# F.4-5 Maths

1.The floor plan of an apartment which is formed by two squares. If the perimeter and the area of the apartment are 16 m and 13 m 2 respectively, find the lengths of sides of the two squares.

2. Peter pays \$45 for x mangoes with unit price \$y. If the unit price is reduced by \$1, he can buy 2 more mangoes and saves \$1. Find the values of x and y.

3.It is given that the line L: x + y + k = 0 intersects the quadratic curve C: y = 2 x 2 - 4 x + k .Find the range of possible values of k.

4. The quadratic curve C: y = px 2 + 2x + 6 touches the line L: y = 8 – 2x at one point A.

(a) Find the value of p.

(b) Find the coordinates of A.

5.

3x-2y-1=0---------(1)

5x^2-y^2=1----------(2)

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1.

Let a m and b m be the lengths of sides of the two squares.

(a > b)

4a + 4b - 2b = 16 …… 

a² + b² = 13 …… 

From :

4a + 2b = 16

2a + b = 8

b = 8 - 2a …… 

Put  into :

a² + (8 - 2a)² = 13

a² + 64 - 32a + 4a² = 13

5a² - 32a + 51 = 0

(a - 3)(5a - 17) = 0

a = 3 or a = 17/5

a = 3 or a = 3.4

Put a = 3 into  :

b = 8 - 2(3)

b = 2

Put a = 3.4 into  :

b = 8 - 2(3.4)

b = 1.2

The lengths of the sides of the two square is:

2 m and 3 m or 1.2 m and 3.4 m

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2.

xy = 45 …… 

(x + 2)(y -1) = 45 - 1 …… 

From :

x = 45/y …… 

Put  into :

[(45/y) + 2](y - 1) = 44

45 - (45/y) + 2y - 2 = 44

2y - 1 - (45/y) = 0

y[2y - 1 - (45/y)] = 0

2y² - y - 45 = 0

(y - 5)(2y + 9) = 0

y = 5 or y = -9/2 (rejected)

Put y = 5 into :

x = 45/5

x = 9

Hence, x = 9, y = 5

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3.

L: x + y + k = 0 …… 

C: y = 2x² - 4x + k …… 

From :

y = -x - k …… 

 = :

2x² - 4x + k = -x - k

2x² - 3x + 2k = 0

Since the equation has two real roots, thus determinant Δ > 0

(3)² - 4(2)(2k) > 0

9 - 16k > 0

-16k > -9

16k < 9

k < 9/16

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4.

(a)

C: y = px² + 2x + 6 …… 

L: y = 8 - 2x …… 

 = :

px² + 2x + 6 = 8 - 2x

px² + 4x - 2 = 0 …… 

Since the equation has double roots, thus determinant Δ = 0

(4)² - 4(p)(-2) = 0

16 + 8p = 0

p = -2

(b)

Put p = -2 into :

-2x² + 4x - 2 = 0

x² - 2x + 1 = 0

(x - 1) = 0

x = 1 (double roots)

Put x = 1 into :

y = 8 - 2(1)

y = 6

Coordinates of A = (1, 6)

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5.

3x - 2y - 1 = 0 …… 

5x² - y² = 1 …… 

From :

2y = 3x - 1

y = (3x - 1)/2 …… 

Put  into:

5x² - [(3x - 1)/2]² = 1

5x² - (3x - 1)²/4 = 1

20x² - 9x² + 6x - 1 = 4

11x² + 6x - 5 = 0

(11x - 5)(x + 1) = 0

x = 5/11 or x = -1

Put x = 5/11 into :

y = [3(5/11) - 1]/2

y = 2/11

Put x = -1 into :

y = [3(-1) - 1]/2

y = -2

x = 5/11, y = 2/11 or x = -1, y = -2

資料來源： 土扁
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