Algebra
Algebra in NEB Class 12 covers counting (permutations and combinations), the binomial theorem, complex numbers, matrices and determinants, sequences and series, quadratic equations, and linear programming. Questions reward exact formula use and careful case analysis.
Core rules
- Permutations: P(n,r)=n!/(n-r)!; combinations C(n,r)=n!/[r!(n-r)!]. Circular arrangements of n distinct objects: (n-1)!; with repeated objects divide by the factorial of repetitions.
- Binomial expansion: (a+b)^n = sum from r=0 to n of C(n,r)a^(n-r)b^r. The general term is T_(r+1)=C(n,r)a^(n-r)b^r; the term independent of x has exponent 0.
- Complex numbers: i²=-1; modulus |a+bi|=√(a²+b²); conjugate a-bi; divide by multiplying by the conjugate.
- Matrices: multiplication requires columns of the first matrix to equal rows of the second. A^(-1) exists only if det A ≠ 0.
- AP: a_n=a+(n-1)d, S_n=(n/2)[2a+(n-1)d]. GP: a_n=ar^(n-1), S_n=a(r^n-1)/(r-1) for r ≠ 1.
Worked example
Find the term independent of x in (2x² - 1/x)^9. The general term is T_(r+1)=C(9,r)(2x²)^(9-r)(-x^(-1))^r=C(9,r)2^(9-r)(-1)^r x^(18-3r). Set 18-3r=0, so r=6. The term is C(9,6)2³(-1)^6=84×8=672.
Traps
- Using r as the term number rather than the index starting at 0.
- Losing the sign from a negative second term.
- Mixing up P(n,r) and C(n,r) in committee versus arrangement questions.
- Forgetting that a determinant is a scalar, while a matrix is an array.
- In linear programming, testing a corner that does not satisfy all constraints.