## 1. If m and n are whole numbers such that mn = 169, then the value of (m - 1) n + 1 is: a. 1 b. 13 c. 169 d. 1728 2. The simplified form of x9/2 . √y7 is: x7/2 . √y3 a. x2/y2 b. x2 . y2 c. xy d. x2/y 3. If √(3 + ³√x) = 2, then x is equal to : a. 1 b. 2 c. 4 d. 8 4. If x is an integer, find the minimum value of x such that 0.00001154111 x 10x exceeds 1000. a. 8 b. 1 c. 7 d. 6 5. Which among the following is the greatest? a. 23^2 b. 22^3 c. 32^3 d. 33^3 6. Solve for m if 49(7m) = 3433m + 6 a. -8/6 b. -2 c. -4/6 d. -1 7. Solve for 2y^√2^2 = 729. a. ±3 b. ±1 c. ±2 d. ±4 8. √[200√[200√[200……..∞]]] = ? a. 200 b. 10 c. 1 d. 20 9. If a and b are positive numbers, 2a = b3 and b a = 8, find the value of a and b. a. a = 2, b = 3 b. a = 3, b = 2 c. a = b = 3 d. a = b = 2 10. If 44m + 2 = 86m - 4, solve for m. a. 7/4 b. 2 c. 4 d. 1 11. If 2x x 162/5 = 21/5, then x is equal to: a. 2/5 b. -2/5 c. 7/5 d. -7/5 12. If ax = by = cz and b2 = ac, then y equals : a. xz/x + z b. xz/2(x + z) c. xz/2(x - z) d. 2xz/(x + z) 13. If 7a = 16807, then the value of 7(a - 3) is: a. 49 b. 343 c. 2401 d. 10807 14. If 3x - 3x - 1 = 18, then the value of x x is: a. 3 b. 8 c. 27 d. 216 15. If 2(x - y) = 8 and 2(x + y) = 32, then x is equal to: a. 0 b. 2 c. 4 d. 6 16. If ax = b, by = c and cz = a, then the value of xyz is: a. 0 b. 1 c. 1/abc d. abc 17. 125 x 125 x 125 x 125 x 125 = 5? a. 5 b. 3 c. 15 d. 2 18. If 52n - 1 = 1/(125n - 3), then the value of n is: a. 3 b. 2 c. 0 d. -2 19. If x = 5 + 2√6, then (x - 1) is equal to: √x a. √2 b. 2√2 c. √3 d. 2√3 20. Number of prime factors in 612 x (35)28 x (15)16 is : (14)12 x (21)11 a. 56 b. 66 c. 112 d. None of these Answer & Explanations 1. Exp: Clearly, m = 13 and n = 2. Therefore, (m - 1) n + 1 = (13 - 1)3 = 12³ = 1728. 2. Exp: x9/2 . √y5 is: = x(9/2 - 5/2) . y(7/2 - 3/2) = x2. y 2 x7/2 . √y3 3. Exp: On squaring both sides, we get: 3 + ³√x = 4 or ³√x = 1. Cubing both sides, we get x = (1 x 1 x 1) = 1 4. Exp: Considering from the left if the decimal point is shifted by 8 places to the right, the number becomes 1154.111. Therefore, 0.00001154111 x 10x exceeds 1000 when x has a minimum value of 8. 5. Exp: 23^2 = 29 22^3 = 28 32^3 = 38 33^3 = 327 As 327 > 38, 29 > 28 and 327 > 29. Hence 327 is the greatest among the four. 6. Exp: 49(7m) = 3433m + 6 Þ 727m Þ (73)3m + 6 Þ 7 2 + m = 79m + 18 Equating powers of 7 on both sides, m + 2 = 9m + 18 -16 = 8m Þ m = -2. 7. Exp: 3y^√2^2 = 729 3y^2 = 34 (√22 = (21/2) 2 = 2) equating powers of 2 on both sides, y2 = 4 Þ y = ±2 8. Exp: Let √[200√[200√[200……..∞]]] = x ; Hence √200x = x Squaring both sides 200x = x² Þ x (x - 200) = 0 Þ x = 0 or x - 200 = 0 i.e. x = 200 As x cannot be 0, x = 200. 9. Exp: 2a = b3 ….(1) ba = 8 …..(2) cubing both sides of equation (2), (ba)3 = 8 3 b3a = (b3)a = 512. from (1), (2a)a = (23)3. comparing both sides, a = 3 substituting a in (1), b =2. 10. Exp: 44m + 2 = (23)6m - 4 => 4 4m + 2 = 218m - 12 Equating powers of 2 both sides, 4m + 2 = 18m - 12 => 14 = 14m => m = 1. 11. Exp: 2x x 162/5 = 21/5 => 2x x (24)2/5 = 21/5 => 2x x 28/5 = 21/5. => 2(x + 8/5) = 21/5 => x + 8/5 = 1/5 => x = (1/5 - 8/5) = -7/5. 12. Exp: Let ax = by = cz = k. Then, a = k 1/x, b = k1/y, c = k1/z. Therefore, b² = ac => (k1/y)2 = k1/x x k1/z => k2/y = k (1/x + 1/z) Therefore, 2/y = (x + z)/xz => y/2 = xz/(x + z) => y = 2xz/(x + z). 13. Exp: 7a = 16807, => 7a = 75, a = 5. Therefore, 7(a - 3) = 7(5 - 3) = 7² = 49. 14. Exp: 3x - 3x - 1 = 18 => 3x - 1 (3 - 1) = 18 => 3x - 1 = 9 = 3² => x - 1 = 2 => x = 3. 15. Exp: 2(x - y) = 8 = 2³ => x - y = 3 ---(1) 2(x + y) = 32 = 25 => x + y = 5 ---(2) On solving (1) & (2), we get x= 4. 16. Exp: a1 = cz = (by)z = b yz = (ax)yz = axyz. Therefore, xyz = 1. 17. Exp: 125 x 125 x 125 x 125 x 125 = (5³ x 5³ x 5³ x 5³ x 5³) = 5(3 + 3 + 3 + 3 + 3) = 515. 18. Exp: 52n - 1 = 1/(125n - 3) => 52n - 1 = 1/[(53)n - 3] = 1/[5 (3n - 9)] = 5(9 - 3n). => 2n - 1 = 9 - 3n => 5n = 10 => n = 2. 19. Exp: x = 5 + 2√6 = 3 + 2 + 2√6 = (√3)² + (√2)² + 2 x √3 x √2 = (√3 + √2)² Also, (x - 1) = 4 + 2√6 = 2(2 + √6) = 2√2 (√2 + √3). Therefore, (x - 1) = 2√2 (√3 + √2) = 2√2. √x (√3 + √2) 20. Exp: 612 x (35)28 x (15)16 = (2 x 3)12 x (5 x 7)28 x (3 x 5)16 = (14)12 x (21)11 (2 x 7)12 x (3 x 7) 11 = 212 x 312 x 528 x 728 x 3 16 x 516 = 2(12 - 12) x 3(12 + 16 - 11) x 5 (28 + 16) x 7(28 - 12 - 11) 212 x 712 x 311 x 711 = 20 x 317 x 544 x 7-5 = 317 x 544 75 Number of prime factors = 17 + 44 + 5 = 66 I. Laws of Indices: i. am * an = am+n ii. am/an = am-n iii. (am)n =amn iv. (ab)n = anbn v. (a/b)n = an/bn vi. a0= 1 1. To find √ (a + √b) write it in the form m + n + 2√mn, such that m + n = a and 4mn = b, then √ (a + √b) = ±(√m + √n) 2. (√a.√a.√a….∞) = a 3. If (√a + √a + √a……..∞) = p, then p (p - 1) = a. 4. If a + √b = c + √d, then a = c and b = d. Examples: 1. Simplify: (i) (81)3/4 (ii) (1/64)-5/6 (iii) (256) -1/4 Solution:(i) (81)3/4 =(34)3/4 =3 3=27. (ii) (1/64)-5/6 = 645/6 = (26) 5/6= 25 = 32 (iii) (256)-1/4 =( 1/256)1/4 = [( 1/4)4] 1/4 =1/4 2. If x=3+2√2, then the value of ( √x- (1/√x)) is:........ Solution: Exercise Questions 1.The value of (√8)1/3 is: a.2 b. 4 c. 2 d. 8 Answer: Option c. (√8)1/3 = (81/2)1/3= 81/6 = (23)1/6= 21/2= √2. 2. The value of 51/4 * (125)0.25 is: a. √5 b.5√5 c.5 d.25 Answer: Option c 50.25 * (53)0.25 = 51 = 5. 3. The value of (32/243)-4/5 is: a. 4/9 b. 9/4 c. 16/81 d. 81/16 Answer: Option d. (32/243)-4/5 = (243/32)4/5 = [(3/2)5] 4/5 = 81/16 4. (1/216)-2/3 ÷ (1/27)-4/3 = ? a. 3/4 b. 2/3 c. 4/9 d. 1/8 Answer: Option c. (1/216)-2/3 ÷ (1/27)-4/3 = 2162/3 ÷ 274/3 = (63)2/3 ÷ (33)4/3 = 4/9 5. (2n+4 -2.2n)/(2.2n+3) = 2-3 is equal to: a. 2n+1 b. -2n+1 + 1/8 c. 9/8 - 2n d. 1 Answer: Option d. (2n+4 -2.2n)/2.2n+3 + 1/23= 7/8 + 1/8 = 1 6. If 5√5 * 53 ÷ 5-3/2 = 5a+2 , the value of a is: a. 4 b. 5 c.6 d. 8 Answer: Option a 53/2 * 53 ÷ 5-3/2 = 5a+2 53/2 + 3 + 3/2 = 5a+2 3/2 + 3 + 3/2 = a+2 a+2=6; a=4 7. If √2n =64, then the value of n is: a. 2 b. 4 c. 6 d. 12 Answer: Option d √2n =64 => 2n/2 = 64= 26 n/2=6; n=12 8.The simplified form of (x7/2 /x5/2).√y3 /√y )is : a.x2/y b. x3/y2 c. x6/y3 d. xy Answer: Option d (x7/2 /x5/2 ). (√y3 /√y) = x 7/2 -5/2. y3/2 - 1/2 = xy

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