The value of (1001)3 is
  • a)
    1003003001
  • b)
    1003033001
  • c)
    100300301
  • d)
    103003001
Correct answer is option 'A'. Can you explain this answer?

SITA DEVI answered  •  11 hours ago
The value of (1001)= 1001 × 1001 × 1001
= 1003003001
Or,
1001 = 1000 + 1
Cubing
(1001)3 = (1000 + 1)3
We know that, (a + b)3 = a3 + b+ 3ab (a + b)
⇒ (1001)3 = 10003 + 1 + 3000 × (1000 + 1)
⇒ (1001)= 1000000000 + 1 + 3003000
⇒ (1001)= 1003003001

If ‘a’ and ‘b’ are two odd positive integers, by which of the following is (a4 – b4) always divisible?
  • a)
    3
  • b)
    6
  • c)
    8
  • d)
    12
Correct answer is option 'C'. Can you explain this answer?

VIJAYATA DEVI answered  •  yesterday
We know that, Summation and Subtraction between two odd integers gives an even integer.
Given, a and b are two odd positive integers.
∴ a + b = 2k1 ……(1)
And a – b = 2k........(2)
Where, k1 and k2are two integers.
(1) + (2) ⇒ 2a = 2(k1 + k2) ⇒ a = k1 + k2
(1) - (2) ⇒ 2b = 2(k1 - k2) ⇒ b = k1 - k2
∴ a2 + b2
= (k1 + k2)2 + (k1 - k2)2
= k12 + k22 + 2k1k2 + k12 + k22 - 2k1k2
= 2(k12 + k22) ………..(3)
Given expression is,
a4 – b4
= (a2 – b2)(a2 + b2)
= (a + b) (a - b)(a2 + b2)
Putting values from (1), (2) and (3)
= 2k× 2k2 × 2(k12 + k22)
= 8× k× k2 × (k12 + k22)
∴ a4 – b= 8× k× k2 × (k12 + k22) is always divisible by 8.

Jais and his father has an age difference of 35 years now. After 5 years, the sum of their age is 125. What will be the age of Jais and his father after 12 years from now?
  • a)
    51 & 85
  • b)
    40 & 75
  • c)
    52 & 87
  • d)
    45 & 70
Correct answer is option 'C'. Can you explain this answer?

PRIYAM BHARDWAJ answered  •  yesterday
Let the present age of Jais be X years and the age of his father be Y years.
Presently, Jais and his father has an age difference of 35 years.
 → Y – X = 35. ----------- (1)
After 5 years, the sum of their ages will be 125.
→ (X + 5) + (Y + 5) = 125
→ X + Y + 10 =125
→ Y + X = 115. ----------- (2)
Adding equation (1) and (2),
Y – X + Y + X = 35 + 115
→ 2Y = 150
→ Y = 75.
Similarly, X = Y – 35 = 75 – 35 = 40.
After 12 years, Jais age is 52 (40 + 12) years and his father’s age will be 87 (75 + 12) years.
Hence, the correct answer is 52, 87.

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