OXFORD CAMBRIDGE And RSA Exercise
OXFORD CAMBRIDGE AND RSA EXAMINATIONS Advanced Subsidiary General Certificate of Education Advanced General Certificate of Education
MATHEMATICS
Core Mathematics 4
Monday
4724
Afternoon 1 hour 30 minutes
23 JANUARY 2006
Additional materials: 8 page answer booklet Graph paper List of Formulae (MF1)
TIME
1 hour 30 minutes
INSTRUCTIONS TO CANDIDATES
• • • •
Write your name, centre number and candidate number in the spaces provided on the answer booklet. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures unless a different degree of accuracy is specified in the question or is clearly appropriate. You are permitted to use a graphical calculator in this paper.
INFORMATION FOR CANDIDATES
• • …show more content…
dx
[2]
(ii) Show that the equation of the tangent to the curve at (p2 , 2p) is
py = x + p2 .
[2]
(iii) Find the coordinates of the point where the tangent at (9, 6) meets the tangent at (25, −10). [4] 6 (i) Show that the substitution x = sin2 θ transforms
1
x dx to 1−x
2 sin2 θ dθ .
[4]
(ii) Hence find
0
x dx. 1−x
[5]
7
The expression
11 + 8x is denoted by f(x). (2 − x)(1 + x)2
(i) Express f(x) in the form
(ii) Given that | x | < 1, find the first 3 terms in the expansion of f(x) in ascending powers of x.
A B C + + , where A, B and C are constants. 2 − x 1 + x (1 + x)2
[5] [5]
4724/Jan06
3 8 (i) Solve the differential equation
giving the particular solution that satisfies the condition y = 4 when x = 5.
(ii) Show that this particular solution can be expressed in the form
dy 2 − x = , dx y − 3
[5]
where the values of the constants a, b and k are to be stated.
(iii) Hence sketch the graph of the particular solution, indicating clearly its main features. 9
(x − a)2 + (y − b)2 = k,
[3] [3]
Two lines have vector equations r= where a is a constant.
4 2 −6
MATHEMATICS
Core Mathematics 4
Monday
4724
Afternoon 1 hour 30 minutes
23 JANUARY 2006
Additional materials: 8 page answer booklet Graph paper List of Formulae (MF1)
TIME
1 hour 30 minutes
INSTRUCTIONS TO CANDIDATES
• • • •
Write your name, centre number and candidate number in the spaces provided on the answer booklet. Answer all the questions. Give non-exact numerical answers correct to 3 significant figures unless a different degree of accuracy is specified in the question or is clearly appropriate. You are permitted to use a graphical calculator in this paper.
INFORMATION FOR CANDIDATES
• • …show more content…
dx
[2]
(ii) Show that the equation of the tangent to the curve at (p2 , 2p) is
py = x + p2 .
[2]
(iii) Find the coordinates of the point where the tangent at (9, 6) meets the tangent at (25, −10). [4] 6 (i) Show that the substitution x = sin2 θ transforms
1
x dx to 1−x
2 sin2 θ dθ .
[4]
(ii) Hence find
0
x dx. 1−x
[5]
7
The expression
11 + 8x is denoted by f(x). (2 − x)(1 + x)2
(i) Express f(x) in the form
(ii) Given that | x | < 1, find the first 3 terms in the expansion of f(x) in ascending powers of x.
A B C + + , where A, B and C are constants. 2 − x 1 + x (1 + x)2
[5] [5]
4724/Jan06
3 8 (i) Solve the differential equation
giving the particular solution that satisfies the condition y = 4 when x = 5.
(ii) Show that this particular solution can be expressed in the form
dy 2 − x = , dx y − 3
[5]
where the values of the constants a, b and k are to be stated.
(iii) Hence sketch the graph of the particular solution, indicating clearly its main features. 9
(x − a)2 + (y − b)2 = k,
[3] [3]
Two lines have vector equations r= where a is a constant.
4 2 −6