (C) 0 1 log ca abc (D) 1 has the value equal to : ab (A) 1/2 (B) 1 (C) 2 (D) 4 E-4. (log210) . (log280) – (log25) . (log2160) is equal to : (A) log25 (B) log220 (C) log210 log E-5. The ratio 2 21/ 4 a 7 (A) a2 − a − 1 log27 ( a2 +1) (D) log216 3 −3 − 2a 4log49 a − a −1 (B) a2 + a − 1 simplifies to : (C) a2 − a + 1 (D) a2 + a + 1 E-6. Let x = 2log 3 and y = 3log 2 where base of the logarithm is 10, then which one of the following holds good? (A) 2x < y (B) 2y < x (C) 3x = 2y (D) y = x E-7. E-8. If loga(ab) = x, then logb(ab) is equal to 1 x (A) (B) x 1+ x 10 ( logp logq (logr x) (A) rq/r E-9. E-11. (D) x x −1 ) = 1 and log log (log x) = 0 then 'p' equals ) q( r p (B) rq (D)rr/q (C) 1 3   1 (D)    log   10   log10(log23) + log10(log34) + log10(log45) + ........ + log10(log10231024) simplifies to (A) a composite (B) a prime number (C) rational which is not an integer (D) an integer (B) log10   1  (C)    log10   2 The sum of all the solutions to the equation 2 log10 x – log10(2x – 75) = 2 (A) 30 E-12. x 1− x Which one of the following is the smallest? (A) log10 E-10. (C) (B) 350 (C) 75 (D) 200 2 If the solution of the equation logx(125x) . log25 x = 1 are  and  ( < ). Then the value of 1/ is : (A) 5 (B) 25 (C) 125 (D) 625 Fundamentals of Mathematics-I E-13. 9 2 + logx 5 is : 4 (C) Even number (D) Divisible by 3 The positive integral solution of the equation logx 5 + logx5x = (A) Composite number (B) Prime number E-14. , where p  2, p  N ; n  N when simplified is The expression logp (A) independent of p (C) dependent on both p and n E-15. If logx log18 ( 2 + (A) 8 8 (B) independent of p and of n (D) positive ) = 31 . Then the value of 1000 x is equal to (B) 1/8 (C) 1/125 (D) 125 E-16. If 32 log3 x − 2 x − 3 = 0, then the number of values of 'x' satisfying the equation is (A) zero (B) 1 (C) 2 (D) more than 2 E-17. Number of real solutions of the equation log10 ( − x ) = log10 x 2 is : (A) zero E-18. (B) exactly 1 The correct graph of y = x logx (A) x2 (C) exactly 2 (D) 4 (C) (D) is (B) Section (F) : Logarithmic inequalities F-1. The solution set of the inequality log  sin   3 1  (A)  ,2  2  F-2. (x 2 − 3x + 2)  2 is  5  (B)  1,   2 1   5 (C)  , 1   2,  2   2 (C) (− 2, − 1) (B) (1, 2) F-3. Solution set of the inequality 2 − log2 (x2 + 3x)  0 is : (A) [− 4, 1] (B) [− 4, − 3)  (0, 1] (C) (−  − 3)  (1, ) (D) (−  − 4)  [1, ) F-4. If log0.5 log5 (x2 – 4) > log0.51, then ‘x’ lies in the interval (A) (– 3, – 5 )  ( 5 , 3) (D)  The set of all solutions of the inequality (1/ 2)x (A) (– , 0)  3 (D)  1,   2 (B) (– 3, – 5 )  ( 5 , 2) (C) ( 5 , 3 5 ) F-6. (1, 2) If log0.3 (x − 1) < log0.09 (x − 1), then x lies in the interval (A) (2, ) F-5. (D) (B) (– , 1) 2 − 2x < 1/4 contains the set (C) (1, ) The set of all the solutions of the inequality log1 – x (x – 2)  – 1 is (A) (– , 0) (B) (2, ) (C) (– , 1) (D) (3, ) (D)  Fundamentals of Mathematics-I PART - III : MATCH THE COLUMN 1. 2. Match the set P in column one with its super set Q in column II Column –  (set P) (A) [32n – 8n – 1 : n  N} (B) {23n – 1 : n  N} (C) {32n –1 : n  N} (D) {23n – 7n – 1 : n  N} Column-I (A) If a = 3 ( 8+2 7 − ) 8−2 7 , b = Column –  (set Q) (p) {49 (n – 1) : n  N (q) {64 (n –1) :  N} (r) {7n : n  N} (s) {8n : n  N} Column-II (42)(30) + 36 (p) –1 then the value of logab is equal to (B) If a = 4 + 2 3 – 4 − 2 3 , b = 11 + 6 2 – 11 − 6 2 , then the value of logab is equal to (q) 1 (C) If a = 3 + 2 2 , b = 3 − 2 2 then the value of logab is equal to (r) 2 (D) If a = (s) 3 2 (t) None (E) (F) 3. 7 − 72 − 1 , then the value of logab is equal to The number of zeroes at the end of the product of first 20 prime numbers, is The number of solutions of 22x – 32y = 55, in which x and y are integers, is Column-I Column-II (A) When the repeating decimal 0.363636............. is written as a rational fraction in the simplest form, the sum of the numerator and denominator is (p) 4 (B) Given positive integer p, q and r with p = 3q · 2r and 100 < p < 1000. The difference between maximum and minimum values of (q + r), is (q) 0 (C) If log8a + log8b = (log8a)(log8b) and logab = 3, then the value of 'a' is (r) 15 (s) 16 (D) 4. 7 + 72 − 1 , b = If P = 3 Column-I log3 2 – 2 log2 3 ( ) then value of P is Column-II (A) Anti logarithm of 0.6 to the base 27 has the value equal to (p) 5 (B) Characteristic of the logarithm of 2008 to the base 2 is (C) The value of b satisfying the equation, loge2 · logb625 = log1016 · loge10 is (q) 7 (D) Number of naughts after decimal before a significant figure (r) 9 (s) 10 100 5 comes in the number   , is 6 (Given log102 = 0.3010 and log103 = 0.4771) Fundamentals of Mathematics-I  Marked questions are recommended for Revision * Marked Questions may have more than one correct option. PART - I : ONLY ONE OPTION CORRECT TYPE 1. Let A1, A2 and A3 be subsets of a set X. Which one of the following is correct ? (A) A1  A2  A3 is the smallest subset of X containing elements of each of A 1, A2 and A3 (B) A1  A2  A3 is the smallest subset of X containing either A1 or A2  A3 but not both (C) The smallest subset of X containing A1  A2 and A3 equals the smallest subset of X containing both A1 and A2  A3 only if A2 = A3 (D) None of these 2. Let A, B, C be distinct subsets of a universal set U. For a subset X of U, let X ' denote the complement of X in U. Consider the following sets : 1. ((A  B)  C)  B) = B  C 2. (A   B )  (A  B  C) = (A  (B  C)) Which of the above statements is/are correct ? (A) 1 only (B) 2 only (C) Both 1 and 2 (D) Neither 1 nor 2 3. In an examination of a certain class, at least 70% of the students failed in Physics, at least 72% failed in Chemistry, at least 80% failed in Mathematics and at least 85% failed in English. How many at least must have failed in all the four subjects ? (A) 5% (B) 7% (C) 15% (D) Cannot be determined due to insufficient data 4. If 6 x2 − 5 x − 3 x2 − 2 x + 6 (A) 36 & 81 4  4, then the least and the highest values of 4x2 are: (B) 9 & 81  1 and  + (C) 0 & 81 (D) 9 & 36 1 is an odd integer then number of possible values of  is  (B) 2 (C) 3 (D) 4 5. If 6. If logab = 2; logbc = 2 and log3c = 3 + log3a then (a + b + c) equals (A) 90 (B) 93 (C) 102 (D) 243 The sum of the solutions of the equation 9 x – 6 · 3x + 8 = 0 is (A) log3 2 (B) log3 6 (C) log3 8 (D) log3 4 7. 8.  +1 (A) 1 2  x2 + 3x + 2  x(x3 + 1) log2 8   + 3x − x+2  (x + 1)(x2 − x + 1)  The expression: reduces to (x − 1)(log2 3)(log3 4)(log4 5)(log5 2) (A) 9. x +1 x −1 (B) x2 + 3x + 2 (log2 5)x − 1 (C) If a, b, c are positive real numbers such that alog3 7 ( 2 (log3 7) of a (A) 489 +b 2 (log7 11) +c 2 (log11 25) (B) 469 ) equals 3x x −1 = 27 ; (C) 464 (D) x blog7 11 = 49 and clog11 25 = (D) 400 11 . The value Fundamentals of Mathematics-I 10. Consider the statement : x ( – x) < y ( – y) for all x, y with 0 < x < y < 1. The statement is true (A) if and only if   2 (B) if and only if   2 (C) if and only if   –1 (D) for no values of  11. The set of values of x satisfying simultaneously the inequalities (x − 8) (2 − x) log0.3 2x − 3 − 31 > 0 is : (A) a unit set (C) an infinite set 12. 13. 14. If f ( x) = (B) 2 1 − log(2x + 3) ( x2 − x + 1) log(3 x − 2) (2x + 3) (D) (–2, –1)  (– 1, 0]  (4, ) is defined is equal to- (C) 3 (D) 4 log4 {log3 {log2 (x 2 − 2x + a)}} is defined xR, then the set of values of 'a' is (A) [9, ) 15. (3 x − 4 x ) · n(x + 2)  0 is x 2 − 3x − 4 (B) (–2, 0]  (4, ) (C) (–1, 0]  (4, ) Number of integers for which (A) 1  0 and (B) an empty set (D) a set consisting of exactly two elements. The solution set of the inequality (A) (–, 0]  (4, )  10   7 (log2 5 − 1)    (B) [10, ) (C) [15, ) (D) [2, ) Let X and Y be two sets. Statement-1 X  (Y  X)' =  Statement-2 If X  Y has m elements and X  Y has n elements then symmetric difference X  Y has m – n elements. (A) Both the statements are true. (B) Statement- is true, but Statement- is false. (C) Statement- is false, but Statement- is true. (D) Both the statements are false. PART - II : SINGLE AND DOUBLE VALUE INTEGER TYPE 1. Let U be set with number of elements in it is 2009 and A, B are subsets of U with n (A  B) = 280. If n(A  B) = x13 + x32 = y13 + y32 for some positive integers x1, x2 , y1, y2 then find the value of (x1 + x2 + y1 + y2). 2. Let U be set with number of elements in it is 2009. A is a subset of U with n (A) = 1681 and out of these 1681 elements, exactly 1075 elements belong to a subset B of U. If n (A – B) = m2 + p1 p2 p3 for some positive integer m and distinct primes p1, p2, p3 then for least m find (p1 + p2 + p3) 3. Let A = {(x, y) : x  R, y  R, x3 + y3 = 1], B = {(x, y) : x  R, y  R, x – y = 1} and C = {(x, y) : x  R, y  R, x + y = 1}. If A  B contains 'p' elements and A  C contains 'q' elements then find (q – p). 4. In a class of 42 students, the number of students studying different subjects are 23 in Mathematics, 24 in Physics, 19 in Chemistry, 12 in Mathematics and Physics 9 in Mathematics and Chemistry, 7 in Physics and Chemistry and 4 in all the three subjects. Then find number of students who have taken exactly one subject. 5. Find the sum of all the real solutions of the inequality 6. Find the value of (log312)(log372) – log3(192).log36 (x 2 + 2)( x 2 − 16 ) (x 4 + 2)(x 2 − 9) 0 Fundamentals of Mathematics-I 7. ( Let x = log1 3 5 ) (log125 343) (log49 729) and y = 25 3 log289 11 log28 17 log1331 784 , then find the value of x2 + y2 is 8. If c(a – b) = a(b – c) then find the value of log(a + c) + log(a − 2b + c) log(a − c) (Assume all terms are defined) 9. If logb a. logc a + loga b. logc b + loga c. logb c = 3 ( where a, b, c are different positive real numbers  1), then find the value of a b c. 10. If 4A + 9B = 10C, where A = log164, B = log39 & C = logx83, then find x. 11. Find the natural number, x , which satisfies the equation log10 (x² − 12 x + 36) = 2 12. Find the value of x satisfying the equation log 1 (x − 1) + log 1 (x + 1) − log 2 2 1 (7 − x) = 1 2 13. Find the non negative square root of product of reciprocal of roots of the equation log 102 x + log10x2 = log 102 2 − 1 14. If log(2x + 3) (6x2 + 23x + 21) = 4 – log(3x + 7) (4x2 + 12x + 9) then find the value of 64 x2 15. Let a, b, c, d are positive integers such that log ab = 3 5 and logcd = . If 2 4 (a – c) = 9, find the value of (b − d) . 16. If the product of all solutions of the equation form as ( 2009 ) x 2010 = ( 2009 ) logx (2010) can be expressed in the lowest m then the value of (m – n) is n 17. If the complete solution set of the inequality (log10 x)2  log10 x + 2 is (0, a]  [b, ) then find the value of ab. 18. The complete solution set of the inequality log x +1 4 x+ 2 19. Find the number of integers which do not satisfy the inequality 20. If log10 2 = 0.3010 and log10 3 = 0.4771, then find sum of the number of digits in 615 and the number of zeros immediately after the decimal in 3–100 . 1  1 log4 ( x + 3) , is (–a, ), then determine ‘a’. log1/2 (x + 5)2 > log1/2 (3x – 1)2. PART - III : ONE OR MORE THAN ONE OPTIONS CORRECT TYPE 1. A and B are two sets such that n(A) = 3 and n(B) = 6, then (A) minimum value of n(A  B) = 6 (B) minimum value of n(A  B) = 9 (C) maximum value of n(A  B) = 6 (D) maximum value of n(A  B) = 9 2. In a survery, it was found that 21 persons liked product A, 26 liked product B and 29 liked product C. If 14 persons liked products A and B, 12 liked products C and A, 13 persons liked products B and C and 8 liked all the three products then which of the following is (are) true ? (A) The number of persons who liked the product C only = 12 Fundamentals of Mathematics-I (B) The number of persons who like the products A and B but not C = 6 (C) The number of persons who liked the product C only = 6 (D) The number of persons who like the products A and B but not C = 12 3. Let a > 2, a  N be a constant. If there are just 18 positive integers satisfying the inequality (x – a)(x – 2a)(x – a2) < 0 then which of the option(s) is/are correct? (A) 'a' is composite (B) 'a' is odd (C) 'a' is greater than 8 (D) 'a' lies in the interval (3, 11) 4. Let N = 5. 6. log3 135 log3 5 – . Then N is : log15 3 log405 3 (A) a natural number (B) a prime number 5 Values of x satisfying the equation log52 x + log5x   = 1 are x 1 (A) 1 (B) 5 (C) 25 8. (D) 3 (B) no prime solution (D) one integral solution 9   2 (log3 x) − 2 log3 x + 5    x The equation = 3 3 has (A) exactly three real solution (C) exactly one irrational solution (B) at least one real solution (D) complex roots. The solution set of the system of equations log3x + log3y = 2 + log32 and log27(x + y) = (A) {6, 3} 9. (D) an integer The equation logx2 16 + log2x64 = 3 has : (A) one irrational solution (C) two real solutions 7. (C) a rational number (B) {3, 6} Consider the quadratic equation, quantities are irrational. (A) sum of the roots (C) sum of the coefficients (C) {6, 12} 2 is : 3 (D) {12, 6} (log108)x2 – (log105)x = 2(log210)–1 – x. Which of the following (B) product of the roots (D) discriminant 10. If loga x = b for permissible values of a and x then identify the statement(s) which can be correct? (A) If a and b are two irrational numbers then x can be rational. (B) If a rational and b irrational then x can be rational. (C) If a irrational and b rational then x can be rational. (D) If a rational and b rational then x can be rational. 11. Which of the following statements are true (A) log2 3 < log12 10 (C) log326 < log29 1 If  log0.1 x  2, then 2 1 (A) maximum value of x is 10 1 (C) minimum value of x is 10 12. (B) log6 5 < log7 8 (D) log1615 > log1011 > log76 (B) x lies between 1 and 100 (D) minimum value of x is 1 10 1 100 Fundamentals of Mathematics-I PART - IV : COMPREHENSION Comprehension # 1 (1 to 3) In a group of 1000 people, there are 750 people, who can speak Hindi and 400 people, who can speak Bengali. 1. 2. 3. Number of people who can speak Hindi only is (A) 300 (B) 400 (C) 500 (D) 600 Number of people who can speak Bengali only is (A) 150 (B) 250 (C) 50 (D) 100 Number of people who can speak both Hindi and Bengali is (A) 50 (B) 100 (C) 150 (D) 200 Comprehension # 2 (4 to 6) 1 4 + = 3. 1 + log4 x 5 − 4log4 x B denotes the value of the product of m and n, if 2m = 3 and 3n = 4. 3 C denotes the sum of the integral roots of the equation log 3x   + (log3x)2 = 1. x The value of A + B equals (A) 10 (B) 6 (C) 8 (D) 4 Let 4. 5. 6. A denotes the sum of the roots of the equation The value of B + C equals (A) 6 (B) 2 (C) 4 (D) 8 The value of A + C  B equals (A) 5 (B) 8 (C) 7 (D) 4 Comprehension # 3 (Q.7- to Q.9) A function f(x) = ax(a > 0, a  1, x  R) is called an exponential function. Graph of exponential function can be as follows : Case -  For a > 1 7*. Case -  For 0 < a < 1 Which of the following is correct : (A) (B) Y y=3 Y –x y=3–x y=2–x (C) X (D) y=2 –x X Fundamentals of Mathematics-I Number of solutions of 3x + x – 2 = 0 is/are : (A) 1 (B) 2 8. 9. (C) 3 (D) 4 The number of positive solutions of log1/2x = 7x is/are : (A) 0 (B) 1 (C) 2 (D) 3 * Marked Questions may have more than one correct option.  Marked questions are recommended for Revision PART - I : JEE (ADVANCED) / IIT-JEE PROBLEMS (PREVIOUS YEARS) 1*. Indicate all correct alternatives, where base of the log is 2. [JEE ’89 ] ( 3/ 4) (l og x )² + l o g x − (5 /4 ) The equation x = 2 has : (A) at least one real solution (B) exactly three real solutions (C) exactly one irrational solution (D) complex roots 2. The number log 2 7 is : (A) an integer (C) an irrational number 3. (B) a rational number (D) a prime number Find all real numbers x which satisfy the equation 2 log 2 log 2 x + log 1/ 2 log 2 (2 2 x) = 1. [REE – 1999, 6 ] 4. Solve the equation log 3/ 4 log 8 (x 2 + 7) + log 1/ 2 log 1/ 4 (x 2 + 7) 5. The number of solution(s) of log 4 (x – 1) = log 2 (x – 3) is/are (A) 3 (B) 1 (C) 2 6. Let f(x) = x 2 − 6x + 5 8. = − 2. [REE–2000,5] [IIT-JEE-2002, Scr., (1, 0)/35] (D) 0 Column –  (A) If – 1 < x < 1, then f(x) satisfies (p) 0 < f(x) < 1 (B) If 1 < x < 2, then f(x) satisfies (q) f(x) < 0 (C) If 3 < x < 5, then f(x) satisfies (r) f(x) > 0 (D) If x > 5, then f(x) satisfies (s) f(x) < 1 Let (x0, y0) be the solution of the following equations (2x) n2 = (3y) n3 3 nx = 2 ny . Then x0 is 1 (A) 6 −1 [IIT-JEE 2007, Paper-2, (6, 0), 81] x 2 − 5x + 6 Column –  7. [JEE ’90 ] 1 (B) 3 1 (C) 2  1 1 1 1 4– 4– 4– .... The value of 6 + log 3   3 2 3 2 3 2 3 2 2  [IIT-JEE 2011, Paper-1, (3, –1), 80] (D) 6   is [IIT-JEE 2012, Paper-1, (4, 0), 70]   Fundamentals of Mathematics-I 9*. 10. If 3x = 4x – 1 , then x = 2log3 2 (A) 2log3 2 – 1 (B) [JEE (Advanced) 2013, Paper-2, (3, –1)/60] 2log2 3 1 (C) (D) 1– log4 3 2log2 3 – 1 2 2 – log2 3 1 2 log2 (log2 9) The value of ((log 2 9) )  ( 7) 1 log4 7 is ________.[JEE(Advanced) 2018, Paper-1,(4, –2)/60] PART - II : PREVIOUS YEARS PROBLEMS OF MAINS LEVEL 1. If log p x =  and log q x = , then the value of log p / q x is – (1)  – (2)  [KCET-1997]  (4) –  (3) – 2. If log x a, a x / 2 and log b x are in G.P. Then x is equal to [KCET-1998] (1) log a (log b a) (2) log a (log e a)+ log a log b b (3) – log a (log a b) (4) none of these 3. If log x 256 = 8/5, then x is equal to (1) 64 (2) 16 [KCET-2000] (3) 32 (4) 8 4. If log 2, log(2 x – 1) and log (2 x + 3) are in A.P., then x is equal to [KCET-2000] (1) 5/2 (2) log 2 5 (3) log 2 3 (4) log 3 2 5. The number log 2 7 is (1) an integer (3) an irrational [DCE-2000] (2) a rational (4) a prime number 6. The roots of the equation log 2 (x 2 – 4x + 5) = (x – 2) are (1) 4, 5 (2) 2, – 3 (3) 2, 3 7. If x = 198 !, then value of the expression (1) –1 (2) 0 [KCET-2001] (4) 3, 5 1 1 1 + + .... + equals log2 x log3 x log198 x (3) 1 (4) 198 [DCE-2005] 8. If log 0 . 3 (x – 1) < log 0 . 0 9 (x – 1), then x lies in the intervel (1) (2, ) (2) (1, 2) (3) (–2, –1) 9. If A, B and C are three sets such that A  B = A  C and A  B = A  C, then [AIEEE-2009, (4, – 1), 144] (1) A = C (2) B = C (3) A  B =  (4) A = B 10. Let X = {1, 2, 3, 4, 5}. The number of different ordered pairs (Y, Z) that can formed such that Y  X, Z  X and Y  Z is empty, is : [AIEEE-2012, (4, – 1), 120] (1) 52 (2) 35 (3) 25 (4) 53 11. If X = {4n – 3n – 1 : n  N} and Y = {9(n – 1) : n  N}, where N is the set of natural numbers, then X  Y is equal to [JEE(Main)2014,(4, – 1), 120] (1) X (2) Y (3) N (4) Y – X 12. The sum of all real values of x satisfying the equation ( x 2 – 5x +5)x [DCE-2006] (4) none of these 2 + 4 x – 60 = 1 is JEE(Main)2016,(4,–1), 120] (1) – 4 (2) 6 (3) 5 (4) 3 Fundamentals of Mathematics-I 13. In a class 140 students numbered 1 to 140, all even numbered students opted Mathematics course, those whose number is divisible by 3 opted Physics course and those whose number is divisible 5 opted Chemistry course. Then the number of student who did not opt for any of the three courses is : [JEE(Main) 2019, Online (10-01-19),P-1 (4, – 1), 120] (1) 38 (2) 42 (3) 102 (4) 1 EXERCISE – 1 PART– I Section (A) : A-1. (i) Yes (ii) Yes A-2. (i) {2,3,5,7} A-3. (i) {x : x = A-4. (i) (iii) Finite Singleton & finite (iii) Yes (ii) {3, 4, 5, 6, 7, 8, 9} p , p  , q  N} q (iv) No (ii) {x : x =  + 1,   N} (ii) (iv) Finite and empty Infinite A-5. {, {}, {{}}, A} Section (B) : B-1. (i) B-2. {3, 5, 9} B-4. (i) (ii) {1, 2, 3, 4} B-3. (ii) {1, 2, 3} (iii) {4, 5, 6} d = bc minimum n(A  B) = 0, maximum n(A  B) = 3 minimum n(A  B) = 6, maximum n(A  B) = 9 Section (C) : C-1. 300 C-2. 25 C-3. 3300 Section (D) : (i) (–, –2)(–2, –1/2)(1, ) (ii) ( −, − 1)  ( 5,  ) (iii) [− 2, −1)  (−1, 2  3, 4) (iv) (– , – 2]  (–1, 4) (v) (–, –5),  (1, 2)  (6, ) D-2. (i) (–17/25, –3/8) (ii) ( −, − 20 )  ( 23,  ) (iii) 1   2 , 3 (iv) D-3. (i) 4 (ii) 7 (iii) (ii) –72 D-1. 9 (iv) 2 (v) (iii) (vii) +ve +ve (iv) (viii) +ve – ve (–, –7),  (–4, –2) 0 Section (E) : E-1. (i) 1 E-2. (i) (v) +ve +ve (ii) – ve (vi) – ve (ix) – ve Fundamentals of Mathematics-I E-3. (i) b – 2a (ii) a + 3b (iii) 2b2 + 3a2 ab (iv) 4(2a + b) 1 − a + 2b E-4. (i) 2 (ii) 1 (iii) 7+ 1 196 (iv) 0 E-5. (i) 1 (ii) 89 E-7. –1 E-9. (i) 3 (ii) ±2 (iii) 3 2 (v) 16 (vi) 8 (vii) {1/3} (viii) {–4} (ix) no root (i) (2) (ii) 10 or {10 −5 , 103} (iv) 9 (v) log26 F I (iv) G H J K ( −1, 0) (1, 2) E-10. 1 100 (iv) (iii) Section (F) : F-1. 1 3  1  1 (i)  − , −    ,1 (ii) (1, 2)  (3, 4) (iii) − , 2 2 4 4     (v) (–, 2) 3  (vi)  , 1  (1, 3  4  F-2. 1 F-3. 2 F-4. (i) (− , − 1)  (1, ) (ii) x  (– 2, –1)  (– 1, 0)  (0, 1)  (2, ) PART– II Section (A) : A-1. (D) A-2. (A) A-3. (D) A-4. (A) A-5. (D) A-6. (B) B-3. (D) B-4. (B) B-5. (A) B-6. (B) Section (B) : B-1. (B) B-2. (A) B-7. (A) B-8. (D) C-2. (C) C-3. (C) C-4. (B) D-2. (D) D-3. (B) D-4. (D) D-5. (D) D-6. (D) D-7. (D) (A) E-2. (D) E-3. (B) E-4. (D) E-5. (D) E-6. (D) E-7. (D) E-8. (A) E-9. (A) E-10. (D) E-11. (D) E-12. (C) E-13. (B) E-14. (A) E-15. E-16. (B) E-17. (C) E-18. (B) F-2. (A) F-3. (B) F-4. (A) F-5. (D) F-6. (D) Section (C) : C-1. (B) Section (D) : D-1. (B) Section (E) : E-1. (D) Section (F) : F-1. (C) Fundamentals of Mathematics-I PART– III 1. (A) → (q), (B) → (r), (C) → (s), (D) → (p) 2. (A) → r, (B) → s, (C) → p, (D) → p, (E) → q, (F) → q 3. (A) → r, (B) → p, (C) → s, (D) → q 4. (A) → r, (B) → s, (C) → p, (D) → q EXERCISE – 2 PART - I 1. (A) 2. (B) 3. (B) 4. (C) 5. (B) 6. (B) 7. (C) 8. (A) 9. (B) 10. (A) 11. (A) 12. (D) 13. (B) 14. (A) 15. (A) PART - II 1. 32 2. 52 3. 1 4. 22 5. 0 6. 1 7. 34 8. 2 9. 1 10. 10 11. 16 12. 3 13. 10 14. 4 15. 93 16. 1 17. 10 18. 1 19. 6 20. 59 7*. (BC) 7. (3) PART - III 1. (AD) 2. (AB) 3. (BD) 4. (ABCD) 5. (ABC) 6. (ABCD) 7. (ABCD) 8. (AB) 9. (CD) 10. (ABCD) 11. (BC) 12. (ABD) (A) 6. (B) x = 3 or –3 5. (B) PART - IV 1. (D) 2. (B) 8. (A) 9. (B) 3. (C) 4. (C) 5. EXERCISE – 3 PART– I 1*. (ABCD) 6. (A) → (p), (r), (s) 7. (C) 8. 2. (C) ; 4 3. (B) → (q), (s) 9*. 8 ; 4. (C) → (q), (s) (A, B, C) 10. ; (D) → (p), (r), (s) 8 PART - II 1. (4) 2. (1) 3. (3) 4. (2) 5. (3) 6. (3) 8. (1) 9. (2) 10. (2) 11. (2) 12. (4) 13. (1) Fundamentals of Mathematics-I LOGARITHM TABLE 10 11 12 13 14 15 16 17 18 19 0 0000 1 0043 2 0086 3 0128 4 0170 0414 0453 0492 0531 0569 0792 0828 0864 0899 0934 1139 1461 1761 1173 1492 1790 1206 1523 1818 1239 1553 1847 2122 2148 2304 2330 2355 2380 2405 2788 2810 2833 2625 2856 8 9 0374 123 5 9 13 4 8 12 456 17 21 26 16 20 24 789 30 34 38 28 32 36 0212 0253 0294 0334 16 20 23 15 18 22 27 31 35 26 29 33 0607 0645 0682 0719 0755 4 8 12 4 7 11 0969 1004 1038 1072 1106 3 7 11 3 7 10 14 18 21 14 17 20 25 28 32 24 27 31 1303 1335 1367 1399 1430 36 10 37 10 13 16 19 13 16 19 23 26 29 22 25 29 1614 1644 1673 1703 1732 36 9 36 9 12 15 19 12 14 17 22 25 28 22 25 26 1903 1931 1959 1987 2014 369 368 11 14 17 11 14 17 20 23 26 19 22 25 11 14 16 10 13 16 19 22 24 18 21 23 1875 2095 2601 7 1584 2068 2577 6 1271 2041 2553 5 2175 2201 2227 2253 2279 368 358 2430 2455 2480 2504 2529 358 358 10 13 15 10 12 15 18 20 23 17 20 22 2672 2695 2718 2742 2765 257 247 9 12 14 9 11 14 17 19 21 16 18 21 2923 3139 2945 3160 2967 3181 2989 3201 247 246 9 11 13 8 11 13 16 18 20 15 17 19 2648 2878 20 3010 3032 3054 3075 3096 2900 3118 246 8 11 13 15 17 19 21 3222 3243 3263 3284 3304 3324 3345 3365 3385 3404 246 8 10 12 14 16 18 22 3424 3444 3464 3483 3502 3522 3541 3560 3579 3598 246 8 10 12 14 15 17 23 3617 3636 3655 3674 3692 3711 3729 3747 3766 3784 246 7 9 11 13 15 17 24 3802 3820 3838 3856 3874 3892 3909 3927 3945 3962 245 7 9 11 12 14 16 25 3979 3997 4014 4031 4048 4065 4082 4099 4116 4133 235 7 9 10 12 14 15 26 4150 4166 4183 4200 4216 4232 4249 4265 4281 4298 235 7 8 10 11 13 15 27 4314 4330 4346 4362 4378 4393 4409 4425 4440 4456 235 689 11 13 14 28 4472 4487 4502 4518 4533 4548 4564 4579 4594 4609 235 689 11 12 14 29 4624 4639 4654 4669 4683 4698 4713 4728 4742 4757 134 679 10 12 13 30 4771 4786 4800 4814 4829 4843 4857 4871 4886 4900 134 679 10 11 13 31 4914 4928 4942 4955 4969 4983 4997 5011 5024 5038 134 678 10 11 12 32 5051 5065 5079 5092 5105 5119 5132 5145 5159 5172 134 578 9 11 12 33 5185 5198 5211 5224 5237 5250 5263 5276 5289 5302 134 568 9 10 12 34 5315 5328 5340 5353 5366 5378 5391 5403 5416 5428 134 568 9 10 11 35 5441 5453 5465 5478 5490 5502 5514 5527 5539 5551 124 567 9 10 11 36 5563 5575 5587 5599 5611 5623 5635 5647 5658 5670 124 567 8 10 11 37 5682 5694 5705 5717 5729 5740 5752 5763 5775 5786 123 567 8 9 10 38 5798 5809 5821 5832 5843 5855 5866 5877 5888 5899 123 567 8 9 10 39 5911 5922 5933 5944 5955 5966 5977 5988 5999 6010 123 457 8 9 10 40 6021 6031 6042 6053 6064 6075 6085 6096 6107 6117 123 456 8 9 10 41 6128 6138 6149 6160 6170 6180 6191 6201 6212 6222 123 456 789 42 6232 6243 6253 6263 6274 6284 6294 6304 6314 6325 123 456 789 43 6335 6345 6355 6365 6375 6385 6395 6405 6415 6425 123 456 789 44 6435 6444 6454 6464 6474 6484 6493 6503 6513 6522 123 456 789 45 6532 6542 6551 6561 6571 6580 6590 6599 6609 6618 123 456 789 46 6628 6637 6646 6656 6665 6675 6684 6693 6702 6712 123 456 778 47 6721 6730 6739 6749 6758 6767 6776 6785 6794 6803 123 455 678 48 6812 6821 6830 6839 6848 6857 6866 6875 6884 6893 123 445 678 49 6902 6911 6920 6928 6937 6946 6955 6964 6972 6981 123 445 678 Fundamentals of Mathematics-I LOGARITHM TABLE 50 0 6990 1 6998 2 7007 3 7016 4 7024 5 7033 6 7042 7 7050 8 7059 9 7067 123 123 456 345 789 678 51 7076 7084 7093 7101 7110 7118 7126 7135 7143 7152 123 345 678 52 7160 7168 7177 7185 7193 7202 7210 7218 7226 7235 122 345 677 53 7243 7251 7259 7267 7275 7284 7292 7300 7308 7316 122 345 667 54 7324 7332 7340 7348 7356 7364 7372 7380 7388 7396 122 345 667 55 7404 7412 7419 7427 7435 7443 7451 7459 7466 7474 122 345 567 56 7482 7490 7497 7505 7513 7520 7528 7536 7543 7551 122 345 567 57 7559 7566 7574 7582 7589 7597 7604 7612 7619 7627 122 345 567 58 7634 7642 7649 7657 7664 7672 7679 7686 7694 7701 112 344 567 59 7709 7716 7723 7731 7738 7745 7752 7760 7767 7774 112 344 567 60 7782 7789 7796 7803 7810 7818 7825 7832 7839 7846 112 344 566 61 7853 7860 7868 7875 7882 7889 7896 7903 7910 7917 112 344 566 62 7924 7931 7938 7945 7952 7959 9766 7973 7980 7987 112 334 566 63 7993 8000 8007 8014 8021 8028 8035 8041 8048 8055 112 334 556 64 8062 8069 8075 8082 8089 8096 8102 8109 8116 8122 112 334 556 65 8129 8136 8142 8149 8156 8162 8169 8176 8182 8189 112 334 556 66 8195 8202 8209 8215 8222 8228 8235 8241 8248 8254 112 334 556 67 8261 8267 8274 8280 8287 8293 8299 8306 8312 8319 112 334 556 68 8325 8331 8338 8344 8351 8357 8363 8370 8376 8382 112 334 456 69 8388 8395 8401 8407 8414 8420 8426 8432 8439 8445 112 234 456 70 8451 8457 8463 8470 8476 8482 8488 8494 8500 8506 112 234 456 71 8513 8519 8525 8531 8537 8543 8549 8555 8561 8567 112 234 455 72 8573 8579 8585 8591 8597 8603 8609 8615 8621 8627 112 234 455 73 8633 8639 8645 8651 8657 8663 8669 8675 8681 8686 112 234 455 74 8692 8698 8704 8710 8716 8722 8727 8733 8739 8745 112 234 455 75 8751 8756 8762 8768 8774 8779 8785 8791 8797 8802 112 233 455 76 8808 8814 8820 8825 8831 8837 8842 8848 8854 8859 112 233 455 77 8865 8871 8876 8882 8887 8893 8899 8904 8910 8915 112 233 445 78 8921 8927 8932 8938 8943 8949 8954 8960 8965 8971 112 233 445 79 8976 9882 8987 8993 8998 9004 9009 9015 9020 9025 112 233 445 80 9031 9036 9042 9047 9053 9058 9063 9069 9074 9079 112 233 445 81 9085 9090 9096 9101 9106 9112 9117 9122 9128 9133 112 233 445 82 9138 9143 9149 9154 9159 9165 9170 9175 9180 9186 112 233 445 83 9191 9196 9201 9206 9212 9217 9222 9227 9232 9238 112 233 445 84 9243 9248 9253 9258 9263 9269 9274 9279 9284 9289 112 233 445 85 9294 9299 9304 9309 9315 9320 9325 9330 9335 9340 112 233 445 86 9345 9350 9355 9360 9365 9370 9375 9380 9385 9390 112 233 445 87 9395 9400 9405 9410 9415 9420 9425 9430 9435 9440 011 223 344 88 9445 9450 9455 9460 9465 9469 9474 9479 9484 9489 011 223 344 89 9494 9499 9504 9509 9513 9518 9523 9528 9533 9538 011 223 344 90 9542 9547 9552 9557 9562 9566 9571 9576 9581 9586 011 223 344 91 9590 9595 9600 9605 9609 9614 9619 9624 9628 9633 011 223 344 92 9638 9643 9647 9652 9657 9661 9666 9671 9675 9680 011 223 344 93 9685 9689 9694 8699 9703 9708 9713 9717 9722 9727 011 223 344 94 9731 9736 9741 9745 9750 9754 9759 9763 9768 9773 011 223 344 95 9777 9782 9786 9791 9795 9800 9805 9809 9814 9818 011 223 344 96 9823 9827 9832 9836 9841 9845 9850 9854 9859 9863 011 223 344 97 9868 9872 9877 9881 9886 9890 9894 9899 9903 9908 011 223 344 98 9912 9917 9921 9926 9930 9934 9939 9943 9948 9952 011 223 344 99 9956 9961 9965 9969 9974 9978 9983 9987 9991 9996 011 223 344 Fundamentals of Mathematics-I ANTILOGARITHM TABLE .00 .01 .02 .03 .04 .05 .06 .07 .08 .09 .10 .11 .12 .13 .14 .15 .16 .17 .18 .19 .20 .21 .22 .23 .24 .25 .26 .27 .28 .29 .30 .31 .32 .33 .34 .35 .36 .37 .38 .39 .40 .41 .42 .43 .44 .45 .46 .47 .48 .49 0 1000 1023 1047 1072 1096 1122 1148 1175 1202 1230 1259 1288 1318 1349 1380 1413 1445 1479 1514 1549 1585 1622 1660 1698 1738 1778 1820 1862 1905 1950 1995 2042 2089 2138 2188 2239 2291 2344 2399 2455 2512 2570 2630 2692 2754 2818 2884 2951 3020 3090 1 1002 1026 1050 1074 1099 1125 1151 1178 1205 1233 1262 1291 1321 1352 1384 1416 1449 1483 1517 1552 1589 1626 1663 1702 1742 1782 1824 1866 1910 1954 2000 2046 2094 2143 2193 2244 2296 2350 2404 2460 2518 2576 2636 2698 2761 2825 2891 2958 3027 3097 2 1005 1028 1052 1076 1102 1127 1153 1180 1208 1236 1265 1294 1324 1355 1387 1419 1452 1486 1521 1556 1592 1629 1667 1706 1746 1786 1828 1871 1914 1959 2004 2051 2099 2148 2198 2249 2301 2355 2410 2466 2523 2582 2642 2704 2767 2831 2897 2965 3034 3105 3 1007 1030 1054 1079 1104 1130 1156 1183 1211 1239 1268 1297 1327 1358 1390 1422 1455 1489 1524 1560 1596 1633 1671 1710 1750 1791 1832 1875 1919 1963 2009 2056 2104 2153 2203 2254 2307 2360 2415 2472 2529 2588 2649 2710 2773 2838 2904 2972 3041 3112 4 1009 1033 1057 1081 1107 1132 1159 1186 1213 1242 1271 1300 1330 1361 1393 1426 1459 1493 1528 1563 1600 1637 1675 1714 1754 1795 1837 1879 1923 1968 2014 2061 2109 2158 2208 2259 2312 2366 2421 2477 2535 2594 2655 2716 2780 2844 2911 2979 3048 3119 5 1012 1035 1059 1084 1109 1135 1161 1189 1216 1245 1274 1303 1334 1365 1396 1429 1462 1496 1531 1567 1603 1641 1679 1718 1758 1799 1841 1884 1928 1972 2018 2065 2113 2163 2213 2265 2317 2371 2432 2483 2541 2600 2661 2723 2786 2851 2917 2985 3055 3126 6 1014 1038 1062 1086 1112 1138 1164 1191 1219 1247 1276 1306 1337 1368 1400 1432 1466 1500 1535 1570 1607 1644 1683 1722 1762 1803 1845 1888 1932 1977 2023 2070 2118 2168 2218 2270 2323 2377 2427 2489 2547 2606 2667 2729 2793 2858 2924 2992 3062 3133 7 1016 1040 1064 1089 1114 1140 1167 1194 1222 1250 1279 1309 1340 1371 1403 1435 1469 1503 1538 1574 1611 1648 1687 1726 1766 1807 1849 1892 1936 1982 2028 2075 2123 2173 2223 2275 2328 2382 2432 2495 2553 2612 2673 2735 2799 2864 2931 2999 3069 3141 8 1019 1042 1067 1091 1117 1143 1169 1197 1225 1253 1282 1312 1343 1374 1406 1439 1472 1507 1542 1578 1614 1652 1690 1730 1770 1811 1854 1897 1841 1986 2032 2080 2128 2178 2328 2280 2333 2388 2443 2500 2559 2618 2679 2742 2805 2871 2938 3006 3076 3148 9 1021 1045 1069 1094 1119 1146 1172 1199 1227 1256 1285 1315 1346 1377 1409 1442 1476 1510 1545 1581 1618 1656 1694 1734 1774 1816 1858 1901 1845 1991 2037 2084 2133 2183 2234 2286 2339 2393 2449 2506 2564 2624 2685 2748 2812 2877 2944 3013 3083 3155 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 456 111 111 111 111 112 112 112 112 112 112 112 122 122 122 122 122 122 122 122 122 122 222 222 222 222 222 223 223 223 223 223 223 223 223 233 233 233 233 233 233 234 234 234 334 334 334 334 334 344 344 789 222 222 222 222 222 222 222 222 223 223 223 223 223 233 233 233 233 233 233 333 333 333 333 334 334 334 334 334 344 344 344 344 344 344 445 445 445 445 445 455 455 455 456 456 456 556 556 556 566 566 Fundamentals of Mathematics-I ANTILOGARITHM TABLE .50 0 3162 1 3170 2 3177 3 3184 4 3192 5 3199 6 3206 7 3214 8 3221 9 3228 123 112 456 344 789 567 .51 3236 3243 3251 3258 3266 3273 3281 3289 3296 3304 122 345 567 .52 3311 3319 3327 3334 3342 3350 3357 3365 3373 3381 122 345 567 .53 3388 3396 3404 3412 3420 3428 3436 3443 3451 3459 122 345 667 .54 3467 3475 3483 3491 3499 3508 3516 3524 3532 3540 122 345 667 .55 3548 3556 3565 3573 3581 3589 3597 3606 3614 3622 122 345 667 .56 3631 3639 3648 3656 3664 3673 3681 3690 3698 3707 123 345 678 .57 3715 3724 3733 3741 3750 3758 3767 3776 3784 3793 123 345 678 .58 3802 3811 3819 3828 3837 3846 3855 3864 3873 3882 123 445 678 .59 3890 3899 3908 3917 3926 3936 3945 3954 3963 3972 123 455 678 .60 3981 3990 3999 4009 4018 4027 4036 4046 4055 4064 123 456 678 .61 4074 4083 4093 4102 4111 4121 4130 4140 4150 4156 123 456 789 .62 4169 4178 4188 4198 4207 4217 4227 4236 4246 4256 123 456 789 .63 4266 4276 4285 4295 4305 4315 4325 4335 4345 4355 123 456 789 .64 4365 4375 4385 4395 4406 4416 4426 4436 4446 4457 123 456 789 .65 4467 4477 4487 4498 4508 4519 4529 4539 4550 4560 123 456 789 .66 4571 4581 4592 4603 4613 4624 4634 4645 4656 4667 123 456 7 9 10 .67 4677 4688 4699 4710 4721 4732 4742 4753 4764 4775 123 457 8 9 10 .68 4786 4797 4808 4819 4831 4842 4853 4864 4875 4887 123 467 8 9 10 .69 4898 4909 4920 4932 4943 4955 4966 4977 4989 5000 123 567 8 9 10 .70 5012 5023 5035 5047 5058 5070 5082 5093 5105 5117 124 567 8 9 11 .71 5129 5140 5152 5164 5176 5188 5200 5212 5224 5236 124 567 8 10 11 .72 5248 5260 5272 5284 5297 5309 5321 5333 5346 5358 124 567 9 10 11 .73 5370 5383 5395 5408 5420 5433 5445 5458 5470 5483 134 568 9 10 11 .74 5495 5508 5521 5534 5546 5559 5572 5585 5598 5610 134 568 9 10 12 .75 5623 5636 5649 5662 5675 5689 5702 5715 5728 5741 134 578 9 10 12 .76 5754 5768 5781 5794 5808 5821 5834 5848 5861 5875 134 578 9 11 12 .77 5888 5902 5916 5929 5943 5957 5970 5984 5998 6012 134 578 10 11 12 .78 6026 6039 6053 6067 6081 6095 6109 6124 6138 6152 134 678 10 11 13 .79 6166 6180 6194 6209 6223 6237 6252 6266 6281 6295 134 679 10 11 13 .80 6310 6324 6339 6353 6368 6383 6397 6412 6427 6442 134 679 10 12 13 .81 6457 6471 6486 6501 6516 6531 6546 6561 6577 6592 235 689 11 12 14 .82 6607 6622 6637 6653 6668 6683 6699 6714 6730 6745 235 689 11 12 14 .83 6761 6776 6792 6808 6823 6839 6855 6871 6887 6902 235 689 11 13 14 .84 6918 6934 6950 6966 6982 6998 7015 7031 7047 7063 235 6 8 10 11 13 15 .85 7079 7096 7112 7129 7145 7161 7178 7194 7211 7228 235 7 8 10 12 13 15 .86 7244 7261 7278 7295 7311 7328 7345 7362 7379 7396 235 7 8 10 12 13 15 .87 7413 7430 7447 7464 7482 7499 7516 7534 7551 7568 235 7 9 10 12 14 16 .88 7586 7603 7621 7638 7656 7674 7691 7709 7727 7745 245 7 9 11 12 14 16 .89 7762 7780 7798 7816 7834 7852 7870 7889 7907 7925 245 7 9 11 13 14 16 .90 7943 7962 7980 7998 8017 8035 8054 8072 8091 8110 246 7 9 11 13 15 17 .91 8128 8147 8166 8185 8204 8222 8241 8260 8279 8299 246 8 9 11 13 15 17 .92 8318 8337 8356 8375 8395 8414 8433 8453 8472 8492 2 4 6 8 10 12 14 15 17 .93 8511 8531 8551 8570 8590 8610 8630 8650 8670 8690 2 4 6 8 10 12 14 16 18 .94 8710 8730 8750 8770 8790 8810 8831 8851 8872 8892 2 4 6 8 10 12 14 16 18 .95 8913 8933 8954 8974 8995 9016 9036 9057 9078 9099 2 4 6 8 10 12 15 17 19 .96 9120 9141 9162 9183 9204 9226 9247 9268 9290 9311 2 4 6 8 11 13 15 17 19 .97 9333 9354 9376 9397 9419 9441 9462 9484 9506 9528 2 4 7 9 11 13 15 17 20 .98 9550 9572 9594 9616 9638 9661 9683 9705 9727 9750 2 4 7 9 11 13 16 18 20 .99 9772 9795 9817 9849 9863 9886 9908 9931 9954 9977 2 5 7 9 11 14 16 18 20