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

• • • •

The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 72. Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying larger numbers of marks later in the paper. You are reminded of the need for clear presentation in your answers.

This question paper consists of 3 printed pages and 1 blank page. © OCR 2006 [R/102/2711] Registered Charity Number: 1066969

[Turn over

2 1

Simplify

x3 − 3x2 . x2 − 9

dy in terms of x and y. dx

[3]

2

Given that sin y = xy + x2 , ﬁnd

[5]

3

(i) Find the quotient and the remainder when 3x3 − 2x2 + x + 7 is divided by x2 − 2x + 5.

[4]

(ii) Hence, or otherwise, determine the values of the constants a and b such that, when [2] 3x3 − 2x2 + ax + b is divided by x2 − 2x + 5, there is no remainder. 4 (i) Use integration by parts to ﬁnd (ii) Hence ﬁnd

x sec2 x dx.

[4] [3]

x tan2 x dx.

5

A curve is given parametrically by the equations x = t2 , y = 2t. (i) Find

dy in terms of t, giving your answer in its simplest form. 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

+t

−8 1 −2

and

r=

−2 a −2

+s

−9 2 −5

,

(i) Calculate the acute angle between the lines. (ii) Given that these two lines intersect, ﬁnd a and the point of intersection.

[5] [8]

4724/Jan06

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Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable e ort has been made by the publisher (OCR) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. OCR is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. 4724/Jan06