Consolidated Formula Handbook for BCECE LE Preparation

Engineering aspirants preparing for the Bihar Combined Entrance Competitive Examination (Lateral Entry) frequently struggle to consolidate formulas across Engineering Mathematics and Engineering Mechanics in the final weeks before the examination. This rapid revision handbook compiles every high-yield formula tested in the BCECE LE paper over the last 10 years.

1. Engineering Mathematics Formula Sheet

Matrices and Determinants

  • Characteristic Equation: $|A - lambda I| = 0$
  • Sum of Eigenvalues: $sum lambda_i = ext{Trace}(A) = sum a_{ii}$
  • Product of Eigenvalues: $prod lambda_i = det(A)$
  • Cayley-Hamilton Theorem: Every square matrix satisfies its own characteristic equation: ^n + c_{n-1}A^{n-1} + dots + c_0 I = 0$. Useful for finding ^{-1}$ and higher powers ^k$.
  • Orthogonal Matrix: A^T = A^T A = I implies A^{-1} = A^T$ and $det(A) = pm 1$.

Differential Calculus

  • Leibnitz Theorem for $-th Derivative: $(uv)_n = u_n v + inom{n}{1} u_{n-1} v_1 + inom{n}{2} u_{n-2} v_2 + dots + u v_n$.
  • Euler Theorem for Homogeneous Functions: If = f(x, y)$ is a homogeneous function of degree $, then rac{partial u}{partial x} + y rac{partial u}{partial y} = n u$.
  • Radius of Curvature (Cartesian): $ ho = rac{[1 + (y')^2]^{3/2}}{|y'|}$.

2. Engineering Mechanics Formula Sheet

Coplanar Force Systems & Equilibrium

  • Resultant of Two Concurrent Forces: = sqrt{P^2 + Q^2 + 2PQ cos heta}$ with angle $ analpha = rac{Qsin heta}{P + Qcos heta}$.
  • Lami's Theorem: For three coplanar concurrent forces in equilibrium, $ rac{P}{sinalpha} = rac{Q}{sineta} = rac{R}{singamma}$.
  • Conditions of Rigid Body Equilibrium: $sum F_x = 0$, $sum F_y = 0$, $sum M_O = 0$.

Friction & Wedges

  • Limiting Friction: = mu_s N$.
  • Angle of Friction ($phi$): $ anphi = mu$.
  • Angle of Repose ($ heta$): For an object on the verge of sliding down an inclined plane, $ heta = phi implies an heta = mu$.

3. Rapid Exam Day Advice

Ensure you practice with authentic timed questions to build numerical speed. Technical English must be cleared with 35 marks to ensure your Mathematics and Mechanics marks are evaluated.