NDA Algebra: Sets, Progressions, Complex Numbers, Quadratics and Counting

Owned by UPSC NDA Mathematics syllabus. Exact NDA-relevant scope: Sets and set operations, Venn diagrams, De Morgan laws, Cartesian products, relations and equivalence relations; real numbers; complex numbers; binary number system; arithmetic, geometric and harmonic progressions; quadratic equations with real coefficients; graphical solution of linear inequations in two variables; permutations and combinations; binomial theorem; logarithms and applications.

Sets and set operations, Venn diagrams, De Morgan laws, Cartesian products, relations and equivalence relations; real numbers; complex numbers; binary number system; arithmetic, geometric and harmonic progressions; quadratic equations with real coefficients; graphical solution of linear inequations in two variables; permutations and combinations; binomial theorem; logarithms and applications.

Arithmetic fluency, factorisation, linear equations, exponent laws, basic coordinate graph reading.

• Set, subset, universal set, complement, union and intersection
• Relation and equivalence relation
• Complex number z = a + ib, modulus and argument
• Arithmetic, geometric and harmonic progression
• Permutation, combination and factorial
• Quadratic equation and roots
• Logarithm and base conditions

• 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), r != 1; infinite sum a/(1-r) only for |r| < 1
• Quadratic roots: x = [-b +/- sqrt(b^2 - 4ac)]/(2a), a != 0
• For roots alpha, beta: alpha + beta = -b/a; alpha beta = c/a
• nPr = n!/(n-r)!; nCr = n!/[r!(n-r)!]
• Binomial: (x+y)^n = sum C(n,r)x^(n-r)y^r for non-negative integer n
• log_a(xy)=log_a x + log_a y; log_a(x/y)=log_a x - log_a y; log_a(x^k)=k log_a x with a>0, a!=1, x>0
• For z=a+ib: |z| = sqrt(a^2+b^2)

• Infinite GP formula requires |r| < 1.
• Logarithm arguments must be positive in real-number work; base must be positive and not 1.
• Quadratic formula requires a != 0; discriminant controls real-root nature.
• nPr and nCr standard formulas use integers n >= r >= 0.
• Equivalence relation requires reflexive, symmetric and transitive properties.

1. A sequence 5, 9, 13, ... is an AP with d=4. The 20th term is 5+19x4=81. Method choice: use the nth-term formula, not repeated addition.
2. For x^2-7x+10=0, factorisation gives (x-5)(x-2)=0. If factorisation is not obvious, the quadratic formula is the safer general method.
3. From 8 students, choosing 3 for a committee uses 8C3 because order does not matter; assigning president, secretary and treasurer uses 8P3 because roles make order matter.

• Confusing permutation with combination.
• Using infinite-GP sum when |r| >= 1.
• Dropping logarithm domain conditions.
• Changing sign incorrectly in sum/product of quadratic roots.
• Treating a non-equivalence relation as equivalence after checking only one property.

• Knowledge Gap: The concept, definition or theorem needed to solve the item is not understood.
• Recall Gap: The concept was learned but a formula, condition or relation cannot be retrieved accurately.
• Execution Error: The correct method is known but algebra, arithmetic, sign, substitution, diagram or step execution fails.
• Decision / Selection Error: The candidate chooses an inefficient method, continues a time-heavy path, or attempts a low-confidence item without adequate basis.
• Needs Review: Evidence is insufficient or contradictory; do not mislabel the error until another worked item is checked.
• Time pressure note: Time pressure is treated as a contributing condition, not automatically as the root cause.

First identify the algebra family. Check whether the question is structural (sets / relation), formula-led (progressions), root-led (quadratics), counting-led (P&C) or representation-led (complex numbers). Use the shortest valid identity. If the algebra expands rapidly, look for symmetry, roots, factorisation or a counting complement before committing.

Negative marking is a real exam condition, but this page does not promise a universal attempt threshold. Use method certainty and option evidence before committing to a low-confidence answer.

• Required reviewer: NDA Mathematics faculty with demonstrable subject competence.