1.1 (a) p = 124.68; (b) q = 400; (c) r = 0.04.

 1.2 (a) 756 = 2²3³7; (b) .

 1.3 (a) p = 1.246 × 10³; (b) − 1.57 × 10³; (c) 4.5 × 10^{− 3}.

 1.5 (a) 2^5/23^11/12; (b) 3^{− 1/6}5^12/5; (c) 7^7/32^{− 1/2}.

 1.6 (a) (b)

*1.7 (a) 10.31 = 1010.0101₂…; (b) 1101.01₂ = 13.25.

*1.8 (a) p × q = 32130₄; (b) (p − q) = 11₄.

 1.9 (a) ; (b) ; (c) 3^5/6a^{− 5/3}.

1.10 (a) identity; (b) equation.

1.11 (a) − 1 < x < 4; (b) − 4 < x < −2 or 2 < x < 4; (c) x > −2 or x < −4.

1.12 31.681.

1.13 − 9828a³b¹¹.

1.17 (a) (b) (c) x = y⁶ + 2.

1.18 

1.19 (a) (b) f^{− 1}(x) = 27x³ − 4.

1.20 (a) y = −x + 3; (b) 3y = 2x − 7; (c) 2y + 5x = 13; (d) 5y + 2x = 22.

1.21 

1.22 Area = 12.

1.23 (a) (x − 1)² + (y − 3)² = 4; (b) (x − 1)² + (y − 3)² = 13.

1.24 Centre , radius .

 2.1 3x² − 2x + 1 = 0.

 2.3 

 2.4  and Angle at centre 2.42 rad = 139°.

 2.5 (a) (x − 1)(x² + 2x + 1) − 3.

 2.6 

 2.7 1.526.

 2.8 (a) ; (b) ; (c) .

 2.9 (a) ; (b) ; (c) .

2.11 (a) θ = 0.666 and 2.475 radians; radians. (b) ; .

2.12  for all integer n and k ≠ 1, and for all integer n and k ≠ −1.

2.13 , .

2.15 A = 0.643 radians, C = 1.999 radians, c = 7.59 cm.

2.16 A = 0.809 radians, B = 1.391 radians, C = 0.942 radians.

2.18 (a) log (x²); (b) 0.

2.19 (a) x = 31.765; (b) x = 1.122.

2.20 (a) x = 32.66; (b) x = 3.401.

2.22 (a) or x = 0; (b) (c)

2.23 x = ±1.317.

2.24 t₁t₂ = −1.

2.25 tangent: y = x + 1; normal: y = −x + 3.

 3.1 (a) 1; (b) 0; (c) 4.

 3.2 (a) 10; (b) ; (c) 1.

 3.5 (a) Removable discontinuity at x = 0; (b) Non-removable discontinuity at x = 3; removable discontinuity at x = −3.

 3.6 (a) A = 1; (b) A = 1 and B = 0 (all n), or for n ≥ 2 (all B).

 3.7 (a) 6x² + 4; (b) (c) − 15sin 3x.

 3.8 (a) x³e^x + 3x²e^x; (b) (c) (d) (e) (f)

 3.9 (a) (b) (c) ; (d)

3.10 (a) x^x(ln x + 1); (b) ; (c) ; (d) tan x.

3.11 (a) ; (b) .

3.12 .

3.13 ; .

3.14 x = −8 and , respectively.

3.15 , 7y = 6x − 21.

3.16 Three: f⁽¹⁾, f⁽²⁾ and f⁽³⁾.

3.17 (a) 2^{n − 1}e^2x + ( − 1)^{n + 1}2^{n − 1}e^{− 2x}; (b) , for all n ≥ 1.

3.19 

3.20 (a) Minimum at x = 0, maxima at x = ±1.

3.21 Maxima at minima at

3.24 Approximate solutions: x = −0.8, 1.5 and 3.4.

3.27 Vertex x = 3a, y = 0.

  4.1 Area = 16.41.

  4.2 (b)

  4.3 (a) (b) (c)

  4.4 (a) (b) 2(3 + x − x²)^1/2 + c;

  4.5 (a) (b)

  4.6 (a) (b)

  4.7 (a) 2ln (x − 2) + 3ln (x + 1) + c; (b)

  4.8 (a) (b)

  4.9 (a) (b)

 4.10 (a) tan xln (tan x) − tan x + c; (b)

 4.11 (a) (b)

 4.12 (a) (b) .

 4.13 (a) (b) (c) –1.

 4.15 

 4.16 (a) (b) (c)

 4.17 (b) At x = 0 and all negative integers;

 4.18 564 m.

 4.19 (a) Convergent with value ; (b) convergent with value ; (c) divergent; (d) divergent.

 4.21 (a) Converges for all α < −1; (b) Converges provided β > −1, α < −1.

 4.22 

 4.23 

 4.24  L = 3.0896 (trapezium rule), L = 3.0847 (Simpson's rule).

 4.25 π = 3.14294 for n = 2 and 3.14170 for n = 4. Four intervals are needed.

*4.26 A = 2π, , S = 3π.

*4.27 

*4.28 (a) (b)

*4.29 

*4.30 

 5.1 Sum = 148875.

 5.2  where . No values of r.

 5.3 , convergent as N → ∞.

 5.4 .

 5.5 (a) convergent; (b) divergent; (c) divergent; (d) convergent.

 5.6 (a) |x| < 1; (b) ; (c) x < 0.

 5.7 (a) ; (b) − π; (c) –1; (d)

 5.8 (a) ; (b) 1.

 5.9 , valid for all x.

5.10 3 terms give sin x = 0.56465, the calculator value is 0.56465.

5.11 (a) ; (b)

5.14 

5.15 (a) ; (b)

5.16 0.838.

5.17 First minimum is at 4.50 rad = 258° to the nearest 1°.

5.20 ; valid for |x| < 1.

5.21 ; valid for all x.

5.22 (a) conditionally convergent; (b) absolutely convergent for α ≠ kπ, where k is an integer; (c) conditionally convergent for α = 0.

5.23 (a) conditionally convergent; (b) not convergent; (c) absolutely convergent; (d) absolutely convergent.

  6.1 (a) − (1 + 2i); (b) − 5(1 + 2i); (c) 6(5 + 7i); (d) 4(3 − 4i);

  6.3 

  6.5 (a) r = 1, ; (b) , ; (c) , .

  6.7 (a) circle radius 4 centre (0, 3); (b) circle radius centre (0, 0); (c) converges for all z.

  6.8 (a) and ; (b) and ; (c) and .

  6.9 (a) − 0.101 − 0.346i; (b) 0.417 + 0.161i; (c) 1.272 − 0.786i and − 1.272 + 0.786i.

 6.10 (a) 0.0313i, (b) 1.864 + 0.290i, − 1.183 + 1.468i and − 0.680 − 1.758i, (c) −i.

 6.11 (a) 0.951 + 0.309i, i, − 0.951 + 0.309i, − 0.588 − 0.809i and 0.588 − 0.809i; (b) 0.080 + 0.440i; (c) 0.920 + 0.391i.

 6.12 (a) 0.1080 + 0.4643i; (b) 3i/4; (c) − 0.805 + 1.007i.

 6.13 (a)− 0.266 + 0.320i; (b) 4.248; (c)

 6.14 (a) cos (12θ) + isin (12θ).

 6.15 (a) cos 7θ − isin 7θ.

*6.16 

*6.17 e^2cos xsin (2sin x).

*6.18 

  7.3 , ; x + 2y + 3z = 14.

  7.8 (a); (c) and (d)

  7.9 (a) ; (b) f(x, y) = x²ln (xy) + k, k an arbitrary constant.

 7.10 (a) y(4x + 3y³) − xy²(9x + 16y); (b) –2, (c) .

 7.11 (a) satisfied k = 3; (b) not satisfied; (c) satisfied k = 0; (d) satisfied .

 7.16 

 7.17 

 7.18 maximum , minimum –.

 7.19 maximum (0, 0), saddle point ( − 1, 1).

*7.20 (0, −1), (2, 1), ( − 2, 1).

*7.21 

*7.22 

*7.23 (a) , I(x) = e^{− x} − e^{− 1}; (b)

*7.24 (a)

*7.25 

  8.1 

  8.3 (0, 4, 0).

  8.4 (b) θ = 2.68 rad = 153.4°, direction cosines (0, 0, 1).

  8.7 , area = .

  8.9 .

 8.10 (a) − 6i − 3j; (b) .

 8.12 (a) τ = −9i + 6j + 5k; (b) τ = 3i − 2j; (c) τ_z = 5; (d) .

*8.15 (a) (b)

 8.16 , .

 8.17 , D is (0, 1, 0).

 8.19 x − 2y − 3z = 7, .

 8.20 x + 4y + 6z = 5.

 8.21 θ = 1.52 rad = 87.2°; r = (2i − 3j) + s(i − 2j + k).

 8.22 .

 8.23  where c is a constant.

 9.1 a × b = 7i + 10j − 9k, b · a × c = 52.

 9.2 (a) 16 + 5i; (b) –1.

 9.3 –105.

 9.4 (a) x = 0, 1 or − 2; (b) Δ₂ = (α + β + γ)(α − β)(β − γ)(γ − α).

 9.5 Δ_n = (n + 1)( − 1)ⁿ.

 9.6 (a) no non-trivial solution; (b) x: y: z = −3: 1: 1.

 9.7 α = 3 and 14. For the latter, .

9.13 (a) (A + B)³ = (A³ + A²B + ABA + AB² + BA² + BAB + B²A + B³); (b) AB = BA.

9.15 (b)

9.19  and .

9.20 

9.21 

9.22 m_a = 960 gm, m_b = 120 gm.

9.23 (a) α ≠ 3, independent of the value of β; (b) ; (c) (i) β = 6, a solution exists, but is not unique; (ii) β = 2, no solutions exist.

  10.1 . Normalised eigenvectors are:

Eigenvectors not orthogonal.

  10.7 Eigenvalues: λ = 0, 1. Linearly independent eigenvectors: , (λ = 0) and , (λ = 1).

The matrix is defective.

  10.9 (a) Eigenvalues: . Corresponding normalised eigenvectors:

Eigenvectors not orthogonal.

(b) Eigenvalues: . Corresponding normalised eigenvectors:

 10.10 

*10.12 

*10.15 

*10.16 Principal axes along: 2x + 2y + z = 0, 2x − y − 2z = 0, x − 2y + 2z = 0, shortest distance is 2.

*10.17 (a) oblate spheroid; (b) one-sheet hyperboloid.

*10.19  to the x-axis. The two branches are closest at .

 11.1 (a) 11; (b) .

 11.2 (a) ; (b) .

 11.3 2π.

 11.4 (a) − 4π; (b) − 16.

 11.5 (a) ; (b) 3.

 11.6 1 − ln 2.

 11.7 11.

 11.9 (a) 2ln 2 − 1; (b) .

11.10 27.

11.11 0.

11.12 .

11.13 .

11.14 (a) 11; (b)

11.15 .

11.16 .

11.17 .

11.20 .

11.21 .

11.22 .

11.23 

  12.1  in direction − 2i + j; direction of most rapid decrease at (–3, 2) is 3i + 2j; .

  12.2 (a) 2i − 2j − k; (b) (c) (1, 1, 1) + (2, −2, 1)t.

  12.3 (a) y i − (3z³ − 4xz)j; (b) x²z(3xz³ + 2y²)i + 3x²yz²(2x − 3z²)j − 3xz²(y² + x²z)k; (c) ( − 4x + 9z²)i + (4z − 1)k; (d) − 2yz²i + z²(3z² − 2x)j + 4yz(3z² − x)k; (e) 0.

  12.6 (a) r²sin 2θsin φ;

(b) rsin θsin φ(sin θcos φ − cos θ) e_r + rsin θsin φ (cos θcos φ + sin θ) e_θ + rsin θcos ²φ e_φ;

(c)

(d) .

  12.7 − 2πa⁴.

  12.8 2π.

  12.9 .

 12.10 (a) ; (b) a³.

 12.11 − 1.

 12.12 − y²cosh ²(xz) + c, c constant

 12.13 a = 4; b = 2; c = −1, d constant.

 12.14 114.

 12.15 2.

 12.16 .

 12.17 (a) ; (b)

 12.18 Ma²/4 + Md²/3.

 12.19 1.

*12.23 (b) (c) .

*12.24 (b) ρ₀(r)e^{− σt/ϵ}.

 12.27 (a) 0; (b) 0; (c) no.

  13.1 

  13.2 

  13.3 

  13.5 (a); (b); (d) do not satisfy Dirichlet conditions; (c) does satisfy Dirichlet conditions. (c)

  13.6 

  13.7 

  13.8 

  13.9 , (b) is a valid series, but is an invalid series.

 13.11 

 13.13 

 13.14 

 13.15 

 13.17 

 13.18 

 13.19 (a) (b) (c) .

 13.20  (b) .

*13.22 .

*13.23 (a) I₁ = I₂ = π; (b) .

  14.1 (a) ; (b) .

  14.2 (a) (x − y − 2) − ln (x + y + 1) = 0; (b) .

  14.4 .

  14.5 

  14.6 (a) not exact; (b) exact, y²ln x = c.

  14.7 (a) 2x³ − 6x²y + 3y² − 6y + c = 0; (b) .

  14.8 (a) x²(sin x − xcos x + c); (b) 2ysin x − cos ²x + c = 0.

  14.9 (a) , (b) 2(x + 1)^5/2 + (x + 1)².

 14.10 .

 14.11 (a) ; (b) e^{− 3x}(3cos 2x + 8sin 2x).

 14.14 (a) ; (b) .

 14.15 (a) (Ax + B)e^{− 2x} + 2x²e^{− 2x}; (b)

 14.16 (a) ; (b) .

 14.17 

 14.18 .

 14.19 .

*14.20 .

*14.21 

*14.23 

*14.24 Ax + Bx^{− 3} + xln x + 1.

*14.25 

  15.2 .

  15.3 .

  15.4 .

  15.5 .

  15.6 .

  15.8 .

  15.9 λ_n = 1 + n²π², y_n(x) = A_nsin (nπx).

 15.10 λ²_n = n²π², .

*15.11 

 15.12 

*15.14 

*15.17 

*15.18 k = 3.8317, 7.0156, 10.1735, 13.3237.

 15.20 J₂(2) = 0.35283 to 5 decimal places, 1 extra term for 7 decimal places.

*15.22 , .

  16.1 

  16.2  n > 0.

  16.3 .

  16.5 .

  16.6  where .

  16.7  a_n, b_n arbitrary constants.

  16.9 .

 16.11 

 16.13 

 16.14 .

 16.16 4.878.

*16.17 (a) ; (b)

*16.18 (a) (b) f(x + 3y) + xg(x + 3y); (c) (d) .

*16.19