AP EAMCET: Mathematics (Engineering stream)

AP EAMCET Mathematics (Engineering stream) carries 80 questions and rewards fast recognition of standard forms across Algebra, Trigonometry, Vector Algebra, Measures of Dispersion, Probability, Calculus, Coordinate Geometry and Differential Equations. Most items are one- or two-step; long derivations are rarely needed.

Core rules to keep ready

• Limits: use standard expansions near 0: sin x = x − x³/6 + …, cos x = 1 − x²/2 + …, e^x = 1 + x + x²/2 + …, ln(1+x) = x − x²/2 + … .
• Definite integrals: use ∫₀ᵃ f(x) dx = ∫₀ᵃ f(a−x) dx; an odd function integrated over [−a, a] gives 0.
• Conics: for y² = 4ax, focus (a, 0), directrix x = −a, latus rectum 4a. Ellipse e² = 1 − b²/a²; hyperbola e² = 1 + b²/a².
• Vectors: scalar projection of a on b is (a·b)/|b|; it is signed.
• Probability: P(A∪B) = P(A) + P(B) − P(A∩B); P(A|B) = P(A∩B)/P(B).

Worked example

Find lim_{x→0} (sin x − x)/x³. Direct substitution gives 0/0. Use sin x = x − x³/6 + O(x⁵). Then sin x − x = −x³/6 + O(x⁵), so dividing by x³ gives −1/6 + O(x²) → −1/6. L’Hôpital three times gives the same: numerator derivatives cos x − 1, −sin x, −cos x; at 0 the third is −1, while the denominator’s third derivative is 6, so the limit is −1/6.

Traps

• Applying L’Hôpital when the form is not 0/0 or ∞/∞.
• Dropping the sign of a scalar projection when a·b is negative.
• Swapping the ellipse and hyperbola eccentricity formulas.
• Forgetting to change limits when substituting in a definite integral.