Homework 1
Differential Equations/Linear Algebra, Fall 2024
Due in Canvas on Sunday, August 25, 5pm
In this homework assignment, you will brush up on your high-school algebra and pre-calculus skills. Feel free to
use the refreshers R1 through R4 that are available on Canvas (under the Files tab, in the “Refreshers” folder).
1. Solve the following algebraic equations. Note that some solutions might be complex numbers.
(a) x2 + 2x − 3 = 0
(b) y 3 − 5y 2 + 6y = 0
(c) z 2 + 2z + 5 = 0
(d) λ3 = λ
(e) t4 = −1
Hint: complete the square.
2. In this problem, a is a real number parameter. For which values of a does the
(a) quadratic equation x2 + ax + 9 = 0 have two different solutions?
(b) cubic equation x3 + (a − 1)x2 + (4 − a)x − 4 = 0 have only real solutions?
3. Simplify the following expressions (into a single fraction whose top and bottom doesn’t have common factors).
(a)
5
8
−
4x − 12 3 − x
(b)
3
4
+
y 2 + 4y + 4 y 2 − 4
(c)
z 3 − 6z 2 + 9z
z 3 − 3z 2
1
1
−
u v
(d) u
v
−
v
u
4. Use Partial Fractions decomposition to write the following rational functions as a sum of simpler fractions.
(a)
(b)
(c)
x+3
2x3 − 8x
1
x3 + x2 + x + 1
2x3 − 4x2 − x − 3
x2 − 2x − 3
5. Find simpler expressions for the quantities:
4
(a) eln(x)−ln(x )
(b) ln e2 ln x
1 (c) ln 3x2 − 9x + ln
3x
p
(d) x2 − 4x + 4
1
6. Solve for y in terms of t:
t
4
(b) ety ln 2 = 8
(d) ln(5 − 2y) = t
(a) e5y =
(e) ln(y 2 − 1) − ln(y + 1) = ln(sin(t))
√
(c) e y = t2
7. Suppose that x and y are such that sin(x) = 51 , cos(x) > 0, sin(y) < 0, and cos(y) = − 53 . Evaluate the
following expressions:
(a) cos(2x)
(b) sin(2x − y)
(c) tan(x + y)
Hint: find the concrete values of cos(x) and sin(y) and use some of the trig identities.
8. Evaluate/simplify the following expressions of complex numbers:
(a) |1 + 2i|
(|z| denotes the modulus of z)
(b) 3 − 5i + (1 + i)2
1
(c) Re 3+4i
(Re(z) denotes the real part of z)
√
1+i 3
in standard rectangular form (i.e. a + ib)
(d) write z = √
3+i
9. Evaluate the following linear combinations:
1
4
0
(a) 3
−2
+
−2
−3
1
0
1
(b) 0 − 3 3
2
−1
0
1
(c) 2 x − y 2
1
7
10. Evaluate the following dot products and identify what type of angle (acute/right/obtuse) there is between
the two vectors:
−1
5
(a) 1 · 1
4
2
1
2
2
(b) 2 · −1
1
4
4
1
3
0 1
(c)
0 · 4
−1 0
1
2
2
0
