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Engineering mathematics (CS section 1; DA sections 1-3) interview questions

Interview questions on engineering mathematics (CS section 1; DA sections 1-3): warm-ups, then what each harder question tests and how a good answer starts.

23 questions: 5 warm-ups and 18 with model answers · Levels: Fresher to mid

Warm-up questions

Short questions with short answers, each answered in full.

Fresher

What is the difference between a symmetric and an antisymmetric relation, and can one relation be both?

Answered in the lesson Logic, sets, lattices and groups for GATE CS
Fresher

When do you count placements with n to the power k, and when with stars and bars?

Answered in the lesson Graphs, counting, recurrences, generating functions
Fresher

What do rank(A) and rank([A | b]) tell you about the solutions of Ax = b?

Answered in the lesson Linear systems, determinants, eigenvalues and LU
Fresher

What is the difference between continuity and differentiability at a point, and which one implies the other?

Answered in the lesson Limits, MVT, integrals, Taylor series and optima
Fresher

What is the difference between independent and mutually exclusive events, and can two events be both?

Answered in the lesson Probability and statistics for GATE CS and DA

Questions with model answers

What each question tests, the first step of a good answer and the follow-ups to expect.

Fresher to mid

A NAT asks how many relations on a 4-element set are both reflexive and antisymmetric. How do you get the number, and how do you check it before typing it?
What it tests

Counting relations by independent decisions for each pair, and confirming a closed form on small cases before entering a NAT.

First step

Split the 16 ordered pairs into the 4 diagonal pairs and the 6 unordered pairs of distinct elements, and ask what each property allows for each kind of pair: reflexivity fixes the diagonal, and antisymmetry limits what an unordered pair may contribute.
Follow-ups
- What changes if the relation must be reflexive and symmetric instead?
- How many of these relations are also transitive?
- How do you check the formula in under a minute without a program?
From the lesson Logic, sets, lattices and groups for GATE CS
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In the answer:
- Model answer
- Follow-up answers
- Common mistakes
- Rubric
- Variants
- about 2 minutes
Fresher to mid

An MSQ shows the divisors of 36 ordered by divisibility and offers four options: a lattice, a distributive lattice, a complemented lattice, a Boolean algebra. Which do you mark, and how do you justify each option on its own?
What it tests

Deciding lattice properties option by option, knowing why divisor lattices are distributive, and finding complements with the gcd and the lcm.

First step

Decide each option separately. Any two divisors of 36 have a gcd and an lcm that also divide 36, so the poset is a lattice. For complements, look for an element x that has no partner y with gcd(x, y) = 1 and lcm(x, y) = 36 at the same time.
Follow-ups
- For which n are the divisors of n a complemented lattice?
- Is every complemented lattice distributive?
- In the divisors of 30, what is the complement of 6, and could it have two?
From the lesson Logic, sets, lattices and groups for GATE CS
Model answer, Pro.
In the answer:
- Model answer
- Follow-up answers
- Common mistakes
- Rubric
- Variants
- about 2 minutes
Fresher to mid

You are given a 5 × 5 operation table in which every element appears exactly once in each row and each column, and one row repeats the header. Is it a group? How do you decide quickly?
What it tests

Knowing which group axioms a table shows at a glance and which need work, and that a Latin square with an identity can still fail associativity.

First step

Read off what the table gives at once: closure, since every entry is in the set; an identity, the element whose row repeats the header; and inverses, since the identity appears in every row. That leaves associativity, the one axiom no glance settles.
Follow-ups
- How many checks does associativity need, and how can you cut them down?
- Every group of order 5 is cyclic. How does that help?
- How do you read the order of an element off the table?
From the lesson Logic, sets, lattices and groups for GATE CS
Model answer, Pro.
In the answer:
- Model answer
- Follow-up answers
- Common mistakes
- Rubric
- Variants
- about 3 minutes
Fresher to mid

A simple graph has 20 vertices and exactly 3 connected components. What is the largest number of edges it can have, and how do you show that no graph does better?
What it tests

Turning a structural condition into an extremal count, and proving a bound with an exchange argument instead of guessing the shape.

First step

Fix the component sizes a, b and c with a + b + c = 20. Each component can be at most complete, so the edges are at most C(a, 2) + C(b, 2) + C(c, 2). Ask how that sum changes when you move a vertex from a smaller component to a larger one.
Follow-ups
- What is the fewest number of edges with exactly 3 components?
- How many edges guarantee that a simple graph on 20 vertices is connected?
- Does the answer change for a forest?
From the lesson Graphs, counting, recurrences, generating functions
Model answer, Pro.
In the answer:
- Model answer
- Follow-up answers
- Common mistakes
- Rubric
- Variants
- about 2 minutes
Fresher to mid

A NAT gives a(n) = 6a(n − 1) − 9a(n − 2) with a(0) = 2 and a(1) = 3 and asks for a(5). A candidate writes a(n) = A · 3ⁿ + B · 3ⁿ. What goes wrong, what is the right form, and how do you check the value?
What it tests

Handling a repeated characteristic root, fitting initial values, and checking a closed form by unrolling.

First step

Write the characteristic equation r² − 6r + 9 = 0 and factor it as (r − 3)². A double root needs two independent solutions, and A · 3ⁿ + B · 3ⁿ is only one, (A + B) · 3ⁿ, which cannot fit two initial values.
Follow-ups
- How do you check a(5) in thirty seconds without the closed form?
- What trial solution does a forcing term 3ⁿ need here?
- What changes if the roots are complex?
From the lesson Graphs, counting, recurrences, generating functions
Model answer, Pro.
In the answer:
- Model answer
- Follow-up answers
- Common mistakes
- Rubric
- Variants
- about 2 minutes
Fresher to mid

How many bit strings of length 10 contain no two consecutive 1s? Derive the count in a way that would also handle "no three consecutive 1s".
What it tests

Setting up a recurrence by the last symbols of a string, choosing initial values, and generalising the argument.

First step

Classify the good strings of length n by how they end. A good string ending in 0 is any good string of length n − 1 followed by 0; one ending in 1 must end in 01. Count each class with shorter good strings.
Follow-ups
- What are the initial values, and why do they matter?
- How does the recurrence change for no three consecutive 1s?
- What closed form does the first count have?
From the lesson Graphs, counting, recurrences, generating functions
Model answer, Pro.
In the answer:
- Model answer
- Follow-up answers
- Common mistakes
- Rubric
- Variants
- about 2 minutes
Fresher to mid

For which values of k and m does the system x + y + z = 1, 2x + 3y + 4z = 3, x + 2y + kz = m have no solution, exactly one solution, and infinitely many solutions?
What it tests

Using the determinant to find the critical parameter, then the rank of the augmented matrix to split the critical case, and describing a solution family.

First step

Compute the determinant of the coefficient matrix as a function of k. Wherever it is nonzero there is exactly one solution, whatever m is; only the values of k that make it zero need the augmented matrix.
Follow-ups
- Describe the solutions in the infinite case.
- Solve the system for k = 4 and m = 9.
- Why does m not matter when the determinant is nonzero?
From the lesson Linear systems, determinants, eigenvalues and LU
Model answer, Pro.
In the answer:
- Model answer
- Follow-up answers
- Common mistakes
- Rubric
- Variants
- about 3 minutes
Fresher to mid

A 3 × 3 matrix A has eigenvalues 1, −1 and 2. Find det(A² + 2I) and the trace of A⁻¹, and say which polynomial equation A itself satisfies.
What it tests

Moving eigenvalues through polynomials and inverses, using trace and determinant as sum and product, and applying the Cayley–Hamilton theorem.

First step

Apply each operation to every eigenvalue: if Av = λv, then (A² + 2I)v = (λ² + 2)v and A⁻¹v = v/λ. Then use that the determinant is the product of the eigenvalues and the trace their sum.
Follow-ups
- Is A diagonalisable?
- Express A⁻¹ as a combination of I, A and A².
- What is det(A − 3I)?
From the lesson Linear systems, determinants, eigenvalues and LU
Model answer, Pro.
In the answer:
- Model answer
- Follow-up answers
- Common mistakes
- Rubric
- Variants
- about 2 minutes
Fresher to mid

Your hand LU decomposition of a 3 × 3 matrix and the output of scipy.linalg.lu give different L and U. How do you decide whether your answer is wrong?
What it tests

Knowing what a library returns (a permutation as well as L and U), why it pivots, and how to compare factorisations without trusting either.

First step

Do not compare entries yet. First read exactly what the function returns and which product of its outputs equals A, then test your own L and U on their own, without the library.
Follow-ups
- Why does SciPy reorder the rows at all?
- Which checks prove your own factors right?
- When is a hand LU without row exchanges impossible?
From the lesson Linear systems, determinants, eigenvalues and LU
Model answer, Pro.
In the answer:
- Model answer
- Follow-up answers
- Common mistakes
- Rubric
- Variants
- about 2 minutes
Fresher to mid

P is a real symmetric 4 × 4 matrix with P² = P and trace 2. What are its eigenvalues and rank, what is det(I + P), and what does P do to a vector geometrically?
What it tests

Deriving the eigenvalues of an idempotent matrix, linking trace and rank, and reading a symmetric idempotent matrix as an orthogonal projection.

First step

Take an eigenvector: Pv = λv gives P²v = λ²v, and P² = P makes that λv, so λ² = λ and every eigenvalue is 0 or 1. The trace is the sum of the eigenvalues, so it counts the 1s.
Follow-ups
- Why does symmetry matter, beyond P² = P?
- What are the eigenvalues and rank of I − P?
- Can the length of Px exceed the length of x?
From the lesson Vector spaces, projections, quadratic forms and SVD
Model answer, Pro.
In the answer:
- Model answer
- Follow-up answers
- Common mistakes
- Rubric
- Variants
- about 2 minutes
Fresher to mid

A is a 5 × 7 matrix whose null space has dimension 4. What is its rank? Can Ax = b be solved for every b in R⁵? When Ax = b is solvable, how many parameters do its solutions have?
What it tests

Applying rank-nullity to the right side of the matrix, relating the rank to the column space, and describing solution sets.

First step

Rank-nullity counts columns: rank + nullity = 7, the number of unknowns. Then compare the rank, the dimension of the column space, with 5, the dimension of the space that b lives in.
Follow-ups
- What is the dimension of the null space of Aᵀ?
- What is the smallest nullity a 5 × 7 matrix can have?
- How would the answers change for a 7 × 5 matrix of the same rank?
From the lesson Vector spaces, projections, quadratic forms and SVD
Model answer, Pro.
In the answer:
- Model answer
- Follow-up answers
- Common mistakes
- Rubric
- Variants
- about 2 minutes
Fresher to mid

Find the singular values and the rank of A = [[1, 2], [2, 4], [0, 0]], and say what its best rank-1 approximation is.
What it tests

Computing singular values from AᵀA, reading the rank off them, and applying the best low-rank approximation result.

First step

Form the Gram matrix AᵀA, which is 2 × 2 here, and find its eigenvalues from its trace and determinant; the singular values are their square roots.
Follow-ups
- How do singular values relate to eigenvalues for a symmetric matrix?
- What are the 2-norm and the Frobenius norm of A?
- What does numpy.linalg.svd return for this matrix, and in which shapes?
From the lesson Vector spaces, projections, quadratic forms and SVD
Model answer, Pro.
In the answer:
- Model answer
- Follow-up answers
- Common mistakes
- Rubric
- Variants
- about 2 minutes
Fresher to mid

Find the global maximum and minimum of f(x) = |x − 1| + |x − 3| on [0, 4], and say where each is reached. Why does setting f′(x) = 0 not work here?
What it tests

Applying the closed-interval method to a function with corners, reading a piecewise-linear function, and knowing where critical points come from.

First step

List the candidates: the endpoints 0 and 4, and the corners 1 and 3, where f′ does not exist. Between and beyond the corners f is linear, so write its slope on each piece before evaluating anything.
Follow-ups
- Where is the minimum reached, at a point or on an interval?
- What changes on the interval [2, 5]?
- How does this relate to the median of the points 1 and 3?
From the lesson Limits, MVT, integrals, Taylor series and optima
Model answer, Pro.
In the answer:
- Model answer
- Follow-up answers
- Common mistakes
- Rubric
- Variants
- about 2 minutes
Fresher to mid

A candidate applies L'Hôpital's rule to the limit of (2x + cos x)/x as x → ∞, gets (2 − sin x)/1 and concludes that the limit does not exist. Is the conclusion right? What is the limit, and what exactly does the rule promise?
What it tests

Knowing the hypotheses and the one-way conclusion of L'Hôpital's rule, and evaluating a limit by splitting it instead.

First step

Before using the rule, try simplifying the fraction by dividing each term by x, and recall exactly what the rule promises: its hypotheses, its conclusion, and the direction in which it works.
Follow-ups
- Give a limit where L'Hôpital's rule applies but takes many steps that a series avoids.
- Can the rule be applied to the limit of (x + 1)/(x + 2) as x → 0?
- Is the limit of x sin(1/x) as x → 0 a case for the rule?
From the lesson Limits, MVT, integrals, Taylor series and optima
Model answer, Pro.
In the answer:
- Model answer
- Follow-up answers
- Common mistakes
- Rubric
- Variants
- about 2 minutes
Fresher to mid

How many terms of the Maclaurin series of eˣ do you need to be sure that the error at x = 0.5 is below 10⁻⁴? Justify the number with an error bound, not with a calculator comparison.
What it tests

Using the Lagrange form of the remainder, bounding a derivative on an interval, and turning the bound into a number of terms.

First step

Write the remainder after the degree-n polynomial in Lagrange's form and bound the derivative in it on [0, 0.5]. Every derivative of eˣ is eˣ, so look for the smallest n that pushes the bound below 10⁻⁴.
Follow-ups
- What bound do you get if you use e^0.5 < 2 instead of the exact value?
- How many terms would x = 2 need for the same accuracy?
- Why does the error at x = −0.5 behave differently?
From the lesson Limits, MVT, integrals, Taylor series and optima
Model answer, Pro.
In the answer:
- Model answer
- Follow-up answers
- Common mistakes
- Rubric
- Variants
- about 2 minutes
Fresher to mid

A spam filter flags 96 % of spam and 2 % of legitimate mail, and 30 % of incoming mail is spam. An email is flagged. What is the probability that it is spam, and how would the answer change if spam made up 3 % of the mail?
What it tests

Applying Bayes' theorem with the law of total probability, keeping P(A | B) and P(B | A) apart, and seeing how the base rate drives the answer.

First step

Count out of 1,000 emails: 300 spam, of which 288 are flagged, and 700 legitimate, of which 14 are flagged. The question asks for the share of spam among the flagged mail, not the share of spam that is flagged.
Follow-ups
- What is the probability that an email that was not flagged is spam?
- The filter is run twice, independently. What changes after two flags?
- Which number should the filter's builders report to users?
From the lesson Probability and statistics for GATE CS and DA
Model answer, Pro.
In the answer:
- Model answer
- Follow-up answers
- Common mistakes
- Rubric
- Variants
- about 2 minutes
Fresher to mid

1,000 keys are hashed independently and uniformly into a table of 1,000 slots. How many slots do you expect to stay empty, and how many pairs of keys do you expect to collide?
What it tests

Using indicator variables and linearity of expectation without assuming independence, and approximating (1 − 1/m)ⁿ.

First step

Write the number of empty slots as a sum of 1,000 indicators, one per slot, each 1 when no key lands there. Its expectation is 1,000 times the probability that a given slot is missed by every key.
Follow-ups
- Are the slots' indicators independent, and does it matter?
- How large must the table be for the expected number of colliding pairs to stay below 1?
- What is the expected number of keys in the slot of a given key?
From the lesson Probability and statistics for GATE CS and DA
Model answer, Pro.
In the answer:
- Model answer
- Follow-up answers
- Common mistakes
- Rubric
- Variants
- about 2 minutes
Fresher to mid

Waiting times at a counter are exponential with a mean of 10 minutes. You have already waited 7 minutes. What is the probability of waiting at least 5 more, and how much longer do you expect to wait?
What it tests

Knowing and using memorylessness, the survival function of the exponential, and contrasting it with a distribution that has memory.

First step

Write the survival function, P(X > t) = e^(−t/10), and the conditional probability P(X > 12 | X > 7) as a ratio of two survival values; then notice what the ratio depends on.
Follow-ups
- What is the median waiting time?
- How do the answers change if the waiting time is uniform on [0, 20] instead?
- Which discrete distribution behaves the same way?
From the lesson Probability and statistics for GATE CS and DA
Model answer, Pro.
In the answer:
- Model answer
- Follow-up answers
- Common mistakes
- Rubric
- Variants
- about 2 minutes

The lessons of this module

Logic, sets, lattices and groups for GATE CS Free core, Pro depth: 7 of 20 minutes free
Graphs, counting, recurrences, generating functions Free core, Pro depth: 8 of 20 minutes free
Linear systems, determinants, eigenvalues and LU Free core, Pro depth: 8 of 20 minutes free
Vector spaces, projections, quadratic forms and SVD Free core, Pro depth: 8 of 20 minutes free
Limits, MVT, integrals, Taylor series and optima Free core, Pro depth: 8 of 20 minutes free
Probability and statistics for GATE CS and DA Free core, Pro depth: 8 of 20 minutes free

All GATE CS & DA Preparation Guide interview questions · The GATE CS & DA Preparation Guide track

Warm-up questions

What is the difference between a symmetric and an antisymmetric relation, and can one relation be both?

When do you count placements with n to the power k, and when with stars and bars?

What do rank(A) and rank([A | b]) tell you about the solutions of Ax = b?

What is the difference between continuity and differentiability at a point, and which one implies the other?

What is the difference between independent and mutually exclusive events, and can two events be both?

Questions with model answers

A NAT asks how many relations on a 4-element set are both reflexive and antisymmetric. How do you get the number, and how do you check it before typing it?

An MSQ shows the divisors of 36 ordered by divisibility and offers four options: a lattice, a distributive lattice, a complemented lattice, a Boolean algebra. Which do you mark, and how do you justify each option on its own?

You are given a 5 × 5 operation table in which every element appears exactly once in each row and each column, and one row repeats the header. Is it a group? How do you decide quickly?

A simple graph has 20 vertices and exactly 3 connected components. What is the largest number of edges it can have, and how do you show that no graph does better?

A NAT gives a(n) = 6a(n − 1) − 9a(n − 2) with a(0) = 2 and a(1) = 3 and asks for a(5). A candidate writes a(n) = A · 3ⁿ + B · 3ⁿ. What goes wrong, what is the right form, and how do you check the value?

How many bit strings of length 10 contain no two consecutive 1s? Derive the count in a way that would also handle "no three consecutive 1s".

For which values of k and m does the system x + y + z = 1, 2x + 3y + 4z = 3, x + 2y + kz = m have no solution, exactly one solution, and infinitely many solutions?

A 3 × 3 matrix A has eigenvalues 1, −1 and 2. Find det(A² + 2I) and the trace of A⁻¹, and say which polynomial equation A itself satisfies.

Your hand LU decomposition of a 3 × 3 matrix and the output of scipy.linalg.lu give different L and U. How do you decide whether your answer is wrong?

P is a real symmetric 4 × 4 matrix with P² = P and trace 2. What are its eigenvalues and rank, what is det(I + P), and what does P do to a vector geometrically?

A is a 5 × 7 matrix whose null space has dimension 4. What is its rank? Can Ax = b be solved for every b in R⁵? When Ax = b is solvable, how many parameters do its solutions have?

Find the singular values and the rank of A = [[1, 2], [2, 4], [0, 0]], and say what its best rank-1 approximation is.

Find the global maximum and minimum of f(x) = |x − 1| + |x − 3| on [0, 4], and say where each is reached. Why does setting f′(x) = 0 not work here?

A candidate applies L'Hôpital's rule to the limit of (2x + cos x)/x as x → ∞, gets (2 − sin x)/1 and concludes that the limit does not exist. Is the conclusion right? What is the limit, and what exactly does the rule promise?

How many terms of the Maclaurin series of eˣ do you need to be sure that the error at x = 0.5 is below 10⁻⁴? Justify the number with an error bound, not with a calculator comparison.

A spam filter flags 96 % of spam and 2 % of legitimate mail, and 30 % of incoming mail is spam. An email is flagged. What is the probability that it is spam, and how would the answer change if spam made up 3 % of the mail?

1,000 keys are hashed independently and uniformly into a table of 1,000 slots. How many slots do you expect to stay empty, and how many pairs of keys do you expect to collide?

Waiting times at a counter are exponential with a mean of 10 minutes. You have already waited 7 minutes. What is the probability of waiting at least 5 more, and how much longer do you expect to wait?