Official SQA Past Papers: Advanced Higher Maths 2005
(a) Givenf(x) = x3 tan2x, where 0 < x < ~,
obtainj'(x).
(b) _ For y = i~~2, where x oj:. -1, determine ~~ in simplified
form.Given the equation 2y2 - 2xy - 4y + x2 = 0 of a curve, obtain
the x-coordinate ofeach point at which the curve has a horizontal
tangent.
)
Write down the Maclaurin expansion of eX as far as the term in x
4.
Deduce the Maclaurin expansion of ex2 as far as the term in
x4
Hence, orothe~wise, find the Maclaurin expansion of eX + x2 as
far as the termin x4
4. The sum, S(n), of the first n terms of a sequence, U1, U2,
u3, is given byS(n) =8n - n2,n ~ 1.
Calculate the values of U1, U2, u3 and state what type of
sequence it is.
Obtain a formula for Un in terms of n, simplifying your
answer.
f3 XUse the substitution U = 1 + x to evaluate 0 -V1 + x dx.
x + y + 2z 12x + A.y + z 0
3x + 3y ,.: 9z 5. 4Explain what happens when A. = 2. 2
Given the matrix A = (. ~ ~ ~) show that A2 + A = kI for some-1
-2 -3'
constant k, where I is the 3 X 3 unit matrix.
Obtain the values of p and q fot which A-1 = pA + qI.
The equations of two planes are x - 4y + 2z = 1 and x - y - z =
-5. By lettingz = t, or otherwise, obtain parametric equations for
the line of intersection of theplanes.
Show that this line lies in the plane with equation
x + 2y - 4z = -11.
-,-llarks4
n 1 1 1~ r(r+ 1)(r+2) = 4 -2(n+ l)(n+ 2)
n 1
~ r(r+l)(r+2)
x3The diagram shows part of the graph of y = --, x::j:. 2.
x-2
3Find the coordinates of the stationary points of the graph of y
= _x_.
x-2
(c) Write down the coordinates of the stationary points of the
graph ofx3y=--+1. 2
x-2
12. Let z = cose + i sine.(a) Use the binomial expansion to
express Z4 in the form u + iv, where u and v
are expressions involving !'line and cos e . 3(b) Use de
Moivre's theorem to write down a second expression for z4. 1
(c) Using the results of (a) and (b), show that
cos4e 2 1t 1t--2- = P cos2e + q sec e + r, where - - < e 0,
show thatf (x) = ---,k-.Jg(x)
stating the value of k.
Hence, or otherwise, find fn dx.
f1/2 -1sin x dx.o
