Mathematics: Algebra, Calculus, Vectors and Probability
Assam CEE Mathematics tests algebra, trigonometry, coordinate geometry, calculus, vectors and probability. Questions are short but hinge on exact domain conditions and formula recognition; one misread condition can change the correct option.
Core rules
- Functions: exclude values that make a denominator zero, put a negative under an even root, or make a logarithm argument non-positive.
- Quadratics: discriminant D = b² − 4ac; equal roots when D = 0; sum of roots = −b/a and product = c/a.
- Trigonometry: sin²θ + cos²θ = 1. The general solution of sinθ = sinα is θ = nπ + (−1)ⁿα.
- Limits and derivatives: L'Hôpital's rule applies only to 0/0 or ∞/∞ forms. d/dx(ln x) = 1/x; use the chain rule for composites.
- Integration: a definite integral has no arbitrary constant; changing the variable requires changing the limits.
- Vectors: a·b = |a||b|cosθ; a×b is perpendicular to both; three vectors are coplanar when their scalar triple product is zero.
- Probability: P(A∪B) = P(A) + P(B) − P(A∩B); conditional probability P(A|B) = P(A∩B)/P(B).
Worked example
Evaluate ∫₀¹ x e^(x²) dx.
Let u = x², so du = 2x dx and x dx = du/2. When x = 0, u = 0; when x = 1, u = 1. The integral becomes ½∫₀¹ e^u du = ½(e − 1). The reasoning is that substitution must transform the integrand, the differential and both limits together; no back-substitution is needed for a definite integral.
Traps that cost marks
- Writing √(x²) = x instead of |x|.
- Evaluating sin⁻¹x outside [−1, 1].
- Adding +C to a definite integral.
- Treating a·b = 0 as parallel vectors; it means perpendicular.
- Using permutations when order does not matter.
- Drawing without replacement as independent events.