Advanced Higher Maths Exam 2004

7/23/2019 Advanced Higher Maths Exam 2004

1/4

X100/701

NATIONAL

QUALIFICATIONS

2004

FRIDAY, 21 MAY1.00 PM - 4.00 PM

MATHEMATICSADVANCED 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.

\y(

SCOTTISHr\

QUALIFICATIONSAUTHORITY


2/4

Official SQA Past Papers: Advanced Higher Maths 2004

(a) Givenf(x) =coszxetanx ,- ~ < x < ~ ,obtainf'(x) and evaluatef'(~).

tan-12x(b) Differentiateg(x) = 1+4x2

Use parametric c!ifferentiation to find tin terms of e .1r

Find the equation of the tangent to the curve at the point where e ="4 .

Given z = 1 + 2i, express zZ(z + 3) in the form a + ib.

, Hence, or otherwise, verify that 1 +2i ISa root of the equation

z3 + 3zz - Sz + 25 =O.

Obtain the other roots of this equation.

1Express ---- in partial fractions.

x2-x-6

11 1

Evaluate' 2 dx.O x -x-6

Write down the 2 x 2 matrix Mt associated with an anti-clockwise rotation of

~ radians about the origin.

Write down the matrix Mzassociated with reflection in the x-axis.

Evaluate Mz Mj and describe geometrically the effect of the transformationrepresented byMz Mt.

Obtain the first three non-zero terms III the Maclaurin expansIon of

f( x ) = eX sinx .

8. Use the Euclidean alg~rithm to show that (231, 17) =1 where (a, b) denotes the

highest common factor of aandb.

Hence find integers x andy such that 231x + .17y = = 1.

9. 4 Use the substitution x = (u - 1)zto obtainJ ~ dx.. (1+ xl .


3/4

10. Determine whether the functionf(x) = = x4 sin2x is ~ ew:a . r

Justify your answer. '

11. A solid is formed by rotating the curve y = = e-2x between x = and x =1 through3600 about the x-axis. Calculate the volume of the solid that is formed. ...

12. Prove by induction that : : 1 1 (xeX) = (x + n)eX for all integers n ~ 1. 5

x-313. The functionf is defined byf(x) = --2 'x 7 : - -2,and the diagram shows part of

x+its graph.

( a )

(b)

(c)

( d )

Obtain algebraically the asymptotes of the graph off.

Prove thatf has no stationary values.

Does the graph of fhave any points of inflexion? Justify your answer.

Sketch the graph of the inverse function,1-1. State the asymptotes and

domain off-1

Find an equation of the plane 1 l'1 containing the points A(1, 0, 3),

B(O, 2, -1) andC(1, 1,0).

Calculate the size of the acute angle between 1l'1 and the plane 1l'2 with

,equation x + y - z = = 0. 3

(b ) Find the point of intersection of plafle ~ and the line

x-ll

4

y-15

5

z -12

2


4/4

16. (a)

(b)

Official SQA Past Papers: Advanced Higher Maths 2004

A mathematical biologist believes that the differential equation

dy 4x -. - - 3y =x models a process. Find the general solution of the

dx

Given that y = 2 when x = 1, find the particular solution, expressing y in

~~~~ 2

(b) The biologist subsequently decides that a better model is given by the

. dy 3 4equatlOn y_. - - x =x .

dx

Giventhaty = 2 when x = 1, obtain y in terms of x.

A geometric sequence of positive terms has first term 2, and the sum of

the first three terms is 266. Calculate the common ratio.

An arithmetic sequence, A, has first term a and common difference 2,

anda geometric sequence, B, has first term a and common ratio 2. The

first four terms of each sequence have the same sum. Obtain the value

of a.

Obtain the smallest value of n such that the sum to n terms for sequence

B is more than twice the sum to n terms for sequenceA. 2

Documents

Advanced Higher Maths Exam 2004