# Program Latih Tubi SPM 2012 - Matematik Tambahan Kertas 2 - Download Now DOC

Document Sample

```					                                                      1                    3472/2

TOPIC: SIMULTANEOUS EQUATIONS

1
1.   Solve the simultaneous equations x +       y = 1 and y2 – 10 = 2x.   [5 marks]
2

2.   Solve the simultaneous equations x + 2y = 1 and x2 + 2y2 + xy = 5.

3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012
2                                          3472/2

TOPIC: TRIGONOMETRIC FUNCTIONS

1. (a) Prove that cosec2 x – 2 sin2 x  cot2 x = cos 2x.                                       [2 marks]

(b) (i) Sketch the graph of y = cos 2x for 0 ≤ x ≤ 2.
(ii) Hence, using the same axes, draw a suitable straight line
x
to find the number of solutions to the equation 3(cosec2 x  2 sin2 x – cot2 x) =     1

for 0 ≤ x ≤ 2.. State the number of solutions.
[6 marks]

2. (a) Sketch the graph of y =  2 sin 2x for 0 ≤ x ≤ 2.                                      [4 marks]

(b) Hence, using the same axis, sketch a suitable graph to find the number of solutions to the

equation     + 2 sin 2x = 0 for 0 ≤ x ≤ 2.. State the number of solutions.             [3 marks]
x

3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012
3                                           3472/2

3. (a) Sketch the graph of y = | 3cos 2x | for 0 ≤ x ≤ 2.                                    [4 marks]

(b) Hence, using the same axis, sketch a suitable graph to find the number of solutions to the
x
equation 2 - |3cos 2x | =      for 0 ≤ x ≤ 2. State the number of solutions.           [3 marks]
2

4. (a) Sketch the graph of y = 1 + 3 sin x for 0 ≤ x ≤ 2.                                     [4 marks]

(b) Hence, using the same axis, sketch a suitable straight line to find the number of solutions to the
equation 6π sin x = 4π – 3x for 0 ≤ x ≤ 2. State the number of solutions.              [3 marks]

3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012
4                                       3472/2

TOPICS: COORDINATE GEOMETRIC AND DIFFERENTIATION

1. A curve has a gradient function px2 – 4x, where p is a constant. The tangent to the curve at the point
(1, 3) is parallel to the straight line y + x – 5 = 0. Find

(a)   the value of p.                                                                     [3 marks]
(b)   the equation of the curve.                                                         [3 marks]

2
2.   A curve with a gradient function 2x –        has a turning point at (k, 8).
x2

(a) Find the value of k.                                                                 [3 marks]
(b) Determine whether the turning point is a maximum or a minimum point.                 [2 marks]
(c) Find the equation of the curve.                                                       [3 marks]

3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012
5                                       3472/2

1. Diagram 2 shows the curve of a quadratic function f(x) = −x2 + kx – 5. The curve has a maximum
point at B(2, p) and intersects the f(x)-axis at point A.

(a) State the coordinates of A.                                                        [1 mark]
(b) By using the method of completing the square, find the value of k and of p.        [4 marks]
(c) Determine the range of values of x, If f(x) ≥ − 5.                                [2 marks]
f(x)

x
0           B(2, p)

A

Diagram 2

2. The gradient function of a curve is hx2 – kx, where h and k are constants. The curve has a turning
point at (3, −4). The gradient of the tangent to the curve at the point x = −1 is 8.

(a) the value of h and of k..                                                         [5 marks]
(b) the equation of the curve.                                                        [3 marks]

3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012
6                                           3472/2

TOPIC: COORDINATE GEOMETRIC

3
1. Solution by scale drawing is not accepted.                 y   P(       , k)
2
Diagram 10 shows a triangle OPQ. Point S lies
on the line PQ.

(a)    A point W moves such that its distance                                 S(3, 1)
from point S is always 2½       units.
Find the equation of the locus of W.
[3 marks]        O                                         x

Q

Diagram 10

(b) It is given that point P and point Q lie on the locus of W. Calculate
(i) the value of k.
(ii) the coordinate of Q.                                                             [5 marks]

(c) Hence, find the area, in unit2, of triangle OPQ.                                         [2 marks]

3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012
7                                           3472/2

2. Solution by scale drawing is not accepted.
Diagram 3 shows the triangle AOB where O is the origin. Point C lies on the straight line AB
.
(a) Calculate the area, in unit2 , of the triangle AOB.                                    [2 marks]
(b) Given AC : CB = 3 : 2, find the coordinate of C.                                       [2 marks]
.
y

A(−3, 4)

C
x
O

B(6,−2)

Diagram 3

(c) A point P moves such that its distance from point A is always twice its distance from point B.
(i)   Find the equation of the locus of P.
(ii) Hence, determine whether this locus intersects the y-axis.                           [6 marks]

3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012
8                                      3472/2
TOPIC: VECTORS

5          2              k
1. Given that AB =  , OB =   and CD =   , find
7       3          5 
                    

(a) the coordinates of A.                                                                  [2 marks]
(b) the unit vector in the direction of OA .                                               [2 marks]
(c) the value of k, if CD is parallel to AB .                                              [2 marks]

2. In diagram 6, ABCD is a quadrilateral. The diagonals
BD and AC intersect at point R. Point P lies on AD.                     D
1        1
It is given that AP = AD, BR = BD ,
3        3
AB = x and AP = y.                                                                      C
P          R
(a) Express in terms of x and y :
(i) BD                                               A                    B
(ii) AR                          [3 marks]                                      Diagram 6

(b) Given that DC  k x  y and AR = h AC , where h and k are constant, find the value of h, and of k.
[4 marks]

3472/2 – ADDITIONAL MATHEMATICS PAPER 2 - PROGRAM LATIH TUBI SPM 2012

```
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
 views: 1200 posted: 10/21/2012 language: pages: 8
Description: Program Latih Tubi SPM 2012 Matematik Tambahan Kertas 2