Engineering Mathematics
Engineering Mathematics carries roughly 13 marks in GATE CH and is the most predictable domain: the questions are short, closed-form, and drawn from linear algebra, calculus, differential equations, probability and numerical methods.
Core rules
- Eigenvalues. For a 2×2 matrix, λ₁+λ₂ = trace and λ₁λ₂ = det - often faster than expanding the characteristic polynomial. Real symmetric matrices have real eigenvalues and orthogonal eigenvectors.
- Exact differentials. M dx + N dy is exact iff ∂M/∂y = ∂N/∂x.
- Second-order ODEs. Solve aλ²+bλ+c = 0: distinct roots give C₁e^(λ₁x)+C₂e^(λ₂x); a repeated root gives (C₁+C₂x)e^(λx).
- PDE classification. B²−4AC > 0 hyperbolic, = 0 parabolic, < 0 elliptic.
- Numerical integration. Trapezoidal is O(h²); Simpson's 1/3 rule is O(h⁴) and needs an even number of intervals.
- Probability. Binomial (np, npq), Poisson (λ, λ) and exponential (1/λ, 1/λ²) give the means and variances you will be asked for.
Worked example
Solve y″ − 3y′ + 2y = 0 with y(0) = 0, y′(0) = 1. The characteristic equation λ²−3λ+2 = 0 gives λ = 1, 2, so y = C₁eˣ + C₂e²ˣ. From y(0) = 0 we get C₁ + C₂ = 0. Since y′ = C₁eˣ + 2C₂e²ˣ, the slope condition gives C₁ + 2C₂ = 1. Hence C₂ = 1, C₁ = −1 and y = e²ˣ − eˣ. Check: y(0) = 1 − 1 = 0 and y′(0) = 2 − 1 = 1, both satisfied.
Traps
- Applying only one initial condition and leaving a free constant in the answer.
- Running Simpson's rule across an odd number of intervals.
- Quoting √λ as the Poisson variance instead of λ.
- Iterating Newton–Raphson where f′ ≈ 0, which flings the next point far away.