Cubes And Cube Roots Test

Result

Cubes And Cube Roots Test - 5

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Question 1
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 √y = 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 2
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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 3
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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 4

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.