Math Master

x² + y² = 185
বা, (x + y)² - 2xy = 185
বা, (x + y)² = 185 + 2xy = 185 + 2.88 = 185 + 176 = 361
∴ x + y = 19 ...........(1)
আবার,
x² + y² = 185
বা, (x - y)² + 2xy = 185
বা, (x - y)² = 185 - 2xy = 185 - 2.88
= 185 - 176 = 9
∴ x - y = 3 .................. (2)
(1)নং + (2)নং ⇒ 2x = 22
∴ x = 11
আবার,
(1)নং - (2)নং ⇒ 2y = 16
∴ y = 8
∴ (xy) = (11, 8)

(x - y, 3) = (0, 2y - x)
∴ x - y = 0
বা, y = x
এবং 2y - x = 3
বা, 2.x - x = 3
∴ x = 3

x + y = 0
∴ y = -x
এখন, 2x - y = 3 = 0
বা, 2x + x + 3 = 0
বা, 3x = -3
∴ x = -1
∴ y = - (-1) = 1

দেওয়া আছে,
3/x = 2
∴ x = 3/2
আবার, 4/y = 3
∴ y = 4/3

∴ (3+y)/(x+4) = {3+(4/3)}/{(3/2)+4}
= {(9+4)/3}/{(3+8)/2}
= (13/3)/(11/2)
= 13/3 × 2/11
= 26/33

(x/3) - (2/y) = 4 ............... (1)
বা, (xy - 6)/3y = 4
বা, xy - 6 = 12y
∴ xy = 12y + 6 .......... (2)
আবার, (x/6) - (4/y) = 1
বা, (xy - 24)/6y = 1
বা, xy - 24 = 6y
বা, 12y + 6 - 24 = 6y
বা, 6y - 18 = 0
বা, 6y = 18
∴ y = 3
(1) নং থেকে পাই,
x/3 - 2/3 = 4
বা, x - 2 = 12
∴ x = 14
∴ (x, y) = (14, 3)

89×89 + 87×87 - 2×89×87
= (89)² + (87)² - 2×89×87
= (89 - 87)²
= 2² = 4

√(5ⁿ) = 3125 = 5⁵
বা, 5ⁿ = (5⁵)²
বা, 5ⁿ = 5¹⁰
∴ n = 10
∴ 5^n-5 = 5^{10 - 5}
= 5⁵ = 3125

প্রশ্নমতে,
50⁷ × 20⁷ = 10⁸ × 10^x
বা, (50×20)⁷ = 10^x+8
বা, (1000)⁷ = 10^x+8
বা, (10³)⁷ = 10^x+8
বা, 10²¹ = 10^x+8
বা, x + 8 = 21
∴ x = 13

log₂1024 + log₄1024 = log₂2¹⁰ + log₄4⁵
= 10 log₂2 + 5 log₄4
= 10 + 5 = 15

log₂√10 - log₂√(5/2)
= log₂(10)^1/2 - log₂(5/2)^1/2
= 1/2 log₂10 - 1/2 log₂5/2
= 1/2 [log₂(5×2) - log₂5/2]
= 1/2 [log₂5 + log₂2 - (log₂5 - log₂2)]
= 1/2 [log₂5 + 1 - log₂5 + 1]
= 1/2 × 2
= 1