Chapter 1 of 8

13%

NDA Mathematics★ ~20% of the Mathematics paper — roughly 22–25 of 120 Q (55–63 of 300 marks)

Algebra — Complex Numbers, Quadratics, Sequences & Binomial Theorem

NDA Algebra is not one trick repeated — it's seven Class 11-12 chapters compressed into a fifth of the Maths paper — complex numbers and De Moivre's theorem, the theory of quadratic equations, sets, AP/GP/HP, the binomial theorem, logarithms, and permutations & combinations. At 2.5 marks a question with a 1/3rd penalty, the highest-yield habit is naming which formula bank a question belongs to before you touch a pencil.

⏱ 34 min readmedium-hard difficulty✓ Free · exam-pattern aligned

Timed practiceMCQs from this chapter with instant scoring

FlashcardsFormulas & traps on spaced repetition

NDA PYQsSolved previous-year questions

By the end of this chapter you'll be able to…

1Reduce powers of i using mod-4 cyclicity, and compute the modulus, argument, and polar form of a complex number
2Apply De Moivre's theorem to find nth roots of unity, and use the cube-root-of-unity identities (1+ω+ω²=0, ω³=1) to simplify expressions in ω
3Use the discriminant to classify the nature of quadratic roots, and construct a quadratic equation from a given sum and product of roots
4Solve two- and three-set cardinality word problems using inclusion-exclusion and De Morgan's laws, and count subsets/proper subsets correctly
5Identify AP/GP/HP, apply the correct nth-term and sum formula for each, and use the AM ≥ GM ≥ HM chain (with AM×HM=GM²) to solve two-number mean problems
6Expand a binomial using the general term to find a specific coefficient or the term independent of x, and locate the middle term(s) for both even and odd n
7Apply logarithm laws and change of base, and use the digit-counting formula (⌊n log N⌋+1) for large-power questions
8Distinguish permutations from combinations by whether order matters, and solve restricted-selection and circular-arrangement problems using the complement method

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Why this chapter matters in NDA

Algebra carries 20% of NDA Mathematics — the single heaviest topic in the paper, ahead of Trigonometry (18%), Analytical Geometry (15%), and every other topic individually. But it isn't one dense topic — it's seven separate Class 11-12 CBSE chapters (complex numbers, quadratic theory, sets, AP/GP/HP, the binomial theorem, logarithms, and permutations & combinations) compressed into a fifth of the paper, each contributing roughly 2-5 questions. That breadth is exactly what makes Algebra the highest-leverage revision target: no single sub-area is individually hard — each reduces to two or three memorisable formula banks — but skipping even one costs 2-5 guaranteed-fast marks. At +2.5/−0.8333 marking with a hard 150-minute, no-calculator ceiling across 120 questions, an algebra question you recognise instantly is worth the same 75 seconds as a hard-won trigonometry identity, which is why 'breadth over depth' is the right prep posture here, not a compromise.

Before you start — revise these

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Class 10 CBSE Algebra & Number Systems

Confidence factoring quadratics, simplifying surds, and working with rational/irrational numbers by hand (no calculator) is assumed — NDA builds directly on top of this without re-teaching it.

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Class 11 CBSE Sets, Relations & Basic Complex Numbers

This chapter compresses seven Class 11-12 NCERT chapters into one; having seen complex numbers, AP/GP, and the binomial theorem at least once before makes the compression tractable — meeting all seven for the first time during NDA revision alone is a much harder ask.

Algebra — NDA Mathematics

SSC-style algebra is an identity-recognition game with roughly ten moves. NDA algebra is different in kind: it's the full CBSE Class 11-12 algebra syllabus — complex numbers, quadratic theory, sets, three kinds of progressions, the binomial theorem, logarithms, and counting — compressed into 150 minutes alongside five other Maths topics. Weight, not depth, is what makes this chapter matter: at weightPct: 20 it is the single heaviest topic in the Mathematics paper, ahead of Trigonometry (18%) and Analytical Geometry (15%).

1. What NDA actually asks

Applied to the 120-question, 300-mark Maths paper (+2.5 correct, −0.8333 wrong — the 1/3rd penalty), Algebra's 20% weight works out to roughly 22-25 questions worth 55-63 marks, spread across seven genuinely distinct sub-areas rather than concentrated in one:

Complex numbers — algebra of $i$ , modulus/argument, De Moivre's theorem, roots of unity.
Theory of quadratic equations — discriminant, nature of roots, sum-product relations, equation construction.
Sets — operations, laws, Venn-diagram cardinality.
Sequences and series — AP, GP, HP, and the AM–GM–HM relationship.
Binomial theorem — expansion, general term, middle term.
Logarithms — laws, change of base, digit-counting applications.
Permutations and combinations — arrangement and selection counting.

No calculator is allowed, so every formula here has to be exam-fast by hand. Each sub-area typically contributes 2-5 questions per paper; the efficient strategy is breadth over depth — know all seven formula banks cold rather than mastering one at the expense of the rest.

2. Complex numbers and De Moivre's theorem

$i = - 1$ , $i^{2} = - 1$ . Powers of $i$ cycle with period 4 — $i^{1} = i, i^{2} = - 1, i^{3} = - i, i^{4} = 1$ — and $i^{n}$ depends only on $n mod 4$ . Any four consecutive powers of $i$ sum to zero.

For $z = a + bi$ : modulus $∣ z ∣ = a^{2} + b^{2}$ ; argument $θ = tan^{- 1} (b / a)$ , adjusted for the quadrant of $(a, b)$ . Polar form: $z = r (cos θ + i sin θ)$ .

De Moivre's theorem, for any rational $n$ : $(cos θ + i sin θ)^{n} = cos (n θ) + i sin (n θ)$

It turns repeated multiplication into a single angle multiplication and is the standard route to $n$ th roots of unity: the $n$ solutions of $z^{n} = 1$ are $z_{k} = cos \frac{2 k π}{n} + i sin \frac{2 k π}{n}$ for $k = 0, 1, \dots, n - 1$ .

Cube roots of unity are the most-tested case: $1, ω, ω^{2}$ with $ω = \frac{- 1 + i 3}{2}$ . Two facts unlock nearly every question on them: $1 + ω + ω^{2} = 0$ and $ω^{3} = 1$ — together they collapse messy-looking expressions in $ω$ to a one-line substitution.

3. Theory of quadratic equations

For $a x^{2} + b x + c = 0$ ( $a \neq = 0$ ), the roots are $x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$ , and the discriminant $D = b^{2} - 4 a c$ reveals the character of the roots before you solve anything:

$D$	Nature of roots
$D > 0$ , perfect square	real, rational, unequal
$D > 0$ , not a perfect square	real, irrational, unequal (conjugate surd pair)
$D = 0$	real, equal ( $x = - b /2 a$ , repeated)
$D < 0$	complex conjugate pair

Sum and product of roots, $α, β$ : $α + β = - \frac{b}{a}$ , $α β = \frac{c}{a}$ . Run it in reverse — the equation with given roots $α, β$ is $x^{2} - (α + β) x + α β = 0.$ Every downstream quantity — $α^{2} + β^{2}$ , $α^{3} + β^{3}$ , $\frac{1}{α} + \frac{1}{β}$ — is built from just these two numbers using the identity kit in Section 9. One more worth knowing directly rather than re-deriving: $∣ α - β ∣ = \frac{D}{∣ a ∣}$ .

4. Sets and Venn diagrams

$n (A \cup B) = n (A) + n (B) - n (A \cap B)$

is the entire engine behind two-set word problems ("neither", "only A", "exactly one"). For three sets: $n (A \cup B \cup C) = n (A) + n (B) + n (C) - n (A \cap B) - n (B \cap C) - n (C \cap A) + n (A \cap B \cap C)$

De Morgan's laws: $(A \cup B)^{'} = A^{'} \cap B^{'}$ and $(A \cap B)^{'} = A^{'} \cup B^{'}$ — "neither" questions are really asking for $n (A^{'} \cap B^{'})$ in disguise.

A set with $n$ elements has $2^{n}$ subsets and $2^{n} - 1$ proper subsets (every subset except the set itself; the empty set is counted as proper). NDA likes to test this exponent relationship directly, often paired with a Venn-diagram cardinality question in the same paper.

5. Sequences and series — AP, GP, HP

Arithmetic progression (AP): $n$ th term $a_{n} = a + (n - 1) d$ ; sum $S_{n} = \frac{n}{2} [2 a + (n - 1) d] = \frac{n}{2} (a + l)$ , where $l$ is the last term.

Geometric progression (GP): $n$ th term $a_{n} = a r^{n - 1}$ ; sum $S_{n} = \frac{a ( r ^{n} - 1 )}{r - 1}$ for $r \neq = 1$ ; sum to infinity $S_{\infty} = \frac{a}{1 - r}$ , valid only when $∣ r ∣ < 1$ — a recurring NDA question type ( $8 + 4 + 2 + 1 + \dots$ -style series).

Harmonic progression (HP): $a, b, c$ are in HP iff their reciprocals $\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in AP. There's no direct HP sum formula — always convert to the reciprocal AP first, solve there, then invert back.

The AM ≥ GM ≥ HM chain, for two positive numbers $a, b$ : $AM = \frac{a + b}{2}$ , $GM = ab$ , $HM = \frac{2 ab}{a + b}$ , and always $AM \geq GM \geq HM$ , equality only when $a = b$ . Useful shortcut: $AM \times HM = GM^{2}$ — given any two of the three means, the third follows without solving for $a, b$ first.

6. Binomial theorem

$(x + y)^{n} = \sum_{r = 0}^{n} (r n) x^{n - r} y^{r}$

The expansion has exactly $n + 1$ terms. The general term — the workhorse for "coefficient of $x^{k}$ " and "term independent of $x$ " questions — is $T_{r + 1} = (r n) x^{n - r} y^{r} .$ Set the exponent you want equal to the target power, solve for $r$ first, then substitute — don't try to eyeball $r$ .

Middle term: if $n$ is even, there's one middle term, the $(\frac{n}{2} + 1)$ th; if $n$ is odd, there are two — the $(\frac{n + 1}{2})$ th and $(\frac{n + 3}{2})$ th.

Sum of binomial coefficients: $(0 n) + (1 n) + \dots + (n n) = 2^{n}$ (put $x = y = 1$ ). The coefficients at odd positions sum to the same total as those at even positions — each is $2^{n - 1}$ (put $x = 1, y = - 1$ and combine with the identity above).

7. Logarithms

$lo g_{a} (mn) = lo g_{a} m + lo g_{a} n lo g_{a} (\frac{m}{n}) = lo g_{a} m - lo g_{a} n lo g_{a} (m^{k}) = k lo g_{a} m$

$lo g_{a} a = 1$ , $lo g_{a} 1 = 0$ . Change of base: $lo g_{a} b = \frac{lo g _{c} b}{lo g _{c} a}$ , with the useful corollary $lo g_{a} b \cdot lo g_{b} a = 1$ .

NDA's favourite application is digit-counting: the number of digits in a positive integer $N$ equals $⌊ lo g_{10} N ⌋ + 1$ . Given $lo g_{10} 2 = 0.3010$ , a typical question asks for the digit count of $2^{30}$ or $2^{50}$ — compute $n lo g_{10} 2$ , take the floor, add 1. No calculator means these standard log values ( $lo g 2, lo g 3, lo g 7$ ) are worth memorising to four decimal places.

8. Permutations and combinations

$^{n} P_{r} = \frac{n !}{( n - r )!}^{n} C_{r} = \frac{n !}{r ! ( n - r )!} = \frac{^{n} P _{r}}{r !}$

$^{n} C_{r} =^{n} C_{n - r}$ ; Pascal's rule: $^{n} C_{r} +^{n} C_{r - 1} =^{n + 1} C_{r}$ . Permutations count order, combinations don't — the single most exam-costly confusion in this sub-area. "Arrange", "seat", "form a number" → permutation; "select", "choose a committee", "pick a team" → combination.

Restricted selection ("at least one woman", "no two together") is almost always faster by complement — total ways minus the ways that violate the restriction — than by summing every valid case separately.

Circular arrangement of $n$ distinct objects: $(n - 1)!$ ways (fix one object to remove the rotational symmetry); halve it to $\frac{( n - 1 )!}{2}$ when clockwise and anticlockwise arrangements count as identical (garlands, bracelets).

9. The x + 1/x identity chain

A slice of NDA algebra overlaps with the identity-drilling style SSC CGL is built around, and it surfaces inside larger quadratic or binomial questions. Given $x + \frac{1}{x} = k$ : $x^{2} + \frac{1}{x ^{2}} = k^{2} - 2 x^{3} + \frac{1}{x ^{3}} = k^{3} - 3 k$ and the three-variable cube identity $a^{3} + b^{3} + c^{3} - 3 ab c = (a + b + c) (a^{2} + b^{2} + c^{2} - ab - b c - c a),$ which collapses to $3 ab c$ whenever $a + b + c = 0$ . NDA doesn't drill these the way SSC does, but keeping them as fast-recall facts saves 60-90 seconds whenever they appear as a hidden step inside a bigger question.

10. Solved PYQ-style examples

Q1. Find $i^{1} + i^{2} + i^{3} + i^{4}$ . Solution. $i - 1 - i + 1 = 0$ — any four consecutive powers of $i$ sum to zero.

Q2. Find $∣3 + 4 i ∣$ . Solution. $3^{2} + 4^{2} = 25 = 5$ .

Q3. For what value(s) of $k$ does $x^{2} + k x + 9 = 0$ have equal roots? Solution. $D = k^{2} - 36 = 0 \Rightarrow k = \pm 6$ .

Q4. Form the quadratic equation whose roots are $2$ and $- 3$ . Solution. Sum $= - 1$ , product $= - 6$ . Equation: $x^{2} - (- 1) x + (- 6) = 0 \Rightarrow x^{2} + x - 6 = 0$ .

Q5. In a survey of 100 people, 60 read newspaper A, 40 read newspaper B, and 20 read both. How many read at least one? Solution. $n (A \cup B) = 60 + 40 - 20 = 80$ .

Q6. The 3rd term of a GP is 12 and the 6th term is 96. Find the first term and common ratio. Solution. $a r^{2} = 12$ , $a r^{5} = 96 \Rightarrow r^{3} = 8 \Rightarrow r = 2$ , so $a = 12/4 = 3$ .

Q7. Find the middle term of $(x + \frac{1}{x})^{6}$ . Solution. $n = 6$ is even, so the middle term is the 4th: $T_{4} = (3 6) x^{3} (\frac{1}{x})^{3} = 20$ . (It's also the constant term — the exponents cancel.)

Q8. How many 4-member committees can be formed from 10 people? Solution. $^{10} C_{4} = \frac{10 \cdot 9 \cdot 8 \cdot 7}{4 !} = 210$ .

11. Common traps

$i^{n}$ sign errors — reduce $n mod 4$ first; don't hand-count through large exponents.
$D = 0$ mistaken for "no real roots" — equal roots ARE real. Only $D < 0$ gives complex conjugate roots.
Confusing proper subsets ( $2^{n} - 1$ ) with all subsets ( $2^{n}$ ) — "proper" excludes only the set itself, not the empty set.
Using $a_{n} = a + n d$ instead of $a + (n - 1) d$ — the single most common AP slip, and it silently shifts every downstream answer.
Applying the GP infinite-sum formula when $∣ r ∣ \geq 1$ — the series diverges; $S_{\infty} = a / (1 - r)$ only holds for $∣ r ∣ < 1$ .
Mixing up $^{n} P_{r}$ and $^{n} C_{r}$ — re-read whether the question implies order (arrangement/rank/seating) before choosing the formula.
Guessing pairs for AM-GM word problems instead of solving the quadratic — two numbers with a given AM and GM satisfy $x^{2} - 2 (AM) x + GM^{2} = 0$ ; solve it rather than hunting for numbers that merely add up right.

12. Training protocol

Because Algebra spans seven distinct chapters, prep should be breadth-first, not depth-first: one formula sheet per sub-area (Sections 2-8), reviewed on rotation rather than mastering complex numbers before touching sets. In the exam, the discriminant sign, the AP/GP $n$ th-term formula, and the $^{n} P_{r}$ vs $^{n} C_{r}$ distinction decide roughly half of all algebra marks — make those three reflexive before anything else. With 150 minutes for 120 questions (~75 seconds/question average across the whole paper), an algebra question that's still unsolved past 90 seconds is a candidate to mark and return to, not to grind through — the 1/3rd penalty punishes a rushed wrong guess more than a skipped question ever does.

Key formulas & results

Everything to memorise for the exam hall, in one card. Screenshot this for revision.

Powers of i (cyclicity)

i = √−1, i² = −1, i³ = −i, i⁴ = 1 — period 4, so iⁿ depends only on n mod 4

Any four consecutive powers of i sum to zero — a fast check for expressions like i¹³+i¹⁴+i¹⁵+i¹⁶.

Modulus, argument, polar form

z = a+bi ⇒ |z| = √(a²+b²), θ = tan⁻¹(b/a) adjusted for quadrant; z = r(cosθ + i sinθ)

Always fix θ's quadrant from the signs of a and b — tan⁻¹ alone only gives the reference angle.

De Moivre's theorem & nth roots of unity

(cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ); roots of zⁿ=1 are z_k = cos(2kπ/n) + i sin(2kπ/n), k=0,…,n−1

Turns repeated multiplication into single angle-multiplication; the standard route to any 'find all nth roots' question.

Cube roots of unity

1, ω, ω² with ω = (−1+i√3)/2; 1+ω+ω² = 0 and ω³ = 1

These two facts alone collapse almost every ω-expression question to a one-line substitution.

Quadratic formula & discriminant

x = (−b ± √(b²−4ac)) / 2a, D = b²−4ac

D>0 perfect square → real rational unequal; D>0 non-square → real irrational unequal; D=0 → real equal; D<0 → complex conjugate pair.

Sum & product of roots / equation construction

α+β = −b/a, αβ = c/a; equation with roots α,β is x² − (α+β)x + αβ = 0

Every downstream quantity (α²+β², α³+β³, 1/α+1/β) is built from just these two numbers.

Root-difference shortcut

|α − β| = √D / |a|

Faster than finding α and β individually and subtracting.

Set union (inclusion-exclusion)

n(A∪B) = n(A)+n(B)−n(A∩B); n(A∪B∪C) = n(A)+n(B)+n(C)−n(A∩B)−n(B∩C)−n(C∩A)+n(A∩B∩C)

The entire engine behind 'at least one', 'exactly one', and 'neither' word problems.

De Morgan's laws & subset count

(A∪B)' = A'∩B', (A∩B)' = A'∪B'; a set with n elements has 2ⁿ subsets and 2ⁿ−1 proper subsets

'Neither' questions are n(A'∩B') in disguise; 'proper' excludes only the full set, not the empty set.

AP — nth term & sum

aₙ = a+(n−1)d; Sₙ = (n/2)[2a+(n−1)d] = (n/2)(a+l)

The single most common slip is aₙ = a+nd — it silently shifts every downstream answer by one term.

GP — nth term & sums

aₙ = ar^(n−1); Sₙ = a(rⁿ−1)/(r−1) for r≠1; S∞ = a/(1−r), valid only for |r|<1

Applying S∞ when |r|≥1 (a divergent series) is a recurring NDA trap.

HP & the AM–GM–HM chain

a,b,c in HP ⇔ 1/a,1/b,1/c in AP; for two positive numbers, AM=(a+b)/2, GM=√(ab), HM=2ab/(a+b), and AM≥GM≥HM with AM×HM=GM²

No direct HP sum formula exists — always convert to the reciprocal AP first. Given any two means, AM×HM=GM² gives the third without solving for a,b.

Binomial expansion & general term

(x+y)ⁿ = Σ C(n,r) x^(n−r) y^r; general term T_(r+1) = C(n,r) x^(n−r) y^r

For 'coefficient of x^k' or 'term independent of x', set the target exponent equal to n−r and solve for r first — never eyeball it.

Middle term rule

n even → one middle term, the (n/2 + 1)th; n odd → two middle terms, the ((n+1)/2)th and ((n+3)/2)th

Always check the parity of n before deciding whether to expect one or two middle terms.

Sum of binomial coefficients

C(n,0)+C(n,1)+…+C(n,n) = 2ⁿ; sum of odd-position coefficients = sum of even-position coefficients = 2^(n−1)

Put x=y=1 for the first identity; combine with x=1,y=−1 for the odd/even split.

Logarithm laws, change of base & digit count

log_a(mn)=log_a m+log_a n; log_a(m/n)=log_a m−log_a n; log_a(mᵏ)=k·log_a m; log_a b = log_c b / log_c a; digits in N = ⌊log₁₀N⌋+1

log_a b · log_b a = 1 is the useful change-of-base corollary. Memorise log 2, log 3, log 7 to four decimals — no calculator means these are recalled, not computed.

Permutations, combinations & Pascal's rule

ⁿPᵣ = n!/(n−r)!; ⁿCᵣ = n!/(r!(n−r)!) = ⁿPᵣ/r!; ⁿCᵣ = ⁿC_(n−r); ⁿCᵣ + ⁿC_(r−1) = ⁿ⁺¹Cᵣ

Permutations count order, combinations don't — the single most exam-costly confusion in this sub-area.

Circular arrangement & restricted selection

n distinct objects in a circle: (n−1)! ways; halve to (n−1)!/2 when clockwise/anticlockwise are identical (garlands, bracelets)

'At least one', 'no two together' style restrictions are almost always faster by complement (total − violating cases) than by direct casework.

x + 1/x identity chain

Given x+1/x=k: x²+1/x²=k²−2, x³+1/x³=k³−3k; also a³+b³+c³−3abc=(a+b+c)(a²+b²+c²−ab−bc−ca), which collapses to 3abc when a+b+c=0

A hidden step inside larger quadratic/binomial questions — keep these as instant-recall facts, not re-derivations.

NDA test series

Ready to be tested on this?

Algebra — Complex Numbers, Quadratics, Sequences & Binomial Theorem shows up in every NDA paper — sit a timed one and find out where you actually stand.

22–26Qs in paper

Very highpriority

Full NDA mocks, real clock
Mathematics-only sectionals & chapter tests
All-India rank + topic-level analysis

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Traps NDA sets — and how to dodge them

These are the exact option-traps and misreads that cost marks under negative marking.

WATCH OUT

✗ Sign errors reducing iⁿ for large n

✓ Reduce n mod 4 first, don't hand-count through the exponent. i¹=i, i²=−1, i³=−i, i⁴=1, and the pattern repeats exactly every 4 powers.

WATCH OUT

✗ Treating D=0 as 'no real roots'

✓ Equal roots ARE real (x=−b/2a, repeated). Only D<0 gives a genuinely non-real (complex conjugate) pair — don't conflate 'equal' with 'absent'.

WATCH OUT

✗ Confusing ω and ω² in cube-root-of-unity questions, or forgetting ω³=1

✓ Keep 1+ω+ω²=0 and ω³=1 written on scratch space before simplifying — most ω-expressions collapse in one substitution once these are fixed in front of you.

WATCH OUT

✗ Using aₙ=a+nd instead of aₙ=a+(n−1)d for an AP

✓ The nth term is 'a plus (n−1) jumps of d', not n jumps — write out the first few terms (a, a+d, a+2d,…) if unsure which term you're actually on.

WATCH OUT

✗ Applying S∞=a/(1−r) when |r|≥1

✓ Check |r|<1 before reaching for the infinite-sum formula — a series with r≥1 or r≤−1 diverges and has no finite sum, regardless of what the formula spits out.

WATCH OUT

✗ Mixing up ⁿPᵣ and ⁿCᵣ

✓ Re-read the question for whether order matters: 'arrange/seat/rank/form a number' → permutation; 'select/choose/pick a committee' → combination. Decide this before writing any formula.

WATCH OUT

✗ Guessing number pairs for AM-GM word problems instead of solving the quadratic

✓ Two numbers with a given AM and GM satisfy x²−2(AM)x+GM²=0 — solve it directly rather than hunting for integers that happen to fit.

WATCH OUT

✗ Eyeballing r in the binomial general term instead of solving for it algebraically

✓ Set the target exponent equal to n−r (or whatever exponent condition is given), solve for r first, then substitute into C(n,r) — guessing r under time pressure is a frequent source of off-by-one errors.

WATCH OUT

✗ Confusing proper subsets (2ⁿ−1) with all subsets (2ⁿ)

✓ 'Proper' excludes only the full set itself — the empty set still counts as a proper subset. Don't subtract twice.

WATCH OUT

✗ Forgetting the '+1' in the digit-counting formula, or using an imprecise log value

✓ Digits in N = ⌊log₁₀N⌋+1, not just the floor. And use the exact memorised log value (e.g. log 2 = 0.3010) — a rounded guess can shift the floor and silently change the answer.

Exam-pattern practice

PYQ-style questions with full solutions. Work through them as a readiness check — mark yourself honestly and get your gap report at the end.

Readiness check

Are you exam-ready for Algebra — Complex Numbers, Quadratics, Sequences & Binomial Theorem?

14 problems from this chapter. Try each one, reveal the worked solution, mark yourself honestly — get your gap report at the end.

14 questions~10 min worth ~5 marks in NDA exams

5-minute revision

The whole chapter, distilled. Read this the night before the exam.

•iⁿ cycles with period 4 — reduce n mod 4; any 4 consecutive powers of i sum to 0.
•|z|=√(a²+b²); De Moivre's: (cosθ+i sinθ)ⁿ=cos nθ+i sin nθ; nth roots of unity via z_k=cos(2kπ/n)+i sin(2kπ/n).
•Cube roots of unity: 1+ω+ω²=0, ω³=1 — nearly every ω-question collapses with these two facts.
•Discriminant D=b²−4ac: D=0 → real EQUAL roots (not 'no roots'); D<0 → complex conjugate pair.
•α+β=−b/a, αβ=c/a; equation from roots: x²−(sum)x+(product)=0; |α−β|=√D/|a|.
•n(A∪B)=n(A)+n(B)−n(A∩B); for three sets, also subtract pairwise intersections and add back the triple intersection.
•A set with n elements has 2ⁿ subsets and 2ⁿ−1 proper subsets (empty set counts as proper).
•AP: aₙ=a+(n−1)d, Sₙ=(n/2)(a+l). GP: aₙ=ar^(n−1), S∞=a/(1−r) only for |r|<1.
•HP has no direct sum formula — invert to the reciprocal AP first. AM≥GM≥HM always; AM×HM=GM².
•Binomial general term T_(r+1)=C(n,r)x^(n−r)y^r — solve for r from the target exponent before substituting, never guess it.
•n even → 1 middle term; n odd → 2 middle terms. ΣC(n,r)=2ⁿ; the odd- and even-position coefficient sums are each 2^(n−1).
•Change of base: log_a b = log_c b/log_c a; digits in N = ⌊log₁₀N⌋+1 — memorise log 2, log 3, log 7 to 4 decimals.
•ⁿPᵣ counts order, ⁿCᵣ doesn't; ⁿCᵣ=ⁿC_(n−r); circular arrangement = (n−1)!, halved when reflections are identical.
•Restricted selection ('at least one', 'no two together') → solve by complement (total − violating), not exhaustive casework.
•x+1/x=k ⇒ x²+1/x²=k²−2, x³+1/x³=k³−3k; a³+b³+c³−3abc=(a+b+c)(a²+b²+c²−ab−bc−ca), collapses to 3abc when a+b+c=0.
•Breadth beats depth: seven formula banks, each worth only 2-5 marks — rotate revision across all seven rather than mastering one at the expense of the rest.

NDA question blueprint

How this topic is asked, tier by tier — so you can prep to the pattern.

Typical weightage: 55–63 of 300 Mathematics marks (~22–25 questions × 2.5 marks, no partial credit)

Question style	Marks each	Typical count	What it tests
Complex numbers & De Moivre's theorem	2.5	~3-4	i-power reduction, modulus/argument, De Moivre's theorem, nth roots of unity, cube-root-of-unity identities
Theory of quadratic equations	2.5	~3-4	Discriminant/nature of roots, sum-product relations, equation construction, root-difference shortcut
Sets	2.5	~2-3	Two- and three-set inclusion-exclusion, De Morgan's laws, subset counting
Sequences & series (AP/GP/HP)	2.5	~3-4	nth-term and sum formulas, GP infinite sum, AM-GM-HM relation
Binomial theorem	2.5	~3	General term, coefficient/term-independent-of-x questions, middle term, sum of coefficients
Logarithms	2.5	~2-3	Log laws, change of base, digit-counting applications
Permutations & combinations	2.5	~3-4	ⁿPᵣ vs ⁿCᵣ, restricted selection via complement, circular arrangement

Prep strategy

Week 1: build one formula sheet per sub-area (complex numbers, quadratics, sets, sequences, binomial, logs, P&C) and drill each independently — this chapter rewards recall speed, not deep problem-solving, so treat it like seven short vocabulary lists rather than one long topic.
Week 2: run timed mixed sets pulling 2-3 questions from each sub-area per sheet — the real exam interleaves all seven, so revision should stop separating them by week 2.
Final week: drill the three highest-leverage reflexes specifically — discriminant sign, AP/GP nth-term formula, and ⁿPᵣ vs ⁿCᵣ — since together they decide roughly half of all algebra marks; then run a full no-calculator timed set at 75 seconds/question to simulate real pacing pressure.

Exam-hall strategy

Battle-tested tips from mentors and toppers for this topic under the sectional clock.

Name the sub-area before you touch a pencil — 'this is a De Moivre question' or 'this is a sum-product-of-roots question' — because each of the seven areas has its own 15-30 second formula-recall path once identified.
Treat this as seven separate formula sheets, not one big topic — revise on rotation (complex numbers one day, quadratics the next, …) rather than mastering one sub-area before touching the next.
The discriminant sign, the AP/GP nth-term formula, and the ⁿPᵣ vs ⁿCᵣ distinction alone decide roughly half of all algebra marks — drill these three until they're reflexive before polishing anything else.
For binomial 'coefficient of x^k' questions, always solve for r algebraically from the exponent condition before substituting — guessing r is the single most common source of an off-by-one wrong answer.
No calculator is allowed — keep log 2, log 3, log 7 memorised to four decimals, and keep i¹…i⁴ and the ω, ω² relations as instant recall, not re-derivable facts.
An algebra question unsolved past 90 seconds is a candidate to mark and return to, not to grind through — at −0.8333 per wrong answer, a rushed guess costs more than a skipped question.
For restricted P&C selections ('at least one', 'no two together'), reach for the complement (total − violating) by default — it's faster than direct casework in the overwhelming majority of NDA questions.

Beyond the exam

Where this skill shows up in the job you're competing for — and in life.

Radar, sonar and signal phase analysis

Complex numbers in polar form (r, θ) are the natural language for describing wave amplitude and phase — radar and sonar signal processing, directly relevant to Naval and Air Force branches, represents oscillating signals as complex exponentials and combines them using exactly the modulus-argument arithmetic this chapter teaches.

Ballistic trajectory and range calculations

A projectile's range, maximum height, and time of flight all fall out of solving a quadratic in time or angle — the discriminant condition for 'does this trajectory reach the target' is a direct real-world instance of the D≥0 check drilled in this chapter.

Cryptographic key spaces and mission team selection

The size of a cryptographic key space is a counting problem (permutations of possible keys), and forming patrol teams, duty rosters, or committees from a larger pool is a textbook combinations problem — both scale up the exact ⁿPᵣ/ⁿCᵣ logic taught here to real operational planning.

Sound intensity, seismic and signal-strength scales

Decibel (sound), Richter (seismic), and signal-to-noise-ratio scales are all logarithmic — a small change in the logged value corresponds to a large change in the underlying physical quantity, the same log-law and change-of-base machinery applied to instrumentation read out in the field.

Where else this topic is tested

Prepare once, score in every exam that asks it.

CDS (Combined Defence Services) Elementary MathematicsVery high — near-identical syllabus, same seven sub-areas, same MCQ style

AFCAT (technical/numerical sections)Medium — lighter treatment, mostly quadratics, sets, and basic P&C

JEE Main (Complex Numbers, Sequences & Series, Binomial Theorem, P&C)Conceptual overlap — same formula banks at significantly higher computational difficulty

CUET MathematicsHigh — same NCERT-level content and question style

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Questions aspirants ask

Pulled from the Q&A community and mentor sessions.

About the same theoretical depth as Class 11-12 NCERT, but compressed and sped up — NDA won't ask you to prove De Moivre's theorem or derive the AM-GM inequality, only apply it fast under a strict per-question time budget with no calculator. The compression, not the difficulty, is what makes this chapter feel heavy.

Quadratic theory (discriminant + sum-product) and permutations & combinations tend to repeat most reliably across years, with complex numbers (cube roots of unity, i-power reduction) close behind. But because each sub-area is only 2-5 questions, skipping any one caps your algebra score at roughly 80-85% even if you ace the rest — breadth genuinely outperforms depth here.

Those three (to four decimal places) cover the overwhelming majority of NDA's digit-counting and log-simplification questions, since most integers factor into small primes. log 5 = 1 − log 2 is free once you have log 2, and log 10 = 1 always — you rarely need anything beyond that set.

The aₙ=a+nd vs a+(n−1)d slip in AP problems and the ⁿPᵣ vs ⁿCᵣ mix-up in counting problems are neck-and-neck for most common — both are one-symbol errors that produce a clean-looking wrong answer, which is exactly why they're dangerous: nothing about the wrong number looks obviously off.

Memorise them as instant-recall facts. NDA is pure MCQ — you're never asked to show a derivation, only to spot when an identity applies and apply it within seconds. Re-deriving under exam pressure costs exactly the time the formula bank exists to save.