## C4 EDEXCEL STYLE 20130226

6666/01
Edexhell GCE

Core Mathematics C4

Wednesday 26 February 2013 – China Time: GMT + 08:00

Materials required for examination
Brains in gear; pencil, pen with blue or black ink; appropriate calculator.

Candidates may use any calculator allowed by the regulations of the Joint Council for Qualifications. Calculators must not have the facility for symbolic algebra manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them.

Instructions to Candidates    Answer all of each question.
When a calculator is used, the answer should be given to an appropriate degree of accuracy.

Information for Candidates
A booklet ‘Mathematical Formulae and Statistical Tables’ is not provided.
Full marks may be obtained for answers to all seven questions.
The marks for individual questions and the parts of questions are shown in square brackets: e.g. [2]. This paper was largely based on the published paper for January 2012. The total mark for that paper is 75.

You must ensure that you’re all here. There are 7 questions in this question paper. Some parts of questions are clearly labelled.
You should show sufficient working to make your methods clear. Answers without working will be marked NWNM.

1.           The curve S has the equation 3x + 4y² + 5x3y = 12x². The curve goes through point P, at (1,1).  Find the gradient of the curve at P.                                                  [5]

Hence find the equation of the normal to the curve at P in the form ax+by=c, for integers a,b & c.                                                                                                                         [3]

Q1 out of 8

2.       Show that the differential of (1+sinθ)/cosθ w.r.t. θ is (1-sinθ)-1                            [3]

Hence show that ∫secθ dθ = ln |secθ + tanθ|                                                                 [3]

Evaluate the integral between π/4 and π/3, expressing the exact result as a sum of real numbers in surd form.                                                                                                    [3]

Q2 out of 9

3.       This is the parametric curve described by x=√(2t); y=2t-t².

Find the maximum of the curve.                                                                                     [3]

Find the volume of revolution formed by rotating the positive segment of the curve through 2π around the x-axis.                                                                                                     [6]

Q3 out of 9

4.   For the curve above in Q3, here is a table corresponding points for x & y

x       0     0.5       1       1.5     2

y      0    0.23    0.75    0.98    0

Write down the explicit Cartesian (non-parametric form) of the equation for the curve.   [2]

Improve the figures for x=0.5 and 1.5 to four d.p.                                                             [2]

Use the trapezium rule to estimate the area, to three d.p.                                                [3]

Calculate the area under the curve                                                                                   [2]

Hence calculate the error to the estimate for the area                                                      [1]

Q5 out of 10

5.   Lines r1 and r2  are defined as follows:

r= 2i-3j+6k + λ(2i+3j+k);  r= i+6j-2k + μ(i-2j+3k);

Show that r1 and r2 meet and find where that is. Call this point A.                                    [5]

Find the acute angle between r1 and r2 to 1 d.p. in degrees.                                            [2]

B is at position 5i-2j+10k. Show that this is on r2.                                                             [1]

Find the minimum distance from B to line r1, to 3 sig.fig.                                                  [4]

Q5 out of 12

6.          Find the integral of cos² θ  w.r.t θ                                                                      [3]

The closed curve  r=  2 + cosθ is to be drawn, where θ is the angle anti-clockwise from the positive horizontal axis and r is the positive displacement outwards in that direction.

By evaluating ∫r²dθ between 0 and π, find the area inside the curve.                            [6]

Q6 out of 9

7.       Express 1 / (1+x)(3-x) in partial fractions.                                                            [3]

Check that your expression works for x=0 and x=1                                                       [1]

Hardwood trees of a certain valuable type grow in length from 1 to 3 metres in line with the differential equation dx/dt = k (1+x)(3-x).  Solve this, given that initially x=0 and write an explicit equation in x, which will be of the form

(a-E)/(b-E) where E = e-ckt and a,b,c  are constants.                                                      [6]

Given that k = ln3 - ln2, show that x=1/3 when t=1/4                                                         [2]

Express (1+x)-1 and (3-x)-1 as expansions up to and including the x²  term.

Combine the two expansions so as to integrate the approximation. Evaluate this expression at x=1/3. [It is close to 0.1].                                                                           [4]

Using your value of k, find t when x=1/3; compare your results, writing the error in t. State the range of values of x for which your approximation is valid.                                       [3]

Q7 out of 18

Q1-7 out of 75

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