California Institute of Technology
Transfer Admissions
Sample Exam Questions
THIS IS NOT THE ACTUAL EXAM. These sample questions are provided for transfer
applicants to prepare for the types of questions that will appear on the actual exam.
On the actual exam we expect to see all work, as we are interested in methods as well as
answers.
PLEASE NOTE: The actual exam will be CLOSED BOOK: No reference materials,
calculators, electronic devices, or other aids are allowed. If an applicant requires special
testing accommodations, they must contact the Office of Undergraduate Admissions by theapplication deadline.

Caltech Transfer Admissions
Mathematics Examination
Sample Questions

Basic Calculus

1) Find the maximum and minimum values of the function

f (x) =sin2 (x)√_ + cos(x)2on the interval [0, π] . Describe the intervals in which f is increasing or decreasing. Sketch the graphof f .

Linear Algebra
1) The vectors (−1, 0, 1) and (0, −1, 0) are solutions of the linear systemx − 2y + 3z = 2
2x + y + z = −1
x − y + 2z = 1a) Find all solutions of the systemb) What is the solution vector of the shortest length?
c) Find a nonzero vector which is perpendicular to all solution vectors of the same system.

Vector Calculus
1) Let C be the helix in R3 parametrized by α(t) = √12 (cos(t), sin(t), t), 0 ≤ t ≤ 2π. Find the arc lengths(t), the unit tangent vector T (t) and the curvature k(t) (this is defined as ||T 0 (t)||). Evaluate s(π),
T (π), k(π).

Differential Equations
1) For which real values of a do there exist solutions of the differential equation
y 00 + 2y 0 + ay = 0which satisfy the conditions y(0) = y(π) = 0 but which are not identically zero? For each such a give an appropriate
non-zero solution

Caltech Transfer Admissions
Physics Examination
Sample Questions

PROBLEM 1
Your job is to fire a cannon and hit a target a distance d away. The cannon's muzzle
velocity is V. The cannon and the target are on the same level.

targetVθda)

For a given angle θ, what is the muzzle velocity V(θ ) that is required to hit thetarget?

b) What is the minimum muzzle velocity V0 that is required to hit the target? For
what angle θ is it achieved?

c)Suppose V =achieved.

2 V0. Determine the firing angles θ for which success will beCaltech Transfer Admissions
Physics Examination
Sample Questions
PROBLEM 2A linear spring has a length D. When a mass m is hung on one end, the spring has anequilibrium length D + l. While it is hanging motionless with an attached mass m, asecond mass m is dropped from a height l onto the first one. The masses collide
inelastically and stick together. The figure below shows the system at the time of thecollision. Subsequently it oscillates with an amplitude A about its new equilibrium
position.

Caltech Transfer Admissions
Physics Examination
Sample Questions
PROBLEM 2 (continued)(a) Explain why the new equilibrium length of the spring is D + 2l, as indicated in(b) What is the period of the resulting motion?

(c) Find the amplitude A of the motion. Express your answer in terms of l.

(d) How long after the collision do the joined masses reach the lowest point of theiroscillation? Express your answer in terms of l and g.

Caltech Transfer Admissions
Physics Examination
Sample Questions
PROBLEM 3Consider the following electrical circuit, containing a capacitor, two resistors, and abattery of voltage V:AC

R1R2BatteryBgrounda) In steady state what is the charge on the capacitor and what is the current througheach of the resistors?

b) Starting from the steady state operation of part a), the battery is disconnected attime t = 0. What is the voltage at point A for all t > 0?

c) Now suppose that C = 100 pF and R1 = R2 = 10 MΩ. Calculate the amount of timein which the voltage at A decreases by a factor of e.