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UPSC NDA & NA Mathematics Flashcards
55 question-and-answer cards covering Mathematics as it is examined in UPSC NDA & NA. 24 of them are printed below, taken from across the deck — no signup, no paywall on the preview.
24 sample cards from the Mathematics deck
Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.
How many permutations of n objects exist when p, q, … are alike of certain kinds?
n! / (p! · q! · …), dividing by the factorials of the counts of indistinguishable objects.
State two key identities relating combinations.
nCr = nC(n−r), and Pascal's rule: nCr + nC(r−1) = (n+1)Cr.
In how many ways can n distinct objects be arranged in a circle?
(n − 1)! ways (circular permutations), since rotations are considered identical.
State the Binomial Theorem for a positive integral index n.
(a + b)ⁿ = Σ (from r=0 to n) nCr · aⁿ⁻ʳ · bʳ.
Write the general (r+1)th term in the expansion of (a + b)ⁿ.
T(r+1) = nCr · aⁿ⁻ʳ · bʳ.
How many terms are in the expansion of (a + b)ⁿ, and what is the sum of the binomial coefficients?
There are n + 1 terms. The sum of all binomial coefficients (setting a = b = 1) is 2ⁿ.
How do you find the middle term(s) in the binomial expansion of (a + b)ⁿ?
If n is even, there is one middle term: the (n/2 + 1)th term. If n is odd, there are two middle terms: the ((n+1)/2)th and ((n+3)/2)th terms.
Define a square matrix, a diagonal matrix, and a scalar matrix.
Square matrix: equal number of rows and columns. Diagonal matrix: a square matrix with all off-diagonal entries zero. Scalar matrix: a diagonal matrix with all diagonal entries equal.
What is an identity matrix and a null (zero) matrix?
Identity matrix I: a diagonal matrix with all diagonal entries 1 (acts as multiplicative identity). Null matrix: a matrix with every entry 0 (additive identity).
Define symmetric and skew-symmetric matrices.
Symmetric: A = Aᵀ (aᵢⱼ = aⱼᵢ). Skew-symmetric: A = −Aᵀ (aᵢⱼ = −aⱼᵢ), forcing all diagonal entries to be 0.
What condition must hold for matrix multiplication AB to be defined, and what is the order of the product?
The number of columns of A must equal the number of rows of B. If A is m×n and B is n×p, then AB is m×p.
State the transpose properties (AB)ᵀ and (A + B)ᵀ.
(AB)ᵀ = BᵀAᵀ and (A + B)ᵀ = Aᵀ + Bᵀ.
How is the determinant of a 2×2 matrix [[a, b], [c, d]] computed?
det = ad − bc.
State two properties of determinants regarding row interchange and a scalar multiple of a row.
Interchanging two rows (or columns) changes the sign of the determinant. Multiplying one row (or column) by k multiplies the determinant by k.
What is the value of a determinant if two rows (or columns) are identical or proportional?
The determinant equals zero.
State the property det(AB) and det(kA) for an n×n matrix.
det(AB) = det(A)·det(B), and det(kA) = kⁿ·det(A) for an n×n matrix.
Define the cofactor of an element and the adjoint (adjugate) of a matrix.
The cofactor Cᵢⱼ = (−1)^(i+j)·Mᵢⱼ, where Mᵢⱼ is the minor. The adjoint is the transpose of the cofactor matrix: adj(A) = [Cᵢⱼ]ᵀ.
State the formula for the inverse of a matrix A using its adjoint.
A⁻¹ = adj(A) / det(A), valid only when det(A) ≠ 0.
When is a matrix invertible (non-singular), and what is the key identity A·adj(A)?
A is invertible iff det(A) ≠ 0 (non-singular). The identity is A·adj(A) = adj(A)·A = det(A)·I.
State Cramer's Rule for solving a system of linear equations AX = B.
For a non-singular system, xᵢ = det(Aᵢ)/det(A), where Aᵢ is A with its ith column replaced by B. Requires det(A) ≠ 0.
How is a system of linear equations solved using the matrix inverse method?
Write the system as AX = B; if det(A) ≠ 0, the solution is X = A⁻¹B.
In a system AX = B, what do the cases det(A) ≠ 0 and det(A) = 0 indicate about solutions?
If det(A) ≠ 0: a unique solution exists. If det(A) = 0: the system has either no solution (inconsistent) or infinitely many solutions (dependent).
For a homogeneous system AX = 0, when does a non-trivial solution exist?
A non-trivial (non-zero) solution exists if and only if det(A) = 0. If det(A) ≠ 0, only the trivial solution X = 0 exists.
What is the relationship between nCr coefficients and the rows of Pascal's triangle?
The entries in the nth row of Pascal's triangle (starting from row 0) are the binomial coefficients nC0, nC1, …, nCn.
What this deck covers
The Mathematics deck follows the UPSC NDA & NA Mathematics syllabus — 7 chapters and 35 topics — so questions land on material that is genuinely examinable rather than trivia around it. That works out to roughly 7.9 cards per chapter.
Answers are written to be recallable, not just readable — averaging about 92 characters, which is long enough to carry the reasoning and short enough to say out loud.
A deck like this earns its keep on the second and third pass. Read the syllabus first so you know the shape of the subject, then use the cards to find the specific facts that have not stuck.
Mathematics flashcards FAQ
How many Mathematics flashcards are in this UPSC NDA & NA deck?
55 cards. This page previews 24 of them, sampled evenly across the deck so you can judge the difficulty before installing anything.
Are these UPSC NDA & NA flashcards free?
Yes. The preview here is free to read with no signup, and the full 55-card deck is free inside the Examius app.
What do the Mathematics cards cover?
They follow the UPSC NDA & NA Mathematics syllabus — 7 chapters and 35 topics — so the questions track what is actually examinable.
How should I use these flashcards?
Read the syllabus first so you know the shape of the subject, then drill the deck. Examius schedules each card with spaced repetition, so cards you keep missing come back sooner and ones you know drift further apart.