Maths T STPM 2014 Sem 1 Trial King George

2014-1-NS-KING GEORGE

Section B [15 marks]

Answer any one question in this section.

[5 marks]

Section A [45 marks]

1. The function f is defined by
:    ,   0

11x + 9 in partial fractions. f ( x)

)

 ( )*

(

(c)     ) If the point O is the origin and point T is a point such that OPTQ is a parallelogram, prove [15 marks] that when m changes, the equation of the locus of T is    4  4.

(b)    

8. The straight line y = mx – 2 intersects the curve    4 at two different points, P(, ,  ) and Q(, ,  . Show that  (a) m ' 0 and m >

(b)Express

Answer allquestions in this section.

[1 marks] [2 marks] [3 marks]

7. Given that f ( x) = x 3 + px 2 + x + q, where p and q are constants, is divisible by ( x − 1). When divided by ( x − 2), the remainder is 17. Find the values of % and &. [4 marks] (a)Show that f ( x ) = 0 has only one real root. Find the set of values of  such that [6 marks] f ( x) > 0 .

(a) State the range of f. (b) Find
 . (c) Sketch the graph of f and
 . 

[4 marks]

2. Find the expansion of 9   as far as term in   and state the range of values of x for which the expansion is valid. Hence, obtain the value of √9.05correct to four decimal places. [6 marks] 3. Find the inverse of matrix A by using elementary row operations 1 0 2   A = 3 1 6   8 9 2

[4 marks]

Hence, solve the simultaneous equations x +2z = 1 3x + y +6z = 2 8x +9y +2z = -1

[2 marks] [4 marks] [2 marks]

4. Find the roots, denoted by  and , of the equation    2  4  0 in polar form. [10 marks] Using de Moivre’s theorem, show that      . 5. The equation of an ellipse is 4      8  12  0. (a) Obtain the standard form of the equation of the ellipse. (b) Find the centre, foci and vertices of the ellipse. (c) Sketch the ellipse.

6. The vector v = ai + bj + ck is perpendicular to the vectors i – 2k and 2i + j – k. (a) Find a and b in terms of c. [4 marks] [3 marks] (b) Given that |#| = √56 , and that c is positive, find c.