trial stpm mathematics t1 johor

8/8/2019 Trial STPM Mathematics T1 JOHOR

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*This question paper is CONFIDENTIAL until the examination is over. CONFIDENTIAL*

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1 Given that the shortest distance from point 1,3 to the straight line03 kyx is 2. Find the values ofk.

[3 marks]

2 Given that xeyxsin , show that .022

2

2

ydx

dy

dx

yd

[4 marks]

3 The equation of a curve is given by 03422 xyyx .

(a)Show thatdx

dy=

xy

xy

2

2

[3 marks]

(b) Find the coordinates of each of points on the curve where the tangent is

parallel to they-axis. [4 marks]

4 Ifx is so small that 3x and the higher powers ofx may be neglected, show that

2

2

11

1

1xx

x

x

.

By taking

24

1x , show that 23

1152

5525. The approximate value for 23 can

also be obtained by putting16

7x in the binomial expansion. State, with reason, which

value ofx will give a better approximation to 23 .

[7 marks]

5 Show that 12322

1

3212

1

rr

rr.

Hence, find the sum of the series

.3212

1...

53

1

31

1

nn

[7 marks]


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6 Find the non-zero values ofx that satisfy the equation

0212314118 xxx .

Giving your answers correct to three decimal places.

[8 marks]

7 Let I

4

1 )4(

1dx

xx.

Use the substitution xu to show that I

2

1)4(

2du

uu.

Hence, show that I 3ln2

1 .

[8 marks]

8 Find the domain of the function 42:f xx .

Find 1f and state its domain and range.

Sketch the graphs of f and 1f on the same diagram.

[8 marks]

9 A point P moves on a curve such that the difference of its distance from the point

)0,4( and )0,4( is a constant and equals to 102 . Find the equation of the locus ofP.

Show that the line 2 xy is a tangent to the curve.[8 marks]

10 The gradient function of a curve is given by

2

12

6

x. )9,2(P is a point on the

curve. The normal to the curve at P meets they-axis at Q and thex-axis atR.

(a)Find the coordinates of the mid point ofQR. [4 marks](b)Find the equation of the curve and sketch the curve. [6 marks](c)A point (x,y) moves along the curve in such a way that thex-coordinate

increases at a constant rate of 0.03 units per second. Find the rate of change of the

y-coordinate as the point passes through P. [2 marks]


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11 The matrix M is given by

M =

110

213

02a

, where 2a .

Find in terms ofa,

(a)M [2 marks](b)cofactor of element 1. [1 mark](c)adj M [3 marks](d)M1 [2 marks]

Hence, solve the simultaneous equations

123 yx

923 zyx

5 zy [5 marks]

12 and are the roots of the equation 02 cbxax . Show that

a

b

and .a

c [4 marks]

(a) If ,and3 cab express a in terms of c . [5 marks](b)Show that 2 2 2 2(2 ) 0c x ac b x a has roots .1and1

22

[6 marks]

trial stpm mathematics t1 johor

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