2007年 STPM Math T paper1 , further math T paper 1 考题...

img3nius

..
1 Express the infinite recurring decimal 0.725 ( = 0.7252525...) as a fraction in its lowest terms.

2 If y = x/(1+x^2) , show that x^2dy/dx=(1-x^2)y^2.

3 If loga (x/a^2) = 3loga 2 - loga (x-2a), express x in terms of a.

4 Simplify
(a) (squaroot 7 - squaroot 3)^2/2(squaroot 7 + squaroot 3)  ,

(b) 2(1+3i)/(1-3i)^2 , where i = squaroot -1

5 The coordinates of the points P and Q are (x,y) and (x/(x^2+y^2),y/(x^2+y^2) respectively ,
where x not = 0 and y not = 0. If Q moves on a circle with centre (1,1) and radius 3 ,show that the locus of P is also a circle.Find the coordinates of the centre and radius of the circle.

6 Find
(a) inte (x^2+x+2)/(x^2+2)  dx  ,

(b) inte x/e^(x+1) dx

7 Find the constants A , B , C and D such that
    (3x^2+5x)/(1-x^2)(1+x)^2 = A/(1-x) + B/(1+x) + C/(1+x)^2 + D/(1+x)^3 .

8 The function f is defined by
                squaroot(x+1) ,    -1《x<1
          f(x)=     |x|-1     ,   otherwise.

  (a) Find lim x>-1^- f(x) , lim x>-1^+ f(x) , lim x>1^- f(x), lim x>1^- f(x)

  (b) Determine wherther f is continuous at x = -1 and x = 1.

9

10

11

12 Find the coordinates of the stationary points on the curve y = x^3/(x^2-1) and determine thier nature.
   Sketch the curse.
   Determine the number of real roots of the equation x^3 = k(x^2-1),where k is Real Number , when k varies.

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Futher Mathematics T paper 1
1 If z = cosA + i sinA , show that 1/(1+z^2) = 1/2 (1-i tanA) and express 1/(1-z^2)in a similar form.

2 Show that inte x=pi/4 , 0   sec x(sec x+tan x)^2 dx = 1+ sqrt 2

3 Using the substitution x = e^z , show that the differential equation

      x^2 d^2y/dx^2 + px dy/dx + qy = 0 ,
where p and q are constants , can be transformed into the differential equation
      d^2y/dx^2 + r dy/dx + sy = 0 ,
where r and s are constants to be determined in terms of p and q.

4 The matrices A and B are given by

          1  2  3  4                  4  2  3  4
          5  1  1  3                  5  4  1  3
      A = 2  0  8  0          ,  B =  2  0 11  0
         -1 -1  3  9                 -1 -1  3 12

Given that 1 is an eigenvector of the matric A , find its correspoding eigenvalue.
           1
           1
           1
Hence , find the eigenvalue of the matric B corresponding to the eigenvector 1 .
                                                                             1
                                                                             1
                                                                             1

5 Show that
      sec^-1 x + cosec^-1 x = pi/2   ,
where 0《sec^-1 x《 pi , sec^-1 x not = pi/2 , -pi/2《 cosec^-1 x《 pi/2 and cosec^-1 not = 0.
Hence , find the value of x such that
      sec^-1 x cosec^-1 x = -3/2

6 A sequence a0,a1,a2... is defined by a0=1 and a(r+1) = 2a(r)+ b for r》0 , where b = real number .Express a(r) in terms of r and b , and verify your result by using mathematical induction.                               (a(r) is a to the term r 不是 a x r= =)

7 Solve the recurrence relation
                 a(r) - a(r-1) - 6a(r-2) = (-2)^r   ,
wherea0 = 0 and a1 = 2.                这题也是term r 不是 x r

8

9

10

11

12 Find the roots of the equation (z-ia)^3=i^3 , where a is a real constant.
(a) Show that the points representing the roots of the above equation form an equilateral triangle.

(b) Solve the equation [z-(1+i)]^3=(2i)^3.

(c) If w is a root of the equation ax^2+bx+c=0 , where a,b,c is real number and a not = 0, show that its conjugate w* is also a roots of this equation.Hence , obtain a polynomial equation of degree six with three of its roots also the roots of the equation (z-i)^3=i^3 .
  大家讨论下, 剩余的之后再放上来。

[ 本帖最后由 img3nius 于 29-11-2007 08:51 PM 编辑 ]

~Lucifer~

第一题是说要把= 0.7252525...表达成分数吗？我算到359/495。0.7252525=0.7+0.252525....,然后janjang geometri求0.252525...的value...
第二题y = x/(1+x^2)，dy/dx=1/(1+x^2)^3。x^2dy/dx=x^2/(1+x^2)^3。(1-x^2)y^2=x^2(1-x^2)/(1+x^2)^2,darab(1+x^2),变成x^2/(1+x^2)^3。
第4题上面下面multiply (squaroot 7 - squaroot 3),a)2squaroot 7 - 3squaroot 3,b是-

我解简单的，难的留给版主吧

[ 本帖最后由 ~Lucifer~ 于 22-11-2007 09:28 PM 编辑 ]

img3nius

原帖由 ~Lucifer~ 于 22-11-2007 06:28 PM 发表
第一题是说要把= 0.7252525...表达成分数吗？我算到359/495

是的
...我也拿到一样的答案
把步骤一起放上来吧
方便讨论

dunwan2tellu

第三题 simplified 后 solve quadratic equation 得到 x=-2a or x=4a . 不过因为 x-2a > 0 和 a > 0 (根据 log 的 definition,base 一定要 positive), 所以 只能 x=4a

第五题 Q 经过的 locus 的 equation 是 (p-1)^2 + (q-1)^2 = 9
把 p = x/(x^2 + y^2) , q = y/(x^2 + y^2) 带入然后 simplified 得到

x^2 + y^2 + 2x/7 + 2y/7 = 1/7 , 也就是 locus of P . 而且是一个 circle

第 12 题 dy/dx = x^2(x^2-3)/(x^2-1)^2

x < -sqrt[3],x>sqrt[3] 时 y increasing . 其他时候 y decreasing .

-sqrt[27]/2 < k < sqrt[27]/2 时候只有 1 个 real root
k > sqrt[27]/2 , k< -sqrt[27]/2 时有 3 个 distinct real roots
k = +-sqrt[27]/2 有 3 个 real roots,其中两个 repeated root

[ 本帖最后由 dunwan2tellu 于 22-11-2007 07:26 PM 编辑 ]

TCLGT

第五题怎样simplified? 我做得很复杂

不懂做么我做到：

x^2 + y^2 + x/3 + y/3 = 0
不过也是circle啦

在simplify的时候先不要expand, 先把它化成:
[x-(x^2 + y^2)]^2...再expand
然后会看到factor x^2 + y^2 在两边
约掉就是了

TCLGT

哦哦。。。我overlook了，哈哈哈。。   没有看到x^2 + y^2 ，谢谢提醒
我的答案和版主一样。。

[ 本帖最后由 TCLGT 于 23-11-2007 09:01 AM 编辑 ]

TCLGT

我也来试试一题，有错告诉我
6 Find
(a) inte (x^2+x+2)/(x^2+2)  dx  ,

inte (x^2+x+2)/(x^2+2)  dx =inte 1+x/(x^2+2)  dx
                                               =x+(1/2)ln |x^2+2| +c

(b) inte x/e^(x+1) dx

inte x/e^(x+1) dx=e^-1 inte xe^-x dx
                             =e^-1(-xe^-x- inte-e^-x)
                             =-(x+1)e^-(1+x)

今年partial fraction 就这样出一题了 =。= 炸到   不知道他们是怎样给分的

[ 本帖最后由 TCLGT 于 23-11-2007 09:04 AM 编辑 ]

img3nius

我把further math
的题目也放上来
好多不会做@@

dunwan2tellu

1 If z = cosA + i sinA , show that 1/(1+z^2) = 1/2 (1-i tanA) and express 1/(1-z^2)in a similar form.

z^2+1 = (cos^2 A - sin^2 A + 1) + i *2 sin A cos A
      = (2cos^2 A) + i*2sin AcosA
      = 2cosA(cosA+isinA)
      =2cosA*z
1/(z^2+1) = 1/(2cosA*z)
          =1/2 * (cosA-isinA)/cosA
         =....  

2 Show that inte x=pi/4 , 0   sec x(sec x+tan x)^2 dx = 1+ sqrt 2

substitute t = sec x + tan x , dt = sec x * t dx ...

4) v = (1 1 1 1)' 那么 let its corresponding eigenvalue = k ,

Av = kv ....
观察到 B = A + 3I
如果 B 的 eigenvalue corresponding to v is q , 那么
Bv = qv
但是 Bv = (A+3I)v = (Av+3v)=（k+3)v
=>q = k+3 ...

img3nius

原帖由 dunwan2tellu 于 29-11-2007 07:43 PM 发表


z^2+1 = (cos^2 A - sin^2 A + 1) + i *2 sin A cos A
      = (2cos^2 A) + i*2sin AcosA
      = 2cosA(cosA+isinA)
      =2cosA*z
1/(z^2+1) = 1/(2cosA*z)
          =1/2 * (cosA-isinA)/cosA
...

eigen 是什么来的
我读的资料上没看过.....

dunwan2tellu

这里有一些 eigenvalue 和 eigenvector 的资料

Eigenvalue,eigenvector

if A is nxn matrix . 那么 k is an eigenvalue of A with corresponding eigenvector x 当

Ax = kx  , x = (n x 1) vector

通常要找 k = ? ，用的方法是

det(A - kI) = 0 来找 k .

[ 本帖最后由 dunwan2tellu 于 29-11-2007 09:24 PM 编辑 ]

dunwan2tellu

12 Find the roots of the equation (z-ia)^3=i^3 , where a is a real constant.

[(z-ia)/i]^3 = 1 ==> (a + iz)^3 = -1
let w = ia + z => w^3 = -1
=> w^3 = cis(1 + 2k)pi
=> w = cis(1/3 + 2k/3)pi , k=0,1,2
=>a + iz = cis(1/3+2k/3)pi
=> z =  [cis(1/3+2k/3)pi - a ]/i
     = ....

(a) Show that the points representing the roots of the above equation form an equilateral triangle.

root = z1,z2,z3 的话要证明

|z1-z2|=|z1-z3|=|z2-z3| 或

|z1-z2|=|z1-z3| 和 | Arg[(z1-z2)/(z1-z3)] | = pi/6

(b) Solve the equation [z-(1+i)]^3=(2i)^3.

method 大致上和上面用样 i.e [(z-(i+1))/2i]^3 = 1
=> [(i(z-1) + 1)/2]^3 = 1
=>w^3 = 1
....

(c) If w is a root of the equation ax^2+bx+c=0 , where a,b,c is real number and a not = 0, show that its conjugate w* is also a roots of this equation.Hence , obtain a polynomial equation of degree six with three of its roots also the roots of the equation (z-i)^3=i^3 .

先证明 (w^2)* = (w*)(w*) , 然后整个 equation aw^2 + bw + c = 0 take its conjugate .=> (aw^2)* + (bw)* + (c*) = (0*)
注意 a* = a,b*=b,c*=c,0*=0 ....