Math 113 Midterm 1 – Practice Exam
These problems form a good review for the first midterm. While these don’t cover all of our
topics from chapters 1-2, this is a representative sample of the format and difficulty level
of the first midterm. This practice exam is intended for roughly a 70 minute exam period;
yours is 50 minutes.
Rules for the exam:
• You are allowed 1 side of a normal sheet of paper for formulas.
• Exam is closed-book, closed notes; no outside materials.
• No calculators or computers for the exam; your cell phones need to be turned off. (This
means you will not need to do Maple for the exam. I may give a take-home Maple
supplement to the second exam and/or the final.)
• I will distribute the exam a couple of minutes early and give you a few minutes of grace
period at the end.
• The exam covers section 11.3-11.5 and chapter 12.
This practice exam does not cover all concepts which might be on the exam; merely knowing
how to do these 7 problems will not ensure success on Wednesday.
Problem 1. The unit cube in the first octant of R3 has opposite vertices O = (0, 0, 0) and
~ makes with the edge lying on the x-axis.
P = (1, 1, 1). Find the angle that the diagonal OP
Problem 2. Using a polar coordinates integral, find the area of the region inside the circle
r = 4 sin θ that lies above the line y = x.
Problem 3. (a) If lines ` and m lie in parallel planes, then they are skew.
TRUE
or
FALSE
(b) For any dimension n, it is possible to define the dot product a · b of two vectors in Rn .
TRUE
or
FALSE
(c) The parametric equations of a line in R3 are unique.
TRUE
or
FALSE
(d) The dot product of two unit vectors equals the cosine of the angle between them.
TRUE
or
FALSE
or
FALSE
(e) A sphere is not a quadric surface
TRUE
Problem 4: (a) Define a cylinder.
(b) Identify the quadric surface z 2 − x2 − y 2 = 1.
Problem 5. Find the plane that passes through (−1, 2, 1) and contains the line of intersection of the planes x + y − z = 2 and 2x − y + 3z = 1.
Problem 6. For each pair of lines below, are they parallel, the same, intersecting, or skew?
` : x(t) = 2 − 4t
y(t) = 2t
z(t) = 1 + 6t
m : x(s) = 4 + 2s
y(s) = 1 − s
z(s) = 4 − 3s
n : x(r) = 3 − 3r
y(r) = 5 + 5r
z(r) = 2 + r
Problem 7. Find the projection of v =< −10, 5, 1 > onto w =< 4, −2, 2 >.
Problem 8. Find the equation of the plane that contains the point (2, 1, 2) and the line
`(t):
x = 4+t
y = −2 − 3t.
z = 1 + 4t