Midterm 2009

950/1, 954/1

MIDTERM STPM 2009

SEKOLAH MENENGAH KEBANGSAAN METHODIST (ACS) KLANG PEPERIKSAAN PERTENGAHAN TAHUN 2009 MATHEMATICS S/T Kertas 1(PAPER 1) TIGA JAM (Three hours) JANGAN BUKA KERTAS SOALAN INI SEHINGGA DIBERITAHU Instructions to candidates: 1. Answer all questions. 2. All necessary working should be shown clearly. 3. Non-exact numerical answers may be given correct to three significant figures, or one decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. Kertas soalan ini mengandungi 2 halaman bercetak. Disediakan

Disemak oleh

Disahkan oleh,

…………………. Puan Rabiah Idris

…………………. Mr Chan Seuk Kong (Ketua Panitia)

…………………… Mr Ong Beng Hwa (Ketua Bidang)

1. Using the algebraic laws of sets, show that, for any sets A and B, ( A − B )′ ∩ A = A ∩ B . dy 4 − 3x = 2. Given y = x 2 − x , show that . Hence evaluate dx 2 2 − x

[4] 7 4

∫ 1

4 − 3x dx . 2−x

[5]

3. If ( x + yi) 2 = 5 − 12i , write two equations of x and y. Hence, find the square root of [6] 5 − 12i in the form of a + bi . 2 d y dy 4. Given y = e 2 x cos x . Find values of k and h such that [6] =k + hy . 2 dx dx 3 2 5. Show that the equation x + 5 = has a root between 0.5 and 0.6. [2] x By taking 0.5 as the first approximation, find this root correct to three significant figures by using the Newton-Raphson method. [5]

6. Express

1 1 in partial fractions. Hence, expand in (1 + x) (2 + x) (1 + x) (2 + x)

ascending powers of x until and including the term in x2.

[8]

1 1 2 7. If y = x + , express x + 2 in term of y. x x Show that equation x 4 − 2 x 3 − 6 x 2 − 2 x + 1 = 0 can be transformed into a quadratic equation in term of y. Hence, solve the equation x 4 − 2 x 3 − 6 x 2 − 2 x + 1 = 0 .

[8]

8. Given that f (x) is polynomial of degree more than three. When f (x) is divided by ( x − 1), ( x + 1), and ( x − 2) , the remainders are 3, 1 and -2 respectively. Find the remainder when f (x) is divided by ( x − 1)( x + 1)( x − 2) . [8] 0 0 0  1 a 0     9. The matrices P and Q are given as P =  − 1 − 2 0  , Q =  b − 5 0   −1 − 3 1  c − 3 − 2     a a. If P = mQ+nI where , b, c, m and n are real numbers and I is the 3x3 identity matrix, find a, b, c, m and n. b. Find the real number s and t such that P 2 = sP + tI , where I is the 3x3 identity matrix. Deduce that P 4 = −5P + 6 I [10]

 1 1 − 1 − 5 −5 5      6 − 4  . Find AB, 10. Matrices A and B are given as A =  2 3 2  and B =  8 5 4 1  − 7 1 1     hence find A−1 . [3]

A certain food stall sells three kind of food, which are ‘nasi lemak’, ‘laksa’ and ‘lontong’. The price of each serving of the food is fixed with the price of each serving of ‘lontong’ equals to the total cost of ‘nasi lemak’ and ‘laksa’. A group of students pays RM16 for 2 servings of ‘nasi lemak’, 3 servings of ‘laksa’ and 2 servings of ‘lontong’. Another group of students pays RM19 for 5 servings of ‘nasi lemak’, 4 servings of ‘laksa’ and one serving of ‘lontong’. Assuming that x, y and z are prices (in RM) of each serving of ‘nasi lemak’, ‘laksa’ and ‘lontong’ respectively. Obtain a system of linear equations based on the information given. Rewrite the equation in the matrix form; then solve the equation using the method of matrix to obtain the price of each serving of ‘nasi lemak’, ‘laksa’ and ‘lontong’ respectively. [8]

2 3 n −1 11. Given S n = a + ar + ar + ar + ... + ar . Show that S n =

The first 3 terms of a geometric sequence are 2, −

a (1 − r n ) . 1− r

[3]

1 1 and . Find the sum to infinity of 2 8

the series. [2] Find the smallest n such that the difference between the sum of the first n terms and the sum to infinity is less than 10−5 [7]

12. Find the coordinates of all the points of intersection between the line y = 12 − 4 x and the curve y = 12 − x 3 . Sketch on the same diagram, the line y = 12 − 4 x and the curve y = 12 − x 3 . Calculate the area of the region enclosed by the line y = 12 − 4 x and the curve

y = 12 − x 3 . The region above is rotated through four right angles about the x-axis. Calculate the volume of solid of revolution.

[15]