XMA1005-03 Calculators: Only Casio fx83wa Calculators To be Provided: Formula Tables. TWO HOURS UNIVERSITY OF MANCHESTER INSTITUTE OF SCIENCE AND TECHNOLOGY SESSION 2002/2003 FIRST SEMESTER Mathematics 1R1

Answer THREE questions

1. (a) Find the roots of the quadratic equation x2 + 5x 14 = 0. [2 marks] (b) Find the asymptotes of the rational function R(x) = x3 + 5x2 14x + 5 x2 + 5x 14 [3 marks] (c) Solve the equation 2x+1 + 23x = 17. [5 marks] (d) (i) Find a positive number R and an angle such that 2 cos x + 12 sin x = R sin(x + ) [5 marks] (ii) Hence or otherwise nd all values of x which satisfy the equation 2 cos x + 12 sin x = 4. [5 marks]

XMA1005-03

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2. (a) Dierentiate the following functions with respect to x. (i) y = x8 ln x [2 marks] (ii) y= sin x x2 + 1 [2 marks] (iii) y = sin(x2 1) [2 marks]
(b) Find the equation of the tangent to the graph of the function y = sin(x2 1) at x = 1. (c) For the polynomial function y = x3 3x (i) nd the turning points, and determine whether they are minima or maxima, [5 marks] (ii) nd the points of intersection with the x-axis, [3 marks] (iii) sketch the graph in the range 2 x 2. [3 marks] [3 marks]

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3. (a) Find a function whose derivative with respect to x is (a) Integrate with respect to x (i) (ii) (iii)

1. x2 [2 marks]

sin(2x + 1) dx

[2 marks]

(x1/2 1)1/2 dx (hint: substitute u = (x1/2 1)) [4 marks] x1/2

x cos(2x) dx

[5 marks]

(b) Write

1 in partial fractions, and use this to nd the area x(x + 1) under the curve whose equation is y= 1 x(x + 1)
for x in the range 1 x 3. Give your answer to 4 decimal places. [7 marks]

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4. (a) Let A, B, C have coordinates (2, 1, 2), (1, 0, 2), (1, 1, 3) respectively. Find (i) the position vector of A. (ii) the vector u = AB. (iii) the vector v = AC. (iv) the length of AC. (v) the vector and scalar products of u and v. (b) The vector equation of a line L1 is r = 1 + 2 and the Cartesian equation of a line L2 is x3 y+1 z1 = =. 3 Find the Cartesian equation of L1 and the vector equation of the line L2. Find the point where these two lines meet, and decide whether or not they are orthogonal. [12 marks]

[8 marks]

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Approved Scientific Calculators for Examinations
CANON F720 CANON F720i CASIO fx-82AU CASIO fx-82AU PLUS CASIO fx-82MS CASIO fx-82NASER CASIO fx-82TL CASIO fx-83WA CASIO fx-85MS CASIO fx-85WA CASIO fx-95MS CASIO fx-100AU CASIO fx-100S CASIO fx-115WA General requirements
CASIO fx-350MS CASIO fx-350TL CASIO fx-570AD CASIO fx-580 CASIO fx-901 CASIO fx-911W CASIO fx-911Z CASIO fx-992S CITIZEN SR-135 CITIZEN SR-260 HEWLETT-PACKARD HP-8S HEWLETT-PACKARD HP-10S OFFICE ONE 720 OFFICE ONE 3000
RSB FB 350MS SHARP EL-509R SHARP EL-509V SHARP EL-520VA SHARP EL-520WG SHARP EL-W531HA SHARP EL-531LH SHARP EL-531RH SHARP EL-531VH SHARP EL-531WH TEXET Albert (TX 890) TEXET Albert (TX 895) TEXET Albert (TX 842)
All calculators must be silent (except in approved cases for blind candidates), hand-held, and battery- or solar-powered. Instruction booklets or cards (eg reference cards) on the operation of calculators are not permitted in the examination centre. Candidates are expected to familiarise themselves with the calculators operation beforehand. Features of approved calculators In addition to the features of a basic (four operation) calculator, a scientific calculator may include some or all of the following: fraction keys (for fraction arithmetic) a percentage key a key memory access keys an EXP key and a sign change (+/) key 2 square (x ) and square root () keys logarithm and exponential keys (base 10 and base e) x y a power key (a , x or similar) trigonometrical function keys with an INVERSE key for the inverse functions a capacity to work in both degree and radian mode a reciprocal key (1/x) n n permutation and/or combination keys ( Pr , Cr) cube and/or cube root keys parentheses keys statistical operations such as mean and standard deviation metric or currency conversion.
Features that are not permitted Calculators with the following features will not be approved: programmable (any calculator that can have a sequence of operations stored and then executed automatically is considered programmable and hence not allowed) capable of storing alphanumeric data input by a user (this does not exclude calculators with memories that are used to store intermediate numerical results obtained during calculations and required later) capable of storing, manipulating or graphing functions entered in symbolic form (this includes calculators with a graphic display capacity) capable of performing hard-wired numerical routines for operations such as differentiation and definite integration, and the solution of equations capable of performing hard-wired symbolic manipulations such as addition of algebraic expressions, binomial expansion and symbolic differentiation.

Scientific Calculator