Engineering Mathematics

Study lessons

1. 1

   Algebra and Trigonometry: Equations, Progressions, Complex Numbers, and TrianglesFree sample

   Master the algebra and trigonometry toolkit, equations, exponents and logarithms, progressions, complex numbers, identities, and the laws of sines and cosines, that underpins nearly every computation on the MSTE paper.

   15 min
2. 2

   Calculus and Differential Equations: Derivatives, Optimization, Areas, Volumes, and First-Order ODEs

   Build the calculus toolkit the CELE rewards most, differentiation rules, maxima and minima, related rates, integration for area and volume, and first-order differential equations, with every worked example carried through to a clean numeric answer.

   15 min
3. 3

   Probability, Statistics, and Engineering Economy

   Master the counting rules, probability laws, descriptive statistics, and money-over-time formulas the MSTE paper turns into fast, exact points, with every peso and every fraction worked to a clean number.

   15 min
4. 4

   Analytic Geometry: Lines, Distance, and the Conic Sections

   Turn geometry into algebra on the coordinate plane, distances, midpoints, slopes and lines, the circle, and the parabola, ellipse, and hyperbola, unified by eccentricity, with every formula walked to a clean number.

   16 min
5. 5

   Matrices, Determinants, and Systems of Equations

   From matrix arithmetic to determinants, inverses, Cramer's rule, Gaussian elimination, and a first look at eigenvalues, the linear-algebra toolkit that solves the simultaneous equations behind structural stiffness, statics, and survey adjustments.

   15 min
6. 6

   Vectors and Three-Dimensional Geometry

   Work with vectors in space, addition and resolution, the dot and cross products, magnitude and direction cosines, unit vectors, and the equations of lines and planes with their distances, each rule carried all the way to a clean number.

   15 min
7. 7

   Engineering Economy: Interest, Annuities, and Depreciation

   A dedicated, formula-by-formula tour of the money-over-time toolkit the MSTE paper rewards, interest, effective rates, annuities and gradients, depreciation, rate of return, benefit-cost, capitalized cost, and break-even, with every peso carried to an exact number.

   15 min
8. 8

   Solid Geometry and Mensuration

   Compute areas of plane figures and the surface areas and volumes of prisms, cylinders, pyramids, cones, spheres, zones and segments, and frustums, unified by the prismatoid formula and the theorems of Pappus, with every result carried to a number.

   15 min
9. 9

   Advanced and Spherical Trigonometry

   Push trigonometry past the basics with the sum, difference, double-angle, and half-angle identities, general equation solving, the many formulas for the area of a triangle, and the spherical triangle with its laws of sines and cosines and the spherical excess that fixes its area.

   15 min
10. 10

    Sequences, Series, and the Binomial Theorem

    Arithmetic, geometric, and harmonic sequences and their sums, the sum to infinity of a geometric series, the binomial theorem with its general term, and the power-series ideas that let you approximate values by hand.

    15 min
11. 11

    Partial Derivatives and Multiple Integrals

    Extend single-variable calculus to functions of several variables, using partial derivatives, the total differential for error propagation, the gradient for rates and directions, and double integrals for area, volume, and centroids, with every result carried to a number.

    15 min
12. 12

    Numerical Methods

    The board's toolkit for problems with no clean closed form, root finding by bisection and Newton-Raphson, linear interpolation between tabulated points, the trapezoidal and Simpson rules for area under a curve, and the finite-difference formulas that turn derivatives into arithmetic, each carried through to a finished number.

    15 min
13. 13

    Complex Numbers, Polar Form, and the De Moivre Theorem

    Move fluently between the rectangular, polar, and Euler forms of a complex number, multiply and divide by handling moduli and angles, raise to powers and pull out every root with the De Moivre theorem, and add phasors the way an AC network demands.

    15 min
14. 14

    Advanced Integration Techniques

    Master the four integration methods the CELE leans on most, integration by parts, trigonometric integrals and trigonometric substitution, partial fractions, and improper integrals, with every worked example carried through to a clean number.

    15 min
15. 15

    Multivariable Calculus and Optimization

    Push functions of several variables to their extremes the way the CELE tests it, from partial derivatives and the total differential for error estimation to locating and classifying critical points with the second-derivative test and solving side-constrained problems with Lagrange multipliers, every rule carried to a number.

    15 min
16. 16

    Statistics: Regression, Correlation, and Sampling

    Fit a least-squares line to paired data, measure the fit with the correlation coefficient, then move to sampling distributions, the standard error, confidence intervals, and a first hypothesis test, every step carried to a finished number.

    15 min
17. 17

    Laplace Transforms and Applications

    Turn calculus into algebra with the Laplace transform, covering the standard transform pairs, linearity, the first shifting theorem, transforms of derivatives, inverse transforms by partial fractions, the unit step, and initial value problems, each worked to a clean number.

    15 min
18. 18

    Fourier Series and Harmonic Analysis

    Build a Fourier series from a0, an, and bn over one period, use even and odd symmetry to drop half the work, handle half-range expansions and harmonics, and read the value a series takes at a jump, with the square wave and sawtooth carried to numbers.

    15 min
19. 19

    Higher-Order Differential Equations

    Solve the linear constant-coefficient ODEs the CELE loves, build homogeneous solutions from the characteristic roots (real, repeated, and complex), add a particular solution by undetermined coefficients, and pin down the constants with initial or boundary conditions, each worked example carried through to a clean number with units.

    15 min
20. 20

    Probability Distributions and Reliability

    Model discrete outcomes with the binomial and Poisson laws, read areas under the normal curve with z-scores, then turn a failure rate into a probability of survival and combine components in series and parallel, every step carried to a finished number.

    15 min
21. 21

    Partial Differential Equations

    Master the partial differential equations the CELE samples, name the order and linearity, recognize the heat, wave, and Laplace equations, classify them as parabolic, hyperbolic, or elliptic with the B^2 - 4AC test, and solve by separation of variables into product solutions pinned down by boundary and initial conditions, with every worked example carried to a clean number with units.

    15 min
22. 22

    Eigenvalues and Eigenvectors

    Eigenvalues and eigenvectors from the characteristic equation det(A - lambda I) = 0 through 2x2 and 3x3 problems, the trace and determinant shortcuts, diagonalization, and how these same roots deliver the buckling loads, natural frequencies, and principal stresses behind civil engineering, every worked example carried to a finished number with units.

    15 min
23. 23

    Optimization and Lagrange Multipliers

    Turn design questions into calculus, locating unconstrained maxima and minima with the first and second derivative tests, running the applied max-min setup from picture to number, and handling side conditions with Lagrange multipliers where grad f equals lambda times grad g, every result carried to a value with units.

    15 min
24. 24

    Line and Surface Integrals

    Work line integrals of scalar and vector fields, work and circulation, conservative fields and path independence, Green theorem, divergence and curl, flux, and the divergence theorem, with every rule carried to a number.

    15 min
25. 25

    Systems of Differential Equations

    First-order linear systems in the matrix form x' = A x solved through the eigenvalues and eigenvectors of A, the node, saddle, and spiral classification of the equilibrium read straight off those eigenvalues, coupled tank and mixing models, and the trick of recasting a higher-order ODE as a first-order system, with every worked example carried to a finished number with units.

    15 min
26. 26

    Taylor and Power Series

    Taylor and Maclaurin series, the radius and interval of convergence, the standard series for e^x, sin x, cos x, and ln(1+x), the Lagrange and alternating-series error bounds, and approximating a value to a stated number of terms by hand.

    15 min
27. 27

    Multiple Integrals in Polar and Cylindrical Coordinates

    Set up and evaluate multiple integrals in polar, cylindrical, and spherical coordinates using the r dr dtheta rule, the Jacobian of each transformation, and applications to area, volume, and centroids, with every result carried to a number with units.

    15 min
28. 28

    Vector Differential Operators

    Master the del operator with the gradient of a scalar field, the divergence and curl of a vector field, the Laplacian, the standard vector identities, and the conservative and solenoidal fields they define, each rule carried all the way to a number with units.

    15 min

Section practice test

35 test sets covering the whole Engineering Mathematics section, 1050 original questions with explanations. After you finish the lessons above, prove it here. Randomized every run.