S5_Mock01_E

If p > q > 0 and k < 0, which of the following must NOT be true?
Ans:
3p + k < 3q + k <=> 3p < 3q <=> p < q (I is false)
kp + 2 < kq + 2 <=> kp < kq <=> p > q (II is true) (e.g. -2(3) < -2(2) <=> 3 > 2)
3k/p^2 < 3k/q^2 <=> 1/p^2 > 1/q^2 <=> p^2 < q^2 <=> p < q (III is false)

If x - 2 is a factor of x3 + ax2 – 6b, then 12b - 8a - 9 =

Ans:
2^3 + a*2^2 - 6b = 0
-6b + 4a + 8 = 0
12b - 8a - 16 = 0
12b - 8a - 9 = 7

If (x + 1) is an odd number, which of the following is also an odd number?

Ans:
2(x + 1) is even as even*odd = even.
(x + 1)(x + 2) is a pair of consecutive integers which are one even and one odd, so it is even.
(x - 1)^2 = (x - 1)(x - 1) = odd*odd = odd, because
odd numbers are in the 2k + 1 form, where k is an integer, and
(2k + 1)^2 = 2(2k^2 + 2k) + 1 which is the 2k + 1 form.
Even numbers alternate with odd numbers, so
x is even and bears the 2k form, and so x^2 = (2k)^2 = 2(2k^2) which is also the 2k form.

A mixture of 6 kg coffee of brand X and 4 kg coffee of brand Y costs $57/kg. If coffee of brand X costs $45/kg, then coffee of brand Y costs

Ans:
Let X be the price/kg of brand X and
Y be the price/kg of brand Y.
6X + 4Y = 570 (per 10 kg)
X = 45
Y = (570 - 45*6)/4 = 75

If 90o < x < y < 180o, which of the following is/are true?

Ans:
Let Lx be the length of the side on the x-axis of a right triangle and
Ly be the length of the side on the y axis of a right triangle
cosx < cosy <=> Lx of angle x > Lx of angle y <=> x > y (I is false)
tanx < tany <=> Ly/Lx of angle x > Ly/Lx of angle y <=> Ly of angle x > Ly of angle y AND Lx of angle x < Lx of angle y <=> x < y (II is true)
sinx < siny <=> Ly of angle x < Ly of angle y <=> x > y (III is false)

(a - b)^2 = a*(a - b) - b*(a - b) = a^2 + b^2 - 2ab = a^2 + b^2 + (2ab - 2ab) - 2ab = a^2 + b^2 + 2ab - 2ab - 2ab = (a + b)^2 - 4ab.

For Q1
I:3p + k < 3q + k
3p< 3q
p<q
Since p > q > 0, so I is false
II:kp + 2 < kq + 2
kp< kq
p>q (you have to change sign because k<0)
Since p > q > 0, so II is true
III:3k/p2 < 3k/q2
3k/p^2<3k/q^2 (you have to change sign because k<0)
1/p^2>1/q^2 (no need change sign because p^2 and q^2 >0)
q^2>p^2
q>p
Since p > q > 0, so III is false.
But you should be careful that the condition will change if either p or q is negative
For example, (-4)^2 will larger than 3^2, but -4<3

For Q2:
Since x - 2 is a factor of x^3 + ax^2 – 6b
Let f(x) be x^3 + ax^2 – 6b
f(2)=2^3+a(2^2)-6b=0
so 8+4a-6b=0, then -6b+4a=-8, then 12b-8a=16
12b - 8a - 9 = 16-9 =7

For Q3, to get an odd number by x times y, both x and y should be odd numbers
For A: 2(x+1) --> 2 is an even, so false
For B: (x+1)(x+2)--> (x+2) is an even, so false
For C: (x-1) is an odd number, (x-1)^2=(x-1)(x-1), both are odd numbers, so true
For D: x is an even, (x)(x), both are even numbers, so false

For Q4:Let $x be the cost of coffee y.
[(6)(45)+4x]/10 =57
so x=$75

For Q5, you should know their graphs
I: cos x is decreasing when 90<x<180
Because 90 < x < y < 180
cos x should be > cos y, so false
II tan x is increasing when 90<x<270
Because 90 < x < y < 180
so tanx<tany true
III sin x is decreasing when 90<x<180
Because 90 < x < y < 180
sin x should be > sin y (false)

For the extra question,
(a-b)^2=a^2-2ab+b^2
If we plus 2ab and then minus 2ab, the value is unchanged 這招叫無中生有
=a^2-2ab+b^2+2ab-2ab
=(a^2+2ab+b^2)-4ab
=(a+b)^2-4ab
The reason for doing this is that we would like to make
sum of roots and product of roots
a+b and ab
(a-b) then we cant use sum of roots to cal the value
It is useful in handling quadratic questions