Mathematics am015

MATHEMATICS AM015





Exercise















































NUR HIDAYAH HANIM AHAMAD
Penang Matriculation College

LIST OF MATHEMATICAL FORMULAE

SENARAI RUMUS MATEMATIK



For the quadratic equation ax  bx c  0:
2

2

Bagi persamaan kuadratik ax  bx c  0:



 b  b  4 ac
2
x 
2 a




For completing the square of a quadratic equation ax  bx c  0:
2
Bagi melengkapkan kuasa dua persamaan kuadratik ax  bx c  0:
2



 b  2  b  2 c
 x        0
 2a   2a  a





For an arithmetic series:

Bagi suatu siri aritmetik:


T  a  n (  d ) 1
n



n
S   a (2  n   d ) 1
n
2

For a geometric series:

Bagi suatu siri geometri:



T n  ar n  1




a (1 r n )
S  , r  1

n
1 r
n
( a r  1)
S  , r  1
n r  1

LIST OF MATHEMATICAL FORMULAE

SENARAI RUMUS MATEMATIK




Differentiation

Pembezaan



 f x h   f   x
f    limx 
h 0 h




If   x  f u   x  v  ,x


then   x  f  u'   x  v'   x





If   x  u    ,x v x
f
    x 
then   x  f  u' x v v' x u
    x




u   x
If   x  ,
f
v   x
    x 
    x
v x u' u x v'
f 
then   x  2
 v    x  




If y  f   andu u  g  ,x

dy dy du
then  
dx du dx





If y  ln f  ,x

dy 1
then  . f    x
dx f   x

Extra Tutorial 1



2
9
1. (a) Simplify 4lnk  3lnm 5lnkm Ans:ln k m

(b) Without using calculator, find the value of log 16 log 36 . Ans: 4
4
6
log y
m
2. Show that log y  z . (Hint: Use y  x )
x
log x
z

log
2
3. Show:  x  y  2  log x   y  log y x    2log x  2log y
4. Express 91 40 3 in the form a b 3 , where and are integers to be determined.
Ans: a  4,b  5


5
5. Solve the equation log x  3 log 3  x Ans: 1.732,9
2


6. Solve 3log 8 log y 8 y  9, where y  1. Ans: 1.975

Extra Tutorial 2



1. Express 3x  6  9in the form p  x q  4and hence state the values of and .

Ans: p   3,q  2


2. Find the solution sets of



(a)  x x  10  4x  7 Ans: , 7  1,  




:
(b) 2 3x  8 Ans: x x   10 or x  2


 3 
3  
3. Determine the solution set of 4x  3  x Ans:  5  ,1



Extra Tutorial 3

1. A high-rise building has forty floors. A cleaning company charges RM8 to clean the first
floor and extra RM40 for each floor above preceding floor.
(a) Find the cleaning cost for the twenty-fifth floor. Ans: RM1040

(b) Find the cleaning cost for the whole building. Ans:RM34400

2. The sum of the first three consecutive terms, and of an arithmetic sequence is RM



45. If , find the value of . Ans: 15

3. The sum of the first two terms of a geometric sequence is 4 and the sum of the first four


terms is 5. Find the common ratio if is positive. Ans:

4. You buy a television and agree to pay as follows:

First month: RM 150
Second month: RM 140
Third month: RM 130
.
.
.
Fifteenth month: ?

(a) What is the fifteenth payment? Ans: RM 10

(b) What is the total payment for the purchase? Ans: RM1200

5. Initially Jimmy saves RM 100 in his personal savings account and in each succeeding
month, he saves more than a precious month. Calculate his total savings after 15
months.

Ans: RM2157.86

6. Geeta saved RM 5 in February and twice as muh in each successive month compared to
the preceeding month. Find the amount of money she had at the end of December of the
same year. Ans: RM10235

Extra Tutorial 4

 2  1 1 

1. (a) If P    x 0  2 , find the value of x if the determinant of is 6.

 
 4  3 5 
Ans: 5

 1 4  14  6 4


(b) Given 2 5    2 m 0     19  9 5 . Find the values of and .



  n  1 1    
 3 6   24  12 6 
Ans: = −2, = 3
 a b c 


2. A 3 3 matrix of the form 0 d e is said to be upper triangular. Find an upper


 
 0 0 f 
 4  6 2 


T
triangular matrix with positive diagonal elements such that X X   6 10  3


 
 2  3 10 
 2  3 1


T
where X is the transpose of X . Ans: 0 1 0


 
 0 0 3 
3. Matrices A and B are given as
 2 2  x y
A    ,B   
  1 1   1 0 

1
Where and are real numbers. Find the diagonal matrix such that ADA  B.
  2 0
Ans: D   
 0 2 


4. Products , and are assembled from three components , and C in different
proportions. Each product consists of two components , three components and two
components ; each product consists of one component , two components and one
component , and each product of consists of one component , one component and
two components . A total of 500 components , 750 components and 600
components are used.

By letting , and represent the number of products , and assembled, obtain a
matrix equation to represent the given information. Hence, find the number of
products , and assembled. Ans: x  150, y  100,z  100

Extra Tutorial 5

1. The functions f and g are defined by

2
f   x   2 x   3  5,x  
3
x  5
g   x  ,x  
5
2
(a) Show that   x is one-to-one and find the expression of f  1   x . Sketch the
f
graph of   x and f  1   x on the same axes. Ans: refer to
f
lecturer

(b) Find an expression of g f Ans: x   3 ,x   3




f
g
x
2. Given   x  4x  1 and   3 x .
(a) Write f as a piecewise function hence find the function
h   3x  f   2x  g   x Ans: refer to lecturer

(b) Sketch the graph of ℎ Ans: refer to lecturer


(c) If g t x      x  5 , find the function and determine the value of such that

t  1   6x  . Ans: 2,4


 kx  5e x
3. Given that f x  ln   and the inverse function f  1 : x  . Find the
:
x
 4x l   me  6
possible values of , and . Ans: l  5,m  4,k 
6
4. The function f and g are defined as follows

1
: f x  ln ,x  x 0
2


g : x  , x x  0

(a) Sketch the graph of f and explain why the inverse function of f, f  1   x exists

Ans: refer to lecturer

(b) Find f  1   x and state its domain Ans: e 2x ;x


x
(c) Find the composite function g f  1 Ans: e

Extra tutorial 6

3
2
1. (a) Find a constant if x  1 is a factor of 2x  7x  kx  3. With this value of

3
2
solve the equation 2x  7x  kx  3 0.
1
Ans: k  2,x   1, ,3
2
(b) Find the values of the constant A,B, C and D such that
3
4
x  Ax  5x  x D   x  4  x  x B  C x    1 .
2

2
2

Ans: A   1,B  1,C  5,D 
9
P
P
2. When   x is divided by x   3 , the remainder is 2a . When   x is divided by
x   2 the remainder is 3a . Find the remainder when   x is divided by x  3 x   2 .
P
Ans: a x    1
P
3. Determine the polynomial   x which has the following properties

(a) P   x is of degree 3
P
(b) x   1 is a factor of   x
(c) P   0  and   1P   10
4
(d) P   x has a remainder 16 when divided by x   2
2
3
Ans: 3x  x  8x 
4
x  3x A B C
2
4. If    , find , and .


1 x 1 x  2 1 x 1 x 1 x  2
Ans: A  1,B  0,C   1
x  1 x   2
5. Express in partial fraction.
x  1  x  2  2


2 2 x
Ans: 
x  1 x  2 2

Extra Tutorial 7

1. Find the value of the following limits (if they exist).


16 x 2
(a) lim Ans:32
x 4 2  x
x  1
(b) lim Ans: 

x 1 1 x
x  2  7 1
(c) lim Ans:
x 5 x  5 2 7

2. Determine the following limits at infinity


2 x 2 1
(a) lim Ans: 
2
x 3x  1 3

1 3x 2
(b) lim Ans: 3
x x  5
3. The function g is defined by

5x  1, x  1
g   x   2
 mx , x  1

g
Find the value of if limit of   x exists. Ans: 4

4. Given

 x  2
 x  , x  0
2
  4 0  
h   x   8 kx , x 2

3
 x , 2  x  4
 2x
 , x  4
 x  1
2
(a) lim h  ,x Ans:64
x 4 

1
(b) lim h  ,x Ans: 
x 2 4

(c) limh  ,x Ans: 0
x

 
(d) the value of iflimh x exists Ans: 0
x 2

Extra Tutorial 8

dy
1. Find using the differentiation from the first principle of the following function
dx

1 3
y  Ans: 
3x  2 3x   2 2


dy
2. Find for each of the following function:
dx


3 5x 41
(a) y  Ans: 

4 7x 4 7x  2
12x   3 2x  5 ln  5
 2x 
(b) y  3 x ln 2x   5 Ans:
2 x 2x   5


(c) y  2e  2x  4e 2x  5 Ans: 4e  2x  8e 2x  5


3. Given y  2mx n e  2x and when x  0, y  2and dy  6. Find the values of the
dx
constant and . Ans: m  1,n  2

4. Differentiate the following with respect to .

2x
dy 2e  4xy
2x
(a) 2x  2  1 y  e Ans: 
dx 2x   1
2

3xy  2 3 2y 2
(b) xy  2 ln 2 3x  Ans:
2xy 2 3x 
dy
4
x
5. Given that y  2 . By taking the natural logarithmic form, find in the terms of x.
dx
2
d y
Hence or otherwise, find .
dx 2
2

d y

Ans:  4x 2 ln2 4x 4 ln2 3y 
dx 2

Extra Tutorial 9

2x  4 dy 1
1. If y  , find the point(s) on the curve where   .
3x  2 dx 4


Ans: 2,0 , 5,    4  
 3 

2
f
2. Find the equation of the tangent line to the curve   x  x  3 x at the point x  3. What
1 397
is the equation of the normal? Ans: y  33x  63; y   x 
33 11

3. A company produces and sells pots each year with cost function,   1000 5C x   x and

demand function,   60 0.004p x   x where x is the number of pots and ( )C x and   x
p
are in ringgit. Determine

(a) the revenue function and maximum revenue

2
Ans:   60R x  x  0.004 ;RM 225000
x
(b) the profit function, maximum profit and the number of pots that need to be
produced and sold each year to achieve maximum profit

Ans:   55x  x  0.004x  1000;RM 188062.50;6875pots
2

(c) the selling price to ensure maximum profit

Ans: RM 32.50
(d) the maximum profit if each pot sold is taxed RM3

Ans: RM 168000



1,3
3
4. Given 2x  y  5is the equation of the tangent to the curve 2x y  py at the point  
2
where and are constants. Find the values of and .
2
Ans: p   ,q  12
3

3
5. Given the equation of a curve is y  3x  2 x .
(a) Find the maximum and minimum points of the curve


Ans: Max pt 2,4 ;min pt 0,0

(b) Find the coordinates of points on the curve where the gradient of the tangent to
the curve is −24. Ans: 4, 16 ; 2,20   