Basic Maths & Logarithm Exercise # 2

BASIC MATHS & LOGARITHM Exercise # 2 1. [Multiple Correct Choice Type Questions] Part # I Which of the following are correct ? (A) log319 . log1/73.log41/7 > 2 1 (C) if m = 4 log4 7 and n =   9 (D) log 2. 5   sin   . log 5 4. 5 simplifies to an irrational number 3 x2 log x 3log10 3 10 If x1 and x2 are the solution of the equation x (B) x1 . x2 = x1 + x2 If x (B) 3 / 4 (lo g 3 x ) 2  lo g 3 x  5 / 4 Let N = 1 5 (C) 5 (D) none of these = 3 , then x has (B) one irrational value (D) none of these     4  (B) 2 sin2      4  (C) 1  sin 2 (D) 2 cos2  (B) a prime number (C) a rational number (D) an integer If ax = b, by = c, cz = a and x = log b a 2 ; y  log c b3 & z  log a c k , where a, b, c > 0 & a, b, c  1, then k is equal to (A) 10. (D) log (x1 . x2) = 0 log 3 135 log 3 5 – . Then N is : log15 3 log 405 3 (A) a natural number 9. (C) log x2 x1 = –1 If sin2  = sin cos then cos2 has the value equal to : (A) 1 + sin 2 8. = 100 3 10 then - If logkx. log5k = logx5, k  1, k > 0, then x is is equal to - (A) one positive integral value (C) two positive rational values 7. (D) 2  1   log 2 y  = log2x and log2x.(log3(x + y)) = 3 log3x is The solution set of the system of equations, log12x   log x 2  (A) x = 6 ; y = 2 (B) x = 4 ; y = 3 (C) x = 2 ; y = 6 (D) x = 3 ; y = 4 (A) k 6. then n = m4 y = x, then ' y ' cannot take the value(s) : x (B) 0 (C) 1 (A) x1x2 = 1 5.  5  2 log 3 7 If x & y are real numbers and (A) – 1 3. sin (B) log5 (1/23) lies between – 2 & – 1 1 5 (B) 1 6 (C) log 64 2 (D) log 32 2 Which of the following statements are true (A) log2 3 < log12 10 (B) log6 5 < log7 8 (C) log326 < log29 (D) log1615 > log1011 > log76 11. 12. If a  0, then the inequation |x – a| + |x + a| < b (A) has no solutions if b  2 |a|  b b  (B) has a solution set  ,  if b > 2 |a|  2 2  b b  (C) has a solution set  ,  if b < 2 |a|  2 2 (D) All above Which of the following when simplified reduces to an integer ? (A) 13. 14. 2log 6 log12  log 3 (B) (C) log5 16  log5 4 log5 128 1 (D) log1/4    16   8  log 8  2  x  The equation = 3 has  log8 x 2 (B) one natural solution (C) two real solution 5 Values of x satisfying the equation log52 x + log5x   = 1 are x 1 (A) 1 (B) 5 (C) 25 (D) one irrational solution (D) 3 16. If y = log7–a (2x2 + 2x + a + 3) is defined  x  R, then possible integral value(s) of a is/are (A) – 3 (B) – 2 (C) 4 (D) 5 17. If |z1 + z2|2 = |z1|2 + |zz|2 then (A) z1 is purely real z2 (B) (C) z1 z 2  z 2 z1 = 0 18. 3 1 2 + – log 3 2 log 9 4 log 27 8 (C) – log8 log4 log2 16 20. z1 is purely imaginary z2 (D) amp  z1 may be equal to 2 z2 Which of the following when simplified, vanishes ? (A) 19. 2 Difference of squares of two distinct odd natural numbers is always a multiple of. (A) 4 (B) 3 (C) 6 (D) 8 (A) no integral solution 15. log 32 log 4 2 (B) log2   + log4 3 9   4 (D) log10 cot 1° + log10cot 2° + log10 cot 3° + ......... + log10 cot 89° Which of the following when simplified, reduces to unity ? 2 (A) log105 . log1020 + log10 2 (B) 2 log 2  log 3 log 48  log 4 (C) – log5 log3 (D) 1 log 6 5 9 3 2  64     27  The inequality [2 – x] + 2 [x – 1]  0 is satisfied by (where [.] denotes greatest integer function) : (A) x  {0} (B) x  W (C) x  N (D) x  [1, ) BASIC MATHS & LOGARITHM 21. 22. 23. The number N = 1  2 log 3 2 + log26 2 when simplified reduces to (1  log 3 2) 2 (A) a prime number (C) a real which is less than log3 (B) an irrational number (D) a real which is greater than log76 The equation log x2 16 + log2x64 = 3 has : (A) one irrational solution (C) two real solutions (B) no prime solution (D) one integral solution If 1  log0.1 x  2, then 2 1 10 (A) maximum value of x is (C) minimum value of x is 24. (B) x lies between 1 10 (D) minimum value of x is If p, q  N satisfy the equation x x  x = x {x} > 1 are (where {.} denotes the fractional part function) 3 1 2 (B) –1 + P P P 26. The expression, logp logp 1 100 (B) twin prime (D) if logqp is defined then logpq is not & vice versa 1 Values of x satisfying   3 (A)  1 10 then p & q are - (A) relatively prime (C) coprime 25. 1 and 100 (C) 2 + P ....... P 1 3 9 (D) e 2 where p  2, p  N, when simplified is - n radical sign (A) independent of p, but dependent on n (B) independent of n, but dependent on p (C) dependent on both p & n (D) negative 27. The solution set of the system of equations log3x + log3y = 2 + log32 and log27(x + y) = (A) {6, 3} 28. (B) {3, 6} (C) {6, 12} (D) {12, 6} (C) log10log109 (D) log10sin25° Which of the following numbers are positive ? 1 (A) log 2   log 3  2  29. 2 is : 3 2 (B) log2   3 9   2  (log 3 x )  2 log 3 x  5   The equation x  (A) exactly three real solution (C) exactly one irrational solution 2 / 3 = 3 3 has (B) at least one real solution (D) complex roots. 30. Let y = cos x  cos 2x  cos3x  cos 4x  cos5x  cos 6x  cos 7x then which of the following hold good? sin x  sin 2x  sin 3x  sin 4x  sin 5x  sin 6x  sin 7x (A) The value of y when x = /8 is not defined. (B) The value of y when x = /16 is 1. (C) The value of y when x = /32 is 2 1. (D) The value of y when x = /48 is 2  3 . [Assertion & Reason Type Questions] Part # II These questions contains, Statement I (assertion) and Statement II (reason). (A) (B) (C) (D) Statement-I is true, Statement-II is true ; Statement-II is correct explanation for Statement-I. Statement-I is true, Statement-II is true ; Statement-II is NOT a correct explanation for statement-I. Statement-I is true, Statement-II is false. Statement-I is false, Statement-II is true. 1. Statement-I : Minimum value of |x – 2| + |x – 5| + |x + 3| is 8. Statement-II : If a < b < c, then the minimum value of |x – a| + |x – b| + |x – c| is |b – a| + |b – c|. 2. Statement-I : If a = y2, b = z2, c = x2, then 8logax3 . logb y3 .logcz3 = 27 Statement-II : logba. logcb = logca, also logba = 3. 1 log a b Statement-I : [x] + [– x] = x2 – 5x + 6 has only two real solution.  1 , x  I Statement-II : [x] + [–x] =  0 , x  I  4. Statement-I : If log (log 5 x) 5 = 2, then x = 5 5 Statement-II : logxa = b, if a > 0, then x = a1/b 5. Statement-I : The equation log 1 (5  x 2  |x | 2 (15  x ) has real solutions. ) = log (3  x2 ) Statement-II : log1/ba = –logba (where a,b > 0 and b  1) and if number and base both are greater than unity then the number is positive. EXERCISE - 2 : PART # I 1. 10. 19. 28. ABC BC ABC AB 2. 11. 20. 29. 1. A AB AB ABCD ABCD 2. 3. 12. 21. 30. B AC AD CD BD 3. A 4. ACD 5. BC 13. AD 14. BC 22. ABCD 23. ABD 6. ABC 15. ABC 24. ACD PART - II 4. A 5. D 7. BCD 16. BCD 25. ACD 8. ABCD 9. BC 17. BCD 18. ABCD 26. AD 27. AB