Analytics Math Exam

MATH 17 MIDTERM EXAM August 27, 2013 MATH 17 MIDTERM EXAM August 27, 2013

I. EASY QUESTIONS: Perform as indicated. Show all I. EASY QUESTIONS: Perform as indicated. Show all
necessary solutions. [2 pts each] necessary solutions. [2 pts each]

1. Find the conjugate and modulus of 1. Find the conjugate and modulus of

z = 3 − (7 + 5i). z = 3 − (7 + 5i).

2. Let A(2, 1), B(4, 5) and C(−3, 4). Find the distance 2. Let A(2, 1), B(4, 5) and C(−3, 4). Find the distance
between C and the midpoint of A and B. between C and the midpoint of A and B.

3. Let X = {(⊗, ), (⊕, ), ( , ), ( , )}. Find the 3. Let X = {(⊗, ), (⊕, ), ( , ), ( , )}. Find the
domain and range of X. domain and range of X.
√ √
4. Let f : Df → [0, +∞) given by f (x) = x − 1 4. Let f : Df → [0, +∞) given by f (x) = x − 1
where Df = [1, +∞). Show that f is onto. where Df = [1, +∞). Show that f is onto.

5. Let f (x) = 3x + 2 and g(x) = x2 − 1. Find (f ◦ g)(2) 5. Let f (x) = 3x + 2 and g(x) = x2 − 1. Find (f ◦ g)(2)
if it is defined. if it is defined.

II. Find the solution set of the following. For no. 10, II. Find the solution set of the following. For no. 10,
sketch also the solution set. [4 pts each] sketch also the solution set. [4 pts each]
2x − 1 x + 2 2x − 1 x + 2
1. − =0 1. − =0
2x + 1 x − 1 2x + 1 x − 1
√ √ √ √
2. 2 3x − 3 − x = 4 2. 2 3x − 3 − x = 4

3. (2 + i)z = 5i7 z − (2 − i) 3. (2 + i)z = 5i7 z − (2 − i)

4. |x−1 + 2|2 + 4|x−1 + 2| = 12 4. |x−1 + 2|2 + 4|x−1 + 2| = 12

5. 2x3 − 3x2 + 2x − 3 = 0 5. 2x3 − 3x2 + 2x − 3 = 0
x+2 x−2 x+2 x−2
6. ≤ 6. ≤
x x−4 x x−4
7. 2|3 − 2x| > −10 + x 7. 2|3 − 2x| > −10 + x
 
 3x − 5y =3  3x − 5y =3
8. 2x − 5y + z = 4 8. 2x − 5y + z = 4
3y − 2z = −4 3y − 2z = −4
 

y − x2 = x − 3 y − x2 = x − 3
 
9. 9.
2x − y = 1 2x − y = 1

 y + 4x ≤ x2 + 3  y + 4x ≤ x2 + 3
 

10. y − 5 < 2x 10. y − 5 < 2x
2y > 3 2y > 3
 

III. Prove the following statements. III. Prove the following statements.

1. Let a ∈ Z. If 3a + 2 is odd, then 5a − 5 is even. 1. Let a ∈ Z. If 3a + 2 is odd, then 5a − 5 is even.
. [4 pts] . [4 pts]

2. Let x > −1 be a real number. Prove that (1 + x)n ≥ 2. Let x > −1 be a real number. Prove that (1 + x)n ≥
1 + nx for all integers n ≥ 1. [6 pts] 1 + nx for all integers n ≥ 1. [6 pts]

IV. Find the domain, range, intercepts, asymptotes and IV. Find the domain, range, intercepts, asymptotes and
symmetries of y(x2 + 1) = 4x. Sketch its graph. [10 pts] symmetries of y(x2 + 1) = 4x. Sketch its graph. [10 pts]