step i and ii - pure mathematics - 2004-2011

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OXFORD CAMBRIDGE AND RSA EXAMINATIONSSixth Term Examination Papersadministered on behalf of the Cambridge Colleges

MATHEMATICS I 9465W e d n e s d a y 30 JUNE 2004 A f t e r n o o n 3 hours

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TIME 3 hours

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• You are advised to concentrate on no more than six questions. Little credit will be given tofragmentary answers.

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MATHEMATICS II 9470Friday 2 JULY 2004 Morning 3 hours

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TIME 3 hours


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OXFORD CAMBRIDGE AND RSA EXAMINATIONS

Sixth Term Examination Papersadministered on behalf of the Cambridge Colleges

MATHEMATICS II 9470Friday 1 JULY 2005 Morning 3 hours

Additional materials: Answer paper Graph paper Formulae booklet


TIME 3 hours


• Write your name, Centre number and candidate number in the spaces on the answer paper/ answer booklet.





• You are advised to concentrate on no more than six questions. Little credit will be given to fragmentary answers.




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STEP I, 2006 2

Section A: Pure Mathematics

1 Find the integer, n, that satisfies n2 < 33127 < (n + 1)2. Find also a small integer m suchthat (n + m)2 � 33127 is a perfect square. Hence express 33127 in the form pq, where p andq are integers greater than 1.

By considering the possible factorisations of 33127, show that there are exactly two values ofm for which (n + m)2 � 33127 is a perfect square, and find the other value.

2 A small goat is tethered by a rope to a point at ground level on a side of a square barn whichstands in a large horizontal field of grass. The sides of the barn are of length 2a and the ropeis of length 4a. Let A be the area of the grass that the goat can graze. Prove that A 6 14⇡a2

and determine the minimum value of A.

3 In this question b, c, p and q are real numbers.

(i) By considering the graph y = x2 + bx + c show that c < 0 is a su�cient condition forthe equation x2 + bx + c = 0 to have distinct real roots. Determine whether c < 0 is anecessary condition for the equation to have distinct real roots.

(ii) Determine necessary and su�cient conditions for the equation x2 + bx + c = 0 to havedistinct positive real roots.

(iii) What can be deduced about the number and the nature of the roots of the equationx3 + px + q = 0 if p > 0 and q < 0?

What can be deduced if p < 0 and q < 0? You should consider the di↵erent cases thatarise according to the value of 4p3 + 27q2 .

STEP I, 2006 3

4 By sketching on the same axes the graphs of y = sinx and y = x, show that, for x > 0:

(i) x > sinx ;

(ii)sinx

x⇡ 1 for small x.

A regular polygon has n sides, and perimeter P . Show that the area of the polygon is

P 2

4n tan⇣⇡

n

⌘ .

Show by di↵erentiation (treating n as a continuous variable) that the area of the polygonincreases as n increases with P fixed.

Show also that, for large n, the ratio of the area of the polygon to the area of the smallestcircle which can be drawn around the polygon is approximately 1.

5 (i) Use the substitution u2 = 2x + 1 to show that, for x > 4,Z

3(x� 4)

p2x + 1

dx = ln✓p

2x + 1� 3p2x + 1 + 3

◆+ K ,

where K is a constant.

(ii) Show thatZ ln 8

ln 3

2ex

pex + 1

dx =712

+ ln23

.

6 (i) Show that, if (a , b) is any point on the curve x2� 2y2 = 1, then (3a + 4b , 2a + 3b) alsolies on the curve.

(ii) Determine the smallest positive integers M and N such that, if (a , b) is any point onthe curve Mx2 �Ny2 = 1, then (5a + 6b , 4a + 5b) also lies on the curve.

(iii) Given that the point (a , b) lies on the curve x2 � 3y2 = 1 , find positive integers P , Q,R and S such that the point (Pa + Qb , Ra + Sb) also lies on the curve.

STEP I, 2006 4

7 (i) Sketch on the same axes the functions cosec x and 2x/⇡, for 0 < x < ⇡ . Deduce thatthe equation x sinx = ⇡/2 has exactly two roots in the interval 0 < x < ⇡ .

Show thatZ

⇡

⇡/2

��x sinx� ⇡

2

�� dx = 2 sin ↵ +3⇡2

4� ↵⇡ � ⇡ � 2↵ cos ↵� 1

where ↵ is the larger of the roots referred to above.

(ii) Show that the region bounded by the positive x-axis, the y-axis and the curve

y =��|ex � 1|� 1

��

has area ln 4� 1.

8 Note that the volume of a tetrahedron is equal to

13 ⇥ the area of the base ⇥ the height.

The points O, A, B and C have coordinates (0, 0, 0), (a, 0, 0), (0, b, 0) and (0, 0, c), respectively,where a, b and c are positive.

(i) Find, in terms of a, b and c, the volume of the tetrahedron OABC.

(ii) Let angle ACB = ✓. Show that

cos ✓ =c2

q(a2 + c2)(b2 + c2)

and find, in terms of a, b and c, the area of triangle ABC.

Hence show that d, the perpendicular distance of the origin from the triangle ABC, satisfies

1d2

=1a2

+1b2

+1c2

.

STEP II, 2006 2


1 The sequence of real numbers u1, u2, u3, . . . is defined by

u1 = 2 , and un+1 = k � 36

un

for n > 1, (⇤)

where k is a constant.

(i) Determine the values of k for which the sequence (⇤) is:

(a) constant;

(b) periodic with period 2;

(c) periodic with period 4.

(ii) In the case k = 37, show that un

> 2 for all n. Given that in this case the sequence (⇤)converges to a limit `, find the value of `.

2 Using the series

ex = 1 + x +x2

2!+

x3

3!+

x4

4!+ · · · ,

show that e > 83 .

Show that n! > 2n for n > 4 and hence show that e < 6724 .

Show that the curve with equation

y = 3e2x + 14 ln(43 � x) , x < 4

3

has a minimum turning point between x = 12 and x = 1 and give a sketch to show the shape

of the curve.

STEP II, 2006 3

3 (i) Show that�5 +

p24

�4 +1

�5 +

p24

�4 is an integer.

Show also that0.1 <

15 +

p24

<219

< 0.11 .

Hence determine, with clear reasoning, the value of�5 +

p24

�4correct to four decimal

places.

(ii) If N is an integer greater than 1, show that⇣N +

pN2 � 1

⌘k

, where k is a positive

integer, di↵ers from the integer nearest to it by less than�2N � 1

2

��k.

4 By making the substitution x = ⇡ � t , show thatZ

⇡

0xf(sin x)dx = 1

2⇡

Z⇡

0f(sin x)dx ,

where f(sinx) is a given function of sinx.

Evaluate the following integrals:

(i)Z

⇡

0

x sinx

3 + sin2 xdx ;

(ii)Z 2⇡

0

x sin x

3 + sin2 xdx ;

(iii)Z

⇡

0

x�� sin 2x

��

3 + sin2 xdx .

5 The notation bxc denotes the greatest integer less than or equal to the real number x. Thus,for example, b⇡c = 3 , b18c = 18 and b�4.2c = �5 .

(i) Two curves are given by y = x2 + 3x � 1 and y = x2 + 3bxc � 1 . Sketch the curves,for 1 6 x 6 3 , on the same axes.

Find the area between the two curves for 1 6 x 6 n, where n is a positive integer.

(ii) Two curves are given by y = x2 + 3x � 1 and y = bxc2 + 3bxc � 1 . Sketch the curves,for 1 6 x 6 3 , on the same axes.

Show that the area between the two curves for 1 6 x 6 n, where n is a positive integer, is

16(n� 1)(3n + 11) .

STEP II, 2006 4

6 By considering a suitable scalar product, prove that

(ax + by + cz)2 6 (a2 + b2 + c2)(x2 + y2 + z2)

for any real numbers a, b, c, x, y and z. Deduce a necessary and su�cient condition on a, b,c, x, y and z for the following equation to hold:

(ax + by + cz)2 = (a2 + b2 + c2)(x2 + y2 + z2) .

(i) Show that (x + 2y + 2z)2 6 9(x2 + y2 + z2) for all real numbers x, y and z.

(ii) Find real numbers p, q and r that satisfy both

p2 + 4q2 + 9r2 = 729 and 8p + 8q + 3r = 243 .

7 An ellipse has equationx2

a2+

y2

b2= 1. Show that the equation of the tangent at the point

(a cos ↵, b sin↵) is

y = �b cot ↵

ax + b cosec ↵ .

The point A has coordinates (�a,�b), where a and b are positive. The point E has coordinates(�a, 0) and the point P has coordinates (a, kb), where 0 < k < 1. The line through E parallelto AP meets the line y = b at the point Q. Show that the line PQ is tangent to the aboveellipse at the point given by tan(↵/2) = k.

Determine by means of sketches, or otherwise, whether this result holds also for k = 0 andk = 1.

8 Show that the line through the points with position vectors x and y has equation

r = (1� ↵)x + ↵y ,

where ↵ is a scalar parameter.

The sides OA and CB of a trapezium OABC are parallel, and OA > CB. The point E on OAis such that OE : EA = 1 : 2, and F is the midpoint of CB. The point D is the intersectionof OC produced and AB produced; the point G is the intersection of OB and EF ; and thepoint H is the intersection of DG produced and OA. Let a and c be the position vectors ofthe points A and C, respectively, with respect to the origin O.

(i) Show that B has position vector �a + c for some scalar parameter �.

(ii) Find, in terms of a, c and � only, the position vectors of D, E, F , G and H. Determinethe ratio OH : HA.

Sixth Term Examination Papers

MATHEMATICS 1 9465MONDAY 23 JUNE 2008 Afternoon

Time: 3 hoursAdditional materials: Answer paper Graph paper Formulae booklet



Please read this page carefully, but do not open this question paper until you are told that you may do so.Write your name, Centre number and candidate number in the spaces on the answer booklet.

Begin each answer on a new page.


Each question is marked out of 20. There is no restriction of choice.

You will be assessed on the six questions for which you gain the highest marks.

You are advised to concentrate on no more than six questions. Little credit will be given for fragmentary answers.

You are provided with Mathematical Formulae and Tables.

Electronic calculators are not permitted.*9203342829*

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1 What does it mean to say that a number x is irrational?

Prove by contradiction statements A and B below, where p and q are real numbers.

A: If pq is irrational, then at least one of p and q is irrational.

B: If p + q is irrational, then at least one of p and q is irrational.

Disprove by means of a counterexample statement C below, where p and q are real numbers.

C: If p and q are irrational, then p + q is irrational.

If the numbers e, ⇡, ⇡2, e2 and e⇡ are irrational, prove that at most one of the numbers ⇡ +e,⇡ � e, ⇡2 � e2, ⇡2 + e2 is rational.

2 The variables t and x are related by t = x +p

x2 + 2bx + c , where b and c are constants andb2 < c. Show that

dx

dt=

t� x

t + b,

and hence integrate1p

x2 + 2bx + c.

Verify by direct integration that your result holds also in the case b2 = c if x + b > 0 but thatyour result does not hold in the case b2 = c if x + b < 0 .

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3 Prove that, if c > a and d > b, then

ab + cd > bc + ad . (⇤)

(i) If x > y, use (⇤) to show that x2 + y2 > 2xy .

If, further, x > z and y > z, use (⇤) to show that z2 + xy > xz + yz and deduce thatx2 + y2 + z2 > xy + yz + zx .

Prove that the inequality x2 + y2 + z2 > xy + yz + zx holds for all x, y and z.

(ii) Show similarly that the inequality

s

t+

t

r+

r

s> 3

holds for all positive r, s and t.

[Note: The final part of this question di↵ers (though not substantially) from what appearedin the actual examination since this was found to be unsatisfactory (though not incorrect) ina way that had not been anticipated.]

4 A function f(x) is said to be convex in the interval a < x < b if f 00(x) > 0 for all x in thisinterval.

(i) Sketch on the same axes the graphs of y = 23 cos2 x and y = sinx in the interval

0 6 x 6 2⇡.

The function f(x) is defined for 0 < x < 2⇡ by

f(x) = e23 sin x.

Determine the intervals in which f(x) is convex.

(ii) The function g(x) is defined for 0 < x < 12⇡ by

g(x) = e�k tan x.

If k = sin 2↵ and 0 < ↵ < ⇡/4, show that g(x) is convex in the interval 0 < x < ↵, andgive one other interval in which g(x) is convex.

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5 The polynomial p(x) is given by

p(x) = xn +n�1Pr=0

ar

xr ,

where a0, a1, . . . , an�1 are fixed real numbers and n > 1. Let M be the greatest value of��p(x)

�� for |x| 6 1. Then Chebyshev’s theorem states that M > 21�n.

(i) Prove Chebyshev’s theorem in the case n = 1 and verify that Chebyshev’s theorem holdsin the following cases:

(a) p(x) = x2 � 12 ;

(b) p(x) = x3 � x .

(ii) Use Chebyshev’s theorem to show that the curve y = 64x5+25x4�66x3�24x2+3x+1has at least one turning point in the interval �1 6 x 6 1.

6 The function f is defined by

f(x) =ex � 1e� 1

, x > 0,

and the function g is the inverse function to f, so that g(f(x)) = x. Sketch f(x) and g(x) onthe same axes.

Verify, by evaluating each integral, that

Z 12

0f(x) dx +

Zk

0g(x) dx =

12(p

e + 1),

where k =1p

e + 1, and explain this result by means of a diagram.

7 The point P has coordinates (x, y) with respect to the origin O. By writing x = r cos ✓ andy = r sin ✓, or otherwise, show that, if the line OP is rotated by 60� clockwise about O,the new y-coordinate of P is 1

2(y �p

3 x). What is the new y-coordinate in the case of ananti-clockwise rotation by 60� ?

An equilateral triangle OBC has vertices at O, (1, 0) and (12 , 1

2

p3), respectively. The point P

has coordinates (x, y). The perpendicular distance from P to the line through C and O is h1;the perpendicular distance from P to the line through O and B is h2; and the perpendiculardistance from P to the line through B and C is h3.

Show that h1 = 12

��y �p

3 x�� and find expressions for h2 and h3.

Show that h1 + h2 + h3 = 12

p3 if and only if P lies on or in the triangle OBC.

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8 (i) The gradient y0 of a curve at a point (x, y) satisfies

(y0)2 � xy0 + y = 0 . (⇤)

By di↵erentiating (⇤) with respect to x, show that either y00 = 0 or 2y0 = x .

Hence show that the curve is either a straight line of the form y = mx + c, wherec = �m2, or the parabola 4y = x2.

(ii) The gradient y0 of a curve at a point (x, y) satisfies

(x2 � 1)(y0)2 � 2xyy0 + y2 � 1 = 0 .

Show that the curve is either a straight line, the form of which you should specify, or acircle, the equation of which you should determine.

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Sixth Term Examination Papers

MATHEMATICS 2 9470WEDNESDAY 25 JUNE 2008 Morning

Time: 3 hoursAdditional materials: Answer paper Graph paper Formulae booklet



Please read this page carefully, but do not open this question paper until you are told that you may do so.Write your name, Centre number and candidate number in the spaces on the answer booklet.



Each question is marked out of 20. There is no restriction of choice.



You are provided with Mathematical Formulae and Tables.

Electronic calculators are not permitted.*1906533762*


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1 A proper factor of an integer N is a positive integer, not 1 or N , that divides N .

(i) Show that 32× 53 has exactly 10 proper factors. Determine how many other integers ofthe form 3m × 5n (where m and n are integers) have exactly 10 proper factors.

(ii) Let N be the smallest positive integer that has exactly 426 proper factors. Determine N ,giving your answer in terms of its prime factors.

2 A curve has the equationy3 = x3 + a3 + b3 ,

where a and b are positive constants. Show that the tangent to the curve at the point (−a, b) is

b2y − a2x = a3 + b3 .

In the case a = 1 and b = 2, show that the x-coordinates of the points where the tangentmeets the curve satisfy

7x3 − 3x2 − 27x− 17 = 0 .

Hence find positive integers p, q, r and s such that

p3 = q3 + r3 + s3 .

3 (i) By considering the equation x2 + x − a = 0 , show that the equation x = (a− x)12 has

one real solution when a > 0 and no real solutions when a < 0 .

Find the number of distinct real solutions of the equation

x =°(1 + a)x− a

¢13

in the cases that arise according to the value of a.

(ii) Find the number of distinct real solutions of the equation

x = (b + x)12

in the cases that arise according to the value of b .

Milosz Wielondek

Mathematics I 2009

4 The sides of a triangle have lengths p − q, p and p + q, where p > q > 0 . The largest andsmallest angles of the triangle are α and β, respectively. Show by means of the cosine rulethat

4(1− cos α)(1− cos β) = cosα + cos β .

In the case α = 2β, show that cosβ = 34 and hence find the ratio of the lengths of the sides of

the triangle.

5 A right circular cone has base radius r, height h and slant length �. Its volume V , and thearea A of its curved surface, are given by

V = 13πr2h , A = πr� .

(i) Given that A is fixed and r is chosen so that V is at its stationary value, show thatA2 = 3π2r4 and that � =

√3 r.

(ii) Given, instead, that V is fixed and r is chosen so that A is at its stationary value, find hin terms of r.

6 (i) Show that, for m > 0 ,Z

m

1/m

x2

x + 1dx =

(m− 1)3(m + 1)2m2

+ lnm.

(ii) Show by means of a substitution thatZ

m

1/m

1xn(x + 1)

dx =Z

m

1/m

un−1

u + 1du .

(iii) Evaluate:

(a)

Z 2

1/2

x5 + 3x3(x + 1)

dx ;

(b)

Z 2

1

x5 + x3 + 1x3(x + 1)

dx .

7 Show that, for any integer m,Z 2π

0ex cos mxdx =

1m2 + 1

°e2π − 1

¢.

(i) Expand cos(A + B) + cos(A−B). Hence show that

Z 2π

0ex cos x cos 6xdx = 19

650

°e2π − 1

¢.

(ii) EvaluateZ 2π

0ex sin 2x sin 4x cos xdx .

8 (i) The equation of the circle C is

(x− 2t)2 + (y − t)2 = t2,

where t is a positive number. Show that C touches the line y = 0 .

Let α be the acute angle between the x-axis and the line joining the origin to the centreof C. Show that tan 2α = 4

3 and deduce that C touches the line 3y = 4x .

(ii) Find the equation of the incircle of the triangle formed by the lines y = 0, 3y = 4x and4y + 3x = 15 .

Note: The incircle of a triangle is the circle, lying totally inside the triangle, thattouches all three sides.


1 Two curves have equations x4 + y4 = u and xy = v , where u and v are positive constants.State the equations of the lines of symmetry of each curve.

The curves intersect at the distinct points A, B, C and D (taken anticlockwise from A).The coordinates of A are (α,β), where α > β > 0. Write down, in terms of α and β, thecoordinates of B, C and D.

Show that the quadrilateral ABCD is a rectangle and find its area in terms of u and v only.Verify that, for the case u = 81 and v = 4, the area is 14.

2 The curve C has equationy = asin(πex) ,

where a > 1.

(i) Find the coordinates of the stationary points on C.

(ii) Use the approximations et ≈ 1+ t and sin t ≈ t (both valid for small values of t) to showthat

y ≈ 1− πx ln a

for small values of x.

(iii) Sketch C.

(iv) By approximating C by means of straight lines joining consecutive stationary points,show that the area between C and the x-axis between the kth and (k + 1)th maxima isapproximately ≥a2 + 1

2a

¥ln

≥1 +

°k − 3

4)−1¥

.

Milosz Wielondek

Mathematics II 2009

3 Prove thattan

°14π − 1

2x¢≡ secx− tanx . (∗)

(i) Use (∗) to find the value of tan 18π . Hence show that

tan 1124π =

√3 +

√2− 1√

3−√

6 + 1.

(ii) Show that √3 +

√2− 1√

3−√

6 + 1= 2 +

√2 +

√3 +

√6 .

(iii) Use (∗) to show that

tan 148π =

q16 + 10

√2 + 8

√3 + 6

√6 − 2−

√2−

√3−

√6 .

4 The polynomial p(x) is of degree 9 and p(x)− 1 is exactly divisible by (x− 1)5.

(i) Find the value of p(1).

(ii) Show that p�(x) is exactly divisible by (x− 1)4.

(iii) Given also that p(x) + 1 is exactly divisible by (x + 1)5, find p(x).

5 Expand and simplify (√

x− 1 + 1)2 .

(i) EvaluateZ 10

5

px + 2

√x− 1 +

px− 2

√x− 1√

x− 1dx .

(ii) Find the total area between the curve

y =

px− 2

√x− 1√

x− 1

and the x-axis between the points x = 54 and x = 10.

(iii) EvaluateZ 10

54

px + 2

√x− 1 +

px− 2

√x + 1 + 2√

x2 − 1dx .

6 The Fibonacci sequence F1, F2, F3, . . . is defined by F1 = 1, F2 = 1 and

Fn+1 = F

n

+ Fn−1 (n > 2).

Write down the values of F3, F4, . . ., F10.

Let S =∞X

i=1

1F

i

.

(i) Show that1F

i

>1

2Fi−1

for i > 4 and deduce that S > 3 .

Show also that S < 323 .

(ii) Show further that 3.2 < S < 3.5 .

7 Let y = (x− a)nebx

√1 + x2 , where n and a are constants and b is a non-zero constant. Show

thatdy

dx=

(x− a)n−1ebxq(x)√1 + x2

,

where q(x) is a cubic polynomial.

Using this result, determine:

(i)Z

(x− 4)14e4x(4x3 − 1)√1 + x2

dx ;

(ii)Z

(x− 1)21e12x(12x4 − x2 − 11)√1 + x2

dx ;

(iii)Z

(x− 2)6e4x(4x4 + x3 − 2)√1 + x2

dx .

8 The non-collinear points A, B and C have position vectors a, b and c, respectively. Thepoints P and Q have position vectors p and q, respectively, given by

p = λa + (1− λ)b and q = µa + (1− µ)c

where 0 < λ < 1 and µ > 1. Draw a diagram showing A, B, C, P and Q.

Given that CQ×BP = AB×AC, find µ in terms of λ, and show that, for all values of λ, thethe line PQ passes through the fixed point D, with position vector d given by d = −a + b + c .What can be said about the quadrilateral ABDC?

© UCLES 2010

91**4023334091* Sixth Term Examination Papers 9465 MATHEMATICS 1 Afternoon

MONDAY 21 JUNE 2010 Time: 3 hours

Additional Materials: Answer Paper Formulae Booklet Candidates may not use a calculator INSTRUCTIONS TO CANDIDATES Please read this page carefully, but do not open this question paper until you are told that you may do so.

Write your name, centre number and candidate number in the spaces on the answer booklet.


INFORMATION FOR CANDIDATES Each question is marked out of 20. There is no restriction of choice.



You are provided with a Mathematical Formulae Booklet.

Calculators are not permitted. Please wait to be told you may begin before turning this page. _____________________________________________________________________________ This question paper consists of 7 printed pages and 1 blank page.

[Turn over

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1 Given that5x2 + 2y2 − 6xy + 4x− 4y ≡ a (x− y + 2)2 + b (cx + y)2 + d ,

find the values of the constants a, b, c and d.

Solve the simultaneous equations

5x2 + 2y2 − 6xy + 4x− 4y = 9 ,

6x2 + 3y2 − 8xy + 8x− 8y = 14 .

2 The curve y =≥x− a

x− b

¥ex, where a and b are constants, has two stationary points. Show that

a− b < 0 or a− b > 4 .

(i) Show that, in the case a = 0 and b = 12 , there is one stationary point on either side of

the curve’s vertical asymptote, and sketch the curve.

(ii) Sketch the curve in the case a = 92 and b = 0 .

3 Show thatsin(x + y)− sin(x− y) = 2 cos x sin y

and deduce thatsinA− sinB = 2 cos 1

2(A + B) sin 12(A−B) .

Show also thatcos A− cosB = −2 sin 1

2(A + B) sin 12(A−B) .

The points P , Q, R and S have coordinates (a cos p, b sin p), (a cos q, b sin q), (a cos r, b sin r)and (a cos s, b sin s) respectively, where 0 6 p < q < r < s < 2π, and a and b are positive.

Given that neither of the lines PQ and SR is vertical, show that these lines are parallel if andonly if

r + s− p− q = 2π .

2

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4 Use the substitution x =1

t2 − 1, where t > 1, to show that, for x > 0,

Z1p

x (x + 1)dx = 2 ln

°√x +

√x + 1

¢+ c .

[Note: You may use without proof the resultZ

1t2 − a2

dt =12a

lnØØØØt− a

t + a

ØØØØ + constant. ]

The section of the curvey =

1√x− 1√

x + 1

between x = 18 and x = 9

16 is rotated through 360o about the x-axis. Show that the volume

enclosed is 2π ln 54 .

5 By considering the expansion of (1 + x)n where n is a positive integer, or otherwise, showthat:

(i)

µn0

∂+

µn1

∂+

µn2

∂+ · · · +

µnn

∂= 2n ;

(ii)

µn1

∂+ 2

µn2

∂+ 3

µn3

∂+ · · · + n

µnn

∂= n2n−1 ;

(iii)

µn0

∂+

12

µn1

∂+

13

µn2

∂+ · · · +

1n + 1

µnn

∂=

1n + 1

°2n+1 − 1

¢;

(iv)

µn1

∂+ 22

µn2

∂+ 32

µn3

∂+ · · · + n2

µnn

∂= n (n + 1) 2n−2 .

6 Show that, if y = ex, then

(x− 1)d2y

dx2− x

dy

dx+ y = 0 . (∗)

In order to find other solutions of this differential equation, now let y = uex, where u is afunction of x. By substituting this into (∗), show that

(x− 1)d2u

dx2+ (x− 2)

du

dx= 0 . (∗∗)

By settingdu

dx= v in (∗∗) and solving the resulting first order differential equation for v,

find u in terms of x. Hence show that y = Ax + Bex satisfies (∗), where A and B are anyconstants.

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7 Relative to a fixed origin O, the points A and B have position vectors a and b, respectively.(The points O, A and B are not collinear.) The point C has position vector c given by

c = αa + βb ,

where α and β are positive constants with α + β < 1 . The lines OA and BC meet at thepoint P with position vector p and the lines OB and AC meet at the point Q with positionvector q. Show that

p =αa

1− β,

and write down q in terms of α, β and b.

Show further that the point R with position vector r given by

r =αa + βb

α + β,

lies on the lines OC and AB.

The lines OB and PR intersect at the point S. Prove thatOQ

BQ=

OS

BS.

8 (i) Suppose that a, b and c are integers that satisfy the equation

a3 + 3b3 = 9c3.

Explain why a must be divisible by 3, and show further that both b and c must also bedivisible by 3. Hence show that the only integer solution is a = b = c = 0 .

(ii) Suppose that p, q and r are integers that satisfy the equation

p4 + 2q4 = 5r4 .

By considering the possible final digit of each term, or otherwise, show that p and q aredivisible by 5. Hence show that the only integer solution is p = q = r = 0 .

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© UCLES 2010

91**4023334091* Sixth Term Examination Papers 9470 MATHEMATICS 2 Morning

Wednesday 23 JUNE 2010 Time: 3 hours

Additional Materials: Answer Paper Formulae Booklet Candidates may not use a calculator INSTRUCTIONS TO CANDIDATES Please read this page carefully, but do not open this question paper until you are told that you may do so.



INFORMATION FOR CANDIDATES Each question is marked out of 20. There is no restriction of choice.




Calculators are not permitted. Please wait to be told you may begin before turning this page. _____________________________________________________________________________ This question paper consists of 6 printed pages and 2 blank pages.

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1 Let P be a given point on a given curve C. The osculating circle to C at P is defined to bethe circle that satisfies the following two conditions at P : it touches C; and the rate of changeof its gradient is equal to the rate of change of the gradient of C.

Find the centre and radius of the osculating circle to the curve y = 1− x + tanx at the pointon the curve with x-coordinate 1

4π.

2 Prove thatcos 3x = 4 cos3 x− 3 cos x .

Find and prove a similar result for sin 3x in terms of sinx.

(i) Let

I(α) =Z

α

0

°7 sinx− 8 sin3 x

¢dx .

Show thatI(α) = −8

3c3 + c + 53 ,

where c = cos α. Write down one value of c for which I(α) = 0.

(ii) Useless Eustace believes thatZ

sinn xdx =sinn+1 x

n + 1

for n = 1, 2, 3, . . . . Show that Eustace would obtain the correct value of I(β) , wherecos β = −1

6 .

Find all values of α for which he would obtain the correct value of I(α).

3 The first four terms of a sequence are given by F0 = 0, F1 = 1, F2 = 1 and F3 = 2. Thegeneral term is given by

Fn

= aλn + bµn , (∗)

where a, b, λ and µ are independent of n, and a is positive.

(i) Show that λ2 + λµ + µ2 = 2, and find the values of λ, µ, a and b.

(ii) Use (∗) to evaluate F6.

(iii) Evaluate∞X

n=0

Fn

2n+1.

2

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4 (i) Let

I =Z

a

0

f(x)f(x) + f(a− x)

dx .

Use a substitution to show that

I =Z

a

0

f(a− x)f(x) + f(a− x)

dx

and hence evaluate I in terms of a.

Use this result to evaluate the integrals

Z 1

0

ln(x + 1)ln(2 + x− x2)

dx andZ π

2

0

sinx

sin(x + π

4 )dx .

(ii) Evaluate Z 2

12

sinx

x°sinx + sin 1

x

¢ dx .

5 The points A and B have position vectors i+ j+k and 5i− j−k, respectively, relative to theorigin O. Find cos 2α, where 2α is the angle \AOB.

(i) The line L1 has equation r = λ(mi + nj + pk). Given that L1 is inclined equally to OAand to OB, determine a relationship between m, n and p. Find also values of m, n andp for which L1 is the angle bisector of \AOB.

(ii) The line L2 has equation r = µ(ui + vj + wk). Given that L2 is inclined at an angle αto OA, where 2α = \AOB, determine a relationship between u, v and w.

Hence describe the surface with Cartesian equation x2 + y2 + z2 = 2(yz + zx + xy).

6 Each edge of the tetrahedron ABCD has unit length. The face ABC is horizontal, and P isthe point in ABC that is vertically below D.

(i) Find the length of PD.

(ii) Show that the cosine of the angle between adjacent faces of the tetrahedron is 1/3.

(iii) Find the radius of the largest sphere that can fit inside the tetrahedron.

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7 (i) By considering the positions of its turning points, show that the curve with equation

y = x3 − 3qx− q(1 + q) ,

where q > 0 and q �= 1, crosses the x-axis once only.

(ii) Given that x satisfies the cubic equation

x3 − 3qx− q(1 + q) = 0 ,

and thatx = u + q/u ,

obtain a quadratic equation satisfied by u3. Hence find the real root of the cubic equationin the case q > 0, q �= 1.

(iii) The quadratic equationt2 − pt + q = 0

has roots α and β. Show that

α3 + β3 = p3 − 3qp .

It is given that one of these roots is the square of the other. By considering the expression(α2 − β)(β2 − α), find a relationship between p and q. Given further that q > 0, q �= 1and p is real, determine the value of p in terms of q.

8 The curves C1 and C2 are defined by

y = e−x (x > 0) and y = e−x sinx (x > 0),

respectively. Sketch roughly C1 and C2 on the same diagram.

Let xn

denote the x-coordinate of the nth point of contact between the two curves, where0 < x1 < x2 < · · · , and let A

n

denote the area of the region enclosed by the two curvesbetween x

n

and xn+1. Show that

An

= 12(e2π − 1)e−(4n+1)π/2

and hence find∞X

n=1

An

.

4

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© UCLES 2011

91**4023334091* Sixth Term Examination Papers 9465 MATHEMATICS 1 Morning

FRIDAY 24 JUNE 2011 Time: 3 hours

Additional Materials: Answer Booklet Formulae Booklet INSTRUCTIONS TO CANDIDATES Please read this page carefully, but do not open this question paper until you are told that you may do so.



Write the numbers of the questions you answer in the order attempted on the front of the answer booklet. INFORMATION FOR CANDIDATES Each question is marked out of 20. There is no restriction of choice.

All questions attempted will be marked.

Your final mark will be based on the six questions for which you gain the highest marks.



Calculators are not permitted. Please wait to be told you may begin before turning this page. _____________________________________________________________________________ This question paper consists of 7 printed pages and 1 blank page.

[Turn over

� ��

2


1 (i) Show that the gradient of the curvea

x+

b

y= 1, where b != 0, is "ay2

bx2.

The point (p, q) lies on both the straight line ax + by = 1 and the curvea

x+

b

y= 1 ,

where ab != 0. Given that, at this point, the line and the curve have the same gradient,show that p = ±q .

Show further that either (a" b)2 = 1 or (a+ b)2 = 1 .

(ii) Show that if the straight line ax + by = 1, where ab != 0, is a normal to the curvea

x" b

y= 1, then a2 " b2 = 1

2 .

2 The number E is defined by E =

! 1

0

ex

1 + xdx .

Show that ! 1

0

xex

1 + xdx = e" 1" E ,

and evaluate

! 1

0

x2ex

1 + xdx in terms of e and E.

Evaluate also, in terms of E and e as appropriate:

(i)

! 1

0

e1!x1+x

1 + xdx ;

(ii)

! !2

1

ex2

xdx .

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3

3 Prove the identity4 sin ! sin(13" ! !) sin(13" + !) = sin 3! . (")

(i) By di!erentiating ("), or otherwise, show that

cot 19" ! cot 2

9" + cot 49" =

#3 .

(ii) By setting ! = 16"!# in ("), or otherwise, obtain a similar identity for cos 3! and deduce

thatcot ! cot(13" ! !) cot(13" + !) = cot 3! .

Show thatcosec 1

9" ! cosec 59" + cosec 7

9" = 2#3 .

4 The distinct points P and Q, with coordinates (ap2, 2ap) and (aq2, 2aq) respectively, lie onthe curve y2 = 4ax. The tangents to the curve at P and Q meet at the point T . Show thatT has coordinates

!apq, a(p+ q)

". You may assume that p $= 0 and q $= 0.

The point F has coordinates (a, 0) and # is the angle TFP . Show that

cos# =pq + 1#

(p2 + 1)(q2 + 1)

and deduce that the line FT bisects the angle PFQ.

5 Given that 0 < k < 1, show with the help of a sketch that the equation

sinx = kx (")

has a unique solution in the range 0 < x < ".

Let

I =

$ !

0

%% sinx! kx%% dx .

Show that

I ="2 sin$

2$! 2 cos$! $ sin$ ,

where $ is the unique solution of (").

Show that I, regarded as a function of $, has a unique stationary value and that this stationaryvalue is a minimum. Deduce that the smallest value of I is

!2 cos"#2.

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4

6 Use the binomial expansion to show that the coe!cient of xr in the expansion of (1 ! x)!3

is 12(r + 1)(r + 2) .

(i) Show that the coe!cient of xr in the expansion of

1! x+ 2x2

(1! x)3

is r2 + 1 and hence find the sum of the series

1 +2

2+

5

4+

10

8+

17

16+

26

32+

37

64+

50

128+ · · · .

(ii) Find the sum of the series

1 + 2 +9

4+ 2 +

25

16+

9

8+

49

64+ · · · .

7 In this question, you may assume that ln(1 + x) " x! 12x

2 when |x| is small.

The height of the water in a tank at time t is h. The initial height of the water is H and waterflows into the tank at a constant rate. The cross-sectional area of the tank is constant.

(i) Suppose that water leaks out at a rate proportional to the height of the water in thetank, and that when the height reaches !2H, where ! is a constant greater than 1, theheight remains constant. Show that

dh

dt= k(!2H ! h) ,

for some positive constant k. Deduce that the time T taken for the water to reach height!H is given by

kT = ln

!1 +

1

!

",

and that kT " !!1 for large values of !.

(ii) Suppose that the rate at which water leaks out of the tank is proportional to#h (instead

of h), and that when the height reaches !2H, where ! is a constant greater than 1, theheight remains constant. Show that the time T " taken for the water to reach height !His given by

cT " = 2#H

!1!

#!+ ! ln

!1 +

1#!

""

for some positive constant c, and that cT " "#H for large values of !.

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5

8 (i) The numbers m and n satisfym3 = n3 + n2 + 1 . (!)

(a) Show that m > n. Show also that m < n+ 1 if and only if 2n2 + 3n > 0 . Deducethat n < m < n+ 1 unless "3

2 ! n ! 0 .

(b) Hence show that the only solutions of (!) for which both m and n are integers are(m,n) = (1, 0) and (m,n) = (1,"1).

(ii) Find all integer solutions of the equation

p3 = q3 + 2q2 " 1 .

5


naugha

© UCLES 2011

91**4023334091* Sixth Term Examination Papers 9470 MATHEMATICS 2 Afternoon

MONDAY 20 JUNE 2011 Time: 3 hours

Additional Materials: Answer Booklet Formulae Booklet INSTRUCTIONS TO CANDIDATES Please read this page carefully, but do not open this question paper until you are told that you may do so.



Write the numbers of the questions you answer in the order attempted on the front of the answer booklet. INFORMATION FOR CANDIDATES Each question is marked out of 20. There is no restriction of choice.

All questions attempted will be marked.

Your final mark will be based on the six questions for which you gain the highest marks.



Calculators are not permitted. Please wait to be told you may begin before turning this page. _____________________________________________________________________________ This question paper consists of 10 printed pages and 2 blank pages.

[Turn over

��

2


1 (i) Sketch the curve y =!1" x+

!3 + x .

Use your sketch to show that only one real value of x satisfies

!1" x+

!3 + x = x+ 1 ,

and give this value.

(ii) Determine graphically the number of real values of x that satisfy

2!1" x =

!3 + x+

!3" x .

Solve this equation.

2 Write down the cubes of the integers 1, 2, . . . , 10 .

The positive integers x, y and z, where x < y, satisfy

x3 + y3 = kz3 , (#)

where k is a given positive integer.

(i) In the case x+ y = k, show that

z3 = k2 " 3kx+ 3x2 .

Deduce that (4z3 " k2)/3 is a perfect square and that 14k

2 ! z3 < k2 .

Use these results to find a solution of (#) when k = 20.

(ii) By considering the case x+ y = z2, find two solutions of (#) when k = 19.

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3

3 In this question, you may assume without proof that any function f for which f !(x) ! 0 isincreasing; that is, f(x2) ! f(x1) if x2 ! x1 .

(i) (a) Let f(x) = sinx! x cosx. Show that f(x) is increasing for 0 " x " 12! and deduce

that f(x) ! 0 for 0 " x " 12! .

(b) Given thatd

dx(arcsinx) ! 1 for 0 " x < 1, show that

arcsinx ! x (0 " x < 1).

(c) Let g(x) = x cosecx for 0 < x < 12!. Show that g is increasing and deduce that

(arcsinx)x"1 ! x cosecx (0 < x < 1).

(ii) Given thatd

dx(arctanx) " 1 for x ! 0, show by considering the function x"1 tanx that

(tanx)(arctanx) ! x2 (0 < x < 12!).

4 (i) Find all the values of ", in the range 0# < " < 180#, for which cos " = sin 4". Henceshow that

sin 18# =1

4

!"5! 1

".

(ii) Given that4 sin2 x+ 1 = 4 sin2 2x ,

find all possible values of sinx , giving your answers in the form p+ q"5 where p and q

are rational numbers.

(iii) Hence find two values of # with 0# < # < 90# for which

sin2 3#+ sin2 5# = sin2 6# .

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4

5 The points A and B have position vectors a and b with respect to an origin O, and O, A and Bare non-collinear. The point C, with position vector c, is the reflection of B in the line throughO and A. Show that c can be written in the form

c = !a! b

where ! =2a.b

a.a.

The point D, with position vector d, is the reflection of C in the line through O and B. Showthat d can be written in the form

d = µb! !a

for some scalar µ to be determined.

Given that A, B and D are collinear, find the relationship between ! and µ. In the case! = !1

2 , determine the cosine of !AOB and describe the relative positions of A, B and D.

6 For any given function f, let

I =

![f !(x)]2 [f(x)]ndx , (")

where n is a positive integer. Show that, if f(x) satisfies f !!(x) = kf(x)f !(x) for some constantk, then (") can be integrated to obtain an expression for I in terms of f(x), f !(x), k and n.

(i) Verify your result in the case f(x) = tanx . Hence find

!sin4 x

cos8 xdx .

(ii) Find !sec2 x (secx+ tanx)6 dx .

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5

7 The two sequences a0, a1, a2, . . . and b0, b1, b2, . . . have general terms

an = !n + µn and bn = !n ! µn ,

respectively, where ! = 1 +"2 and µ = 1!

"2 .

(i) Show thatn!

r=0

br = !"2 +

1"2an+1 , and give a corresponding result for

n!

r=0

ar .

(ii) Show that, if n is odd,2n!

m=0

"m!

r=0

ar

#= 1

2b2n+1 ,

and give a corresponding result when n is even.

(iii) Show that, if n is even, "n!

r=0

ar

#2

!n!

r=0

a2r+1 = 2 ,

and give a corresponding result when n is odd.

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6

8 The end A of an inextensible string AB of length ! is attached to a point on the circumferenceof a fixed circle of unit radius and centre O. Initially the string is straight and tangent to thecircle. The string is then wrapped round the circle until the end B comes into contact withthe circle. The string remains taut during the motion, so that a section of the string is incontact with the circumference and the remaining section is straight.

Taking O to be the origin of cartesian coordinates with A at (!1, 0) and B initially at (!1,!),show that the curve described by B is given parametrically by

x = cos t+ t sin t , y = sin t! t cos t ,

where t is the angle shown in the diagram.

A

By

xt

O

Find the value, t0, of t for which x takes its maximum value on the curve, and sketch thecurve.

Use the area integral

!ydx

dtdt to find the area between the curve and the x axis for ! ! t ! t0.

Find the area swept out by the string (that is, the area between the curve described by B andthe semicircle shown in the diagram).

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