X100/701NATIONALQUALIFICATIONS2006

FRIDAY, 19 MAY1.00 PM - 4.00 PM MATHEMA TICS

ADVANCED HIGHER

1. Calculators may be used in this paper.

2. Candidates should answer all questions.

3. Full credit will be given only where the solution contains appropriate working.

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QUALIFICATIONSAUTHORITY

Official SQA Past Papers: Advanced Higher Maths 2006

1. Calculate the inverse of the matrix ( 2 x).-1 3

Differentiate, simplifying your answers:

(a) 2 tan -1 .J1 + x, where x > -1;1+lnx---, where x > O.3x

~ . 1Express the complex number z = -i+--. in the form z = x + iy, stating the

1-zvalues of x and y.

Find the modulus and argument of z and plot z and z on an Argand diagram.

Given xy - x = 4, use implicit differentiation to obtain dy in terms of x and y.dx

H b d2y. f dence a taln --2 m terms 0 x an y.

dx

F dS 12x3 -6x d

In 4 2 X.X -x +1

For all natural numbers n, prove whether the following results are true or false.

(a) n3 - n is always divisible by 6.

(b) n3 + n + 5 is always prime.

8. Solve the differential equation

d2y dy-+2-+2y=0dx2 dx '

given that when x = 0, y = 0 and dy = 2.dx

2x '- y + 2z = 1X + Y -'- 2z = 2x - 2y + 4z = -1.

. - . -10. 1Jneamountx micrognims of an impurity removed per kg of a substance by a

chemical process depends on the temperature Toe as follows:

At what temperature in the given range should the process be carried out toremove as mtlch:impurityper kg as possible? 4

Showthatl+cot2B= cosec28, where 0< 8 < ~.

By expressing y =cor'-lx as x = cot y, obtain dy in terms 6f x.dx

The diagram shawso part of the graph of a function! which satisfies the followingconditions:

(i) ! is an even function;(ii) two of the asymptotes of the graph'y =l(x).are.y = x and x = 1.

Copy the diagrain and cbmplete the graph. Write down equations for the other. two asymptotes. 3

/

13. The square matrices A and Bare such that AB= BA. Prove by induction thatAnB = BAn for all integers n ~ 1. 5

Determine whetherf(x) = x2 sinx is odd, even or neither. Justify youranswer.

Use integration by parts to find Jx2 sin x dx.. n n

Hence find the area bounded by y = x2 sin x, the lines x = -4' x =4 andthe x-axis.

15. Obtain.an equation for the plane passing through the point P (1, 1, 0) which isperpendicular to the line L given by

x+1 y-2 z2 1 -1

Find the coordinates of the point Q where the plane and L intersect. 4

Hence, or otherwise; obtain the shortest distance from P to L and explain whythis is the shortest distance. 2, 1

x(x +1) x(x +1)2 d x(x +1)3 h 2, 2 an 3 ' were x < .

(x-2) (x-2) (x-2)

(a) Obtain expressions for the common ratio and the nth term of the sequence. 3

(b) Find an expression for the sum of the first.n terms of the sequence. 3

(c) Obtain the range of values of x for which the sequence has a sum to infinityand find an expression for the sum to infinity. 4

(b) Bywriting cos4x = cos X cos3x andusiJdgintegration by parts, show that

17f:/4 4 1 17f:/4 2 2.. cos x dx = - + 3 sin x cos x dx.o 4 0r7f:/4. . 2 n +2

(c) Show that Jo CQ,~X dx =-g-.

(d) Hence, using the above results, show that

17f:/4 4. . 3n + 8cos x dx = ..o 32