1. Differentiate a compound function
For x > 0, find the derivative of log(1 + x²).
Answer: 2x/(1 + x²).
Explanation: Apply the chain rule: the outer logarithm contributes the reciprocal of its input and the inner square contributes 2x.
Interview prep / Pure Math
25 maths interview questions for quants with answers
25 original quant maths questions with concise solutions in calculus, series, optimisation, linear algebra, combinatorics and differential equations.
How to approach these questions
Quant maths interviews often use a short calculation to test whether you understand the structure behind it. Derivatives become sensitivities, positive-definite matrices become well-behaved risk objectives and projections become least-squares fits. Work through these calculations without a computer, then explain an assumption or sanity check for each answer.
Use structure before algebra
Look for symmetry in a matrix, a square to complete or a recognised series. These observations can reveal both the answer and a check on it.
Keep assumptions attached
Convergence, invertibility, differentiability and positivity determine whether a familiar formula applies. Name the condition at the point where you use it.
Translate back to the application
After finding a derivative or eigenvalue, explain what it says about sensitivity, curvature or risk. Fluency is more useful when connected to a quantitative decision.
Sample questions, answers and explanations
Try each question before reading its answer. These original teaching examples remain free to read on this page.
Calculus and local approximation
Differentiate and integrate carefully, including the chain rule and endpoint behaviour.
2. Exponential sensitivity
Find the derivative of exp(3x).
Answer: 3exp(3x).
Explanation: The inner linear function contributes the factor 3. This same chain rule appears in discount and growth sensitivities.
3. Integrate an exponential
Evaluate the integral of exp(−2x) from 0 to infinity.
Answer: 1/2.
Explanation: The antiderivative is −exp(−2x)/2. The upper endpoint vanishes, which requires a positive decay rate.
4. Integration by parts
Evaluate the integral of x exp(−x) from 0 to infinity.
Answer: 1.
Explanation: Take u = x and integrate exp(−x). The boundary term vanishes, leaving the unit exponential integral.
5. Local exponential approximation
Give the second-order Taylor approximation to exp(x) near zero.
Answer: 1 + x + x²/2.
Explanation: All derivatives of exp(x) equal 1 at zero. The omitted remainder is of order x³ locally.
Series, limits and optimisation
Check convergence and curvature instead of relying only on a stationary point or a formal sum.
6. A geometric series
Sum 1 + 0.5 + 0.5² + … .
Answer: 2.
Explanation: The infinite geometric sum is 1/(1 − r) for |r| < 1. A finite partial sum has a different formula.
7. Differentiate a geometric sum
Evaluate the sum of k/2^k for k = 1,2,… .
Answer: 2.
Explanation: Differentiate the geometric series within its convergence interval to obtain sum k r^k = r/(1 − r)², then set r = 1/2.
8. A basic limit
Evaluate (exp(x) − 1)/x as x tends to zero.
Answer: 1.
Explanation: Use the derivative of exp at zero or its first-order expansion. Direct substitution only gives an indeterminate form.
9. Convexity from curvature
Is f(x) = x² + 3x + 1 convex on the real line?
Answer: Yes, strictly convex.
Explanation: Its second derivative is 2 everywhere. A positive second derivative makes the derivative strictly increasing.
10. Minimise a quadratic
Find the minimiser of x² − 6x + 10 over real x.
Answer: x = 3, with minimum value 1.
Explanation: Complete the square as (x − 3)² + 1, or set the derivative to zero and check positive curvature.
Multivariate calculus and eigenvalues
Gradients, Hessians and eigendirections expose the geometry of quadratic models.
11. A multivariate gradient
Find the gradient of f(x,y) = x² + xy + 2y².
Answer: (2x + y, x + 4y).
Explanation: Differentiate while holding the other coordinate fixed. The cross term contributes to both components.
12. Check positive definiteness
Is the Hessian of f(x,y) = x² + xy + 2y² positive definite?
Answer: Yes; it is [[2,1],[1,4]].
Explanation: Its first leading principal minor is 2 and determinant is 7, both positive. This makes the quadratic strictly convex.
13. A constrained minimum
Minimise x² + y² subject to x + y = 1.
Answer: x = y = 1/2, giving value 1/2.
Explanation: Equal partial derivatives from the Lagrange equations imply x = y. The positive-definite quadratic ensures the stationary feasible point is the minimum.
14. Eigenvalues of a symmetric matrix
Find the eigenvalues of [[2,1],[1,2]].
Answer: 3 and 1.
Explanation: Vectors (1,1) and (1,−1) are eigenvectors. Symmetry also guarantees real eigenvalues and orthogonal eigenspaces.
15. Trace and eigenvalues
A 3 × 3 matrix has eigenvalues 1, 2 and 5, counted with multiplicity. What are its trace and determinant?
Answer: Trace 8; determinant 10.
Explanation: The trace is the sum and determinant the product of eigenvalues. These identities do not require distinct eigenvalues.
Linear algebra for risk and regression
Rank, projections and positive semidefiniteness are practical constraints on a model, not just matrix vocabulary.
16. Dependent columns
What is the rank of [[1,2],[2,4]]?
Answer: 1.
Explanation: The second column is twice the first and the first is nonzero. A zero determinant indicates singularity but does not alone distinguish rank zero from rank one.
17. Invert a diagonal matrix
Find the inverse of diag(2,5).
Answer: diag(1/2,1/5).
Explanation: Each nonzero diagonal entry can be inverted independently. A zero diagonal entry would make the matrix singular.
18. Project onto a direction
Project (3,1) onto the line spanned by (1,1) in Euclidean space.
Answer: (2,2).
Explanation: Use [v·u/(u·u)]u. The residual (1,−1) is orthogonal to the direction, providing a quick check.
19. Covariance matrix constraint
Can [[1,2],[2,1]] be a covariance matrix?
Answer: No.
Explanation: Its eigenvalues are 3 and −1, violating positive semidefiniteness. Equivalently, the implied covariance exceeds the product of standard deviations.
20. Choose a subset
How many unordered groups of 3 can be selected from 8 distinct items?
Answer: 56.
Explanation: Use C(8,3) = 8 × 7 × 6/(3 × 2 × 1). Divide out ordering because each group is counted six times otherwise.
Counting and further techniques
Round out the toolkit with combinations, modular arithmetic, an ODE and a useful norm bound.
21. Arrange repeated letters
How many distinct arrangements does AABBC have?
Answer: 30.
Explanation: Start with 5! permutations and divide by 2! for each repeated pair. Identical letters do not create new arrangements.
22. Modular arithmetic
What is the remainder of 2^10 on division by 7?
Answer: 2.
Explanation: Since 2³ has remainder 1, write 2^10 = (2³)³ × 2. Repeated cycles avoid a large integer calculation.
23. Solve a growth equation
Solve y′ = 2y with y(0) = 3.
Answer: y(x) = 3exp(2x).
Explanation: Separate variables or recognise exponential growth. The initial condition fixes the multiplicative constant.
24. Complex square
Compute (1 + i)².
Answer: 2i.
Explanation: Expand and use i² = −1. The real terms cancel, which also agrees with doubling the complex argument.
25. Bound a dot product
Vectors u and v have Euclidean norms 3 and 4. How large can |u·v| be?
Answer: At most 12.
Explanation: Cauchy–Schwarz bounds the absolute dot product by the product of norms. Equality requires the vectors to be linearly dependent.
111 pure math questions from the bank
Each question has its own page with a short answer and a credit to its source. Hints and full worked solutions are in the question bank with a free account.
- Choosing the drift that makes an exponential walk fairEasy
- Angle between two vectors after an orthogonal mapEasy
- Seventh derivative at zero of a polynomial times a GaussianMedium
- Last digits of a huge power of eleven mod 50Medium
- Column count from solvability of two linear systemsMedium
- Eigenvalues of a 30 by 30 matrix with constant off-diagonalMedium
- Growth rate of the eigenvalue of a rank-one matrixHard
- Speed of a ladder top as its oily foot slides outEasy
- Area growth of a pond ripple after ten secondsEasy
- Nonzero eigenvalue and its eigenspace for a projectorMedium
- Rank and nullity of projection onto one vectorMedium
- Surface-area growth of an expanding sphereEasy
- Volume shared by two perpendicular unit cylindersMedium
- Leading coefficient of a sum of thousandth powersMedium
- Power of fifteen reduced modulo seventeenEasy
- Radius where a circle's area and perimeter grow equallyEasy
- When a fast-growing colony outruns its life supportEasy
- Volume rate of a widening, shortening cylinderEasy
- Perimeter rate of a triangle with changing sidesEasy
- Rank and nullity of a 3D rotation matrixMedium
- Recovering unknown entries from a matrix squareEasy
- Largest area of a revolved quarter-circle arcMedium
- Growth rate of a walker's shadow from a street lampMedium
- Derivative of log x raised to the power x at eMedium
- Eigenvalue sum after a shift, inverse and cubeEasy
- Incircle area of a 16-16-24 isosceles triangleMedium
- Base c making a Poisson exponential a martingaleMedium
- Sum of squared eigenvalues of a 2x2 matrixEasy
- Trace of the exponential of a diagonal matrixMedium
- Scaled integral of 1/(1+x^n) just above 1Hard
- Vectors shared by two different eigenspacesEasy
- Angle rate for a 50-ft ladder sliding on oilMedium
- Distance between two orthonormal vectorsEasy
- General integer solution of 3x + 7y = 10000Easy
- Integers below a million with digit sum 19Hard
- Four positive integers from three pairwise productsEasy
- Monotone grid paths hitting at least one power-upEasy
- Three-digit codes with nonzero digits summing to 10 or lessEasy
- Point sequences reaching 4-4 under deuce rulesEasy
- Distinct ages of three children between 7 and 15Easy
- Medal outcomes for six sprinters when ties allowedMedium
- Right-and-down paths across a 6x6 grid and N by MEasy
- Seating four of ten guests at a round tableEasy
- Team assignments with two people barred from researchEasy
- Photo line-up with the top three in the middleEasy
- Period of the over-under shuffle on 14 cardsEasy
- Eigenvalues of the cubed inverse of A minus 3IMedium
- Locks and keys so any three of five can openMedium
- Growth rate of a walker's shadow from a lamp postEasy
- Biggest circle inside a triangle with sides 16, 16, 24Easy
What our question-bank data shows
Our bank summary dated contains 121 Pure Math entries in the bank. Counts describe entries in our curated practice bank, including its source and classification choices. They do not measure employer interview frequency, candidate success rates or how often any one question is asked.
- Calculus: 26 entries, or 21.5% of this topic's bank. Use this coverage to identify areas for practice alongside the technique-based examples above.
- Linear Algebra: 21 entries, or 17.4% of this topic's bank. Use this coverage to identify areas for practice alongside the technique-based examples above.
- Algebra & Number Theory: 17 entries, or 14.0% of this topic's bank. Use this coverage to identify areas for practice alongside the technique-based examples above.
The public examples above were written for these guides. They illustrate the techniques and are not a reproduction or ranking of the bank's sourced questions. The dashboard's broader bank combines sourced questions, worked answers where available and references for book entries without a reproduced solution.
Subtopic coverage in the bank
| Subtopic | Bank entries |
|---|---|
| Calculus | 26 |
| Linear Algebra | 21 |
| Algebra & Number Theory | 17 |
| Combinatorics | 12 |
| Series & Limits | 12 |
| Integration | 9 |
| Discrete Math | 8 |
| Differential Equations | 7 |
| Complex Analysis | 4 |
| Markov Chains | 2 |
| Geometry | 2 |
| Optimisation | 1 |
Bank difficulty labels: 65 easy · 43 medium · 13 hard.
Lessons behind the techniques
Use these lesson previews to find the definitions and derivations behind the worked examples.
- The Derivative as a Limit: Rates of Change and the Derivative Function
- The Chain Rule
- Eigenvalues and Eigenvectors: Ax = λx
- Matrix Algebra: Operations, Rules, and the Transpose
- Series and the Geometric Series: Partial Sums and Convergence
- Counting and Combinatorics
Continue with 121 pure math questions in the practice bank.
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