Cubes And Cube Roots Test

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Cubes And Cube Roots Test - 7

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Weekly Quiz Competition

Question 1
1 / -0

Find the value of 57³ - 57² - 27³.

A
162261

B
169872

C
146711

D
146625

Solution

57³ - 57² - 27³
= 185193 - 3249 - 19683
= 162261
Question 2
1 / -0

If , then find the value of x.

A
340

B
170

C
3400

D
1700

Solution
Question 3
1 / -0

If , then find the value of x.

A
1.5

B
2.5

C
3.5

D
4.5

Solution

So, x = 3.5
Question 4
1 / -0

How many Rubik's cubes of side 7 cm can be stored in a cubical box of side 21 cm?

A
81

B
54

C
27

D
19

Solution

Side of the Rubik's cube = 7 cm
So, volume of the Rubik's cube = 7³= 343 cm³Side of the cubical box = 21 cm
So, volume of the cubical box = 21³ = 9261 cm³Number of Rubik's cube that can be stored inside the cubical box = == 27
Question 5
1 / -0

What is the smallest number that can be multiplied by 5400 to make it a perfect cube?

A
5

B
3

C
2

D
9

Solution

5400 = 2 × 2 × 2 × 3 × 3 × 3 × 5 × 5
So, if we multiply 5400 by another 5, it will become 27000, which is a perfect cube.
Question 6
1 / -0

If the cube root of 512 is subtracted from the square root of 961, then what will be the result?

A
22

B
23

C
24

D
25

Solution

= 31
= 8
31 - 8 = 23
Question 7
1 / -0

Solve the following equation for the value of x:

A
3

B
9

C
27

D
81

Solution

Or, x = 3⁴ = 81
Question 8
1 / -0

Three numbers are in the ratio 2 : 3 : 5. The sum of their cubes is 10240. The numbers are

A
2, 3 and 5

B
4, 6 and 10

C
6, 9 and 15

D
8, 12 and 20

Solution

Let the numbers be 2a, 3a and 5a.
According to the question:
8a³+ 27a³+ 125a³= 10240
160a³= 10240
a³= 64
a = 4
Therefore, the numbers are:
2a = 2 × 4 = 8
3a = 3 × 4 = 12
5a = 5 × 4 = 20
Question 9
1 / -0

The cube of a two-digit number contains

A
four digits in every case

B
three digits in every case

C
either three digits or four digits

D
anywhere from four digits to six digits

Solution

The cube of a two-digit number contains anywhere from four digits to six digits.
For example: Cube of the smallest two-digit number 10³= 1000 (four digits), and cube of the largest two-digit number 99³= 970299 (six digits)
Question 10
1 / -0

The cube of an even natural number is always ___________.

A
even

B
odd

C
either even or odd

D
prime

Solution

The cube of an even natural number is always even.
For example: 2³ = 8 (even), 4³ = 64(even), 6³ = 216 (even)
Question 11
1 / -0

If each side of a cubical gift box is 4.5 cm, then the volume of the box is ___________ m³.

A
9.1125 × 10^-6

B
91.125 × 10^-6

C
0.91125 × 10^-6

D
911.25 × 10^-6

Solution

Each side of the cubical gift box = 4.5 cm
Volume of the cubical gift box = 4.5 × 4.5 × 4.5 × 10^-6= 91.125 × 10^-6 m³.
Question 12
1 / -0

The units digit of the cube root of 21952 is

A
6

B
8

C
3

D
9

Solution

21952 = 4 x 4 x 4 x 7 x 7 x 7
So, cube root of 21952 = 4 x 7 = 28
Hence, units digit is 8.
Question 13
1 / -0

The volume of a cubical box of side d is twenty-five times d. Find the value of d, such that d ≠ 0 and d ≠ - 5.

A
5

B
4

C
6

D
7

Solution

According to the question:
d³= 25d
d³ - 25d = 0
d(d - 5²) = 0
d(d - 5)(d + 5) = 0
d = 0, d = 5, d = -5
Since d ≠ 0 and d ≠ -5, therefore the only possible value of d is 5.
Question 14
1 / -0

Two Rubik's cubes have their volumes in the ratio of 64 : 729. Find the ratio of the area of each face of the first cube to area of each face of the second cube.

A
12 : 27

B
16 : 49

C
14 : 27

D
16 : 81

Solution

Let the volume of the first Rubik's cube with side a₁ be 64 cu. units
Let the volume of the second Rubik's cube with side a₂ be 729 cu. units.
So, a₁³_: a₂³
= 64 : 729
So, a₁ : a₂ = 4 : 9

Area of each face of first cube = a₁²Area of each face of second cube = a₂²
Required ratio = a₁² : a₂² = 16 : 81
Question 15
1 / -0

Find the smallest number by which 192 must be divided to obtain a perfect cube.

A
3

B
4

C
6

D
8

Solution

Now, 192 = 2 × 2 × 2 × 2 × 2 × 2 × 3

When we group the prime factors of 192 into triples, 3 is left out.
∴ 192 is not a perfect cube.
Now, 192/3 = 2 × 2 × 2 × 2 × 2 × 2
Or, 64 = 2 × 2 × 2 × 2 × 2 × 2
64 is a perfect cube.
Thus, the required number is 3.
Question 16
1 / -0

The difference between the squares of two numbers is 275. If the square root of the smaller of the two numbers is 5, then find the cube of the larger number.

A
27000

B
64000

C
216000

D
1728000

Solution

Let the two numbers be x and y, where x > y.
According to the question:
x² - y² = 275 and = 5
Therefore, y = 25 (smaller number) and x² = 275 + y² = 275 + 625 = 900
So, x = 30
Hence, the larger number is 30 and its cube is 27000.
Question 17
1 / -0

Naveen has a square piece of land, having an area of 529 square yards. Knowing naveen is an expert in dealing with cubes, his friend, Hitesh, challenged him to tell the cube of the length of his square yard. What did naveen answer?

A
12167 cu. yards

B
13824 cu. yards

C
15625 cu. yards

D
9261 cu. yards

Solution

Area of square yard, A = 529 sq. yards

Length of square yard, L = = 23 yards

Cube of length of square yard = 23³ = 12167 cu. yards
Question 18
1 / -0

Rahul wants to fit the cuboids of dimensions 2 units 3 units 5 units in a cube. The minimum number of cuboids that he can fit into the cube is

A
800

B
600

C
900

D
1000

Solution

Dimensions of cuboid = 2 units 3 units 5 units
Dimensions of the required cube = 2 2 2 3 3 3 5 5 5
So, the minimum number of cuboids that can fit in the cube = 2 2 3 3 5 5 = 900
Question 19
1 / -0

A rectangular metallic cuboid, with dimensions 14 m × 10 m × 5 m, is melted, and some other molten alloy is added to it. This mixture is then cooled down and turned into a solid cube, having integral sides. What is the minimum amount of metal that is added and what is the side of this cube?

A
25 m³,5 m

B
30 m³,7 m

C
29 m³, 9 m

D
27 m³,8 m

Solution

Volume of the metallic cuboid = 14 m × 10 m × 5 m = 700 m³
A certain amount of molten alloy is added to make it a perfect cube.
Now, 729 is the closest perfect cube greater than 700.
Hence, amount of molten alloy added to metal = 729 - 700 = 29 m³

Volume of the new metallic cube = 729 m³
So, side of the cube = = 9 m
Question 20
1 / -0

The people of Belgam village decide to store rainwater for irrigation purpose. They construct a cuboidal tank of length 24 m and breadth 13 m, with a total capacity of 13,416 m³. Find the height of the tank.

A
23 m

B
37 m

C
43 m

D
47 m

Solution

Let the length, breadth and height of the tank be L, B and H, respectively.

Now, volume of the tank = 13,416 m³
So,
13,416 = L × B × H
13,416 = 24 × 13 × H

H = = 43 m

Therefore, the height of the tank is 43 m.
Question 21
1 / -0

Which of the following statements is INCORRECT?

A. Three numbers are in the ratio 2 : 1 : 3 and the sum of their cubes is 972. The numbers are 4, 7 and 11.
B. The digit at the units place of the cube root of a three-digit number of the form xy8 is 2.
C. The smallest number by which 1800 should be divided to make it a perfect cube is 225.

A
A

B
B

C
C

D
None of these

Solution

(A) Let the numbers be 2x, x and 3x.
Sum of the cubes of the numbers = 972
Therefore, x³+ (2x)³+ (3x)³= 972
36x³ = 972
x³= 27
x = 3
So, the numbers are:
3
2x = 2 × 3 = 6
3x = 3 × 3 = 9

Hence, statement (A) is incorrect.

(B) The units digit of a number of the form xy8 is 8. Now, cube root of 8 is 2. Hence, the digit at the units place of the cube root of a three-digit number of the form xy8 is 2.
Hence, statement (B) is correct.

(C) 1800 = 2³ × 5² × 3²
To make it a perfect cube, it must be divided by 5²× 3², i.e. 225.
Hence, statement (C) is correct.
Question 22
1 / -0

Find the cube roots of the following:

(i) 0.000027 = _______
(ii) 1.520875 = _______
(iii) 3.511808 = _________
(iv) 49.027896 = ________

A
(i) 0.03, (ii) 1.15, (iii) 1.52, (iv) 3.66

B
(i) 0.03, (ii) 1.5, (iii) 2.52, (iv) 7.66

C
(i) 0.09, (ii) 12.15, (iii) 1.38, (iv) 3.06

D
(i) 0.03, (ii) 1.56, (iii) 1.23, (iv) 4.21

Solution

(i) = 0.03

(ii) = 1.15

(iii) = 1.52

(iv) = 3.66

Question 23

1 / -0

Match the following:

Column – I	Column – II
P. The smallest number that should be subtracted from 260 to make it a perfect cube is	114
Q. The smallest number that should be subtracted from 4634 to make it a perfect cube is	44
R. The smallest number that should be added to 1097 to make it a perfect cube is	538
S. The smallest number that should be added to 13710 to make it a perfect cube is	234

(P) 538, (Q) 44, (R) 114, (S) 234

(P) 234, (Q) 114, (R) 44, (S) 538

(P) 44, (Q) 538, (R) 234, (S) 114

(P) 114, (Q) 538, (R) 44, (S) 234

Solution

(P) 6³ ≤ 260 ≤ 7³
The nearest perfect cube less than 260 = 216
So, 260 - 216 = 44

(Q) 16³ ≤ 4634 ≤ 17³
The nearest perfect cube less than 4634 = 4096
So, 4634 - 4096 = 538

(R) 10³ ≤ 1097 ≤ 11³
The nearest perfect cube greater than 1097 = 1331
So, 234 should be added to 1097.

(S) 23³ ≤ 13710 ≤ 24³
The nearest perfect cube greater than 13710 = 13824
So, 114 should be added.

Question 24
1 / -0

Evaluate .

A
2

B
3

C
4

D
1

Solution

-

= ÷ -

= - = 1
Question 25
1 / -0

Read the statements given below carefully:

(1) Cube root of 175.616 is a rational number.
(2) Cube of an even number is always odd.

Which of the two statements is/are correct?

(A) Only statement (1)
(B) Only statement (2)
(C) Both statements (1) and (2)
(D) Neither statement (1) nor statement (2)

A
(A)

B
(D)

C
(C)

D
(B)

Solution

Statement (1):
= 5.6

It is a rational number. Hence, (1) is true.

Statement (2):
Cube of an even number is always even. Hence, (2) is false.