Exam Paper-analysis Quantitative B(nov 2005)

UNIVERSITI MALAYA PEPERIKSAAN IJAZAH SARJANA MUDA EKONOMI SESI AKADEMIK 2005/2006: SEMESTER 1 ESEE1102: ANALISIS KUANTITATIF B November 2005

MASA: 2 jam

ARAHAN KEPADA CALON: Calon dikehendaki menjawab SEMUA soalan. Kertas ini merupakan 60 peratus daripada penilaian kursus. Kalkulator mini boleh digunakan. Kertas graf akan disediakan. Senarai formula adalah terlampir.

(Kertas soalan ini mengandungi 5 soalan dalam 6 halaman yang dicetak)

ESEE1102/2

(1)

Sejenis kemeja-t yand baru dijual dengan harga RM15 sehelai. Kos tetapnya adalah RM60,000 dan kos seunit ialah RM2. Pengeluar berupaya mengeluarkan sebanyak 10,000 kemeja-t dan boleh menjual semua kemeja-t yang dikeluarkan. (i)

Dapatkan fungsi hasil, R(x) dan fungsi kos, C(x) dalam sebutan x di mana x adalah kuantiti.

(ii)

Nyatakan domain dan julat bagi fungsi-fungsi yang diperolehi di (i).

(iii)

Cari jumlah kuantiti keluaran apabila fungsi hasil sama dengan fungsi kos dan terangkan jawapan anda.

(iv)

Dapatkan fungsi keuntungan, U(x). Apakah keuntungan atau kerugian apabila 4,000 helai kemeja-t dijual?

(v)

Apakah keluaran dan jualan yang diperlukan untuk memperolehi keuntungan sebanyak RM50,000?

(vi)

Dapatkan fungsi kos purata, C(x) , iaitu, kos purata untuk sehelai kemeja-t.

(vii)

Apakah had fungsi kos purata bila x → 0+ dan bila x → 10,000 Terangkan jawapan anda.

(viii)

Dapatkan fungsi kos sut, C’(x) dan tentukan C’(1000). Terangkan jawapan anda.

–

?

(20 markah)

(2)

(a)

Perubahan harga minyak kelapa sawit dari tahun 2001 hingga 2003 boleh ditulis dalam bentuk fungsi seperti di bawah: h(t) = 4t3 – 2t2 + 1.5

;

0≤t≤2

di mana h(t) adalah dalam ribu RM per metrik ton, t ialah tahun dan t=0 mewakili tahun 2001. (i)

Dapatkan titik-titik maximum dan minimum (jika ada) bagi fungsi tersebut.

(ii)

Dapatkan titik lengkuk balas untuk fungsi tersebut. Di manakah fungsi ini cekung ke atas dan ke bawah? (6 markah)

ESEE1102/3

(b)

Interaksi di antara dua industri dalam ekonomi, Keluli dan Arang, diberikan dalam matriks input-output: Keluli

(a)

Permintaan akhir

Industri: Keluli

50

80

100

Arang

60

70

120

120

100

Lain-lain

(3)

Industri Arang

(i)

Dapatkan matriks input-output (A).

(1 markah)

(ii)

Dapatkan (I-A) -1.

(2 markah)

(iii)

Dapatkan matrik output (X) jika permintaan akhir berubah menjadi 200 bagi Keluli dan 300 bagi Arang. (1 markah)

Fungsi permintaan untuk sesuatu produk boleh ditulis sebagai q=

288 ln( p 2 )

;

p dan q > 0

di mana q ribu unit dijual pada harga RM p seunit. (i)

Tentukan keanjalan permintaan, E(p).

(ii)

Jika permintaan adalah 120 unit, gunakan keanjalan permintaan untuk menentukan sama ada satu pertambahan harga akan menaikan atau menurunkan hasil. Terangkan. (7 markah)

(b)

Seorang penjual kereta menawarkan Farah pilihan di antara dua jenis pinjaman: (i)

RM3,000 bagi 3 tahun pada kadar faedah ringkas 12% setahun.

(ii)

RM3,000 bagi 3 tahun pada kadar 10% setahun dikompaun bulanan.

Manakah di antara pinjaman tersebut mempunyai kos terendah? (3 markah)

ESEE1102/4

(4)

Persamaan permintaan untuk suatu barangan diberikan oleh D(x) = 30 – x2

;

x≥0

dan persamaan penawarannya diberi oleh S(x) = 3x + 2

;

x≥0

(i)

Tentukan kuantiti dan harga tandingan.

(ii)

Dapatkan lebihan pengguna dan lebihan pengeluar.

(iii)

Lakarkan kedua fungsi D(x) dan S(x) atas satu graf dan tunjukkan lebihan pengguna serta lebihan pengeluar.

(iv)

Tentukan luas kawasan di antara D(x) dan S(x). (10 markah)

(5)

(a)

Permudahkan

1 − 1 

0   1  4  6

4  2 − 5 5

6   0   (2 markah)

(b)

(i)

Nyatakan sistem persamaan linear berikut dalam bentuk matriks. x – y – z - 2 = 0, x + y + 2z + 5 = 0, 2x + z + 3 = 0.

(ii)

(1 markah)

Selesaikan sistem persamaan di atas dengan menggunakan kaedah penurunan. (2 markah)

ESEE1102/5

(c)

Sebuah kilang mengeluarkan sejenis aloi yang terbentuk daripada gabungan dua jenis logam, X dan Y. Unit keluaran aloi tersebut, z boleh ditulis sebagai fungsi di bawah: z(x, y) = 60x + 20y + 2xy – x2 – 3y2

di mana x dan y adalah unit logam X dan Y masing-masing. Kilang itu berupaya menjual semua keluarannya pada harga RM4 seunit. Kos seunit logam X dan Y adalah RM200 dan RM50 masingmasing. (i)

Apakah fungsi keuntungan P(x,y)?

(ii)

Cari Px, Py, Pxx, Pyy dan Pyx .

(iii)

Berapakah unit logam X dan Y mesti digunakan untuk memaksimumkan keuntungan? (5 markah)

ESEE1102/6 RUMUSAN Pembezaan (a ) y = cx n ,

Pengkamiran y ' = ncx n − 1

(b) u = f ( x), v = g ( x) (i ) y = uv dy dv du = u. + v. dx dx dx u (ii ) y = v du dv v. − u. dy dx = dx 2 dx v y = g (u ), u = h(u ) dy dy du = . dx du dx (d ) y = u n , u = f ( x)

(a) ∫ x n dx =

xn +1 +k n +1

(n ≠ 1)

1 (b) ∫ dx = ln x + k x (c) ∫ e x dx = e x + k 1 (d ) ∫ e ax + b dx = e ax + b + k a x a (e) ∫ a x dx = +k ln a

(c )

dy du = nu n − 1 dx dx (e) y = log u, u = f ( x) a dy log a e du = dx u dx ( f ) y = ln u, u = f ( x) dy 1 du = . dx u dx ( g ) y = au , u = f ( x)

( f ) ∫ ( ax + b ) n dx = ( g )∫

1 ( ax + b ) n + 1 +k a n +1

1 1 dx = ln ( ax + b ) + k ax + b a

( h ) ∫ udv = uv − ∫ vdu N

(i )Amaun Anuiti = ∫ Pe rt dt 0 1

( j )Koefisien ketidakseimbangan = 2 ∫ [ x − f ( x )]dx 0

dy du = a u ln a dx dx p f '( p ) (h) E(p) = f ( p)

Matematik Kewangan Faedah ringkas Faedah Jumlah ringkas

I = Prt S = P(1+rt)

Diskaun ringkas Diskaun Nilai kini

D = Sdt P = S(1-dt)

( a ≠ 0 , n ≠ − 1)

Faedah Kompaun Amaun kompaun S = P(1+i)n Nilai kini P = S(1+i)-n Anuiti

Amaun

 (1 + i )n − 1  S = R  i  

Nilai kini

1 − (1 + i )− n  A= R  i  

“”””””””””””””””””””OOOOOOOOOOOOOOOO”””””””””””””””””

UNIVERSITI OF MALAYA EXAMINATION FOR THE DEGREE OF BACHELOR OF ECONOMICS ACADEMIC SESSION 2005/2006: SEMESTER 1 ESEE1102: QUANTITATIVE ANALYSIS B November 2005 TIME ALLOWED: 2 hours

INSTRUCTION TO CANDIDATES:

Candidates are required to answer ALL questions. This paper constitutes 60 percent of the course assessment. Calculators are allowed for this examination. Graph papers will be provided. A formulate sheet is appended.

(This paper contains 5 questions in 6 printed pages)

ESEE1102/2

(1)

A new t-shirt is sold at RM15 a piece. Its fixed cost is RM60,000 and the cost for a unit is RM2. The producer is able to produce as many as 10,000 t-shirts and can sell all products. (i)

Obtain the revenue function, R(x) and cost function, C(x) written in x where x is the quantity.

(ii)

State the domain and range for the functions obtained in (i).

(iii)

Find total quantity produced when the revenue function equals the cost function and interpret your answer.

(iv)

Obtain the profit function, U(x). What is the profit or loss when 4,000 t-shirts are sold?

(v)

How much is to be produced and sold in order to obtain a profit of RM50,000?

(vi)

Write the average cost function, C(x) , that is the average cost of producing a piece of t-shirt.

(vii)

What is the limit of the average cost function when x → 0+ and when x → 10,000 – ? Explain your answers.

(viii)

Write the marginal cost function, C’(x) and find C’(1000). Interpret your answer. (20 marks)

(2)

(a)

Price movement of crude palm oil from year 2001 to 2003 can be represented by the function as below: h(t) = 4t3 – 2t2 + 1.5

;

0≤t≤2

where h(t) is in thousand RM per metric ton, t is the year and t = 0 corresponds to year 2001. (i)

Obtain the maximum and minimum points (if exists) for the given function.

(ii)

Obtain the inflection point for the given function. Where does the function concave upward and downward? (6 marks)

ESEE1102/3

(b)

The interaction of two industries in the economy, Steel and Coal are given in the following input-output matrix: Steel

(a)

Final Demand

Industry: Steel

50

80

100

Coal

60

70

120

120

100

Others

(3)

Industry Coal

(i)

Find the input-output matrix (A).

(1 mark)

(ii)

Find (I-A) -1.

(iii)

Find the output matrix (X) if the final demand changes to 200 for Steel and 300 for Coal. (1 mark)

(2 marks)

Demand function for a certain product can be written as: q=

288 ln( p 2 )

;

p and q > 0

where q is in thousand of units sold at price RM p per unit. (i)

Determine elasticity of demand, E(p).

(ii)

If demand is 120 units, using the elasticity of demand, determine if an increase in price will lead to an increase or decrease in revenue. Explain. (7 marks)

(b)

A car salesman offers Farah a choice between two types of loans: (i)

RM 3,000 for 3 years at the simple rate of interest of 12% annually.

(ii)

RM 3,000 for 3 years at the rate of 10% annually compounded monthly.

Which of the two options offers the lowest cost? (3 marks)

ESEE1102/4

(4)

The demand equation for a commodity is given as D(x) = 30 – x2

;

x≥0

while its supply equation is given as S(x) = 3x + 2

;

x≥0

(i)

Determine the quantity and price at equilibrium.

(ii)

Obtain the consumers’ surplus and producers’ surplus.

(iii)

Sketch function D(x) and S(x) on the same graph and show the consumer surplus and production surplus.

(iv)

Determine the area between D(x) and S(x). (10 marks)

(5)

(a)

Simplify

1 − 1 

(b)

(i)

0   1  4  6

4  2 − 5 5

(2 marks)

Write the following system of linear equations in matrix form x – y – z - 2 = 0, x + y + 2z + 5 = 0, 2x + z + 3 = 0.

(ii)

6   0  

(1 marks)

Solve the above system of equations by the method of reduction. (2 marks)

ESEE1102/5

(c)

A factory produces one type of alloy by combining two types of steel, X and Y. Unit of alloy produced, z can be written as such: z(x, y) = 60x + 20y + 2xy – x2 – 3y2

where x and y are the unit of steel X and Y respectively. The factory is able to sell all produce at RM4 per unit. Unit cost for steel X and Y is RM200 and RM50 respectively. (i)

What is profit function P(x,y)?

(ii)

Find Px, Py, Pxx, Pyy dan Pyx .

(iii)

How many units of steel X and Y must be used to maximize profit? (5 marks)

ESEE1102/6 FORMULA Differentiation (a ) y = cx n ,

Integration y ' = ncx n − 1

(b) u = f ( x), v = g ( x) (i ) y = uv dy dv du = u. + v. dx dx dx u (ii ) y = v du dv v. − u. dy dx = dx 2 dx v y = g (u ), u = h(u ) dy dy du = . dx du dx (d ) y = u n , u = f ( x)

(a) ∫ x n dx =

xn +1 +k n +1

(n ≠ 1)

1 (b) ∫ dx = ln x + k x (c) ∫ e x dx = e x + k 1 (d ) ∫ e ax + b dx = e ax + b + k a x a (e) ∫ a x dx = +k ln a

(c )

dy du = nu n − 1 dx dx (e) y = log u, u = f ( x) a dy log a e du = dx u dx ( f ) y = ln u, u = f ( x) dy 1 du = . dx u dx ( g ) y = au , u = f ( x)

( f ) ∫ ( ax + b ) n dx = ( g )∫

1 ( ax + b ) n + 1 +k a n +1

( a ≠ 0 , n ≠ − 1)

1 1 dx = ln ( ax + b ) + k ax + b a

( h ) ∫ udv = uv − ∫ vdu N

(i ) Annuity amount = ∫ Pert dt 0 1

( j ) Coefficient of inequality = 2 ∫ [ x − f ( x)]dx 0

dy du = a u ln a dx dx p f '( p ) (h) E(p) = f ( p)

Finance Mathematics Simple Interest Interest Simple sum

I =Prt S = P(1+rt)

Simple discount Discount Present value

D = Sdt P = S(1-dt)

Compound Interest Compound amount S = P(1+i)n Present value P = S(1+I)-n Annuities

 (1 + i )n − 1  Amount S = R  i   1 − (1 + i )− n  Present value A = R   i  

“”””””””””””””””””””OOOOOOOOOOOOOOOO”””””””””””””””””