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A S S O C I A T I O N O F C A N A D A L A N D S S U R V E Y O R S - B O A R D O F E X A M I N E R S W E S T E R N C A N A D I A N B O A R D O F E X A M I N E R S F O R L A N D S U R V E Y O R S A T L A N T I C P R O V I N C E S B O A R D O F E X A M I N E

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ASSOCIATION OF CANADA LANDS SURVEYORS - BOARD OF EXAMINERS WESTERN CANADIAN BOARD OF EXAMINERS FOR LAND SURVEYORS ATLANTIC PROVINCES BOARD OF EXAMINERS FOR LAND SURVEYORS ________________________________________________________________________

SCHEDULE I / ITEM 1 MATHEMATICS

February 2001 (1990 Regulations) (Closed Book) Time: 3 hours

Marks Note: This examination consists of 10 questions on 2 pages.

1.a) What is the limit of f(x) = x e-x as x goes to infinity? 5 b) What is the limit of g(x) = x-1 sin x as x goes to zero? 5 2.a) Differentiate f(x) = esin x with respect to x. 5 b) Differentiate g(x) = xsin x with respect to x. 5 3.a) Give the first three terms in the Taylor series expansion in x for (1 + 3x)-1. 5 b) What is an expression for the remainder after the previous three terms in

the series expansion for (1 + 3x)-1? 5

4.a) Set up the integral to compute the area inside an ellipse with semi-major

axis a and semi-minor axis b. 5

b) Set up the integral to compute the arclength around an ellipse with semi-

major axis a and semi-minor axis b. 5

5.a) Given three points in two-dimensional Cartesian space, how can one

check for collinearity? 5

b) Given four points in three-dimensional Cartesian space, how can one

check for coplanarity? 5

6.a) Given a small square matrix A of order 3 with elements aij = i + j for

i = 1, 2, 3 and j = 1, 2, 3, what is the principal property of A? 5 b) What is the determinant of the matrix A of order 3 with elements aij = i + j

for i = 1, 2, 3 and j = 1, 2, 3? 5

7.a) For a system of linear algebraic equations with zero right-hand side, what can be done for a solution? Illustrate the situation with three simple equations.

5 b) For a system of linear algebraic equations with nonzero right-hand side

which are not independent, what can be done for a solution? Illustrate the situation with three simple equations.

5 8.a) What is the method of separation of variables for solving differential

equations? Illustrate with a simple differential equation. 5 b) What is the general solution of d2x/dt2 - ax = 0 for arbitrary scalar a? 5

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BOARD OF EXAMINERS FOR CANADA LANDS SURVEYORS

WESTERN CANADIAN BOARD OF EXAMINERS FOR LAND SURVEYORS ATLANTIC PROVINCES BOARD OF EXAMINERS FOR LAND SURVEYORS ________________________________________________________________________

SCHEDULE I / ITEM 1 MATHEMATICS (continued) February 2001 9.a) Given two vectors a = (1, 5, 7)T and b = (3, 2, 1)T , where the superscript

T means transpose, what is a ⋅ b ? 5

b) Given two vectors c = (2, 3, 7)T and d = (3, 1, 2)T , where the superscript

T means transpose, what is c × d ? 5

10.a) Given two linear equations: x + 2y = 8 and 2x + 3y = 13, solve them for x and y using Cramer's rule.

5 b) Given the same two linear equations: x + 2y = 8 and 2x + 3y = 13, solve

them for x and y using Gaussian elimination.

5 Total Marks: 100

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