F.4 MATHS - Ans.fyi 參考答案

F.4 MATHS

2011-01-06 12:31 am

a)√6+√6+√6+√6+√6+ ......<--睇得明?IT IS GIVEN THAT√6+√6+√6+√6+√6 ...... has a specific value. let the value be x. prove that x^2=6+x. hence find the value of √6+√6+√6+√6+√6+ ......b) explore the value of √6√6√6√6√6 ......2. For a quadratic equation ax^2+bx+c=0, where a and b and c are integers,
a)under what condition for a,b and c such that the quadratic equation has rational roots?
b) apart form the condition found in (a) if b= 2am and c = an, where m and n are integers, prove that the quadratic equation has integral roots.

3. if a and b are two roots of the quadratic equation ax^2+bx+c=0, express the two roots of the quadratic equation cx^2+bx+a=0 in terms of a and b

thz

更新1:

LIM我末教BO..係咪A MATHS黎架?? 可唔可以用D 一般數學既方法做? (事關依條係一般數學書既題目,非M2非M1非AMATHS)

更新2:

(b) See http://hk.knowledge.yahoo.com/question/question?qid=7011010401046 同我果個有D唔同喎我最前面果個有開方~ 係開方6X開方6X開方6..... 如果係咁既話做法有D咩唔同?.?

回答 (1)

myisland8132

2011-01-06 1:44 am

✔ 最佳答案

(a) Let a_1=√6 and a_n+1=√(6+a_n).
Then a_n is an increasing sequence which is bounded by an integer M. So, the limit of the sequence exists. Called it x.

Then lim(n->infinity) a_n+1=lim(n->infinity) √(6+a_n)
x=√(6+x)
x^2=6+x
(x-3)(x+2)=0
x=3 or -2

So, the value of the original expression is 3

(b) See http://hk.knowledge.yahoo.com/question/question?qid=7011010401046

The value of the expression is 36

2(a) Since the root of ax^2+bx+c=0 are
x=[-b-√(b^2-4ac)]/2a ; x=[-b+√(b^2-4ac)]/2a

So, if b^2-4ac is equal to m^2, then the quadratic equation has rational roots.

(b) b = 2am and c = an. b^2-4ac=4a^2m^2-4a^2n=>m^2-n is a square
m^2-n=r^2. x=[-b-√(b^2-4ac)]/2a ; x=[-b+√(b^2-4ac)]/2a becomes
x=[-2am-2ar]/2a ; x=[-2am+2ar]/2a or x=-m-r ; x=-m+r

3 α+β=-b/a,αβ=c/a.
Let the two roots of the quadratic equation cx^2+bx+a=0 are s and t
s+t=-b/c=a(α+β)/aαβ=(1/α+1/β)
st=a/c=1/αβ

So s=1/α and t=1/β

2011-01-05 21:33:15 補充：
(a) is PURE MATHS

(b) 那你除回6就可以了。答案就是6啦

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