Trial p1 Stpm 2015 Mt q & Ans

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SMJK SAM TET IPOH PEPERIKSAAN PERCUBAAN PENGGAL 1 STPM 2015 MATHEMATICS (T) 954/1 Paper 1 (1 ½ hours)

Disediakan oleh : En Wang Yaw Weng Cik Ong Siew Eng Disemak oleh :…………………………… En Wang Yaw Weng (Ketua Panitia Matematik)

Disahkan oleh : ……………………………. Pn Ng Sook Chin (PK Tingkatan 6)

A list of mathematical formulae is provided on page 4 of this question paper.

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This question paper consists of 4 printed pages. © SMJK SAM TET 2013 Section A [45 marks] 1

Answer all questions in this section. 1

2

The functions f and g are defined as f : x  3x – 5, x  and g : x  e–2x, x  (i) State the range of g.

[1]

(ii) Sketch, on the same axes, the graphs of the inverse functions of f –1 and g–1.

[3]

(iii) State, giving a reason, the number of roots of the equation f –1(x) = g–1(x).

[1]

1 (iv) Evaluate fg(– 3

), giving your answer to 3 decimal places.

[2]

−√ 6+ 3 √2 i in polar form.

[3]

Express the complex number Hence, (a) find z4,

(b) solve the equation z3 =

3

4

5

[2]

−√6+ 3 √2 i

The matrices A and C are given by A =

( ) 1 1 1 1 2 2 2 1 3

[3]

, C=

( ) 1 0 2 3 1 0 1 1 1

(i) Using elementary row operations, obtain the inverse of A.

[5]

(ii) Find the matrix B satisfying BA = C.

[2]

A geometric progression has positive terms. The sum of the first six terms is nine times the sum of the first three terms. The seventh term is 320. Find the common ratio and the first term.

[4]

Find the smallest value of n such that the sum to n terms of the progression exceeds 10 6.

[3]

Show that the point P(2sec  + 2, tan  – 3) lies on the curve x2 – 4y2 – 4x – 24y – 36 = 0.

[2]

Expressing the equation of the curve in standard form, determine whether it is a parabola, an ellipse or a hyperbola.

[2]

Sketch the curve.

[2]

Find the centre, the vertices, the foci and the equations of asymptotes (if any). 2

[4]

6

Two lines have equations r = ( i + 5j + 2k ) + s ( i – 2j + 3k) and r = (–i – j + 10k ) + t (3i + 4j – 5k) (i) Show that the lines meet, and find the point of intersection.

[4]

(ii) Calculate the acute angle between the lines.

[2]

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

(a) Find the value of a for which (x – 2) is a factor of 3x3 + ax2 + x – 2. Show that, for this value of a, the cubic equation 3x3 + ax2 + x – 2 = 0 has only one real root. [5] (b) Determine the solution set of the inequality

| x+5x |

> 2.

[4]

(c) Obtain the first three terms in the expansion, in ascending powers of x, of 2

(8  3 x) 3 , stating the set of values of x for which the expansion is valid. 2

(7.97) 3 Hence, find

8

, correct your answer to four decimal places.

[6]

The plane  passes through the points A(-2, 3, 5), B(1, -3, 1) and C(4, -6, -7). (i) Find AC x BC.

[3]

Hence, find (ii) the area of the triangle ABC.

[2]

(iii) the equation of the plane  in the form r . n = p.

[2]

The perpendicular from the point D(14, 1, 0) to  meets the plane at the point E. Find (iv) the equation of DE.

[2]

(iv) the coordinates of E.

[3] 3

(vi) the angle between the line AD and the plane .

[3]

MARKING SCHEME TRIAL PENGGAL 1 STPM 2015

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