Algebra and Calculus Sample Paper
HONG KONG EXAMINATIONS AND ASSESSMENT AUTHORITY
HONG KONG DIPLOMA OF SECONDARY EDUCATION EXAMINATION
MATHEMATICS
Extended Part
Module 2 (Algebra and Calculus)
(Sample Paper)
Time allowed: 2 hours 30 minutes
This paper must be answered in English
INSTRUCTIONS
1.
This paper consists of Section A and Section B. Each section carries 50 marks.
2.
Answer ALL questions in this paper.
3.
All working must be clearly shown.
4.
Unless otherwise specified, numerical answers must be exact.
Not to be taken away before the end of the examination session
HKDSE-MATH-M2–1 (Sample Paper)
42
FORMULAS FOR REFERENCE
sin ( A ± B ) = sin A cos B ± cos A sin B
sin A + sin B = 2 sin
A+ B
A− B cos 2
2
cos ( A ± B) = cos A cos B m sin A sin B
sin A − sin B = 2 cos
A+ B
A− B sin 2
2
cos A + cos B = 2 cos
A+ B
A− B cos 2
2
tan ( A ± B) =
tan A ± tan B
1 m tan A tan B
cos A − cos B = −2 sin
2 sin A cos B = sin ( A + B) + sin ( A − B)
A+ B
A− B sin 2
2
2 cos A cos B = cos ( A + B) + cos ( A − B)
2 sin A sin B = cos ( A − B) − cos ( A + B)
********************************************************
Section A (50 marks)
1.
Find
d
( 2 x ) from first principles. dx (4 marks)
2.
A snowball in a shape of sphere is melting with its volume decreasing at a constant rate of
4 cm 3s −1 . When its radius is 5 cm, find the rate of change of its radius.
(4 marks)
3.
The slope at any point ( x, y ) of a curve is given by
dy
= 2 x ln( x 2 + 1) . It is given that the curve dx passes through the point (0,1) .
Find the equation of the curve.
(4 marks)
4.
Find
∫
4
2 1
x − dx . x
(4 marks)
5.
By considering sin
π
7
cos
π
7
cos
2π
3π
π
2π
3π cos , find the value of cos cos cos .
7
7
7
7
7
(4 marks)
6.
Let C be the curve 3e x − y = x 2 + y 2 + 1 .
Find the equation of the tangent to C at the point (1, 1) .
(5 marks)
HKDSE-MATH-M2–2 (Sample Paper)
43
7.
Solve the system of linear equations
x + 7 y − 6 z = −4
3 x − 4 y + 7 z = 13 .
4 x + 3 y + z = 9
HONG KONG DIPLOMA OF SECONDARY EDUCATION EXAMINATION
MATHEMATICS
Extended Part
Module 2 (Algebra and Calculus)
(Sample Paper)
Time allowed: 2 hours 30 minutes
This paper must be answered in English
INSTRUCTIONS
1.
This paper consists of Section A and Section B. Each section carries 50 marks.
2.
Answer ALL questions in this paper.
3.
All working must be clearly shown.
4.
Unless otherwise specified, numerical answers must be exact.
Not to be taken away before the end of the examination session
HKDSE-MATH-M2–1 (Sample Paper)
42
FORMULAS FOR REFERENCE
sin ( A ± B ) = sin A cos B ± cos A sin B
sin A + sin B = 2 sin
A+ B
A− B cos 2
2
cos ( A ± B) = cos A cos B m sin A sin B
sin A − sin B = 2 cos
A+ B
A− B sin 2
2
cos A + cos B = 2 cos
A+ B
A− B cos 2
2
tan ( A ± B) =
tan A ± tan B
1 m tan A tan B
cos A − cos B = −2 sin
2 sin A cos B = sin ( A + B) + sin ( A − B)
A+ B
A− B sin 2
2
2 cos A cos B = cos ( A + B) + cos ( A − B)
2 sin A sin B = cos ( A − B) − cos ( A + B)
********************************************************
Section A (50 marks)
1.
Find
d
( 2 x ) from first principles. dx (4 marks)
2.
A snowball in a shape of sphere is melting with its volume decreasing at a constant rate of
4 cm 3s −1 . When its radius is 5 cm, find the rate of change of its radius.
(4 marks)
3.
The slope at any point ( x, y ) of a curve is given by
dy
= 2 x ln( x 2 + 1) . It is given that the curve dx passes through the point (0,1) .
Find the equation of the curve.
(4 marks)
4.
Find
∫
4
2 1
x − dx . x
(4 marks)
5.
By considering sin
π
7
cos
π
7
cos
2π
3π
π
2π
3π cos , find the value of cos cos cos .
7
7
7
7
7
(4 marks)
6.
Let C be the curve 3e x − y = x 2 + y 2 + 1 .
Find the equation of the tangent to C at the point (1, 1) .
(5 marks)
HKDSE-MATH-M2–2 (Sample Paper)
43
7.
Solve the system of linear equations
x + 7 y − 6 z = −4
3 x − 4 y + 7 z = 13 .
4 x + 3 y + z = 9