Number System - Study Mode

[#616] If p is a positive fraction less than 1, then
Correct Answer

(D) $$frac{2}{p}$$ - p is a positive number

Explanation

Solution: p < 1 ⇒ $$frac{1}{p}$$ > 1 ⇒ $$frac{2}{p}$$ > 2 ⇒ $$frac{2}{p}$$ - p > 2 - p > 0 [∵ p < 1] Hence, $$left( {frac{2}{p} - p}
ight)$$ is a positive number

[#617] Given that (1 2 + 2 2 + 3 2 + ..... + 20 2 ) = 2870, the value of (2 2 + 4 2 + 6 2 + ... + 40 2 ) is :
Correct Answer

(C) 11480

Explanation

Solution: = (2 2 +4 2 +6 2 + ..... + 40 2 ) = (1 × 2) 2 +(2 × 2) 2 +(2 × 3) 2 + ..... + (2 × 20) 2 = 2 2 × (1 2 + 2 2 + 3 2 + ..... + 20 2 ) = (4 × 2870) = 11480

[#618] If the number 357*25* is divisible by both 3 and 5, then the missing digits in the unit's place and the thousandth's place respectively are :
Correct Answer

(D) None of these

Explanation

Solution: Let the unit's place be x and the thousand's place be y. Then, 357y25x is divisible by 5 only when x = 0 or x = 5 Also, this number is divisible by 3 only when sum of its digits is divisible by 3 So, (22 + x + y) must be divisible by 3 ∴ x + y = 2 Taking x = 0, we get y = 2 So, the unit place = 0 and thousand's place = 2

[#619] In the relation x > y + z, x + y > p and z < p, which of the following is necessarily true ?
Correct Answer

(B) x + y > z

Explanation

Solution: x + y > p and p > z ⇒ x + y > z

[#620] (x n - a n ) is divisible by (x - a)
Correct Answer

(A) for all values of n

Explanation

Solution: We know that (x n - a n ) is always divisible by (x - a) for all values of n.