Tiêu đề 01. Logarithm & Modulus Function.pdf ✅

(A) b a –1 (B) b 1– a (C) b 1+ a (D) 1– a b [B] Q.15 If log3(x) = p and log7(x) = q, which of the following yields log21(x) (A) pq (B) p q 1 + (C) –1 –1 p q 1 + (D) –1 –1 p q pq + [C] Q.16 The value of p which satisfies the equation 122p–1 = 5 (3p 7 p ) is (A) n21– n12 n5 – n12     (B) n12 – n21 n12 n5    +  (C) n144 – n21 n5 n12    +  (D) n12 – 5 n21 n12    [C] Q.17 If log10(x – 1)3 – 3 log10(x – 3) = log108, then logx625 has the value equal to- (A) 5 (B) 4 (C) 3 (D) 2 [B] Q.18 If log3x = a and log7x = b, then which of the following is equal to log21x ? (A) ab (B) 1 1 a b ab − − + (C) a b 1 + (D) 1 1 a b 1 − − + [D] Q.19 log abc 1 bc + log abc 1 ca + log abc 1 ab has the value equal to (A) 1/2 (B) 1 (C) 2 (D) 4 [B] Q.20 1 log a log c 1 + b + b + 1 log a log b 1 + c + c + 1 log b log c 1 + a + a = (A) abc (B) abc 1 (C) 0 (D) 1 [D] Q.21 log418 is (A) a prime number (B) a rational number (C) an irrational number (D) None of these [C] Q.22 Given that logpx =  and logqx = , then value of logp/q x equals- (A)  −   (B)   −  (C)   −  (D)  −   [A] Q.23 The value of         log 3 1 5 81 + 36 9 log 27 + 9 7 log 4 3 (A) 49 (B) 625 (C) 216 (D) 890 [D] Q.24 If a2 + 4b2 = 12ab, then log (a + 2b) = (A) 2 1 (log a + log b – log 2) (B) log a/2 + log b/2 + log 2 (C) 2 1 (log a + log b + 4 log 2) (D) 2 1 (log a – log b + 4log 2) [C] Q.25 Let N= 409 81 3 log 3 3 log 9 1 5 6 + ( )             − log 6 25 7 25 log 2 7 125 Then log2N has the value – (A) 0 (B) 1 (C) –1 (D) None of these [A] Q.26 The expression logp n radical sign ____________ log ........... p p p p p p   where p  2, p  N ; n  N when simplified is. (A) Independent of p (B) Independent of p and of n (C) dependent on both p & n (D) positive [A]
Q.27 If xn > xn–1 >...> x2 > x1 > 1 then the value of x1 x2 x3 xn log log log ...log 1 x xn 1 xn  − is equal to- (A) 0 (B) 1 (C) 2 (D) None of these Sol.[A] x1 log x2 log x3 log ... xn 1 log −         − − x n x xn 1 log x n . .x1 . n 2 = ............................................. ................................................ ............................................= x x1 log 1 = 1 Q.28 If [log x (log x) 10] 2 3 2 3 x + − = 1/x2 , then x = (A) 9 (B) 81 (C) 3 (D) 2 [A] Q.29 The number of real solution of the equation log10 (7x – 9)2 + log10(3x–4)2 = 2 is (A) 1 (B) 2 (C) 3 (D) 4 [B] Q.30 The ratio 7 a 1 2 3 2a 4 log a log (a 1) log a 49 2 3 27 1/ 4 2 − − − − + simplifies to (A) a 2 – a – 1 (B) a2 + a –1 (C) a2 – a + 1 (D) a2 + a + 1 [D] Q.31 No. of ordered pair satisfying simultaneously the system of equation x 2 . y 2 = 256 & log10 xy – log10 1.5 = 1 is. (A) zero (B) exactly one (C) exactly two (D) None of these [C] Q.32 The value of log a log log N b b b a is- (A) logbN (B) –logbN (C) logN b (D) –logN b [A] Q.33 If ( ) 2 log x b a –5 log a b x + 6 = 0 where a > 0, b > 0 & ab  1. Then the value of x is equal to (A) log a b 2 (B) log b a 3 (C) log 2 a 2 (D) log 3b a [B] Q.34 The solution set of the inequation log1/3 (x2 + x + 1) + 1 > 0 is (A) (–, –2)  (1, + ) (B) [–1, 2] (C) (–2, 1) (D) (–, + ) [C] Q.35 Find the values of x,       −       5 4 log log x 2 5 2 1 2 1 < 1 (A) – 1 < x < – 5 2 , 5 2 < x < 1 (B) – 1 < x < – 0, 5 2 < x < 1 (C) – 1 < x < – 5 2 , 5 2 < x < 3 (D) None of these [A] Q.36 Find the values of x, 1. log x 1 log x 3 log x 3 2  − − + (A) (0, 10) (B) (–1, 10) (C) (–10, 1) (D) None of these [A] Q.37 log4 (2x2 + x + 1) – log2 (2x – 1)  – tan 4 7 (A) x  – 1 (B) x  1 (C) x  – 1 (D) None of these [B] Q.38 x 5 implies log x 5  - (A) x (0, ) (B) x (0, 1/5) (5, ) (C) x  (1, ) (D) x  (1, 2) [B] Q.39 Number of integral values of x for which the inequality log10       + − x 1 2x 2007  0 holds true, is (A) 1004 (B) 1005 (C) 2007 (D) 2008 [B] Q.40 Set of values of x satisfying the inequality 2 2 2 2 (x x 1)(3x 6) (x 3) (2x 5) (x 7) + + + − + −  0 is [a, b)  (b, c] then 2a + b + c is equal to (A) 0 (B) 2 (C) 5 (D) 7 [A]
Q.41 The number of positive integral solutions of the inequation 5 6 2 3 4 (x 5) (2x 7) x (3x 4) (x 2) − − − −  0 is – (A) 2 (B) 0 (C) 3 (D) 4 [C] Q.42 Number of integral values of x satisfying the inequality 2 6x 10 x 4 3 + −       < 64 27 is (A) 6 (B) 7 (C) 8 (D) Infinite [B] Q.43 Number of ciphers after decimal before a significant figure comes in 100 3 5 −       is - (A) 21 (B) 22 (C) 23 (D) None of these [B] Q.44 If log10 3 = 0.477, the no. of digits in 340 is (A) 18 (B) 19 (C) 20 (D) 21 [C] Q.45 If x1 and x2 are two solutions of the equation log3 |2x –7| = 1 where x1 < x2, then the number of integer(s) between x1 and x2 is/are- (A) 2 (B) 3 (C) 4 (D) 5 [A] Q.46 If | x – 1 | + | x – 2 | + | x – 3 |  6 then. (A) 0  x  4 (B) x  – 2 or x  4 (C) x  0 or x  4 (D) None of these [C] Q.47 The set of real values of x satisfying ||x – 1| – 1|  1 is- (A) [–1, 3] (B) [0, 2] (C) [–1, 1] (D) None of these [A] Q.48 The solution of the inequation log0.1         − + | x 1| x 1 log 2 2 < 0 lies in the interval - (A) (1, ) (B) (–, 1) (C) [1, ) (D) None of these [A] Q.49 The value of [e] – [–] is, where [.] denotes greatest integer function - (A) 5 (B) 6 (C) 7 (D) 8 [B] Q.50 The value of [ 2 ] – [–  ] is, where [.] denotes greatest integer function - (A) 9 (B) 10 (C) 19 (D) –1 [C] Q.51 The value of [e2 ] – [–e 2 ] is, where [.] denotes greatest integer function - (A) 19 (B) 15 (C) 10 (D) 18 [B] Q.52 If y = x3 , 1  x  2 then set of all possible value of [y] is, (where [.] denotes greatest integer function) (A) {1, 2, 3, 4, 5, 6, 7, 8} (B) {1, 2, 3, 4} (C) {1, 4, 8} (D) None of these [A] Q.53 If 1  x < 2 and – 1  y < 0 then the value of [x] + [y] is, where [.] denotes greatest integer function - (A) 10 (B) 0 (C) 17 (D) 12 [B] Q.54 If  is an imaginary fifth root of unity, then log2  + +  +  − 1 1 2 3 = (A) 1 (B) 0 (C) 2 (D) –1 [A] Q.55 log27 is (A) an integer (B) a rational number (C) an irrational number (D) a prime number [C] Q.56 Which is the correct order for a given number  in increasing order : (A) log2, log3, loge, log10 (B) log10, log3, loge, log2 (C) log10, log2, loge, log3 (D) log3, loge, log2, log10 [B]