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480
UNIVERSITY OF LONDON
General Certificate Of Education Examination
JUNE 1973 ADVANCED LEVEL Mathematics (Alternative Syllabus)
MATHEMATICS 1
Three hours
Answer EIGHT questions, of which at least ONE must be chosen from each of the sections A, B and C.
All questions carry equal marks.
Mathematical formulae and tables are provided. Statistical formulae and tables are included with the formulae.
Section A
Answer at least ONE question from this section.
1. Explain what is meant by an odd function. If the function f(x) is defined for all real values of x, prove that f(x) f((x) is an odd function.
The function g(x) is defined as follows:
g(x) = EMBED Equation.3
(a) In two separate diagrams sketch the graphs of g(x) and g((x).
(b) Sketch also the graph of g(x) g((x) and find the set of values of x for which
g(x) g((x) = x + 1/x.
2. An operation * is defined on the set ! of real numbers by
x * y = 3((x3 + y3)
for all x, y ( !.
(a) Is the system (!, *) closed with respect to the operation * ?
(b) Is the operation * commutative and/or associative?
(c) Does the system (!, *) possess an identity element?
(d) Is the system a group?
Give reasons for your answers, using counter-examples if appropriate.
3. The positions of two points A and B in space are defined by the vectors a and b (relative to a given origin O). If C is the point on AB such that AC : CB = ( : ( and C is between A and B, show that its position vector c is given by
(( + () c = (a + (b.
The vertices A, B, C, D of a tetrahedron have position vectors a, b, c, d respectively. Write down
(a) the position vectors of the mid-points P and Q of the edges AB and CD,
(b) the position vector of the mid-point of PQ.
Hence show that the three lines which join the mid-points of pairs of opposite edges of the tetrahedron share a common mid-point.
4. (i) Find, in degrees and minutes, the general solution of the equation
4 cos x + 2 sin x = (5.
(ii) Sketch the curve with parametric equations
x = 1 + 2 cos (, y = 2 + sin (
where 0 ( ( ( 2(.
5. (i) Express
EMBED Equation.3
in the form x + iy, where x and y are real. Find the modulus and argument of p and hence write down the argument of p2.
(ii) State de Moivres theorem for a positive integral index n. Deduce that, if z = cos ( + i sin (, then
z + EMBED Equation.3 = 2 cos ( and zn + EMBED Equation.3 = 2 cos n( .
Express cos6 ( in the form EMBED Equation.3 , where the coefficients ar are independent of (.
6. (i) Find the general solution of the differential equation
EMBED Equation.3 + x sin x cos2 y = 0.
(ii) The differential equation of the motion of a particle moving in a straight line is
EMBED Equation.3 + 4x = 0 ,
where x is the displacement of the particle from a fixed point of the line at time t. If x = 3 and EMBED Equation.3 = 8 when t = EMBED Equation.3 , find the position and velocity of the particle when t = (.
Section B
Answer at least ONE question from this section.
7. If a graph is drawn, plotting the speed of a particle against time, what physical interpretation can be attached to (a) the slope of the graph, (b) the area under the graph?
A train starts from rest at time t = 0 and proceeds with constant acceleration until a speed V is attained at time t = T. The train then carries on at speed V until time t = 5T, when the brakes are applied, producing a constant deceleration; the train comes to rest at time t = 6T. If the average speed of the train was 55 km/h, calculate the maximum speed. Find also, as a fraction of the total distance covered, the distance for which this maximum speed was maintained.
8. In this question, distances are measured in nautical miles and speeds in nautical miles per hour.
A motor boat sets out at 2 p.m. from a point with position vector (4i ( 5j relative to a marker buoy (where i and j are two fixed perpendicular unit vectors) and travels at a steady speed of magnitude (41 in a straight line to intercept a ship S. The ship S maintains a steady velocity vector i + 4j and at 3 p.m. is at a position 3i ( j relative to the buoy. Find the position vector of the ship at 2 p.m., the velocity vector of the motor boat, and the time of interception.
9. A particle is moving in a horizontal circle and is kept in its path by a string tied to a point at a height h above the centre of the circle. Find the period of rotation.
If the tension in the string is three times the weight of the particle, find the length of the string in terms of h.
Section C
Answer at least ONE question from this section.
10. (i) Prove that, if ( is sufficiently small,
2( (2 + cos 2( ) ( 3 sin 2(
is approximately equal to 8( 5/15.
(ii) Write a flow diagram to list the prime numbers between 1000 and 2000.
11. Two players X and Y engage in a series of 12 games. Of these, X wins 5 games, Y wins 4, and 3 are drawn. They then agree to play a second series of 4 games. If the probabilities of X winning, of Y winning, or of a game being drawn are the same in the second series as in the first series, estimate the probability that
(a) X wins all four games,
(b) exactly two games are drawn,
(c) X and Y win alternate games,
(d) X and Y win two games each,
(e) Y wins at least one game.
12. The heights of 100 students, taken from a population of 10 000 students, are given in the following table:
Height in inches65666768697071727374No. of students13121538188311
Find, to the nearest tenth of an inch, the mean height and the standard deviation of these heights.
If the distribution of heights is regarded as normal, and if the above sample was random, how many students from the total population would be expected to have heights
(a) between 67 and 68 inches,
(b) over 72 inches?
Mathematics 2
480
UNIVERSITY OF LONDON
General Certificate Of Education Examination
JUNE 1973 ADVANCED LEVEL
Mathematics (Alternative Syllabus)
MATHEMATICS 2
Three hours
Answer EIGHT questions, of which at least ONE must be chosen from each of the sections A, B and C.
All questions carry equal marks.
Mathematical formulae and tables are provided. Statistical formulae and tables are included with the formulae.
Section A
Answer at least ONE question from this section.
1. Write down a set of axioms that will define a group.
(a) A set S consists of all the 2 ( 2 matrices whose elements consist of two 0s and two ls (arranged in any way). Decide whether S is a group with respect to matrix multiplication and justify your answer.
(b) If H is a finite group of isometries in the Euclidean plane, prove that H contains no translations.
2. (i) Prove, by induction or otherwise, that the sum of the cubes of the first n natural numbers is EMBED Equation.3 n2(n + 1)2.
(ii) If An = 23n + 1 + 3(52n + 1), show that
An + 1 = 25An ( 17(23n + 1).
Prove by induction that, for all positive integers n, An is divisible by 17.
3. The point P1 has position vector p1 = EMBED Equation.3 relative to rectangular axes in the plane. The foot of the perpendicular from P1 onto the line y = mx is the point P2 with position vector p2 = EMBED Equation.3 . Express x2 and y2 in terms of x1, y1 and m.
Find the matrix A, with elements expressed in terms of m, for which p2 = Ap1. Show that
(a) A is symmetric and singular,
(b) A2 = A.
4. (i) By means of the remainder theorem, or otherwise, prove that x + y + z is a factor of x3+y3+z3 ( 3xyz and find the other factor.
(ii) The sum to infinity of the geometric series
a + ar + ar2 + ...
is 10. The sum to infinity of the series formed by the squares of the terms is 100/9. Show that r = 4/5 and find a. Find the sum to infinity of the series formed by the cubes of the terms.
5. (i) Differentiate the following with respect to x
(a) EMBED Equation.3 , (b) EMBED Equation.3 ,
simplifying your answers where possible.
(ii) If y is given implicitly in terms of x by
x2 + 3xy + 3y2 = 3,
obtain an expression for EMBED Equation.3 in terms of x and y.
Find the coordinates of the two points of the curve x2 + 3xy + 3y2 = 3 at which the tangents are parallel to the y-axis.
6. (i) Express
f(x) = EMBED Equation.3
in partial fractions and hence find EMBED Equation.3 .
(ii) Evaluate EMBED Equation.3 .
Section B
Answer at least ONE question from this section.
7. Two particles of equal mass m on a smooth horizontal table are connected by an elastic string of natural length a in which a tension mg would produce an extension a. The particles are held at rest at a distance 3a apart. If the particles are released simultaneously, show that they each start to move with simple harmonic motion of period EMBED Equation.3 and find the time that elapses before they collide.
[Assume that the elastic string has no effect on the motion after the instant when the particles are at a distance a apart.]
8. Show that the path of a projectile whose initial velocity is V at an angle of elevation ( has the equation
y = x tan ( ( EMBED Equation.3 (tan2 ( + 1)
referred to horizontal and vertical axes through the point of projection.
A boy kicks a ball from ground level with a speed of 10.5 m/s so as just to clear a pole 1 m high which is at a horizontal distance of 6 m from the boy. Find the two possible angles of elevation at projection. Hence find the greater of the two possible horizontal distances beyond the pole that the ball can reach on landing.
[Take g to be 9.8 m/s2.]
9. State a set of conditions for a system of coplanar forces to be in equilibrium.
XYZ is a triangle in which XY = 5, YZ = 6, ZX = 7 and T is the foot of the perpendicular from X on to YZ. A force of magnitude P acts along XT and equilibrium is maintained by forces Q and R, parallel to P, acting through Y and Z. Find the magnitudes of Q and R in terms of P.
The directions of P, Q and R are now rotated (about X, Y and Z respectively) through the same angle so as to be at right angles to XY (their magnitudes being unaltered). Prove that their resultant is then a couple and find its moment.
Section C
Answer at least ONE question from this section.
10. Describe Newtons method for obtaining successive approximations to a root of an equation f(x)= 0.
(a) Notwithstanding the fact that the root of f(x) ( 3(x = 0 is trivially x = 0, a student took a nonzero approximation to this root and began to find successive approximations using Newtons method. Investigate what happened and, by reference to a graph of y = 3(x, explain why this method was destined to failure.
(b) Show that the equation x3 + 6x ( 1 = 0 has just one real root and, by Newtons method or otherwise, find it correct to two decimal places.
11. Tabulate, to two places of decimals, the values of ((x3 +9) for integral values of x between 2 and8 inclusive.
Use (a) the trapezium rule and (b) Simpsons rule, with strips of unit width in each case, to calculate approximately the value of
EMBED Equation.3 ((x3+9) dx.
Sketch the graph of y=((x3+9) and, by use of your sketch, determine whether your approximation by the trapezium rule is greater or less than the true value of the integral.
12. (i) Find the probability of obtaining three or more sixes in five throws of a die.
(ii) A manufacturer of machine parts guarantees that a box of his parts will contain at most two defective items. A box holds 20 parts and experience has shown that his factory produces one per cent of defective items. Find an expression for the probability that a box of his parts will satisfy his guarantee.
PAGE
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TE&S 72/255 14/3/3/2260
1973 University of London
TE&S 72/255 14/3/3/2260
1973 University of London
PAGE 1
TE&S 72/256 14/3/3/2260
1973 University of London
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