Number System - Study Mode

[#296] If a and b are positive integers, a > b and (a + b) 2 - (a - b) 2 > 29 Then the smallest value of a is -
Correct Answer

(B) 4

Explanation

Solution: Since (a + b) 2 - (a - b) 2 = 4ab,
So, the given expression should be a multiple of 4 So, the least value of 4ab is 32 and so the least value of ab is 8 Hence, the smallest value of a is 4 and that of b is 2 Hence, a = 4

[#297] The digit in the unit's place of the number (67) 25 - 1 must be -
Correct Answer

(B) 6

Explanation

Solution: Unit digit of (67) 25 = Unit digit of 7 25 Unit digit of 7 4 is 1 and so the unit digit of (7 4 ) 6 is 1 ∴ Unit digit of 7 25 = (1 × 7) = 7 Hence, the unit digit of (7 25 - 1) is (7 - 1) = 6

[#298] 287 × 287 + 269 × 269 - 2 × 287 × 269 = ?
Correct Answer

(D) 324

Explanation

Solution: $$eqalign{
& left( {287 imes 287 + 269 imes 269 - 2 imes 287 imes 269}
ight) cr
& = {left( {287}
ight)^2} + {left( {269}
ight)^2} - left( {2 imes 287 imes 269}
ight) cr
& = {left( {287 - 269}
ight)^2} cr
& = {left( {18}
ight)^2} cr
& = 324 cr} $$ [∵ a 2 + b 2 - 2ab = (a - b) 2 ]

[#299] What number should replace M in this multiplication problem ?
Correct Answer

(A) 0

Explanation

Solution: Clearly, M = 0, since 304 × 4 = 1216

[#300] 397 × 397 + 104 × 104 + 2 × 397 × 104 = ?
Correct Answer

(B) 251001
(F) 251001

Explanation

Solution: Given expression, = (397) 2 + (104) 2 + 2 × 397 × 104 = (397 + 104) 2 = (501) 2 = (500 + 1) 2 = (500) 2 + 1 2 + 2 × 500 × 1 = 250000 + 1 + 1000 = 251001