F4 maths (Quadratic)

Daniel

2009-11-20 4:16 am

1.In a two-digit number, the units digit is greater than the tens digit by 3, and the sum of the squares of the two digits is greater than the two-digit number by 3. Let the units digit be x.
(a)Express, in terms of x,
(i) the tens digit
(ii) the value of the two-digit number.
(b) Hence, find the two-digit number.

2. Joan is 2 years younger than her brother Matthew, and 32 years younger than her father. Three years ago, the product of the ages of Joan and Matthew was exactly 2 times the age of their father. How old is Matthew now?

3. If a and b are the roots of 2x^2-9x+8=0, where a>b, form a quadratic equation with roots (a-2) and (b+2). Leave the coefficients and/or the constant term in radical sign if necessary.

4. If a and b are the roots of x^2-px+q=0, express the following expressions in terms of p and q .

(i) (a+2b)(b+2a)

5. If one root of 3x^2-4x+p=0 is three times the other root, find the value of p.

回答 (1)

用戶7639

2009-11-21 3:50 am

✔ 最佳答案

1. (a)(i)the tens digit=x-3
(ii)the two-digit number =10(x-3)+x
=10x-30+x
=11x-30

(b)x2+(x-3)2=11x-27
2x2-17x+36=0
x=9/2(rej.) or x=4
the two-digit number=14

2.Let the age of Matthew now be x,the age of Joan now be x-2, the age of father now be x+30.

(x-3)[(x-2)-3]=2[(x+30)-3]
(x-3)(x-5)=2(x+27)
x2-8x+15=2x+54
x2-10x-39=0
x=13 or x=-3(rej.)

∴Matthew is 13 years old now.

3.
a+b=9/2
ab=8/2
=4

Sum of roots=(a-2)(b+2)
=ab+2a-2b-4
= ab+2(a-b)-4
=8/2 +2

4.
(i)
Sum of roots=a+b
=p
Product of roots=ab
=q

(a+2b)(b+2a)=ab+2a2+2b2+4ab
=5ab+2(a2+b2)
=5ab+2[(a+b)2-2ab]2
=5p+2(p2-2q)2
=5p+2(p4-4p2q+4q2)
=2p4-8p2q+8q2+5p

5. Letαandα+3 are the roots of 3x2-4x+p=0.

Sum of roots:α+3α=4/3
α=1/3

Product of roots:α(3α)=p/3
3α2=p/3
1/3=p/3
∴p=1

2009-11-20 20:29:08 補充：
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