Mathematics
What the Mathematics section rewards
CUSAT CAT B.Tech mathematics is a multiple-choice test built on the Class 11–12 core: algebra, trigonometry, coordinate geometry, calculus, vectors and 3D geometry, and probability. Most questions are short but combine two ideas, so recognising a standard form matters more than long computation.
Core rules to keep ready
- For a 2×2 matrix A = [[a,b],[c,d]], det = ad − bc; A is invertible only when det ≠ 0, and A⁻¹ = (1/det)[[d,−b],[−c,a]].
- A limit of the form 0/0 or ∞/∞ permits L'Hôpital only after the indeterminate form is confirmed; differentiate numerator and denominator separately.
- For vectors, a·(b×c) is the scalar triple product and equals the signed volume of the parallelepiped; it is zero exactly when the three vectors are coplanar.
- Conditional probability: P(A|B) = P(A∩B)/P(B), with P(B) > 0; independent events satisfy P(A∩B) = P(A)P(B).
- For a conic ax² + 2hxy + by² + ... = 0, the discriminant h² − ab distinguishes ellipse (negative), parabola (zero), hyperbola (positive), ignoring degenerate cases.
Worked example
If A = [[2,1],[1,2]], evaluate A² − 4A + 3I. A² = [[2,1],[1,2]]·[[2,1],[1,2]] = [[5,4],[4,5]]. 4A = [[8,4],[4,8]] and 3I = [[3,0],[0,3]]. So A² − 4A + 3I = [[5−8+3, 4−4+0],[4−4+0, 5−8+3]] = [[0,0],[0,0]]. The zero result is not an accident: A satisfies its characteristic equation λ² − 4λ + 3 = 0, which is the Cayley–Hamilton theorem in action.
Traps that cost marks
- Dropping the negative root when solving trigonometric or modulus equations.
- Applying the determinant formula after a row operation that has already changed the determinant's value.
- Forgetting that a definite integral's value is independent of the dummy variable but not of the limits.
- Mixing up nPr and nCr when order is irrelevant.