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2012 TRIAL STPM BAHARU

MATHEMATICS T

JPS TERENGGANU

Section A [45 marks] Answer all questions in this section. 1.

The function of f and g are define by , a) Find f and state the domain b) Find the inverse of g , and sketch its graph. Hence state the range.

2.

Express

[3m] [5m]

in ascending order powers of x up to and including the term in

By using the substitution x = , find an approximate value of

in the form ,

where p and q are positive integers. 3.

[6m]

The matrix

Given that

.

is a symmetry matrix.

.

Find a, b and c, hence solve the system of equations for

[8m] 4.

If + 2 i and = -3 -3 i, find the modulus and argument of and . Hence, find the modulus and argument of and ,and express and , in polar form. [7m]

5.

6.

The line and intersect the curve respectively where the x coordinates are positive. Find the coordinates of Calculate the perpendicular distance of to

Two line

have vector equation respectively. Find The position vector of their common point The angle between the lines.

at point

, where

[4m] [4m]

is origin

and [4m] [4m]

Page 1

© Jabatan Pelajaran Negeri Terengganu STPM 954/1

2012 TRIAL STPM BAHARU

MATHEMATICS T

JPS TERENGGANU

Section B [15 marks] Answer any one question in this section. 7.

The expression cos x - sin x may be written in the form r cos ( x + ) for all values of x, where r is positive and is acute. a) Determine the values of r and [3m] b) State the minimum and maximum values of cos x - sin x, and determine the corresponding values of x in the interval [3m] c) Sketch the curve y = cos x - sin x for [3m] By drawing an appropriate line on the graph, determine the number of roots of the equation cos x - sin x = in the interval [3m] d). Solve the equation cos x - sin x = 1 [3m]

8.

a)

b)

Given that , = and = . i) find the unit vector that is perpendicular to both vectors and ii) if is a triangle, show that for the smallest angle,

[4m]

. Hence calculate the area of triangle to the 3 significant figures

[6m]

Determine the coordinates where the line the plane

[5m]

intersect

Page 2

© Jabatan Pelajaran Negeri Terengganu STPM 954/1

2012 TRIAL STPM BAHARU 1.

a)

MATHEMATICS T

f

JPS TERENGGANU M1 A1 A1

b)

M1

= ,

A1

y 𝑔−1

D1 for D1 ( all correct)

0

x A1

2. = = ( 1 + 6x +

M1

= ( 1 + 6x + = = 1 + 7x + Substituting x = ,

Page 3

© Jabatan Pelajaran Negeri Terengganu STPM 954/1

2012 TRIAL STPM BAHARU

MATHEMATICS T

JPS TERENGGANU

3.

M1 either one M1 solving A1 all correct A1

From

B1

M1

M1 A1 (all correct)

4.

+2 -3

B1 ( both correct )

arg = =

B1

arg = =

B1 B1

arg

arg =

=0

B1

Page 4

© Jabatan Pelajaran Negeri Terengganu STPM 954/1

2012 TRIAL STPM BAHARU

MATHEMATICS T

JPS TERENGGANU

and

B1

arg

=

B1

Hence, 5

i)

and For

B1

coordinates P

For coordinates Q M1

M1

A1 ) ii)

A1

The straight line OQ B1 The perpendicular distance

M1

A1

,

6.

At the common to m and n

and

M1 A1 M1

The position vector is

ii)

From

A1

……………..

(i)

……………..(ii) The direction vector of

M1 either one

………………..(i)

The direction vector of

………………………..(ii)

M1 either one

Page 5

© Jabatan Pelajaran Negeri Terengganu STPM 954/1

2012 TRIAL STPM BAHARU

MATHEMATICS T

JPS TERENGGANU M1

The angle between line m and n is

7.

(a).

Let

cos x-sin x

= r cos ( x +

/

A1

,r>0

= r cos r sin

--------B1 either one

tan (b).

and r =

Minimum value is -

--------M1,A1(boths are correct )

when

----------B1

x= Maximum value is

when

-----------B1

x=

------------A1 [ boths are correct ]

(c). 1.5 2

1 𝜋 4

3 𝜋 4

7 D1-shape, D1𝜋 4 ), values (max-min

2𝜋

D1 All corrects-

− 2 0

Page 6

© Jabatan Pelajaran Negeri Terengganu STPM 954/1

2012 TRIAL STPM BAHARU

MATHEMATICS T

JPS TERENGGANU

x y

0

0.75

1.5

Number of roots = 3 (d). cos x - sin x = 1,

------- Line-B1,B1,A1

---------B1 ---------B1 x = 0,

8.

a)

-----------A1 (boths)

i.

M1

=2i – 2j –2k

A1 =

A1 A1 M1 either one M1 M1 A1

Area of the triangle

or equivalent .

b)

from

A1

and

…………………(*) sub to eqn

M1(all correct) M1 A1

M1 The coordinate is

A1 Page 7

© Jabatan Pelajaran Negeri Terengganu STPM 954/1

View more...
MATHEMATICS T

JPS TERENGGANU

Section A [45 marks] Answer all questions in this section. 1.

The function of f and g are define by , a) Find f and state the domain b) Find the inverse of g , and sketch its graph. Hence state the range.

2.

Express

[3m] [5m]

in ascending order powers of x up to and including the term in

By using the substitution x = , find an approximate value of

in the form ,

where p and q are positive integers. 3.

[6m]

The matrix

Given that

.

is a symmetry matrix.

.

Find a, b and c, hence solve the system of equations for

[8m] 4.

If + 2 i and = -3 -3 i, find the modulus and argument of and . Hence, find the modulus and argument of and ,and express and , in polar form. [7m]

5.

6.

The line and intersect the curve respectively where the x coordinates are positive. Find the coordinates of Calculate the perpendicular distance of to

Two line

have vector equation respectively. Find The position vector of their common point The angle between the lines.

at point

, where

[4m] [4m]

is origin

and [4m] [4m]

Page 1

© Jabatan Pelajaran Negeri Terengganu STPM 954/1

2012 TRIAL STPM BAHARU

MATHEMATICS T

JPS TERENGGANU

Section B [15 marks] Answer any one question in this section. 7.

The expression cos x - sin x may be written in the form r cos ( x + ) for all values of x, where r is positive and is acute. a) Determine the values of r and [3m] b) State the minimum and maximum values of cos x - sin x, and determine the corresponding values of x in the interval [3m] c) Sketch the curve y = cos x - sin x for [3m] By drawing an appropriate line on the graph, determine the number of roots of the equation cos x - sin x = in the interval [3m] d). Solve the equation cos x - sin x = 1 [3m]

8.

a)

b)

Given that , = and = . i) find the unit vector that is perpendicular to both vectors and ii) if is a triangle, show that for the smallest angle,

[4m]

. Hence calculate the area of triangle to the 3 significant figures

[6m]

Determine the coordinates where the line the plane

[5m]

intersect

Page 2

© Jabatan Pelajaran Negeri Terengganu STPM 954/1

2012 TRIAL STPM BAHARU 1.

a)

MATHEMATICS T

f

JPS TERENGGANU M1 A1 A1

b)

M1

= ,

A1

y 𝑔−1

D1 for D1 ( all correct)

0

x A1

2. = = ( 1 + 6x +

M1

= ( 1 + 6x + = = 1 + 7x + Substituting x = ,

Page 3

© Jabatan Pelajaran Negeri Terengganu STPM 954/1

2012 TRIAL STPM BAHARU

MATHEMATICS T

JPS TERENGGANU

3.

M1 either one M1 solving A1 all correct A1

From

B1

M1

M1 A1 (all correct)

4.

+2 -3

B1 ( both correct )

arg = =

B1

arg = =

B1 B1

arg

arg =

=0

B1

Page 4

© Jabatan Pelajaran Negeri Terengganu STPM 954/1

2012 TRIAL STPM BAHARU

MATHEMATICS T

JPS TERENGGANU

and

B1

arg

=

B1

Hence, 5

i)

and For

B1

coordinates P

For coordinates Q M1

M1

A1 ) ii)

A1

The straight line OQ B1 The perpendicular distance

M1

A1

,

6.

At the common to m and n

and

M1 A1 M1

The position vector is

ii)

From

A1

……………..

(i)

……………..(ii) The direction vector of

M1 either one

………………..(i)

The direction vector of

………………………..(ii)

M1 either one

Page 5

© Jabatan Pelajaran Negeri Terengganu STPM 954/1

2012 TRIAL STPM BAHARU

MATHEMATICS T

JPS TERENGGANU M1

The angle between line m and n is

7.

(a).

Let

cos x-sin x

= r cos ( x +

/

A1

,r>0

= r cos r sin

--------B1 either one

tan (b).

and r =

Minimum value is -

--------M1,A1(boths are correct )

when

----------B1

x= Maximum value is

when

-----------B1

x=

------------A1 [ boths are correct ]

(c). 1.5 2

1 𝜋 4

3 𝜋 4

7 D1-shape, D1𝜋 4 ), values (max-min

2𝜋

D1 All corrects-

− 2 0

Page 6

© Jabatan Pelajaran Negeri Terengganu STPM 954/1

2012 TRIAL STPM BAHARU

MATHEMATICS T

JPS TERENGGANU

x y

0

0.75

1.5

Number of roots = 3 (d). cos x - sin x = 1,

------- Line-B1,B1,A1

---------B1 ---------B1 x = 0,

8.

a)

-----------A1 (boths)

i.

M1

=2i – 2j –2k

A1 =

A1 A1 M1 either one M1 M1 A1

Area of the triangle

or equivalent .

b)

from

A1

and

…………………(*) sub to eqn

M1(all correct) M1 A1

M1 The coordinate is

A1 Page 7

© Jabatan Pelajaran Negeri Terengganu STPM 954/1

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