maths problem
C1.It is given that z=(p+2i)(i-3) .(complex no)
(a)If p is a real number, express z in standard form.
(b)If z is a purely imaginary number, find the value of p.
C2.If Alpha and Beta are the roots of the equation 2x^2+3=3x, find a quadratic equation in x with each of the following sets of roots.
(a)1/Alpha,1/Beta
(b) Alpha/Beta,Beta/Alpha
C3.(a)If the quadratic equation kx^2+(3k-1)x+(2k-1)=0 has two equal real roots and k is a constant, find the value of k.
(b)From the result of (a), solve the equation kx^2+(3k-1)x+(2k-1)=0 .
C4.Given that f(x)=x-3 and g(x)=1/x , where x is not 0 , find the values of the following.
(a)f(4)+g(1/4)(b)f(5)*g(1/2) (c) f(2)/g(2)
C5.In a factory, the total cost of producing n boxes of models is C(n) (in $), where .
(a)Find the total cost of producing 1 500 boxes of models.
(b)It is given that the selling price of each box of model is $120. Express the profit of selling n boxes of models as a function P(n) (in $).
(c)If 2 000 boxes of models are sold, find the profit of the factory.
Update:where C(n)=25000+30n
Comments
C1.
(a)
z = (p + 2i)(i - 3)
= pi - 3p - 2 - 6i
= -(3p + 2) + (p - 6)i
(b)
Since z is a purely imaginary number, then its real part is 0.
-(3p + 2) = 0
p = -2/3
C2.
αand β are the roots of the equation2x²-3x + 3 = 0
Sum of the roots: α + β = 3/2
Product of the roots : αβ = 3/2
(a)
(1/α) + (1/β)
= (α+ β)/αβ
= (3/2)/(3/2)
= 1
(1/α)* (1/β)
= 1/αβ
= 1/(3/2)
= 2/3
The required equation is :
x² - x + (2/3) = 0
i.e. 3x²- 3x + 2 = 0
(b)
(α/β) + (β/α)
= (α² + β²)/αβ
= [(α+ β)²- 2αβ]/αβ
= [(3/2)² - 2(3/2)]/(3/2)
= -1/2
(α/β)* (β/α)
= 1
The required equation is :
x² +(1/2)x + 1 = 0
i.e. 2x² + x + 2 = 0
C3.
(a)
Since kx² + (3k - 1)x + (2k - 1) = 0 has two equal real roots, then Δ = 0
(3k - 1)² -4k(2k - 1) = 0
9k² - 6k+ 1 - 8k² + 4k= 0
k² - 2k+ 1 = 0
(k - 1)² = 0
k = 1 (double roots)
(b)
Put k = 1 into the equation :
(1)x² + [3(1) - 1)x + (2(1) - 1) = 0
x² + 2x+ 1 = 0
(x + 1)² = 0
x = -1 (double roots)
C4.
(a)
f(4) + g(1/4)
= [(4) - 3) + [1/(1/4)]
= 1 + 4
= 5
(b)
f(5) * g(1/2)
= [(5) - 3) * [1/(1/2)]
= 2 * 2
= 4
(c)
f(2) / g(2)
= [(2) - 3] / (1/2)
= -1 * 2
= -2
C5.
The question is incomplete, something has been missed :
"Ina factory, the total cost of producing n boxes of models is C(n) (in $), where ......??????"
2012-07-06 23:40:47 補充：
C5.
(a)
C(1500)
= 25000 + 30(1500)
= 70000
Total cost = $70000 ...... (answer)
(b)
P(n) = 120n ...... (answer)
(c)
P(2000) - C(2000)
= 120(2000) - [25000 + 30(2000)]
= 155000
Profit of the factory = $155000 ...... (answer)