MATHS QUESTIONS!! HELP!!><

34.
The answer : (C) The roots of the equation f(x) + 4 = 0 are real numbers.

The vertex of the quadratic function is (3, -4).
Hence, y = f(x) = a(x - 3)² - 4, where a is a real number.

A. false
f(x) = 0
a(x - 3)² - 4 = 0
(x - 3)² = 4/a
x = 3 ± √(4/a)
x is an integer only when a = 1 or a = 4

B. false
f(x) - 3 = 0
a(x - 3)² - 4 - 3 = 0
(x - 3)² = 7/a
x = 3 ± √(7/a)
The roots may not be rational, as √(7/a) may not be rational.

C. true
f(x) + 4 = 0
a(x - 3)² - 4 + 4 = 0
a(x - 3)² = 0
(x - 3)² = 0
x = 3 (double roots)
The roots are real.

D. false
f(x) + 5 = 0
a(x - 3)² - 4 + 5 = 0
(x - 3)² = -1/a
x = 3 ± √(-1/a)
The roots are real when a < 0.


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38.
The answer : (C) 120°

Let a = 2013

In ΔXBY :
XY² = XB² + BY² (Pythagorean theorem)
XY² = (a/2)² + (a/2)²
XY² = a²/2
XY = a/√2

In ΔYGZ :
YZ² = YG² + GZ² (Pythagorean theorem)
YZ² = (a/2)² + (a/2)²
YZ² = a²/2
YZ = a/√2

In ΔXBG :
XG² = XB² + BG² (Pythagorean theorem)
XG² = (a/2)² + (a)²
XG² = 5a²/4

In ΔXGZ :
XZ² = XG² + GZ² (Pythagorean theorem)
XZ² = 5a²/4 +(a/2)²
XZ² = 3a²/2

In ΔXYZ :
cos∠XYZ = [XY² +YZ² - XZ²] /[2 × XY × YZ]
cos∠XYZ = [(a²/2)+ (a²/2) - (3a²/2)]/ [2 × (a/√2) × (a/√2)]
cos∠XYZ = -(a²/2)/ a²
cos∠XYZ = -1/2
∠XYZ = 180° - 60°
∠XYZ = 120°


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43.
The answer : (D) 4692

Case I : Neither John nor Paul is selected
• Choose 4 from 18 other students. (18C4)

Case II : Either John or Paul is selected
• Choose one from John and Paul. (2C1)
• Then, Choose 3 from 18 other students. (18C3)

Number of different teams
= 18C4 + 2C1 × 18C3
= 3060 + 2 × 816
= 4692

2015-03-03 12:24:30 補充：
43. Alternative method :

Number of different teams
= (Number of teams without restriction) - (Number of team when both of John and Paul are selected)
= 20C4 - 2C2 × 18C2
= 4845 - 1 × 153
= 4692

43.
1st method:
Any teams - Teams(with both John and Paul) = 20C4 - 2C2 x 18C2 = 4692

2nd method:
Teams(without John and Paul)+Teams(with John only)+Teams(with Paul only)
= 18C4 + 1C1x18C3 + 1C1x18C3 = 4692

34.
f(x) = a(x-3)^2-4 (where a is a non-zero constant)
If f(x) + 4 = 0
Then
a(x-3)^2 = 0
x = 3 (repeated root)
3 is a real number, so ans is C

38.
XY = YZ = sq rt[(2013/2)^2+(2013/2)^2] = 2013 / sq rt(2)
XZ = sq rt(XM^2+MZ^2)= sq rt[2013^2 + (2013/sqrt(2))^2] = sq. rt (3/2) x 2013
cos (angle XYZ) = (XY^2+YZ^2-XZ^2)/(2 x XY x YZ)
(angle XYZ) =120degree

2015-03-03 12:38:39 補充：
http://postimg.org/image/hdiebccd1/