Simplification
- The smallest number by which 243000 be divided so that the quotient is a perfect cube is
-
View Hint View Answer Discuss in Forum
243000 = 243 × 1000
= 3 × 3 × 3 × 3 × 3 × 10 × 10 × 10
= 33 × 32 × 103
∴ Required number = 32 = 9Correct Option: C
243000 = 243 × 1000
= 3 × 3 × 3 × 3 × 3 × 10 × 10 × 10
= 33 × 32 × 103
∴ Required number = 32 = 9
- The sum of the cubes of two numbers is 793. The sum of the numbers is 13. Then the difference of the two numbers is
-
View Hint View Answer Discuss in Forum
Let the numbers be a and b where a > b.
According to the question,
a3 + b3 = 793
and a + b = 13
∴ (a + b)3 = a3 + b3 + 3ab (a + b)
⇒ (13)3 = 793 + 3ab ×13
⇒ 2197 = 793 + 39ab
⇒ 39ab = 2197 – 793 = 1404⇒ ab = 1404 = 36 39
∴ (a + b)2 = (a + b)2 – 4ab
= (3)2 = 4 × 36
= 169 – 144 = 25
⇒ a – b = √25 = 5Correct Option: C
Let the numbers be a and b where a > b.
According to the question,
a3 + b3 = 793
and a + b = 13
∴ (a + b)3 = a3 + b3 + 3ab (a + b)
⇒ (13)3 = 793 + 3ab ×13
⇒ 2197 = 793 + 39ab
⇒ 39ab = 2197 – 793 = 1404⇒ ab = 1404 = 36 39
∴ (a + b)2 = (a + b)2 – 4ab
= (3)2 = 4 × 36
= 169 – 144 = 25
⇒ a – b = √25 = 5
- What is the smallest number by which 625 must be divided so that the quotient is a perfect cube ?
-
View Hint View Answer Discuss in Forum
∴ 625 = 5 × 5 × 5 × 5 = 53 × 5
For the smallest cube number,
625 should be divided 5,
625 ÷ 5 = 125 = 53Correct Option: B
∴ 625 = 5 × 5 × 5 × 5 = 53 × 5
For the smallest cube number,
625 should be divided 5,
625 ÷ 5 = 125 = 53
- The value of (1001)3 is
-
View Hint View Answer Discuss in Forum
Look at the pattern :
1001 × 1001 = 1002001
1001×1001 × 1001 = 1003003001Correct Option: A
Look at the pattern :
1001 × 1001 = 1002001
1001×1001 × 1001 = 1003003001
-
If x = √3 + √2 then the value of x3 − 1 is x3
-
View Hint View Answer Discuss in Forum
x = √3 + √2
∴ 1 = 1 x √3 + √2 = √3 − √2 (√3 + √2)(√3 − √2) = √3 − √2 = √3 − √2 3 − 2 ∴ x − 1 = √3 + √2 − √3 + √2 = 2√2 x ∴ x3 − 1 = 
x − 1 
3 + 3 x − 1 x3 x x
= (2√2)3 + 3 × 2√2
= 16√2 + 6√2 = 22√2Correct Option: C
x = √3 + √2
∴ 1 = 1 x √3 + √2 = √3 − √2 (√3 + √2)(√3 − √2) = √3 − √2 = √3 − √2 3 − 2 ∴ x − 1 = √3 + √2 − √3 + √2 = 2√2 x ∴ x3 − 1 = x − 1 3 + 3 x − 1 x3 x x
= (2√2)3 + 3 × 2√2
= 16√2 + 6√2 = 22√2
