Mathematics
TS ECET Mathematics tests diploma-level algebra, trigonometry, coordinate geometry, calculus and differential equations. Questions are objective and formula-driven, so recognise the standard form first, then compute cleanly.
Core rules
- Determinants: expand by cofactors with sign (-1)^(i+j); a matrix is singular exactly when its determinant is zero. Cramer's rule applies only when the determinant is non-zero.
- Partial fractions: factor the denominator completely; use a constant for each distinct linear factor, and a linear numerator for an irreducible quadratic.
- Trigonometry: use compound-angle identities, and remember principal ranges for inverse functions, e.g. sin^-1 x lies in [-pi/2, pi/2].
- Straight lines and circles: y = mx + c, x cos a + y sin a = p; circle (x-h)^2 + (y-k)^2 = r^2.
- Calculus: product, quotient and chain rules; maxima/minima from f'(x)=0 and the sign of f''(x).
- Integration: standard forms, substitution, by parts; for definite integrals change limits when you substitute. Differential equations: separate variables or use an integrating factor for dy/dx + Py = Q.
Worked example
Evaluate the integral from 0 to 1 of x e^x dx. Let u=x, dv=e^x dx, so du=dx, v=e^x. Then the indefinite integral is x e^x - e^x = e^x(x-1). Apply limits: at x=1, e(0)=0; at x=0, 1(-1)=-1. Value = 0 - (-1) = 1.
Traps
- Dropping the constant +C in indefinite integrals.
- Forgetting the negative sign when differentiating cos x or integrating sin x.
- Using degrees in calculus limits; use radians.
- In a definite integral after substitution, keeping the old limits.
- Treating sqrt(x^2) as x without the absolute value.