DecemberRevision - WordPress.com · Part Marks Level Calc. Content Answer U2 OC4 (a) 2 C CN S1 544...

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Higher Mathematics December Revision 1. (a) Given f ( x)= x 3 tan 2x , where 0 < x < π 4 , obtain f ( x) . 3 (b) For y = 1 + x 2 1 + x , where x = 1, determine dy dx in 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 answers where possible. (a) h( x)= sin ( x 2 ) 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 SQA All others c Higher Still Notes

Transcript of DecemberRevision - WordPress.com · Part Marks Level Calc. Content Answer U2 OC4 (a) 2 C CN S1 544...

Page 1: DecemberRevision - WordPress.com · 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= 12 11 (c2) 2 A CN n= 7 hsn.uk.net

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

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Page 2: DecemberRevision - WordPress.com · 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= 12 11 (c2) 2 A CN n= 7 hsn.uk.net

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

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Page 3: DecemberRevision - WordPress.com · 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= 12 11 (c2) 2 A CN n= 7 hsn.uk.net

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

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Page 4: DecemberRevision - WordPress.com · 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= 12 11 (c2) 2 A CN n= 7 hsn.uk.net

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

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Page 5: DecemberRevision - WordPress.com · 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= 12 11 (c2) 2 A CN n= 7 hsn.uk.net

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

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Page 6: DecemberRevision - WordPress.com · 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= 12 11 (c2) 2 A CN n= 7 hsn.uk.net

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

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Page 7: DecemberRevision - WordPress.com · 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= 12 11 (c2) 2 A CN n= 7 hsn.uk.net

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

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Page 8: DecemberRevision - WordPress.com · 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= 12 11 (c2) 2 A CN n= 7 hsn.uk.net

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

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Page 9: DecemberRevision - WordPress.com · 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= 12 11 (c2) 2 A CN n= 7 hsn.uk.net

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

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Page 10: DecemberRevision - WordPress.com · 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= 12 11 (c2) 2 A CN n= 7 hsn.uk.net

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

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Page 11: DecemberRevision - WordPress.com · 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= 12 11 (c2) 2 A CN n= 7 hsn.uk.net

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

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Page 12: DecemberRevision - WordPress.com · 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= 12 11 (c2) 2 A CN n= 7 hsn.uk.net

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

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Page 13: DecemberRevision - WordPress.com · 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= 12 11 (c2) 2 A CN n= 7 hsn.uk.net

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

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Page 14: DecemberRevision - WordPress.com · 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= 12 11 (c2) 2 A CN n= 7 hsn.uk.net

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

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Page 15: DecemberRevision - WordPress.com · 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= 12 11 (c2) 2 A CN n= 7 hsn.uk.net

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

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Page 16: DecemberRevision - WordPress.com · 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= 12 11 (c2) 2 A CN n= 7 hsn.uk.net

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

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Page 17: DecemberRevision - WordPress.com · 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= 12 11 (c2) 2 A CN n= 7 hsn.uk.net

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

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Page 18: DecemberRevision - WordPress.com · 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= 12 11 (c2) 2 A CN n= 7 hsn.uk.net

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

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Page 19: DecemberRevision - WordPress.com · 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= 12 11 (c2) 2 A CN n= 7 hsn.uk.net

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

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Page 20: DecemberRevision - WordPress.com · 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= 12 11 (c2) 2 A CN n= 7 hsn.uk.net

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]

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Page 21: DecemberRevision - WordPress.com · 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= 12 11 (c2) 2 A CN n= 7 hsn.uk.net

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]

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