中4math題..

1)
a = 2 - (1/b)
(1/b) = 2 - a
b = 1/(2 - a)


2)
The answer is a.
a = (b - 1)/(b - 2)
a(b - 2) = b - 1
ab - 2a = b - 1
ab - b = 2a - 1
b(a - 1) = 2a - 1
b = (2a - 1)/(a - 1)


3)
There is no correct answer. The answer should be k ≤ 5.
x2 - 4x + k = 1 has real root.
x2 - 4x + (k - 1) = 0 has real root.
D = (-4)2 - 4(1)(k - 1) ≥ 0
16 - 4k + 4 ≥ 0
20 - 4k ≥ 0
5 - k ≥ 0
-k ≥ -5
k ≤ 5

If the question changes to "no real root", the answer is c.
x2 - 4x + k = 1 has no real root.
x2 - 4x + (k - 1) = 0 has no real root.
D = (-4)2 - 4(1)(k - 1) < 0
16 - 4k + 4 < 0
20 - 4k < 0
5 - k < 0
-k < -5
k > 5


4)
The answer is a.
x2 + 2x - k = 2 has two distinct real roots
x2 + 2x - (k + 2) = 0 has two distinct real roots
D = (2)2 - 4(1)[-(k + 2)] > 0
4 + 4k + 8 > 0
4k + 12 > 0
k + 3 > 0
k > -3


5)
The answer is b.
Let the price of an apple be $a, and that of an orange be $r.
6r + 3a = 42 ... (1)
8r + 5a = 60 ... (2)
4x(1): 24r + 12a = 168 ... (3)
3x(2): 24r + 15a = 180 ... (4)
(4)-(3): 3a = 12
a = 4
Price of an apple = $4
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