practice question on Simplification & Power and Roots — Square, Cube

Square, Cube, indices and Surds

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Question no. 1

The cube root of 1.061208 is :

looks_one 1.022

looks_two 10.22

looks_3 0.102

looks_4 1.02

Option looks_4 1.02

Solution :
(1 + 0.02)³ = 1 + 0.000008 + 3 * 1 * 0.02 (1 + 0.02) = 1.061208

Question no. 2

If x^abc = x^ax^bx^c and x are all positive integers, then a²+b²+ c² = ?

looks_one 12

looks_two 14

looks_3 29

looks_4 can't be determined

Option looks_two 14

Solution :
only one pair a = 1 , b = 2 , c = 3
a + b + c = abc
a²+b²+ c² = 1²+2²+ 3² = 14

Question no. 3

If 16*8ⁿ⁺² =2^m, then m is equal to

looks_one n + 8

looks_two 2n + 10

looks_3 3n + 2

looks_4 3n + 10

option looks_4 3n + 10

Solution :
16*8ⁿ⁺² =2^m
2⁴ * 2 ^{3n + 6} = 2^m
2^{3n + 10} = 2^m
m = 3n + 10

Question no. 4

The least number of four digits which is a perfect square is

looks_one 1024

looks_two 1204

looks_3 1402

looks_4 1420

Option looks_one 1024

Solution :
1024 = 32²

Question no. 5

If 196x⁴ = x⁶, then x³ is equal to which one of the following ?

looks_one x⁶/14

looks_two 14x⁴

looks_3 x²/14

looks_4 14x²

Option looks_4 14x²

Solution :
196x⁴ = x⁶
(14x²)² = (x³)² = 14x² = x³

Question no. 6

The difference of cubes of two consecutive integers

looks_oneis odd or even

looks_twois never divisible by 2

looks_3is always even

looks_4 None of these

Option looks_4 -3²⁷

very soon 6

Question no. 7

If a^x = b,b^y = c and xyz = 1, then what is the value of c^z ?

looks_one a

looks_two b

looks_3 ab

looks_4 a/b

Option looks_one a

Solution :
a^x = b , b^y = c , xyz = 1
c = b^y
c^z = b^zy = (a^x)^zy = a^xyz = a¹ = a

Question no. 8

What is the smallest number that must be added to 1780 to make it a perfect square?

looks_one 39

looks_two 49

looks_3 59

looks_4 69

Option looks_4 69

Solution :
42 * 42 = 1764
43 * 43 = 1849
1849 is the perfect square greater than 1780
So , 1849 - 1780 = 69

Question no. 9

A man plants 15376 apple trees in his garden and arranges them so that there are as many rows as there are apples trees 34. in each row. The number of rows is

looks_one 124

looks_two 126

looks_3 134

looks_4 144

Option looks_one 124

Solution :
No. of rows = no. of apples in each row = a
a * a = 15376
a = 124

Question no. 10

Which of the following numbers is the largest ?

looks_one 3^1/3

looks_two 5^1/5

looks_3 2^1/2

looks_4 7^1/7

Option looks_3 2cos(π/10)

Solution :

Question no. 11

The value of is

looks_one 0.02

looks_two 0.2

looks_3 2

looks_4 None of these

Option looks_two 0.2

Solution :

Question no. 12

The value of

looks_one -1/36

looks_two 1/36

looks_3 -36

looks_4 36

Option looks_4 36

Solution :

Question no. 13

If 4^x=5^y =20^z then z is (x, y, z ≠ 0)

looks_one

looks_two

looks_3 xy

looks_4

Option

Solution :
Take log on both sides
4^x = 20^z
x log 4 = z log 20
x log 4 = z[log 4 + log 5 ]
(x-z)log 4 = z log 5 ...........[1]
5^y =20^z
y log 5 = z [log 4 + log 5]
(y -z) log 5 = z log 4 ⇒ z log 4 = (y -z) log 5 ........[2]
divide 1 and 2

Question no. 14

If , then x =

looks_one 1/3

looks_two 2/3

looks_3 3

looks_4 4/3

Option looks_two 2/3

Solution :

Question no. 15

In a cricket match, the number of runs scored by any team is equal to power of the number of batsmen playing in the team. Six batsmen played in team A and eleven batsmen played in team B. If team A won by 95 runs, then find the runs scored by team A.

looks_one 216

looks_two 220

looks_3 210

looks_4 230

Option looks_one 216

Solution :
team A = 6^x
team B = 11^y
6^x - 11^y = 95
for x = 3
6³ - 11^y = 95
216 - 11^y = 95
-11^y = -121
y = 2
team A = 216