Cengage Maths Solutions Class 11 Trigonometry - Logarithm

1. CENGAGE / G TEWANI MATHS SOLUTIONS CHAPTER LOGARITHM || TRIGONOMETRY  Download Doubtnut Today Ques No. Question CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Exponential Functions |x − 3|3x2−10x+3= 1 Solve: 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Exponential Functions x 1 Solve: ( ) 2 ^ (2 − 2x) < 1/4. 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Exponential Functions (x − 2)x2−6x+8) > 1 Find the smallest integral value of satisfying x 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Exponential Functions Find the number of solutions of equation (2x − 3)2x= 1 4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Functions Find the value of (log)2√31728. 5  Watch Free Video Solution on Doubtnut

2. CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Functions 1 1 Prove that < (log)103 < 6 . 3 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Functions Arrange in decreasing order. (log)25, (log)0.55,(log)75, (log)35 7  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Functions Prove that number is an irrational number. (log)27 8  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Functions Which of the following numbers are positive/negative? (ii) (iii) (log)27 (log)0.23 1 9 (v) (log)1/3( ) (log)43 (log)2((log)29) 5  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Functions Find the value of log tan10logtan20.......... .logtan890 10  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Functions If and , then prove that (log)a3 = 2 (log)b8 = 3 (log)ab = (log)34.

3. 3 a b a 11  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Functions If find in terms of 72x (log)3y = xand(log)2z = x, y and z. 12  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Functions If x(y + z − x) y(z + x − y) z(x + y − z) ,provethatxyyx = logx logy logz 13 = zxyz= xzzx  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Functions Solve : 2(25)x− 5(10x) + 2(4x) ≥ 0. 14  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Functions Find the number of solution to equation (log)2(x + 5) = 6 − x: 15  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Functions 1 Find the number of solutions of the following equations: x2− 4x + 3 − (log)2x = 0 x−(log)0.5x = 1 2 16

4.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of 17 Find the value of (log)2(293 − 2) + (log)2(1233 + 4 + 493).  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of 1 a + b If then find the relation between (log)e( ) = ((log)ea + (log)eb), 18 2 2 aandb.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of x If are real numbers such that xandy ,then find . 19 2log(2y − 3x) = logx + logy y  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of 20 What is logarithm of to the base 3245 2√2?  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of Which is greater: 21 x = (log)35 or y = (log)1725?

5.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of 1 22 , then find x in terms of y. (log)x4 y = 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of 1 1 1 1 If 23 n > 1, thenprovethat + + + = . (log)2n (log)3n (log)53n (log)53!n  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of 24 If then find the value of ax= b, by= c, cz= a, xyz.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of Find the value of 25 ((log)34)((log)45)((log)56)((log)67)((log)78)((log)89).  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of 26 If then prove that y2= xz and ax= by= cz, (log)ab = (log)bc

6.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of 1 1 1 Simplify: 27 + + 1 + (log)abc 1 + (log)bca 1 + (log)cab  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of Suppose and are not equal to 1 and x, y, z = 0 1 1 Find logx + logy + logz = 0. 1 28 1 1 1 the value of + + + ylogz xlogy zlogx ^ (logz) ^ (logx) ^ (logy)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of 29 If then find (log)616∫ermsofa (log)1227 = a,  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of 1+ (log)72 1 −1(log)( )(7) 1 30 Find the value of ( ) + 5 5 49  Watch Free Video Solution on Doubtnut

7. CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of 31 4 ( ) Find the value of 81(1/log53)+ (27log936) + 3 log79  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of 1 3− 2x > 2(log)2x− 3log_ (27)(x2+ 1) 4 32 Prove that 0 74(log)49x− x − 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of Prove that: . √( log)a4√ab+ (log)b4√ab− (log)a4√ b a + (log)b4√ √(log)ab a b 2 33 = {2 if b ≥ a > 1 and 2loga(b) if 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of Which of the following pairs of expression are defined for the same set of values of ? f1(x) = 2(log)2xandf2(x) = (log)10x2f1(x) = (log)2 f1(x) = (log)10(x − 2) + (log)10(x − 3)andf2(x)= (log)10(x x ×andf2(x) = 2 34 − 2)(x − 3).  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations Solve (log)48 + (log)4(x + 3) − (log)4(x − 1) = 2. 35  Watch Free Video Solution on Doubtnut

8. CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations Solve log( − x) = 2log(x + 1). 36  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations Solve (log)2(3x − 2) = (log) x 1 37 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations Solve 2x+227x/ (x−1)= 9 38  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations Solve: (log)2(4.3x− 6) − (log)2(9x− 6) = 1. 39  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations Solve : 6((log)x2 − (log)4x) + 7 = 0. 40  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations 4(log)2logx= logx − (logx)2 Solve: (base is e) 41

9.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations Solve: 4(log) (√x) + 2(log)4x(x2) = 3(log)2x(x3). x 2 42  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations Solve 4(log)9x− 6x(log)92+ 2(log)327= 0 43  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations 1 1 Solve: 4 xlos2√x= (2.x(log)2x) 44 4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations |x − 1|(log)10x^ 2 − (log)10x2= |x − 1|3 Solve: 45  Watch Free Video Solution on Doubtnut

10. CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic InequalitiesCENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities 46 Solve (log)2(x − 1) > 4.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities Solve (log)3(x − 2) ≤ 2. 47  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities Solve : (log)0.3(x2− x + 1) > 0 48  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities Solve 1 < (log)2(x − 2) ≤ 2. 49  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities Solve log2|x − 1| < 1 50  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities Solve (log)0.2|x − 3| ≥ 0.

11. . 51  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities x − 1 Solve : (log)2 > 0 52 x − 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities 3 − x Solve: (log)0.5 < 0 53 x + 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities Solve: (log)3(2x2+ 6x − 5) > 1 54  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities Solve (log)0.04(x − 1) ≥ (log)0.2(x − 1) 55  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities Solve : (log)(x+3)(x2− x) < 1 56  Watch Free Video Solution on Doubtnut

12. CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic InequalitiesCENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities Solve : 2(log)3x − 4(log)x27 ≤ 5 (x > 1) 57  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities log2(x − 1) Solve: (log)x+( ) > 0 58 1 x + 2 x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities Solve: )(x2− 10x + 22) > 0 (log)(log)2( x 59 x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities x2+ 1 Solve: (log)0.1((log)2( ) < 0 60 x − 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities x − 1 Solve: ≤ 1. 61 (log)3(9 − 3x) − 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities

13. log 1 3 Solve: _ (10)a2+ 2 > ( ) 2 62 2(log)10( −a)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Finding Logarithm Write the characteristic of each of the following numbers by using their standard forms: 1235.5 (ii) 346.41 (iii) 62.723 (iv) 7.12345 0.35792 (vi) 0.034239 (vii) 0.002385 (viii) 0.0009468 63  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Finding Logarithm Write the significant digits in each of the following numbers to compute the mantissa of their logarithms: 3.239 (ii) 8 (iii) 0.9 (iv) 0.02 0.0367 (vi) 89 (vii) 0.0003 (viii) 0.00075 64  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Finding Logarithm Find the mantissa of the logarithm of the number 5395 65  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Finding Logarithm Find the mantissa of the logarithm of the number 0.002359 66  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Finding Logarithm Use the logarithm tables to find the logarithm of the following numbers (1)25795 (ii)25.795 67

14.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Antilogarithm Find the antilogarithm of each of the following: 2.7523 (ii) 3.7523 (iii) 5.7523 (iv) 0.7523 1.7523 (vi) 2.7523 (vii) 3.7523 68  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Antilogarithm 1 Evaluate (72.3) if log0.723 = 1. 8591. 3 69  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Antilogarithm Using logarithms, find the value of 6.45 x 981.4 70  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Antilogarithm x = (0.15)20 Let base 10. Assume Find the characteristic and mantissa of the logarithm of to the x . 71 (log)102 = 0. 301and(log)103 = 0.477.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Antilogarithm If 312⋅ 28 then find the number of digits in (log)102 = 0.30103, (log)103 = 0. 47712, 72  Watch Free Video Solution on Doubtnut

15. CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Antilogarithm In the 2001 census, the population of India was found to be population increases at the rate of 2.5% every year, what would be the population in 2011? If the 8. 7 ⋅ 107 . 73  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Antilogarithm Find the compound interest on Rs. 12000 for 10 years at the rate of 12% per annum compounded annually. 74  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Antilogarithm If characteristic reciprocals to the base 10 have the characteristic log10P − (log)10Q. is the number of natural numbers whose logarithms to the base 10 have the is the number of natural numbers logarithms of whose pandQ P , then find the value of −q 75  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Antilogarithm Let denote antilog_32 0.6 and M denote the number of positive integers which have the characteristic 4, when the base of log is 5, and N denote the value of L 76 LM Find the value of 49(1− ( log)72)+ 5− (log)54. . N  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of 77 If ,prove that x = (log)2aa,y = (log)3a2a,z = (log)4a3a 1 + xyz = 2yz  Watch Free Video Solution on Doubtnut

16. CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations xandy:(3x)log3= (4y)logg4,4logx= 3logy. Solve the equations for 78  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of 79 If then find the value of a = (log)1218,b = (log)2454, ab + 5(a − b).  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of 1 1 1 80 If and ,then prove that 1− (log)ax 1− (log)ay 1− (log)az y = a z = a x = a  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of 81 Solve (log)x2(log)2x2 = (log)4x2.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations 3 5 Let be positive integers such that If . a,b, c,d (log)ab = and(log)cd = 82 2 4 then find the value of (a − c) = 9, (b − d).  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations √

17. 83 Solve (base is 10). √log( − x) = log√x2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities If then find the largest possible value of the expression a ≥ b > 1, a b 84 ) + (log)b( ). (log)a( a b  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations Solve : 3(log9x)× 2 = 3√3 85  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities 2x − 3 √(log)2( Solve the inequality ) < 1 86 x − 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities Find the number of solutions of equation 2x+ 3x+ 4x− 5x= 0 87  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations Solve x(log)yx= 2andy(log)xy= 16 88

18.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations Solve (log)2x2 + (log)42x = − 3/2. 89  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations x:(2x)(log)b2= (3x)(log)b3 Solve for: . 90  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of If different positive real numbers (where are (log)ba(log)ca + (log)ab(log)cb + (log)ac(log)bc = 3 ≠ 1), a, b,c 91 then find the value of abc.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of If (log)aN (log)aN − (log)bN = , whereN > 0andN ≠ 1,a, b,c 92 (log)cN (log)bN − (log)cN > 0 and not equal to 1, then prove that b2= ac  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities 2

19. 4(log10a)2+ ((log)2b)2= 1. 93 Given the range of values of are positive numbers satisfying aandb Find aandb.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations Solve: (log)(2x+3)(6x2+ 23x + 21) + (log)(3x+7)(4x2+ 12x + 9) 94 = 4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Finding Logarithm is a rational number (b) an irrational number a prime number (d) none of (log)418 95 these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of The number of integers whose sum is equal to (a)5 (b) 7 (c) 9 (c) 10 lies between two successive N = 6 − (6(log)102 + (log)1031) 96  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Antilogarithm Given that (b) 6602 (c) 6603 (d) 6604 the number of digits in the number is 6601 20002000 log(2) = 0.3010, 97  Watch Free Video Solution on Doubtnut

20. CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of 1 1 98 If , then the value of is 0 (b) 1 (c) 2 (d) 4 (21.4)a= (0.00214)b= 100 − a b  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of 99 The value of (b) (c) (d) none of these −loga logab − log|b| = loga log|a|  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of If of are consecutive positive integers and (b) (c) 2 (d) 1 logb loga then the value a, b,c (log(1 + ac) = 2K, 100 is K  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of a + (log)43 a + (log)83 1 2 1 3 If ,then b is equal to (2) (c) (d) 101 = = b 2 3 3 2 a + (log)23 a + (log)43  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of are such that is equal to 0 (b) 1 (c) 2 (d) none of these log(p − 1) + log(q − 1) If then the value of p > 1andq > 1 log(p + q) = logp + logq, 102  Watch Free Video Solution on Doubtnut

21. CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of 1 + 2(log)32 + ((log)62)2 The value of is 2 (b) 3 (c) 4 (d) 1 103 (1 + (log)32)2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of 1 1 If then is equal to (b) (c) (log)45 = aand(log)56 = b, (log)32 2a + 1 2b + 1 104 1 (d) 2ab + 1 2ab − 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of If 2a + 3b − 1 then in terms of 2a + 5b − 2 is equal to (a) aandb (log)102 = a, (log)103 = b (log)0.72(9. 6) 3a + b − 2 5a + b − 1 105 (b) (c) 5a + b − 2 3a + 2b − 2 2a + 3b − 1 3a + b − 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations There exists a natural number then is divisible by N which is 50 times its own logarithm to the base 10, N 106  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of (log)224 (log)2192 107 The value of is 3 (b) 0 (c) 2 (d) 1 − (log)122 (log)962  Watch Free Video Solution on Doubtnut

22. CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations is equal to: 9 (b) 16 (c) 25 (log)x− 1x(log)x−2(x − 1)(log)x−12(x − 11) = 2, x 108 (d) none of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations 1 + x If (a) (b) f(x) = log( ),then f(x1)f(x) = f(x1+ x2) 1 − x (c) (d) f(x) + f(x + 1) = f(x2+ x) f(x + 2) − 2f(x + 1) + f(x) = 0 109 x1+ x2 f(x1) + f(x2) = f( ) 1 + x1x2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of If then the value of equals a4b5= 1 loga(a5b4) 110  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of 111 The value of is 0 (b) 1 (c) 2 (d) none of these 3(log)45− 5(log)43  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of If then the value of is 1 (b) 2x+y= 6yand3x−1= 2y +1, (log3 − log2)(x − y) 112 3

23. 3 (d) none of these ) (log)23 − (log)32 log( 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations x ( (log)5(log)5log5( ) )=3 3√5(log5)5 2 The value of satisfying the equation (d) 54 1 (b) 3 (c) 18 x 113  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations If then equals odd integer (b) prime number x √(log)2x − 0.5 = (log)2√x, 114 composite number (d) irrational  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of logx logy logz If , then which of the following is/are true? (b) = = zyz = 1 115 b − c xaybzc= 1 xb+cyc+b= 1 c − a a − b (d) xyz = xaybzc  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of if the value of is 9 (b) 12 (c) 15 (d) 30 (log)yx + (log)xy = 2,x2+ y = 12, 116 xy  Watch Free Video Solution on Doubtnut

24. CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Miscellaneous (x + 1)(log)10( x+1)= 100(x + 1), If all the roots lie in the interval (0,100) all the roots lie in the interval [-1,99] none of these then all the roots are positive real numbers 117  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of If 10 (log)2x + (log)x2 = = (log)2y + (log)y2andx ≠ y,thex 118 3 + y = 2 (b) 65/8 (c) 37/6 (d) none of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations 1 If , then 4 (b) 3 (c) 2 (d) 1 (log)10[ ] = x[(log)105 − 1] x = 119 2x+ x − 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations If then is equal to 4 (b) 3 (c) 8 (d) x (log)3{5 + 4(log)3(x − 1)} = 2, (log)216 120  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations (log) 3 (log)

25. 121 If then is equal to x 2x(log)43+ 3(log)4x= 27,  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations Equation (log)4(3 − x) + (log)0.25(3 + x) = (log)4(1 − x) 122 + (log)0.25(2x + 1)has only one prime solution two real solutions no real solution (d) none of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations The 2(log) (bx + 28) = − (log)5(12 − 4x − x2) 1 25 b = 4 or b = − 12 b = 4 or b = − 12 value of for which has coincident roots is the equation b = − 12 b 123 (b) (d) b = − 4 or b = 12  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations If the equation the sum of the solution is (d) x(log)2(2x+ 1) is solved for in terms of where (b) x(log)2(1 − 2x) x + (log)2(1 − 2x) (log)2(1 − 2x) then 2x+ 4y= 2y+ 4x x < 0, y x 124  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities xlog_ x(x + 3)2= 16 The number of solution of is 0 (b) 1 (c) 2 (d) ∞ 125  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations

26. 3 126 The product of roots of the equation is 1 (b) (c) 1/3 (d) = 3 1/4 (log8x)2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities Let a2(log)2x= 15 + 4x(log)2a be a real number. Then the number of roots equation is 2 (b) infinite (c) 0 (d) 1 a > 1 127  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities The number of roots of the equation 0 is 1 (b) 2 (c) 3 (d) (log)3√xx + (log)3x√x = 0 128  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations The number of elements 1 in set of all satisfying the equation x xlog3x2+ ( log3x)2−10= (a)1 (b) 2 (c) 3 (d) 0 is 129 x2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations Number of real values of satisfying the equation x x − 1 ) + (log2x)2= 4 130 ,is (a) (b) (c) (d) 0 log2(x2− x) ⋅ log2( 2 3 7 x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of If ,then equals (a) (b) x xy2= 4and(log)3((log)2x) + (log) ((log) y) = 1 4 8 131 1 1 3 2 (c) (d) 16 64  Watch Free Video Solution on Doubtnut

27. CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic InequalitiesCENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities If x1= 2x2 are the roots of the equation (b) x1= x2 2(c)2x1= x2 with then e2⋅ xln x= x3 x1> x2, x1andx2 132 (d) x2 1= x3 2 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities The (log16x)2− (log)16x + (log)16k = 0 (1) (b) number of real values of the parameter for which k with real coefficients will have exactly one none of these (d) 133 solution is 2 1 4 (c)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities implies xlog5x> 5 134 x ∈  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities If , then (a) S = {x ∈ N:2 + (log)2√x + 1 > 1 − (log) √4 − x2} S = {1} 1 135 2 (b) (d) (d) none of these S = Z S = N  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities If then is equal to S = {x ∈ R:((log)0.60. 216)(log)5(5 − 2x) ≤ 0}, (2.5,∞) [2,2.5) (2, 2.5) S 136 (b) (c) (d) (0,2.5)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities

28. 1 1 4 Solution set of the inequality is (b) (c) ) 0,(log)2( > 1,∞) 2x− 1 1 − 2x−1 3 137 4 ( − 1,∞) (0, (log)2( ) ∪ (1, ∞) 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities If then the least value of is 4 (b) 8 (d) 16 (d) 32 (log)2x + (log)2y ≥ 6, x + y 138  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities 5 1 Which of the following is not the solution of (log)x( ) > 1 − 2 x 139 2 1 2 (a)( , )(b)(1, 2)(c)( ,1)(d)Nonofthese 5 2 5  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities The solution set of the inequality is (log)10(x2− 16) ≤ (log)10(4x − 11) 4, ∞) 11 11 140 (b) (c) (d) ( ,∞) ( , 5) (4,5) 4 4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities x2+ x Solution set of the inequality is (b) (log)0.8((log)6 ) < 0 ( − 4, − 3) x + 4 ( − 3,4) ∪ (8, ∞) ( − 3, ∞) ( − 4, − 3) ∪ (8,∞)

29. 141 (d) ( − 3,4) ∪ (8, ∞) ( − 3, ∞) ( − 4, − 3) ∪ (8,∞)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities 3 Which of the following is not the solution of (log)3(x2− 2) < (log)3( |x| − 1) 2 142 is (b) (d) none of these (√2,2) ( − 2, − √2) ( − √2, 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities The true solution set of inequality is equal to (b) (log)(x+ 1)(x2− 4) > 1 2,∞) 1 + √21 1 − √21 1 + √21 1 + √21 143 (2, ) ( ) ( , ∞) (d) , 2 2 2 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations For the roots of the equation a−3 4 a (log)axa + (log)xa2+ (log)a2a3= 0 a > 0, ≠ 1, a−4 144 a−1 are given (b) (c) (d) 3 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations The real solutions of the equation (d) −(log)10(250) is/are 1 (b) 2 (c) 2x+2. 56−x= 10x^ 2 145 (log)104 − 3  Watch Free Video Solution on Doubtnut

30. CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations . If none of these then is equal to (b) 1/5 (c) 5 (d) x 146 (log)kx log5k = (log)x5,k ≠ 1,k > 0, k  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations x√x= (√x)x, If (b) twin prime (c) coprime (d)if satisfy the equation then are (a)relatively prime is not and vice versa p,q ∈ N pandq 147 is defined, then (log)qp (log)pq  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of Which of . log1020 + ((log)102)2 the following, when 2log2 + log3 simplified, reduces to unity? (a) (b) . (c) (d) −(log)5(log)3√5√9 (log)105 log48 − log4 148 1 64 ( ) (log) √3 6 27 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Finding Logarithm If can be correct. (a)If and are two irrational numbers, then can be rational. (b)If is rational and is irrational, then can be rational. (c)If is irrational and is rational, then can be rational. (d)if x aandb for permissible values of and then identify the statement(s) which (log)ax = b x, a a b x a 149 b x a b are rational, then can be rational. x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations (log) ( − .5) = (log) ( + 1)

31. The equation no prime solution (C) one integral solution (D) no irrational solution has (A) two real solutions (B) 150 (log)x+1(x − .5) = (log)x−0.5(x + 1)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations (A)(log)308 =3(1 − a) If (log)105 = aand(log)103 = b, then b + 1 151 a + b 3 − 2a(C)(log)24332 =1 − a (d) none of these (B)(log)4015 = b  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities (log)b2a (6a(log)eb((log)a2b) The value of independent of (b) a is e(log)ea(log)eb 152 independent of dependent on (d) dependent on b a b  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations √x(log)2√x≥ 2 The inequality is satisfied by (A) only one value of (B) x 153 1 (d) x ∈ (0, ( )] (C)x ∈ [4,∞) x ∈ (1,2) 4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations If is............ then the value of (log)ab = 2,(log)bc = 2,and(log)3c = 3 + (log)3a, c/(ab) 154  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations (log102)3+ log108log105 + (log105)3 The value of is ............ 155  Watch Free Video Solution on Doubtnut

32. CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Finding Logarithm B If (where [ ] represents )] (log)4A = (log)6B = (log)9(A + B), then [4( 156 A the greatest integer function) equals ..............  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations Integral value of which satisfies the equation x 4 = log654 + (log)x16 = (log)√2x − (log)36( )is.. 157 9  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations 2= (log)27x6 The difference of roots of the equation is ....... ((log)27x3) 158  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities Sum of all integral values of satisfying the inequality x 5 (3( )log(12−3x)) − (3logx) > 32 159 is...... 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of 1 160 The least integer greater than is ................ (log)2(15) ⋅ (log) 2 ⋅ (log)3 1 6 6  Watch Free Video Solution on Doubtnut

33. CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of 2 3 The reciprocal of is ............ 161 + (log)4(2000)6 (log)5(2000)6  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities 1 Sum of integers satisfying is...... (log)2(x3) + 2 > 0 √(log)2x − 1 − 162 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities Number of integers satisfying the inequality is..... (log) |x − 3| ≻ 1 1 163 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities Number of integers satisfying the inequality ≤ 10 1 1

34. 1 1 164 is............ 2(log) (x − 1) ≤ 1 − 3 (log)x2−x8 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Functions The value of is........... (log)√4+2√2√4−2√229 165  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of 1 ( ) 4 1 (log) 1 166 The value of is......... 2 5 + (log)√2 + (log) 5 1 √7 + √3 10 + 2√21 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of (log)5250 (log)510 167 The value of is........... N = − (log)505 (log)12505  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations The x + (log)10(2x+ 1) = x(log)105 + (log)106 number of positive integers satisfying 168 is...........  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations

35. The are positive real 2 numbers such that x,y,z 1 then the value of is ( ) (log)2xz = 3,(log)5yz = 6, and(log)xyz = , 169 3 2z ............  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations 1 1 Solve for x:4x− 3x− = 3x+ − 22x−1 170 2 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations Solve the following equation of x : 2(log)xa + (log)axa + 3(log)a2xa = 0, a > 0 171  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations 7 If are in arithmetic progression, (log)32, (log)3(2x− 5)and(log)3(2x− ) 2 172 determine the value of x.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations The solution of the equation is... (log)7(log)5(√x + 5 +√x = 0 173  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities 2(log) − (log) (0.01) > 1 10 2

36. The least value of the expression (c) (d) −0. 01 4 for is (a) (b) 174 2(log)10x − (log)x(0.01). x > 1 10 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Logarithms Laws Of . . If In are in then (b) (a + c), ln(a − c),ln(a − 2b + c) . P.a,b, c a,b, c, are ∈ A P., A P. 175 . . are in (d) are in a2, b2, c2are ∈ A a,b,c G P. H P.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Functions (2x)1n2= (3y)1n3 Let be the solution of the following equations: (x0, y0) 1 1 1 176 The is (b) (c) (d) 6 31nx= 21ny x0 6 3 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Equations 3 ) (log2x)2+ (log2x) − ( 5 x( )= √2 The equation exactly three solutions (2) has at least one real solution complex roots (4) (1) 4 4 177 exactly one irrational solution (3)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Functions 2(log)32 2 1 If , then (b) (d) 3x= 4x−1 x = 2 − (log)23 1 − (log)43 2(log)32 − 1 178 2(log)23 2(log)23 − 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Logarithmic Inequalities 1 < 2−k+3( − a)< 2, Let than .......... An integer satisfying k must be less a = (log)3(log)32. 179  Watch Free Video Solution on Doubtnut

37. CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Finding Logarithm    ⎷(4 − ⎡ ⎣ 1 1 1 The value of is ............... )√4 − 6 + (log) ⋅ ....... 3 180 3√2 3√2 3√2 2  Watch Free Video Solution on Doubtnut  Download Doubtnut to Ask Any Math Question By just a click  Get A Video Solution For Free in Seconds  Doubtnut Has More Than 1 Lakh Video Solutions  Free Video Solutions of NCERT, RD Sharma, RS Aggarwal, Cengage (G.Tewani), Resonance DPP, Allen, Bansal, FIITJEE, Akash, Narayana, VidyaMandir  Download Doubtnut Today