decemberrevision - wordpress.com · part marks level calc. content answer u2 oc4 (a) 2 c cn s1 544...
TRANSCRIPT
Higher Mathematics
December Revision
1. (a) Given f (x) = x3 tan 2x , where 0 < x <π
4 , obtain f′(x) . 3
(b) For y =1+ x2
1+ x, where x 6= −1, determine dy
dxin its simplest form. 3
Part Marks Level Calc. Content Answer U1 OC2
(a) 3 C CN D4, D2 2005 Q1
(b) 3 C CN D4
2.[SQA] Differentiate the following functions with respect to x , simplifying your answerswhere possible.
(a) h(x) = sin(
x2)
cos(3x) . 3
(b) y =ln(x+ 3)
x+ 3, x > −3. 3
Part Marks Level Calc. Content Answer U1 OC2
(a) 3 C CN D4, D3, D6, D2 1999 SY1 Q3
(b) 3 C CN D5, D8
hsn.uk.net Page 1
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
3. Use the substitution x+ 2 = 2 tan θ to obtain∫
1
x2 + 4x+ 8dx . 5
Part Marks Level Calc. Content Answer U1 OC3
5 C CN I5 12 tan
−1( x2 + 1) + c 2002 A6
4. Use the substitution u = 1+ x to evaluate∫ 3
0
x√1+ x
dx . 5
Part Marks Level Calc. Content Answer U1 OC3
5 C CN I5 2005 Q5
5. A solid is formed by rotating the curve y = e−2x between x = 0 and x = 1through 360◦ about the x -axis. Calculate the volume of the solid that is formed. 5
Part Marks Level Calc. Content Answer U1 OC3
5 B CN I8 π
4 (1− 1e4
) 2004 A11
hsn.uk.net Page 2
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
6. A function f is defined by f (x) =x2 + 6x+ 12
x+ 2, x 6= −2.
(a) Express f (x) in the form ax+ b+b
x+ 2stating the values of a and b . 2
(b) Write down an equation for each of the two asymptotes. 2
(c) Show that f (x) has two stationary points.
Determine the coordinates and the nature of the stationary points. 4
(d) Sketch the graph of f . 1
(e) State the range of values of k such that the equation f (x) = k has no solution. 1
Part Marks Level Calc. Content Answer U1 OC4
(a) 2 C CN A7 a = 1, b = 4 2001 A8
(b) 2 C CN F9 x = −2, y = x+ 4
(c) 4 C CN F3 (0, 6) local min, (−4,−2)local max
(d) 1 C CN F10 sketch
(e) 1 C CN F1 −2 < k < 6
[No marking instructions available]
7. Use Gaussian elimination to solve the following system of equations
x + y + 3z = 22x + y + z = 23x + 2y + 5z = 5. 5
Part Marks Level Calc. Content Answer U1 OC5
5 C CN A10 x = 2, y = −3, z = 1 2002 A1
hsn.uk.net Page 3
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
8.[SQA] A car manufacturer is planning future production patterns. Based on estimates oftime, cost and labour, he obtains a set of three equations for the numbers x , y , z ofthree new types of car. These equations are
x + 2y + z = 602x + 3y + z = 853x + y + (λ + 2)z = 105,
where the integer λ is a parameter such that 0 < λ < 10.
(a) Use Gaussian elimination to find an expression for z in terms of λ . 5
(b) Given that z must be a positive integer, what are the possible values for z? 2
(c) Find the corresponding values of x and y for each value of z . 2
Part Marks Level Calc. Content Answer U1 OC5
(a) 5 A/B CN A10 2000 SY1 Q13
(b) 2 C CN CGD
(c) 2 C CN
hsn.uk.net Page 4
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
9. (a) Given f (x) = x(1+ x)10 , obtain f ′(x) and simplify your answer. 3
(b) Given y = 3x , use logarithmic differentiation to obtaindy
dxin terms of x . 3
Part Marks Level Calc. Content Answer U2 OC1
(a) 3 C CN D4 (1+ x)9(1+ 11x) 2003 A1
(b) 3 C CN D16 3x ln 3
•1 first summand•2 second summand•3 complete•4 strategy (e.g. take logs)•5 apply chain rule•6 complete
•1 (1+ x)10 + · · ·•2 · · · + x.10(1+ x)9
•3 (1+ 11x)(1+ x)9
•4 ln y = x ln 3
•5 1ydy
dx= ln 3
•6 dydx
= 3x ln 3
hsn.uk.net Page 5
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
10.[SQA] Differentiate the following with respect to x .
(a) y = x3e−x2, 2
(b) f (x) = tan−1(√x− 1
)
, x > 1, 2
(c) f (x) =x2
cos x, −π
2 < x <π
2 . 2
Part Marks Level Calc. Content Answer U2 OC1
(a) 2 C CN D4, D8, D2, D3 Add: D6 1997 SY1 Q1
(b) 2 C CN D13, D3
(c) 2 C CN D5, D2
hsn.uk.net Page 6
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
11. 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. 4
Part Marks Level Calc. Content Answer U2 OC1
4 C CN F3, D15 2005 Q2
12. A curve is defined by the parametric equations
x = t2 + t− 1, y = 2t2 − t+ 2
for all t . Show that the point A(−1,−5) lies on the curve and obtain an equationof the tangent to the curve at the point A. 6
Part Marks Level Calc. Content Answer U2 OC1
6 C CN D17, Higher y = 5x+ 10 2002 A3
hsn.uk.net Page 7
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
13. Express1
x2 − x− 6 in partial fractions. 2
Evaluate∫ 1
0
1
x2 − x− 6 dx . 4
Part Marks Level Calc. Content Answer U2 OC2
(1) 2 C CN A6 151x−3 − 1
51x+2 2004 A5
(2) 4 C CN I11 15 ln
49
hsn.uk.net Page 8
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
14.[SQA]
(a) By using the substitution u = 2 sin x , or otherwise, evaluate the definiteintegral
∫ π
6
0
cos x
1+ 4 sin2 xdx. 4
(b) Use integration by parts to find
∫
x2 ln x dx. 3
Part Marks Level Calc. Content Answer U2 OC2
(a) 4 C CN I5, I10 1996 SY1 Q8
(b) 3 C CN I12
hsn.uk.net Page 9
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
15. Functions x(t) and y(t) satisfy
dx
dt= −x2y, dy
dt= −xy2.
When t = 0, x = 1 and y = 2.
(a) Expressdy
dxin terms of x and y and hence obtain y as a function of x . 5
(b) Deduce thatdx
dt= −2x3 and obtain x as a function of t for t ≥ 0. 5
Part Marks Level Calc. Content Answer U2 OC2
(a) 5 C CN D17, DE1dydx = y
x , y = 2x 2002 A9
(b) 5 C CN DE1 x = 1√4t+1
hsn.uk.net Page 10
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
16. Define In =∫ 1
0xne−x dx for n ≥ 1.
(a) Use integration by parts to obtain the value of I1 =∫ 1
0xe−x dx . 3
(b) Similarly, show that In = nIn−1− e−1 for n ≥ 2. 4
(c) Evaluate I3 . 3
Part Marks Level Calc. Content Answer U2 OC2
(a) 3 C CN I12 1− 2e 2003 A10
(b) 4 B CN I12 proof
(c) 3 C CN Higher 6− 16e
•1 start to integrate by parts•2 complete integration by parts•3 process limits•4 start to integrate by parts
•5, 6 complete (lose 1 for each error)•7 interpret limits•8 use (b)•9 iterate (b)•10 substitute I1
•1 x∫
e−x dx−∫
(1∫
e−x dx) dx•2 [−xe−x − e−x]10•3 1− 2
e or 0·264•4 xn
∫
e−x dx−∫
(nxn−1∫
e−x dx) dx
•5, 6 [−xne−x]10 + n∫ 10 xn−1e−x dx
•7 −e−1 − (−0) + n∫ 10 xn−1e−x dx
•8 I3 = 3I2 − e−1•9 3(2I1 − e−1) − e−1•10 3(2− 4e−1 − e−1) − e−1
hsn.uk.net Page 11
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
17. Let z = cos θ + i sin θ .
(a) Use the binomial expansion to express z4 in the form u+ iv , where u and vare expressions involving sin θ and cos θ . 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
cos 4θ
cos2 θ= p cos2 θ + q sec2 θ + r, where −π
2 < θ <π
2 ,
stating the values of p , q and r . 6
Part Marks Level Calc. Content Answer U2 OC3
(a) 3 C CN A4 2005 Q12
(b) 1 C CN A22
(c) 6 A CN A23
18. Verify that i is a solution of z4 + 4z3 + 3z2 + 4z+ 2 = 0.
Hence find all the solutions. 5
Part Marks Level Calc. Content Answer U2 OC3
5 C CN A13, A16, A19,A20
±i,−2±√2 2002 A2
hsn.uk.net Page 12
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
19.[SQA] The point A represents −5+ 5i on an Argand diagram and ABCD is a square withcentre −2+ 2i . Find the complex numbers represented by the points B, C and D,giving your answers in the form x+ iy . 4
Part Marks Level Calc. Content Answer U2 OC3
4 C CN A15 1996 SY1 Q2
20. (a) Obtain the sum of the series 8+ 11+ 14+ · · · + 56. 2
(b) A geometric sequence of positive terms has first term 2, and the sum of thefirst three terms is 266. Calculate the common ratio. 3
(c) An arithmetic sequence, A , has first term a and a common difference 2, anda geometric sequence, B , has first term a and common ratio 2. The first fourterms of each sequence have the same sum. Obtain the value of a . 3
Obtain the smallest value of n such that the sum to n terms for sequence B ismore than twice the sum to n terms for sequence A . 2
Part Marks Level Calc. Content Answer U2 OC4
(a) 2 C CN S1 544 2004 A16
(b) 3 C CN S2 r = 11
(c1) 3 B CN S1, S2 a = 1211
(c2) 2 A CN n = 7
hsn.uk.net Page 13
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
21. 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. 3
Obtain a formula for un in terms of n , simplifying your answer. 2
Part Marks Level Calc. Content Answer U2 OC4
5 C CN S1 2005 Q4
22. Define Sn(x) by
Sn(x) = 1+ 2x+ 3x2+ · · ·+ nxn−1,where n is a positive integer.
(a) Express Sn(1) in terms of n . 2
(b) By considering (1− x)Sn(x) , show that
Sn(x) =1− xn
(1− x)2 −nxn
1− x , x 6= 1. 4
(c) Obtain the value of limn→∞
{
2
3+3
32+4
33+ · · ·+ n
3n−1+3
2
n
3n
}
. 3
Part Marks Level Calc. Content Answer U2 OC4
(a) 2 C CN S5 12n(n+ 1) 2002 A10
(b) 4 A CN S2 proof
(c) 3 A CN D1, CGD 54
hsn.uk.net Page 14
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
23. Given that uk = 11− 2k , k ≥ 1, obtain a formula for Sn =n
∑
k=1
uk . 3
Find the values of n for which Sn = 21. 2
Part Marks Level Calc. Content Answer U2 OC4
(1) 3 C CN S5, S1 10n− n2 2003 A2
(2) 2 C CN Higher 3, 7
•1 strategy (e.g. arithmetic series)•2 process•3 complete•4 form equation•5 solve equation
•1 a = 9, d = −2•2 Sn = n
2
(
18+ (n− 1) × (−2)
)•3 Sn = −n2 + 10n•4 −n2 + 10n = 21•5 n = 3, 7
hsn.uk.net Page 15
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
24.[SQA] Let
A =
(
1 0−1 2
)
.
Use induction to show that, for all positive integers n ,
An =
(
1 01− 2n 2n
)
.
Determine whether or not this formula for An is also valid when n = −1. 6
Part Marks Level Calc. Content Answer U3 OC2
6 C CN P3, A25 1998 SY2 Q1
hsn.uk.net Page 16
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
25.[SQA] The matrices A and B are defined by
A =
1 −1 32 −1 94 −8 1
and B =
71 a −634 b −3−12 c 1
where a , b and c are constants.
(a) Find the matrix B− 3A . 2
(b) (i) Verify that AB = I , where I is the 3× 3 identity matrix, provided that
a − b + 3c = 02a − b + 9c = 14a − 8b + c = 0. 3
(ii) Use Gaussian elimination to find the values of a , b and c for whichAB = I . 4
Part Marks Level Calc. Content Answer U3 OC2
(a) 2 C CN A25 1998 SY1 Q14
(bi) 3 C CN A25
(bii) 4 C CN A10
hsn.uk.net Page 17
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
26.[SQA]
(a) Show that if M =
(
cos θ sin θ
sin θ − cos θ
)
then M2 = I where I is the 2× 2 identitymatrix. 1
By choosing two different values of θ , exhibit two matrices A , B such thatA2 = I and B2 = I but (AB)2 6= I . 4
(b) Prove that if C and D are n × n matrices such that C2 = I , D2 = I and Cand D commute, then (CD)2 = I . 2
Part Marks Level Calc. Content Answer U3 OC2
(a) 1 C CN A25 1999 SY2 Q9
(a) 4 A/B CN A25
(b) 2 C CN A26
hsn.uk.net Page 18
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
27.[SQA] The square n× n matrix A satisfies the equation
A2 = 5A− 6I
where I is the n × n identity matrix. Show that A is invertible and express A−1
in the form pA+ qI . 2
Obtain a similar expression for A3 . 2
Part Marks Level Calc. Content Answer U3 OC2
2 C CN A26 1996 SY2 Q4
2 A/B CN A26
hsn.uk.net Page 19
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
28.[SQA] Show that
det
2 2k 11 k− 1 12 1 k+ 1
has the same value for all values of k . 3
Part Marks Level Calc. Content Answer U3 OC2
3 C CN A27 1998 SY2 Q4
29. Expand(
x2 − 2x
)4, x 6= 0
and simplify as far as possible. 5
Part Marks Level Calc. Content Answer U1 OC1
5 C CN A4 x8 − 8x5 + 24x2 − 32x + 16
x42001 A6
[No marking instructions available]
hsn.uk.net Page 20
Questions marked ‘[SQA]’ c© SQAAll others c© Higher Still Notes
Higher Mathematics
30. Determine whether the function f (x) = x4 sin 2x is odd, even or neither.
Justify your answer. 3
Part Marks Level Calc. Content Answer U1 OC4
3 B CN F8 odd 2004 A10
[END OF QUESTIONS]
hsn.uk.net Page 21Questions marked ‘[SQA]’ c© SQA
All others c© Higher Still Notes