這幾題數學題 20點

1.
答案：(B) 22

f(x) = 2x²+ ax + b
= 2[x²+ (a/2)x] + b
= 2[x²+ (a/2)x + (a/4)²] - 2(a/4)² + b
= 2[x + (a/4)]²- (a²/8)+ b

在x = -3 時，f(x) 有最小值 4：
a/4 = 3 ...... [1]
-(a²/8)+ b = 4 ...... [2]

由 [1]：
a = 12

代入 [2] 中：
-(12²/8)+ b = 4
-18 + b = 4
b = 22


2.
答案： C. -3√3

[9√3 + 7√12 - 5√48] / (√5 + 2)(2 - √5)
= [9√3 + 7√(2²×3) - 5√(2⁴×3)] / (2 + √5)(2 - √5)
= [9√3 + 14√3) - 20√3)] / (2² - 5)
= 3√3 / (-1)
= -3√3


3.
答案： A. 323


x⁴ + x²y² + y⁴
= (√5 + 2)⁴ + (√5 + 2)²(√5 - 2)² + (√5 - 2)⁴
= [(√5)⁴ + 4(√5)³(2) + 6(√5)²(2)² + 4(√5)(2)³ + (2)⁴] + [(√5 + 2)(√5 - 2)]² + [(√5)⁴ - 4(√5)³(2) + 6(√5)²(2)² - 4(√5)(2)³ + (2)⁴]
= 2[(√5)⁴ + 6(√5)²(2)² + (2)⁴] + [5 - 4]²
= 2[25 + 120 + 16] + 1
= 323


4.
答案： C. x² + (5/2)x - (5/4)

用長除法，商式
= x² + (5/2)x - (5/4)


5.
答案： D. 21

6x² - 19x + 15 = (3x - 5)(2x -3)
所以 a = 3, b = -5, c = 2, d =-3

ac + bd = (3)(2) + (-5)(-3) = 21


6.
答案： A. 8/3

(43² + 42*43 + 21²) / (24*997 - 933*24)
= (43² + 2*43*21 + 21²) / 24(997 - 933)
= (43 + 21)² / 24*64
= 64² / 24*64
= 64/24
= 8/3


7.
答案： A. -3

設 6x² - 7x + m = (2x - 3)(3x +a)
則 6x² - 7x + m = 6x² + (2a - 9)x - 3a

比較 x 項：
2a - 9 = -7
2a = 2
a = 1

比較常數項：
m = -3a
m = -3


8.
答案： A. 3

a 為 x² + 5x - 4 = 0 之一根
則 a² + 5a - 4 = 0

√[(a + 3)(a + 2) - 1]
= √[a² + 5a + 6 - 1]
= √[(a² + 5a) + 5]
= √[(a² + 5a - 4) + 4 + 5]
= √[0 + 9]
= 3


9.
答案： D. -14/5

x² - 2x - 5 = 0 兩根為 a, b。
兩根之和： a + b = 2
兩根之積： ab = -5

(b/a) + (a/b)
= (b²/ab) + (a²/ab)
= (a² + b²) / ab
= [(a + b)² - 2ab] / ab
= [2² - 2(-5)] / (-5)
= -14/5

1.
f'(x) = 4x + a
在 x=-3 時 f(x)有最小值
即f'(-3) = 4(-3) + a = 0
得出 a = 12
帶回原式
f(-3) = 2(-3)^2 + 12(-3) + b = 4
= 18 - 36 + b = 4
=> b = 22

(B)

2.
9 √ 3+ 7 √ 12- 5 √ 48 / ( √ 5+2)(2- √ 5)
= 9 √ 3+ 7 √ (2^2)*3 - 5 √ (4^2)*3 / ( 2 + √ 5)( 2 - √5 )
= (9+14-20)√ 3 / ( 4 - 5 )
= -3√ 3

(C)

3.
令 X = x^2 = 9 + 4√ 5 ; Y = y^2 = 9 - 4√ 5
則 x^4 +x^2y^2 + y^4
= X^2 + XY + Y^2
= (X + Y)^2 - XY
= 18^2 - (81 - 80)
= 324 - 1
= 323

(A)

4.
1 +5/2-5/4
________________
2+1) 2 + 6 + 0 - 7
2 + 1
_______________
5 + 0
5 +5/2
___________
-5/2- 7
-5/2-5/4
_________
-23/4
(C)

5.
(ax+b)(cx+d) = (ac)x^2 + (ad + bc)x + bd = 6x^2-19x+15
ac + bd = 6 + 15 = 21

(D)

6.
43^2+42 * 43 +21^2 /24 *997 - 933 *24
= 43^2 + 2*21*43 + 21^2 /24 *( 997 - 933 )
= ( 43 + 21 )^2 / 24*64
= 64^2 / 24*64
= 64 / 24
= 8 / 3

(A)

7.
因為6x^2 -7x +m 是2x-3 的倍式，則：
可以使 2x - 3 = 0 之 x 值，也可以使 6x^2 -7x +m = 0
即 6(3/2)^2 - 7(3/2) + m = 0
27/2 - 21/2 + m = 0
3 + m = 0
m = -3

(A)

8.
a為 x^2 +5x -4 = 0 之一根，則：
將a代入方程式會成立：a^2 + 5a - 4 = 0
√ (a+3)(a+2)-1
= √ (a^2 + 5a + 6 - 1)
= √ (a^2 + 5a + 5 )
= √ [(a^2 + 5a - 4) + 9 ]
= √9
= 3

(A)

9.
由於方程式x^2-2x-5=0 之兩根為 a、b
即 x^2 - 2x - 5 = (x-a)(x-b) = x^2 - (a+b) x + ab = 0
比較係數，得到 a + b = 2 ； ab = -5

將題目通分，
b/a + a/b
= (a^2 + b^2) / ab
= [(a + b)^2 - 2ab] / ab
= [(a + b)^2 / ab] - 2
= 2^2/(-5) - 2
= -4/5 - 2
= -14/5

(D)

1.有最大最小值得點 代表微分值等於0
故F'(-3)= -12+a=0
a = 12
又 F(-3) =18-36+b =4
b= 22
選 (a) 22

2.
9 √ 3+ 7 √ 12- 5 √ 48 / ( √ 5+2)(2- √ 5)
= 9 √ 3+ 14√ 3 - 20√ 3 / (4-5)
= 9 √ 3+ 14√ 3 + 20√ 3
= 43√ 3

你題目是否有給錯呢 ?- 5 √ 48 / ( √ 5+2)(2- √ 5)
這段的+-號顛倒 答案就會換成 3√ 3

3.
x^4 +x^2y^2 + y^4 = (x^2 + y^2 )^2 - x^2y^2
x^2 = 9 + 4√ 5
y^2 = 9 - 4√ 5
x^2 + y^2 = 18
x^2y^2 = ( 9 + 4√ 5 )( 9 - 4√ 5 ) = 81 - 80 =1
x^4 +x^2y^2 + y^4 = 18^2 - 1 = 324 - 1 = 323

選(a) 323

4.
(2x^3+ 6x^2-7 ) = (2x+1) (x^2+5/2x-5/4)-33/4

選(c)

5.
6x^2-19x+15 = (ax+b)(cx+d) = acx^2+ ....+bd (係數對照)
ac = 6 bd =15
ac+bd = 21

選(d)

6.題目我稍作修改
( 43^2+42 * 43 +21^2 ) /24 *997 - 933 *24
=(43+21)^2 /24*(997-933)
=64^2 / 24*64
= 8 / 3

選(a)

7.若 6x^2 -7x +m 是2x-3 的倍式
代表 6x^2 -7x +m = (2x-3)q(x)
得到 q(x) = 3x+1
m = -3

選(a)

8.
a為 x^2 +5x -4 =0 之一根
表示 a^2 +5a -4 =0 => a^2 +5a = 4
√ (a+3)(a+2)-1
= √ (a^2 +5a+6)-1
= √ 4+6-1 = √ 9 = 3

選(a)

9.
b/a + a/b = (a^2 + b^2 ) / ab = ( a +b )^2 / ab - 2
根據根與係數
a+b = 2
ab = -5
故 b/a + a/b
= ( a +b )^2 / ab - 2
= - 4 / 5 -2
= -14 / 5

選(d)