Paper: Kenya National Examinations Council (KNEC), Diploma in Electrical and Electronic Engineering, Module III — Engineering Mathematics III, March/April 2025. Paper codes: 2521/303, 2601/303, 2602/303 and 2603/303.

This revision lesson follows the supplied four-page paper and includes all eight questions with independently prepared worked solutions. Candidates are instructed to answer any five questions; all questions carry equal marks. These explanations are study notes, not an official KNEC marking scheme.

Question 1: Numerical methods

1(a) Newton–Raphson method (8 marks)

Question. For 2x3 − 5x + 3 = 0, show that Newton–Raphson gives xn+1 = (4xn3 − 3)/(6xn2 − 5). Starting with x0 = 1.5, determine the root to three decimal places.

Worked answer.

Let f(x) = 2x3 − 5x + 3, so f′(x) = 6x2 − 5.
Newton’s rule xn+1 = xn − f(xn)/f′(xn) simplifies to the stated iteration.

n xn
0 1.500000
1 1.235294
2 1.092472
3 1.025206
4 1.002967
5 1.000051

The polynomial factors as (x − 1)(2x2 + 2x − 3), and this iteration converges to x = 1.

The root to three decimal places is 1.000.

1(b) Newton–Gregory interpolation (12 marks)

Question. The table gives x = 1, 2, 3, 4, 5, 6 and f(x) = 6, 23, 64, 141, 266, 451. Use Newton–Gregory interpolation to estimate f(1.5) and f(5.5).

Worked answer.

The forward differences are Δf = 17, 41, 77, 125, 185;

Δ2f = 24, 36, 48, 60;
and Δ3f = 12 throughout.
With x0 = 1 and h = 1, p = x − 1:

f(x) = 6 + 17p + 12p(p − 1) + 2p(p − 1)(p − 2) = 2x3 + 3x + 1.

Therefore f(1.5) = 12.25 and f(5.5) = 350.25.

Question 2: Matrices and dynamic systems

2(a) Eigenvalues and eigenvectors (10 marks)

Question. Determine the eigenvalues and corresponding eigenvectors of A = [[0, 1], [8, 2]].

Worked answer.

det(A − λI) = λ2 − 2λ − 8 = (λ − 4)(λ + 2), so the eigenvalues are 4 and −2.

  • For λ = 4, an eigenvector is (1,4)T.
  • For λ = −2, an eigenvector is (1,−2)T.

2(b) State-transition matrix (10 marks)

Question. For dx/dt = Cx, where C = [[−2, 0], [0, −3]], determine Φ(t) and show that Φ(0) = I.

Worked answer.

The system matrix is diagonal, so Φ(t) = eCt = [[e−2t, 0], [0, e−3t]].
At t = 0 this becomes [[1,0],[0,1]] = I.

Question 3: Complex variables

3(a) Exponential function (10 marks)

Question. Given f(z) = ez+1, express f(z) as u + jv and show that u and v are harmonic.

Worked answer.

Put z = x + jy.
Then f(z) = ex+1(cos y + j sin y), so u = ex+1cos y and v = ex+1sin y.
For u, uxx = ex+1cos y and uyy = −ex+1cos y;
hence ∇2u = 0.
Similarly vxx = ex+1sin y and vyy = −ex+1sin y, so ∇2v = 0.

3(b) Verify the Cauchy–Riemann equations (5 marks)

Question. Given f(z) = cosh x cos y + j sinh x sin y, show that its real and imaginary parts satisfy the Cauchy–Riemann equations.

Worked answer.

Here u = cosh x cos y and v = sinh x sin y.
Then ux = sinh x cos y = vy, while uy = −cosh x sin y = −vx.

Both Cauchy–Riemann equations hold.

3(c) Image under a bilinear transformation (5 marks)

Question. Find the image of |z| = 2 under ω = (z + 3j)/(z − 2j).

Worked answer and source note. Rearranging gives z = j(2ω + 3)/(ω − 1). Thus |2ω + 3| = 2|ω − 1|. Writing ω = u + jv and expanding yields 20u + 5 = 0, or u = −1/4. Since the pole z = 2j lies on |z| = 2, the circle maps to the straight line Re(ω) = −1/4, not to a finite circle.

Question 4: Fourier series

4(a) Fourier series of a periodic piecewise function (10 marks)

Question. Over one period, f(t) = 1 − t for −1 < t < 0 and f(t) = t − 1 for 0 < t < 1, with f(t + 2) = f(t). Obtain its Fourier series.

Worked answer.

The period is 2, so L = 1.

Direct integration gives a0/2 = 1/2, an = 2[(-1)n − 1]/(n2π2) and bn = 2[(-1)n − 1]/(nπ).
Only odd harmonics remain:

f(t) = 1/2 − 4Σm=0∞{cos((2m + 1)πt)/[(2m + 1)2π2] + sin((2m + 1)πt)/[(2m + 1)π]}.

At jump points, the series converges to the midpoint of the one-sided limits.

4(b) Fourier series of a triangular wave (10 marks)

Question. For the periodic wave shown, determine h(x) and obtain its Fourier series.

Worked answer.

On −π ≤ x ≤ π the graph is h(x) = π − |x|, extended with period 2π.
It is even, with mean π/2 and cosine coefficients an = 2[1 − (−1)n]/(πn2).
Thus

h(x) = π/2 + (4/π)Σm=0∞cos((2m + 1)x)/(2m + 1)2.

Question 5: Surface integrals and Green’s theorem

5(a) Flux through a plane (10 marks)

Question. Evaluate ∬SF·n dS for F = 2xi + yj + 3zk, where S is x + y + z = 1 in the first octant.

Worked answer.

Write z = 1 − x − y.
For the upward orientation, n dS = (1,1,1)dxdy and F·n dS = (3 − x − 2y)dxdy.
The projection is x ≥ 0, y ≥ 0, x + y ≤ 1, so the flux is ∫01∫01−x(3 − x − 2y)dy dx = 1.

5(b) Green’s theorem on a triangular boundary (10 marks)

Question. Evaluate ∮C[xy2dx + xy dy] using Green’s theorem, where C is the counter-clockwise boundary of A(0,0), B(2,0), C(2,2).

Worked answer.

Let P = xy2 and Q = xy.
Then Qx − Py = y − 2xy.
The triangular region is 0 ≤ y ≤ x ≤ 2, so the line integral is ∬R(y − 2xy)dA = ∫02∫0x(y − 2xy)dy dx = −8/3.

Question 6: Multiple integrals

6(a) Double integral in polar coordinates (10 marks)

Question. Use polar coordinates to evaluate ∬Rx2/√(x2 + y2) dxdy, where R is 1 ≤ x2 + y2 ≤ 4, y ≥ 0.

Worked answer.

The region is the upper half-annulus: 1 ≤ r ≤ 2 and 0 ≤ θ ≤ π.

The integrand with the Jacobian becomes r2cos2θ.

Hence the integral is ∫0π∫12r2cos2θ drdθ = (7/3)(π/2) = 7π/6.

6(b) Spherical-coordinate integral (5 marks)

Question. Use spherical coordinates to evaluate ∭V1/(x2 + y2 + z2)dV over the unit sphere x2 + y2 + z2 ≤ 1.

Worked answer.

In spherical coordinates the integrand times the Jacobian is sin φ.

Over the unit ball, 0 ≤ ρ ≤ 1, 0 ≤ φ ≤ π and 0 ≤ θ ≤ 2π.
Thus the integral is ∫02π∫0π∫01sin φ dρdφdθ = 4π.

The singularity at the origin is integrable.

6(c) Line integral along a curve (5 marks)

Question. Evaluate ∫C(x + 2y)dx where C follows y = x2 + 1 from (0,1) to (6,37).

Worked answer.

Use x as the parameter, with 0 ≤ x ≤ 6 and y = x2 + 1.
The integral is ∫06[x + 2(x2 + 1)]dx = [x2/2 + 2x3/3 + 2x]06 = 174.

Question 7: Vector calculus

7(a) Triple integral in cylindrical coordinates (5 marks)

Question. Evaluate the given triple integral of x2 + y2 using cylindrical coordinates, over 0 ≤ z ≤ 1 and the quarter-cylinder x ≥ 0, y ≥ 0, x2 + y2 ≤ 4.

Worked answer.

With 0 ≤ r ≤ 2, 0 ≤ θ ≤ π/2 and 0 ≤ z ≤ 1, the integrand and Jacobian give r3.
The value is ∫01∫0π/2∫02r3drdθdz = 4(π/2) = 2π.

7(b) Work along a straight line (5 marks)

Question. For F = (x2 − y2)i + 2xyj, find the work done moving an object along a straight line from (−2,−1) to (4,2).

Worked answer.

Parameterise r(t) = (−2 + 6t, −1 + 3t), 0 ≤ t ≤ 1.
Along the path y = x/2, so F = (3x2/4, x2) and r′ = (6,3).
Therefore F·r′ = (15/2)x2.
Since ∫01x2dt = 4, the work is 30.

7(c) Scalar potential (10 marks)

Question. Show that the line integral of (x3 + y)dx + (x − y3)dy is independent of path and find a scalar potential.

Worked answer.

Let P = x3 + y and Q = x − y3.
Since Py = 1 = Qx, the field is conservative.
Integrating P with respect to x gives φ = x4/4 + xy + g(y).
Matching φy to Q gives g′(y) = −y3, so a potential is φ(x,y) = x4/4 + xy − y4/4 + C.

Question 8: Diagonalisation and divergence theorem

8(a) Modal matrix, spectral matrix and powers of A (10 marks)

Question. A has eigenvalues 4 and 1 with eigenvectors e1 = (1,3)T and e2 = (1,0)T. Determine the modal matrix M, spectral matrix S, A and A3.

Worked answer.

Using the eigenvectors as columns, M = [[1,1],[3,0]] and S = diag(4,1).
Since M−1 = [[0,1/3],[1,−1/3]], A = MSM−1 = [[1,1],[0,4]].
Therefore A3 = MS3M−1 = [[1,21],[0,64]].

8(b) Divergence theorem on a half-cylinder (10 marks)

Question. Use the divergence theorem to evaluate the outward flux of A = 4xi − 2y2j + z2k over the closed surface bounded by x2 + y2 = 4, y ≥ 0, z = 0 and z = 3.

Worked answer.

∇·A = 4 − 4y + 2z.
The enclosed half-cylinder is 0 ≤ r ≤ 2, 0 ≤ θ ≤ π, 0 ≤ z ≤ 3.

Thus the flux is ∭V(4 − 4y + 2z)dV. The three terms contribute 24π, −64 and 18π, respectively, giving a total outward flux of 42π − 64.

Revision note: these independently prepared explanations are for study and are not the official KNEC marking scheme. Check the supplied scan for the original notation and instructions.

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