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• Graphs/Functions - 2015 Sample Paper 1 JC HL RyanS

Hello,

I am having trouble with Questions 7 and 10! Can someone please help?

Thanks :)

https://www.examinations.ie/schools/Mathematics_Sample_Paper_P1_-_Higher_Level.pdf

1. Dazzla16

Hi RyanS! Here are the worked solutions to those two questions:

7(a)

x^2 + (10 - x)^2 = the area of the two shaded squares. We will try different values for x to find the minimum possible value.

f(1) = (1)^2 + (10 - 1)^2 = 82

f(2) = (2)^2 + (10 - 2)^2 = 68

f(3) = (3)^2 + (10 - 3)^2 = 58

f(4) = (4)^2 + (10 - 4)^2 = 52

f(5) = (5)^2 + (10 - 5)^2 = 50

f(6) = (6)^2 + (10 - 6)^2 = 52

f(7) = (7)^2 + (10 - 7)^2 = 58

f(8) = (8)^2 + (10 - 8)^2 = 68

f(9) = (9)^2 + (10 - 9)^2 = 82

Therefore, the smallest possible value of the total area of the two shaded squares = 50 square units

(You may also draw the graph of the function f(x) = x^2 + (10 - x)^2 and find the minimum value of f(x). The answer will still be 50)

(b)

(before I explain the answer, you have to label the two sides that touches the line d. In my solution, I called them b and c)

In Pythagoras' theorem: a^2 = b^2 + c^2

If we apply this to the right angled triangle, then d^2 = b^2 + c^2

b^2 = area of bigger shaded square

c^2 = area of smaller shaded square

Therefore, area of bigger shaded square + area of smaller shaded square = d^2

10(a)

3 = (2)^2 + a(2) + b

3= 4 + 2a + b

2a + b = -1

-4 = (-5)^2 + a(-5) + b

-4 = 25 - 5a + b

-5a + b = -29

(b)

(You have to solve the simultaneous equations)

2a + b = -1

- -5a + b = -29

____________

7a = 28

a = 4

Sub back in to one of the equations:

2(4) + b = -1

8 + b = -1

b = -9

(c)

y = x^2 + 4x - 9

y = (0)^2 + 4(0) -9

y = -9

The co-ordinate is (0,-9)

(d)

y = x^2 + 4x - 9

0 = x^2 + 4x - 9

x^2 + 4x - 9 = 0

ax^2 + bx + c = 0, a = 1, b = 4, c = -9

(Now use the quadratic formula and sub in the values for a, b and c)

x = -(4) +/- square root((4)^2 - 4(1)(-9)

___________________________

2(1)

x = -4 +/- square root(16 + 36)

_____________________

2

x = -4 + square root(52)

________________

2

or

x = -4 - square root(52)

________________

2

x = 1.6 or x = -5.6

The two points are (1.6,0) and (-5.6,0)

Hope this helps :)

2. RyanS

It does help. Thanks a million!! 😃

3. Me

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