STPM Trial 2009 MathT&S1 Q&A (Pahang)
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Transcript of STPM Trial 2009 MathT&S1 Q&A (Pahang)
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CONFIDENTIAL* 2
Mathematical Formulae for Paper 1 Mathematics T / Mathematics S :Logarithms :
a
x x
b
ba log
loglog =
Series :
)1(2
1
1
+==
nnr n
r
)12)(1(61
1
2 ++==
nnnr n
r
22
1
3 )1(41 +=
=nnr
n
r
Integration :
= dxdxdu
vuvdxdxdv
u
c x f dx x f x f += )(ln)(
)('
ca x
adx
xa+
=
+ 122 tan11
ca x
dx xa
+
=
122 sin
1
Series:
Nnwhere ++
++
+
+=+ ,21
)( 221 nr r nnnnn bbar
nba
nba
naba
1,!
)1()1(!2
)1(1)1( 2
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CONFIDENTIAL* 3
1. Given iiik =
+
2213
, where k is a constant and 12 =i , find
(a) the argument of iik
21
3+
correct to two decimal places, [2 marks]
(b) the value of k . [3 marks]
2. By using seven ordinates in the trapezium rule, find the approximate value of
dx x +41 )12ln( correct to three decimal places. [4 marks]
Give a reason why the trapezium rule gives an estimate which is less than the exactvalue. [1 marks]
3.(a) By using definition of sets, show that RQP RQP )()( . [3 marks]
(b) By using the algebraic laws on sets, prove that RQP RQP = )()( . [3 marks ]
4.(a) Evaluate =
5
3 )1)(2(1
r r r . [2 marks]
(b) Express )1)(2(1
r r in partial fractions.
Hence, find the value of =
49
3 )1)(2(1
r r r . [4 marks]
5.(a) By using the substitution xu +=12 , find the value of dx x x
+8
3 1
1. [5 marks]
[ Hint :)1(2
1
)1(2
1
1
12 +
= x x x
]
(b) Find dxe
x x 3 . [2 marks]
6.(a) Find all the values of k that satisfy the inequality 0)21(3
212
k k
k . [3 marks]
(b) Solve the inequality
x
x
21
112
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CONFIDENTIAL* 4
7. Given that )ln( 22 y x y = where x > 0 and y > 0 ,
(a) show that)12(
22
= y x
ydxdy
[3 marks]
(b) find the small change in x, correct to three decimal places, when the value of y changes from 1 to 1.01. [4 marks]
8. Expand21
211
+
x x
in ascending powers of x up to and including the term 2 x .
[4 m arks]
By finding the range of the values of x where the expansion is valid, explain why
41= x can be used in the above expansion to estimate the value of 2 . [2 marks]
Hence, estimate the value of 2 correct to four decimal places. [2 marks]
9.(a) The polynomial baxax x ++ 23 32 has the factor x 1 and leaves a remainder of 54 when divided by x + 2 . Find the values of a and b .
Using these values of a and b , factorise the above polynomial completely .
Subsequently , find all the real zeroes of the polynomial 4392 246 ++ x x x .[6 marks]
(b) Given)2)(1(
93+
= x x
x y , show that y does not have any real value between
3
1and 3
for all real values of x . [4 marks]
10. A parabola has parametric equations )2( = t t x and )1(2 = t y .
(a) By finding the Cartesian equation of the parabola, determine(i) the vertex,(ii) the focus(iii) the equation of the directrix [5 marks]
(b) Show that the equation of the normal to the parabola at the point when y = 4 is
1105
=+ y x . [5 marks]
(c) Calculate the shortest distance between the vertex of the parabola and thenormal to the parabola at the point when y = 4. [2 marks]
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CONFIDENTIAL* 5
11. The adjoint of matrix
=1104
180
120
A is
032
044
682
k
.
(a) Show that matrix A is non-singular (i.e. A is invertible). [3 marks](b) Find
(i) the value of k , [2 marks](ii) the inverse of matrix A. [2 marks]
The matrix
+
+
11629
802
10413
c
cbcb
ba
where a , b and c are constants, is a symmetric
matrix.
(c) Write a system of equations in terms of a , b and c. [2 marks](d) Hence, find the values of a , b and c using matrix. [4 marks]
12(a) The continuous function )( x f is defined by
= 0.0745
M1 Standardize
A1
(iii) 3 ( ) 98 X P > ( ) 98 X P > ( )105 X P >
= 3 2
2100 98
Z P
> > 2
100105 Z P
= 3 ( )] 21 Z P ( ) 5 . 2 Z P > = 3 [ ] 28413 .0 0.00621= 0.0132
M1 Standardize
M1A1
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