1980 A-Level Pure Mathematics Paper I - M★th revise

1980 A-Level Pure Mathematics Paper I

x1

y1

1. Let V be the set of all 31 real matrices. For any x x 2 , y y2 in V and any real number ,

x

3

y

3

x1 y1

x1

we define x y x 2 y2 , x x 2 . It is known that under this addition and scalar multiplication ,

x3

y3

x

3

0 V forms a real vector space with zero vector 0 0.

0

(a) For a given 3 3 real matrix A , let E {xV : Ax 0}.

(i) Show that E forms a vector subspace of V. (ii) For b in V , suppose we have p in V such that Ap b . Show that , for any y in V ,

Ay b if and only if y p x for somex in E .

x y z 0 (b) (i) Find all solutions to 10x 5y 4z 0

5x 5y z 0

(ii) Suppose x 1 , y 4 , z 2 is a solution to the system of equations 23

x y z b1 10x 5y 4z b2 . 5x 5y z b3.

Find all solutions to the system.

(1980)

2. Let F denote the set of all positive-valued continuous functions on the set R of all real numbers. For any f , gF, define f * g by (f * g)(x) f (x)g(x) x R . It is known that F forms a group under the

operation *. The identity I of this group and the inverse g of f F are given respectively by

I(x) 1 x R , g(x) 1 x R. f (x)

Define a relation ~ in F as follows: For f , gF, f ~ g if there are polynomials p, q in F such that p * f = q * g.

(a) Show that ~ is an equivalence relation on F. (b) Let f / ~ be the equivalence class of f with respect to ~, and let F/~ be the quotient set consisting

of all these equivalence classes. For any f / ~, g / ~ F / ~,define f / ~ g / ~ to be (f * g) / ~.

(i) Show that is well defined on F/~, i.e., if f / ~ f1 / ~ and g / ~ g1 / ~, then f / ~ g / ~ f1 / ~ g1 / ~ .

(ii) Show that F/~ forms a group under .

(1980)

3. (a) If x > 0 and p is a positive integer , show that x p1 1 x p 1 , and that the equality holds only

p 1

p

if x = 1 .

n

(b) Let x1, x2,..., xn be positive numbers and xi n .

i1

(i) Show that , for any positive integer m ,

n

xim

n.

i1

(ii) If

n

xim

n

for some integer m

greater than one , show that

x1 x 2 ... x n 1.

i1

(c) Using (b) , or otherwise, show that , for any positive numbers y1, y2 ,. .., yn , and positive integer m ,

y1m y2m ...yn m y1 y2 ... yn m and that the equality holds only when m =1 or

n

n

y1 y2 ... yn .

(1980)

4. (a) The terms of a sequence y1, y2 , y3 ,... satisfy the relation yk Ayk1 B (k 2) where A , B

are constants independent of k and A 1 . Guess an expression for yk (k 2) in terms of y1 , A , B and k and prove it.

(b) The terms of a sequence x 0 , x1 , x 2 ,... satisfy the relation x k (a b)x k1 abx k2 (k 2),

where a , b are non-zero constants independent of k and a b .

(i) Express xk axk1 (k 2) in terms of (x1 ax0 ), b and k .

(ii) Using (a) or otherwise , express x k ( k 2) in terms of x0 ,x1,a,b and k.

(c) If the terms of the sequence x0 , x1, x 2 ,...

satisfy the relation

xk

1 3

x k1

2 3

x k2

(k 2) ,

express lim xk in terms of x0 and x1.

k

(1980)

5. (a) (i) Let 3 1 and 1 . Show that the expression x3 3uvx (u3 v3) 0 can be factorized as

(x u v) (x u 2v) (x 2u v)

(ii) Find a solution to the following system of equations

u3 v3 6 uv 2

Hence , or otherwise, find the roots of the equations x3 6x 6 0

(b) Given an equation x3 px q 0............ (*)

(i) Show that , if (*) has a multiple root , then 27q2 4p3 0

(ii) Using the method indicated in (a) (ii) , or otherwise , show that , if 27q2 4p3 0 , then (*)

has a multiple root.

(1980)

6. Let a , b be real numbers such that a < b and let m , n be positive integers.

mn

(a) If for all real numbers x , u , [(1 u) x (au b)]mn Ak (x)u k ........... (*)

k0

show that Ak (x) Cmk n (x a)k (x b)mnk for k = 0 , 1 , ... , m + n ,

where

C

m n k

is the coefficient of t k

in the expansion of

(1 t)mn .

(b) By integrating both sides of (*) with respect to x , or otherwise , calculate

b

(x

a)

m

(x

b)

n

dx

.

a

(c)By differentiating both sides of (*) with respect to x , or otherwise , find

dr dx r

{(x

a)m (x

b) n }

at

x = a , where r is a positive integer.

(1980)

7. Let C be the set of complex numbers. A function f : C C is said to be an isometry if it preserves

distance, that is , if f (z1) f (z2) z1 z2 for all z1, z2 C. (a) If f is an isometry, show that g(z) f (z) f (0) is an isometry satisfying g(1) = 1 and g(0) = 0 .

f (1) f (0) (b) If g is an isometry satisfying g(1) = 1 , g(0) = 0 , show that

(i) the real parts of g(z) and z are equal for all z C , (ii) g(i) = i or -i . (c) If g is an isometry satisfying g(1) = 1 , g(0) = 0 and g(i) = i (respectively -i) , show that g(z)=z (respectively z ) for all z C .

(d) Show that any isometry f has the form f(z) = az + b or f(z) = az b with a and b constant

and a 1 .

(1980)

8. N balls are distributed randomly among n cells. Each of the nN possible distributions has probability

nN .

(a) (i) Calculate the probability Pk that a given cell contains exactly k balls.

(ii) Show that the most probable number k 0 satisfies the inequality

N n 1 n

k0

N 1 n

.

N

(iii) Compute the mean number kPk of balls in a given cell and show that it can differ from

k0

k 0 by at most one.

(b) Let A(N, n) be the number of distributions leaving none of the cells empty. Show that

A(N, n1)

N

Ck N A(N k,

n),

where

Ck N is the coefficient of

t k in the expansion of

k 1

(1 t)N . Hence show by mathematical induction (on n), or otherwise, that

n

A(N, n)

(1)i

C

n j

(n

j) N .

j 0

(1980)

1980 A-Level Pure Mathematics Paper II

1. Let P(t) = (x(t), y(t)) be a point on the unit circle with parametric equations

1 t2

x(t)

, y(t)

2t ,

Q

be the point

(a, 0), 0 < a < 1.

An arbitrary line of slope m passing

1t2

1 t2

through Q cuts the circle at the points R = P(t1) and S P(t 2 ) . Let T be the point where RO

meets the line through Q parallel to SO, where O is the origin.

(a) Show that

t1t

2

a a

1, 1

t1

t2

m(a

2 1)

.

(b) Express the coordinates of T in terms of a, t1, t 2

(c)Verify that the locus of T is an ellipse with equation

C is a constant. What is C? (1980)

(1 a 2 )(x a )2 y2 = C, 2

where

2. In a 3-dimensional space with a Cartesian coordinate system , two lines l1 and l 2 are given by the pairs of equations :

l1 :

x 2y 3z 3 0 x 2y 2z 4 0

(a) Let P be the plane

,

l2:

x y z 1 0 2x 3y 5z 2 0

.

( x + 2y + 3z - 3) + (x + 2y + 2z - 4) = 0 and

Q be the

plane (x + y + z ? 1) + (2x + 3y + 5z - 2) = 0 .

Show that P is parallel to Q if and only if there exists m0 such that

1 m 1 2m 0 2 m 2 3m 0 ........................... (*) 3 m 2 5m 1 0

(b) Find the value of m for which there are numbers and satisfying (*) in (a) . Hence

find the equations of the two parallel planes M1 and M2 containing l1 and l 2 respectively.

(c) Find the equation of the plane N containing l1 and perpendicular to M 2 . (d) Let l1 ' be the projection of l1 on M2 ( i.e. l1 ' is the intersection of N and

Find its equation. (1980)

M2 ).

n

3. (a) Let f(z) = a k z k be an n th degree polynomial in the complex variable z with real

k0

coefficients. Show that

(i)

f (z) 2 n

n

rk jak a j cos(k j),

where z = r ( cos i sin) ,

k 0j0

(ii)

1 2

2 0

f (cos i sin)

2

d

n

ak2.

k0

(b) If

C

n k

is

the

coefficient

of

t k

in the binomial expansion of

(1 t)n , show that

 n

k0

Cnk

2

2n

(1 cos)n d 22n 1

0

2 (cost)2n dt.

0

(1980)

4. (a) A right circular cone is inscribed in a sphere of radius a as shown in the figure. Determine the height of the cone if it is to have maximum volume.

(b) Two points A and B lie on the circumference of a circle with centre O and radius a such

that AOB 2 . X is a point on AO produced with OX = x; P is a point on arc AB 3

with

AOP = .

For each

x > 0, let

g(x)

2 3

asin d,

where

r(x, )

is the distance

0 r(x,)

between P and X.

(i) Show that

g(x)

3a

.

x

a

(a

2

x

2

ax)

1 2

(ii) Prove that on (0, ), 0 g(x) 3 . 2

(1980)

5. (a) Let f and g be two continuous functions defined on the real line R and let x0 R ,

show that if f(x) = g(x) for all x R\{x0}, then f (x0 ) g(x0 ).

(b) If a real polynomial p(x) can be written as p(x) (x x0 )m q(x) for some positive integer

m and polynomial q(x) with q(x0 ) 0 , show that the expression is unique , that is , if k is

a positive integer and

h(x) is a polynomial with h(x0 ) 0 such that

p(x) (x x0 )k h(x) , then m = k and q(x) = h(x) for all x R.

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