Quantitative Aptitude-More exercise questions

Surds and Indices-Exercise Questions

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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 ba = 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 xx 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. y2

           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 72 + 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 = 83

           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 => 44m + 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 = k1/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 = 2=> x + y = 5 ---(2)

             On solving (1) & (2), we get x= 4.

16. Exp: a1 = cz = (by)z = byz = (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 316 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.

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