GUJCET Mathematics: Properties Before Long Calculation
GUJCET Mathematics is the third subject for PCM candidates: 40 single-correct MCQs, 1 mark each, −0.25 for a wrong answer. The GSEB Class 12 syllabus supplies the scope - relations and functions, inverse trigonometry, matrices and determinants, continuity and differentiability, applications of derivatives, integration, differential equations, vectors, 3-D geometry, linear programming and probability. The paper is speed-sensitive: a typical question should fall to a standard property, a substitution or a determinant identity rather than lengthy elimination.
Core rules
- Matrices: AB ≠ BA in general; (AB)ᵀ = BᵀAᵀ; a square matrix is invertible only if |A| ≠ 0.
- Determinants: row/column operations preserve the determinant except row interchange (sign change) and row scaling (factor multiplies).
- Calculus: chain rule for composites; integration by parts ∫u dv = uv − ∫v du; for definite integrals use symmetry, split |x| at zero, and change limits with substitution.
- Vectors/3-D: a·b = 0 for perpendicular; a×b gives a normal; shortest distance uses projection.
- Probability: P(A|B) = P(A∩B)/P(B); binomial P(X=r) = ⁿC_r pʳ qⁿ⁻ʳ; without replacement requires hypergeometric reasoning.
Worked example
Evaluate ∫₀¹ x eˣ dx. Use integration by parts with u = x, dv = eˣ dx, so du = dx, v = eˣ. ∫x eˣ dx = x eˣ − ∫eˣ dx = x eˣ − eˣ. Apply limits: [x eˣ − eˣ]₀¹ = (1·e − e) − (0·1 − 1) = 0 − (−1) = 1. The common error is to substitute limits into x eˣ only and forget the second term.
Traps that cost marks
- Cancelling matrices as if they were numbers.
- Forgetting the constant in indefinite integrals when options differ by a constant.
- Ignoring domain restrictions in inverse trigonometric functions.
- Using the wrong principal value for sin⁻¹ or cos⁻¹.
- In 3-D, taking the cross product in the wrong order and reversing the normal.
- In LPP, testing only vertices but missing an unbounded feasible region.
- Misreading “at least”, “exactly” and “at most” in probability.