> Friday 22 October 2021 – Afternoon A Level Further Mathematics B (MEI) Y436/01 Further Pure with Technology. GRADED A+
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Answer all the questions.
1 A family of circles is given by the equation
( ) x a - + 2 2 cos s 2 2 ( ) y a - = in 1 (*)
where the parameter a satisfies 0 2 G a 1 r.
(a) Use a slider (for a) to inv
...
estigate this family of circles. Write down the cartesian equation of
the curve which contains the centre of each circle in the family. [1]
(b) Let b and c be real numbers with 0 G b c 1 1 r. Find and simplify an expression, in terms of
b and c, for the distance between the centre of the circle corresponding to a b = and the centre
of the circle corresponding to a c = . [2]
(c) Hence, or otherwise, find a condition on b and c for the two circles in part (b) to touch. [2]
A curve which every member of a family of curves or lines touches tangentially is called an
envelope of the family.
(d) By tracing the family of curves using a slider (for a), or otherwise, sketch the envelope of the
family (*) in the Printed Answer Booklet. [2]
(e) Write down the equations of the curves which make up the envelope for this family (*). [2]
2 This question is about the family of straight lines which pass through the points (0, a) and ( , 1 a2)
where the parameter a is any real number.
(a) In terms of a, find the equation of the straight line which passes through the points (0, a) and
( , 1 a2). [2]
(b) Let b and c be distinct real numbers. Given that the straight line corresponding to a b = and
the straight line corresponding to a c = are parallel, find b in terms of c. [3]
(c) By tracing the family using a slider (for a), or otherwise, sketch the envelope of this family in
the Printed Answer Booklet. [2]
(d) Determine, in the form y x = h( ), the cartesian equation of the envelope for this family. [5]3
© OCR 2021 Y436/01 Oct21 Turn over
3 (a) (i) Create a program which returns the highest common factor of positive integers m and n.
Write out your program in full in the Printed Answer Booklet. [3]
In the rest of this question the highest common factor of positive integers m and n is
denoted by (m, n).
(ii) Use your program to find (74333, 89817). [1]
(b) Euler’s totient function {( ) n , where n is a positive integer, is defined to be the number of
integers m with 1 G G m n such that ( , m n) . = 1
For example {( ) 6 2 = because ( , 1 6) , = 1 ( , 2 6) , = 2 ( , 3 6) , = 3 ( , 4 6) , = 2 ( , 5 6) = 1 and
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