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1 UNIVERSITI MALAYA UNIVERSITY OF MALAYA PEPERIKSAAN IJAZAH SARJANA MUDA SAINS EXAMINATION FOR THE DEGREE OF BACHELOR OF SCIENCE SESI AKADEMIK 009/010 : SEMESTER ACADEMIC SESSION 009/010 : SEMESTER SCES1101 : MATEMATIK ASAS UNTUK KIMIA BASIC MATHEMATICS FOR CHEMISTRY April/Mei 010 April/May 010 MASA : jam TIME : hours ARAHAN KEPADA CALON : INSTRUCTIONS TO CANDIDATES : Jawab SEMUA soalan. Answer ALL Questions. (Kertas soalan ini mengandungi 4 soalan dalam 6 halaman yang dicetak) (This question paper consists of 4 questions on 6 printed pages)

2 SCES1101/0 Jawab SEMUA soalan. Answer ALL questions. 1. (a) Jika, cari 3 nilai bagi supaya. Seterusnya dengan menggunakan Teorem Cayley-Hamilton nyatakan songsangan bagi matriks. If, find 3 values for such that. Hence using the Cayley Hamilton Teorem deduce the inverse of the matri. (b) Tunjukkan bahawa matriks pantulan bagi garis adalah dimana Show that the matri of reflection of the line denoted by is [Perhatian: gunakan transformasi bagi titik dibawah ialah dan ialah ] Seterusnya cari matriks pantulan dalam garis dan dibawah transformasi ini cari archa bagi persamaan. [Note: use the transformation of under is and is ] Hence find the matri of reflection in the line and under this transformation find the image of equation. (9 markah/ marks)

3 SCES1101/03 (c) Selesaikan sistem dengan menggunakan matriks augmented Solve the following system using the augmented matri Tuliskan matriks augmented dan selesaikan matriks dengan menggunakan teknik baris terturun. Write the augmented matri and solve using the row reduction method.. (a) Tunjukkan bahawa 4 titik yang mempunyai vektor kedudukan dan adalah koplanor. Show that the four points whose position vectors are and are coplanar. (b) Tunjukkan bahawa titik yang mempunyai vektor kedudukan, dan adalah kolinear. Show that the points whose position vectors are, and are collinear. (c) Diberi vektor dan cari nilai bagi Comp, Proj dan Prog. Given the vector and find the value of Comp, Proj dan Prog. (d) Cari persamaan berparameter bagi garis yang melalui dan selari dengan. Cari titik persilangan garis dengan satah. Find the parametric equations for the line through and is parallel to. Find the point of intersection of on the plane.

4 SCES1101/04 (e) Tunjukkan bahawa persamaan bagi satu satah yang melalui titik dan berothogonal dengan vektor ialah dimana. Seterusnya cari persamaan satah yang melalui titik dan satah itu adalah normal kepada vektor Show that the equation of a plane passing through the point and orthogonal to the vector is where. Hence find the equation of the plane passing through the point and the plane is normal to the vector dy 3. (a) Cari bagi d (i) 5 y c 3 dimana c adalah pemalar (ii) y ( 1 3) dy Find given d (i) 5 y c 3 where c is a constant (ii) y ( 1 3) (10 markah/ marks) (b) Kebarangkalian suatu molekul berjisim m dalam gas bersuhu T dan halaju v diberi oleh taburan Mawell-Boltzmann berikut 3 mv m g( v) 4 v e kt kt dg dimana k adalah pemalar Boltzmann. Jika = 0, cari v. dv The probability that a molecule of mass m in a gas at temperature T has speed v given by the Mawell-Boltzmann distribution 3 mv m g( v) 4 v e kt kt dg where k is Boltzmann s constant. If = 0, find v. dv (7 markah/ marks)

5 SCES1101/05 (c) Diberi persamaan f (, y) sin( y ). Carikan Given f (, y) sin( y ). Find f and f. y f dan f. y 4. Selesaikan Solve (a) e e d 5 (b) sin 3 6 cos d (c) Sekiranya terdapat dalam kimia fizikal persamaan, 3 v dv cdt v v dimana c adalah satu pemalar. Selesaikan persamaan ini. If there eist in physical chemistry an equation, v v 3 dv v cdt where c is a constant. Solve this equation.

6 SCES1101/06 (d) Satu contoh persamaan pembeza linear tertib pertama dalam kinetik kimia ialah dy k y ak1e 1 dt k t dimana k 1, k dan a adalah pemalar. Selesaikan persamaan ini. An eample of a linear first-order differential equation in chemical kinetics is dy k y ak1e 1 dt where k 1, k and a are constant. Solve this equation. k t (7 markah/ marks) ~~~~0000~~~~~

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