# 多條中四math數 (20點)

1. Maria takes 5 days more to do a job than John. Together, they can do the

job on 6 days. How long does it take each to do the job alone?

2a. Let y=4x²+mx+4. Find the value of m if the minimum value of y is 0

2b. Let y=4x²+mx+4. Find the value of m if the minimum value of y is attained at x=5

3. P divides a line segment AB of length 8 cm. Find the distance of P from A so that AP²+PB² is a minimum.

4. a piece of wire of length l is cut into pieces, and each is bent to from a square. Find the minimum total area of the two squares.

5. Solve 2cos(2x+75°)+1=0 for 0°≦x≦180°

6. Solve the equation 4sin²x-3=0 such that cos x<0 and 0°≦x≦360°

7. The length and the width of rectangle are x cm and y cm respectively.

(a) The perimeter of the rectangle is 68cm. Write down an equation in x and y.

(b) Suppose its length is increased by 4 am and its width is decreased by 4 cm.

(i) Find, in terms of x and y, the new area of the rectangle.

(ii) Find its original dimensions if the new area is reduced by 40cm².

8. ax³+bx²is divisible by4x-1 and the remainder is 5 when it is divided by x+1. Find the values of a and b. Hence factorize the polynomial.

9. A two-digit number is less than three times the product of its digit by 8, and the digit in the tenth place exceeds the digit in the unit place by 2. Find the number.

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1.

Let John takes x days to do the job alone.

Maria takes 5+x days to do the job alone.

[ 1/x + 1/(5+x) ]*6 = 1

30 + 12x = 5x + x^2

x^2 - 7x -30 = 0

(x-10)(x+3)=0

x = 10 or -3(rej)

So, John takes 10 days,

Maria takes 15 days.

2a. y=4x²+mx+4

y = 4 ( x^2 + m/4 * x ) + 4

= 4 (x + m/8)^2 + 4 - m^2 /16

min value = 4 - m^2 /16 = 0

m^2 = 64

m= +8 or -8

2b.

y = 4 ( x^2 + m/4 * x ) + 4

= 4 (x + m/8)^2 + 4 - m^2 /16

y attaines min value at x = 5

m/8 = -5

m = -40

3.

Let AP = x cm

AP²+PB²

= x^2 + (8-x)^2

= 2(x^2 - 8x) +64

= 2 (x - 4)^2 +32

So, AP²+PB² is a minimum when x=4,

distance of P from A= 4cm

2008-08-02 15:42:52 補充：

4.

Let x = length of the smaller square,

L/4 - x = length of the bigger square,

total area of the two squares

= x^2 + (L/4 - x)^2

= 2 (x^2 - L/4 * x) + L^2 / 16

= 2 (x- L/8)^2 + L^2 /32

So, minimum total area = L^2 /32

2008-08-02 15:47:29 補充：

5.

75°&lE;(2x+750°)&lE;435°

2cos(2x+75°)+1=0

cos(2x+75°) = -1/2

2x+75° = 120° or 240°

x = 22.5° or 82.5°

2008-08-02 15:51:53 補充：

6.

cos x<0 and 0°&lE;x&lE;360°

So, 90°&lE;x&lE;270°

4sin²x-3=0

sin²x = 3/4

sinx = (root 3) / 2 or - (root 3) / 2

x = 120° or 240°

2008-08-02 16:02:46 補充：

7.

(a)

2x + 2y = 68

x + y = 34

(b)

(i)

area

= (x+4)(y-4)

(ii)

Solve the following simultaneous eqt:

xy - (x+4)(y-4) = 40

x + y = 34

After simplying,

x+y =34

x-y = 6

So, x = 20, y = 14,

length and the width of rectangle are 20 cm and 14 cm respectively.

2008-08-02 16:12:18 補充：

8.

ax³+bx²is divisible by4x-1,

a (1/4)³ + b (1/4)² = 0

a / 64 + b / 16 =0

a + 4b = 0

and the remainder is 5 when it is divided by x+1,

a (-1)³ + b (-1)² = 5

b - a = 5

Solve the following simultaneous eqt:

a + 4b = 0

b - a = 5

So, a = -4, b = 1

the polynomial

= -4 x³+ x²

= x² (1-4x)

2008-08-02 16:18:50 補充：

9.

Let x = tens digit,

y = unit digit.

Solve the following simultaneous eqt:

10x + y = 3xy - 8

x - y = 2

Solve the following simultaneous eqt:

xy - (x+4)(y-4) = 40

x + y = 34

After simplying,

3y² - 5y - 28 = 0

(3y+7)(y-4) = 0

y = 4 or -7/3 (rej)

So, y = 4, x = 6,

the number is 64.