F4 maths (Quadratic)

Daniel

2009-11-20 4:45 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9112

2009-11-20 5:02 am

✔ 最佳答案

(1) (a) (i) The tenth digit is x - 3
(ii) The value of the number = 10(x - 3) + x = 11x - 30
(b) x^2 + (x - 3)^2 = 11x - 30 + 3
x^2 + x^2 - 6x + 9 = 11x - 27
2x^2 - 17x + 36 = 0
(x - 4)(2x - 9) = 0
x = 4 or x = 9/2 (rejected)
The number is 11(4) - 30 = 14
2. Let the age of Matthew be x.
Then the age of Joan is x - 2
and the age of the father is (x - 2) + 32 = x + 30
3 years ago, (x - 3)(x - 2 - 3) = 2(x + 30 - 3)
(x - 3)(x - 5) = 2(x + 27)
x^2 - 8x + 15 = 2x + 54
x^2 - 10x - 39 = 0
(x - 13)(x + 3) = 0
x = 13 or x = -3(rejected)
Matthew is 13 years old.
3. Sum of roots a + b = 9/2
Product of roots ab = 8/2 = 4
(a + b)^2 = a^2 + 2ab + b^2 = 81/4
a^2 + 8 + b^2 = 81/4
a^2 + b^2 = 49/4
a^2 - 2ab + b^2 = 49/4 - 8 = 17/4
a - b = √17 / 2
New sum of roots = (a - 2) + (b + 2) = a + b = 9/2
New product of roots = (a - 2)(b + 2)
= ab + 2a - 2b - 4
= 4 + 2(a - b) - 4
= √17
The new equation is x^2 - 9x/2 + √17 = 0 or 2x^2 - 9x + 2√17 = 0
4. Sum of roots a + b = p
Product of roots = ab = q
(i) (a+2b)(b+2a)
= ab + 2a^2 + 2b^2 + 4ab
= ab + 2(a + b)^2
= q + 2p^2
5. Let the smaller root in magnitude be a then the larger root is 3a
Sum of roots = a + 3a = 4/3
4a = 4/3
a = 1/3
Product of roots = (a)(3a) = p/3
1/3 = p/3
p = 1

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