Before a prospective student is invited to Penn for an in-person interview, he or she will be asked to complete a take home written assessment which consists of 12 problems in calculus, linear algebra, real analysis, and abstract algebra. The rules of the assessment are as follows. 

• Students are expected to attempt any 8 of the 12 problems that they choose.
• Students can use any textbook references for assistance.
• Students are not allowed to use the internet or to receive help from anyone while completing the assessment.
• Once they receive the assessment by email, students will have 4.5 hours to complete the assessment and to submit their responses.
• Students will submit their responses by email in a single file. For example, this could be a scanned pdf or a zip file of images of their solutions

A sample assessment can be found here. The specific subject matter the assessment is based on is detailed below.

Limits, derivatives and integrals, fundamental theorem of calculus, intermediate value theorem, mean value theorem, extremum problems, Taylor’s theorem, series, sequences, ordinary differential equations, computing areas and volumes, gradients, double and triple integrals

Good references: Calculus: Early Transcendentals by James Stewart; Thomas’ Calculus by Joel Hass, Christopher Heil and Maurice Weir

Solving linear systems, determinants, trace, Gaussian elimination, matrix multiplication, linear transformations, linear independence, vector spaces, eigenvalues and eigenvectors, diagonalization, Cayley-Hamilton theorem, symmetric matrices, norms, inner products, Gram-Schmidt process, orthogonal matrices

Good references: Linear Algebra Done Right by Sheldon Axler; Linear Algebra by Stephen H. Friedberg, Arnold J. Insel, and Lawrence E. Spence; Introduction to Linear Algebra by Gilbert Strang

Topology of the real line, infimum and supremum, continuity, uniform continuity, epsilon-delta proofs, sequences and series of numbers, differentiability, Riemann integrability, sequences and series of functions, uniform convergence, partial derivatives, multiple integrals, Lagrange multipliers

Good references: Principles of Mathematical Analysis by Walter Rudin; Introduction to Real Analysis by Robert G. Bartle and Donald R. Sherbert; Understanding Analysis by Stephen Abbott

Finite groups, matrix groups, symmetry groups, normal subgroups and quotient groups, homomorphism theorems, abelian groups, polynomial rings, roots and irreducibility, unique factorization of integers and polynomials, greatest common divisors, ideals, integral domains, Euclidean domains, finite fields.

Good references: Algebra by Artin; Topics in Algebra I.N. Herstein; A first course in abstract algebra by John B. Fraleigh