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MA1201 Calc & Basic Linear Algebra 2, Exams of Architecture

City University of Hong Kong Architecture

2024 Exam past paper MA1201 Calc & Basic Linear Algebra 2

Typology: Exams

2023/2024

Uploaded on 03/10/2025

uliana-24 🇭🇰

1 document

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City University of Hong Kong

Course code & title: MA1201 Calculus and Basic Linear Algebra II

Session: Semester B, 2023-2024

Time allowed: Three hours

This paper has 6pages (including this cover page).

1. This paper consists of 7 questions.

2. Attempt ALL questions.

3. Start each question on a new page.

4. Show all working.

This is a closed-book examination.

Candidates are allowed to use the following materials/aids:

•Non-programmable portable battery operated calculator.

Materials/aids other than those stated above are not permitted. Candi-

dates will be subject to disciplinary action if any unauthorized materials

or aids are found on them.

NOT TO BE TAKEN AWAY

Discover Exams of Architecture City University of Hong Kong

Partial preview of the text

Download MA1201 Calc & Basic Linear Algebra 2 and more Exams Architecture in PDF only on Docsity!

City University of Hong Kong

Course code & title: MA1201 Calculus and Basic Linear Algebra II

Session: Semester B, 2023-

Time allowed: Three hours

This paper has 6 pages (including this cover page).

1. This paper consists of 7 questions.

2. Attempt ALL questions.

3. Start each question on a new page.

4. Show all working.

This is a closed-book examination.

Candidates are allowed to use the following materials/aids:

Non-programmable portable battery operated calculator.

Materials/aids other than those stated above are not permitted. Candi-

dates will be subject to disciplinary action if any unauthorized materials

or aids are found on them.

NOT TO BE TAKEN AWAY

1. [15] Evaluate the following integrals.

(a)[5]

R

(cos(2x))

dx.

(b)[5]

R 1

e^4 x^ + e−^4 x^

dx.

(c)[5]

R 1

2 x + 1

x^2 + 6x + 10

dx.

2. [15] Evaluate the following integrals.

(a) [5]

R 1

4 x^2 + 12x + 10

dx.

(b)[5]

R

e^3 x^ cos(2x)dx.

(c)[5]

R π

− π 6 (|x|^ + sin(2x))

2 dx.

3. [20]

(a)[5] Let R be the region bounded by the curve x = −y^2 +2y +3 and

the line x = 1. Find the volume of the solid generated by revolving

the region R about the line y = 3.

(b)[5] Let R be the region bounded by the curves x = 3y

− 2 and

x = − 3 y^2 +10. Find the area size of the surface generated by rotating

the region R about the line x = 10 + b where b is a positive real

number. Your answer should depend on b.

(c)[5] Let R 1 be the region bounded by the curve y = x

− 3 x

+3x− 1

and the curve y = a(x

− 2 x+1) (0 ≤ a ≤ 1), and let R 2 be the region

bounded by the curve y = x^3 − 3 x^2 +3x−1, the curve y = a(x^2 − 2 x+1)

(0 ≤ a ≤ 1), and the line x = 2, where a is the same constant for R 1

and R 2. Let V 1 , V 2 be the volumes of the solids generated by revolving

the regions R 1 , R 2 about the line y = 0, respectively. Find the value

of a that minimizes the value of V 1 + V 2. What is the minimal value

of V 1 + V 2?

(d) [5] Find the area of the surface generated by revolving the curve

x(t) = 2t + cos(2t) and y(t) = sin(2t) + 4 with t ∈ [0, π], about the

line y = 5.

7. [8] Let n be an arbitrary positive integer and a be a real number which

is not equal to 1. Leta⃗ 1 a,⃗ 2 , · · · a,⃗ n be vectors in R

a(⃗ 1 a,⃗ 2 , · · · ,⃗a n

are real valued n + 1-component vectors). For any 1 ≤ i ≤ n, every

component ofa⃗ i is equal to 1 except that the i-th component ofa⃗ i

is equal to a. Is the collection of vectors {a⃗ 1 ,⃗a 2 , · · · a,⃗ n} linearly

dependent? You need to show detailed explanation.

End

Useful Elementary Integrals

Constant and powers

∫

k dx = kx +C. 2.

∫

n dx =

xn+^1

n + 1

+C, n 6 = − 1

ln|x| +C, n = − 1

Exponentials

∫

ex^ dx = ex^ +C. 4.

∫

ax^ dx =

ln a

+C, a 6 = 1 , a > 0.

Trigonometric functions

∫ sin x dx = − cos x +C. 6.

∫ cos x dx = sin x +C.

∫

sec

2 x dx = tan x +C. 8.

∫

csc

2 x dx = − cot x +C.

∫

sec x tan x dx = sec x +C. 10.

∫

csc x cot x dx = − csc x +C.

∫

tan x dx = ln|sec x| +C. 12.

∫

cot x dx = ln|sin x| +C.

∫

sec x dx = ln|sec x + tan x| +C. 14.

∫

csc x dx = ln|csc x − cot x| +C.

∫

sec^3 x dx =

[

sec θ tan θ + ln|sec θ + tan θ |

]

+C.

Algebraic functions

∫ 1

1 + x^2

dx = tan−^1 x +C. 17.

∫ 1 √ 1 − x^2

dx = sin−^1 x +C.

Hyperbolic functions

∫

sinh x dx = cosh x +C. 19.

∫

cosh x dx = sinh x +C.

MA1201 Calc & Basic Linear Algebra 2, Exams of Architecture

Related documents

Partial preview of the text

Download MA1201 Calc & Basic Linear Algebra 2 and more Exams Architecture in PDF only on Docsity!

City University of Hong Kong

Course code & title: MA1201 Calculus and Basic Linear Algebra II

Session: Semester B, 2023-

Time allowed: Three hours

This paper has 6 pages (including this cover page).

1. This paper consists of 7 questions.

2. Attempt ALL questions.

3. Start each question on a new page.

4. Show all working.

This is a closed-book examination.

Candidates are allowed to use the following materials/aids:

Materials/aids other than those stated above are not permitted. Candi-

dates will be subject to disciplinary action if any unauthorized materials

or aids are found on them.

NOT TO BE TAKEN AWAY

1. [15] Evaluate the following integrals.

(a)[5]

R

(cos(2x))

dx.

(b)[5]

R 1

e^4 x^ + e−^4 x^

dx.

(c)[5]

R 1

2 x + 1

x^2 + 6x + 10

dx.

2. [15] Evaluate the following integrals.

(a) [5]

R 1

4 x^2 + 12x + 10

dx.

(b)[5]

R

e^3 x^ cos(2x)dx.

(c)[5]

R π

− π 6 (|x|^ + sin(2x))

2 dx.

3. [20]

(a)[5] Let R be the region bounded by the curve x = −y^2 +2y +3 and

the line x = 1. Find the volume of the solid generated by revolving

the region R about the line y = 3.

(b)[5] Let R be the region bounded by the curves x = 3y

− 2 and

x = − 3 y^2 +10. Find the area size of the surface generated by rotating

the region R about the line x = 10 + b where b is a positive real

number. Your answer should depend on b.

(c)[5] Let R 1 be the region bounded by the curve y = x

− 3 x

+3x− 1

and the curve y = a(x

− 2 x+1) (0 ≤ a ≤ 1), and let R 2 be the region

bounded by the curve y = x^3 − 3 x^2 +3x−1, the curve y = a(x^2 − 2 x+1)

(0 ≤ a ≤ 1), and the line x = 2, where a is the same constant for R 1

and R 2. Let V 1 , V 2 be the volumes of the solids generated by revolving

the regions R 1 , R 2 about the line y = 0, respectively. Find the value

of a that minimizes the value of V 1 + V 2. What is the minimal value

of V 1 + V 2?

(d) [5] Find the area of the surface generated by revolving the curve

x(t) = 2t + cos(2t) and y(t) = sin(2t) + 4 with t ∈ [0, π], about the

line y = 5.

7. [8] Let n be an arbitrary positive integer and a be a real number which

is not equal to 1. Leta⃗ 1 a,⃗ 2 , · · · a,⃗ n be vectors in R

a(⃗ 1 a,⃗ 2 , · · · ,⃗a n

are real valued n + 1-component vectors). For any 1 ≤ i ≤ n, every

component ofa⃗ i is equal to 1 except that the i-th component ofa⃗ i

is equal to a. Is the collection of vectors {a⃗ 1 ,⃗a 2 , · · · a,⃗ n} linearly

dependent? You need to show detailed explanation.

End

Useful Elementary Integrals

[

]

+C.