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# algebra-1

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```GARCIA COLLEGE OF TECHNOLOGY
Kalibo, Aklan
Electrical Engineering Department
EE 412 (ELECTRICAL ENGINEERING CORRELATION 1)
MATHEMATICS SUBJECTS
SET
1. Let U = {1,2,3,4,5…,10}, A = {2,4,6,8,10}, B = {3,6,9}, C = {1,2,3,8,9,10}
determine:
a. A  B
b. A  B
c. A’  C
2. Determine the equivalent set for the following:
a. A  A’
b. U’
c. (A  B)  B
d. A  A’
PRINCIPLE OF INCLUSION AND EXCLUSION/VENN DIAGRAM
3. In a music school of 80 students, 50 plays violin, 40 plays piano, and
10 students do not play any of the two. Determine the number of
students who play piano and violin.
4. In a certain room, a moderator surveyed the job of people and the
result are as follows; 44 are engineer, 64 are dancer, 94 are model, 14
are engineer and dancer, 24 are engineer and model, 44 are dancer and
model, and 4 are engineer, dancer, and model. Determine the total
number of engineer or dancer or model.
5. In a coffee shop that offers four kinds of drinks, mocha, choco, coffee
and vanilla. Thirty ordered mocha, 29 ordered choco, 23 ordered coffee
and 26 ordered vanilla. Eighteen ordered mocha and choco, 9 ordered
mocha and coffee, 15 ordered mocha and vanilla, 15 ordered choco and
coffee, 14 ordered choco and vanilla, 14 ordered coffee and vanilla. Six
ordered mocha, choco and coffee, 9 ordered mocha, choco and vanilla,
7 ordered mocha, coffee, and vanilla, 9 ordered choco, coffee and
vanilla. Assuming that each person ordered one of a kind for each
purchase, determine the total number of people who ordered.
16.
𝑥 5 −2𝑥 4 +𝑥 3 +𝑥+5
(𝑥 2 +1)(𝑥−2)
17.
𝑥 4 −𝑥 3 +14𝑥 2 −2𝑥+22
(𝑥+1)(𝑥 2 +4)(𝑥 2 −2𝑥+5)
𝐴
𝐵𝑥+𝐶
𝐷𝑥+𝐸
= (𝑥+1) + (𝑥 2 +4) + (𝑥 2 −2𝑥+5)
18. Decompose into partial fractions:
2𝑥 4 + 3𝑥 3 + 7𝑥 2 + 10𝑥 + 10
(𝑥 − 1)(𝑥 2 + 3)2
19. Decompose into partial fractions:
𝑥 2 − 4𝑥 + 12
(𝑥 − 3)3
INEQUALITIES
20. 4(x+3) &lt; 10
21. 3 &lt; 2x+3 &lt; 10
22. 2 + x &lt; 3x – 2 &lt; 5x +2
23. 5 -2x &lt; 2x + 1 &lt; -3
24. x^3 + 4x^2 – 11x  30
25. x^2 - 6x + 13  0
THEORY OF EQUATION
26. x^2 + 1 &gt; 0
6. Determine the value of k so that x – 3 is a factor of x^2 - kx - 6.
7. Form an equation of real coefficient with -2 as double root and 1/2 as
simple root.
8. Form an equation with minimum degree with real coefficient having 2
+ 3i, and -3 as simple roots.
9. Form an equation with minimum degree with rational coefficient
having 3 – sqrt 2 as simple roots.
27.
𝑥 2 −49
𝑥 2 −25
28.
𝑥 2 −10𝑥+25
𝑥 2 −14𝑥+48
≤ 0
≤ 0
29. 3x + 1 P5
30. 3x - 1 6
31. x + 1 &gt; x - 2 
10. Determine the least integral upper bound and greatest integral
lower bound for the real roots of the polynomial f(x) = 4x^3 + 8x^2 – 31x
– 35 = 0.
DOMAIN AND RANGE OF FUNCTION
DESCARTES’ RULE OF SIGN
32. Determine the domain and range of f(x) = sinx and f(x) = log x
11. Determine the maximum possible number of positive roots and
negative roots of f(x) = x^6 – 2x^5 + 3x^4 – 2x^3 + x^2 + 5x +3.
33. Determine the domain of the following function:
a. f(x) = sq rt. of (x^2 – 81)
b. f(x) = sq rt. of (4 - x^2)
c. f(x) = sq rt of (x + 4) / (1 – x)
12. Determine the number of possible positive and negative roots for
the polynomial f(x) = x^3 – 9x^2 + 31x – 39. Determine the nature of
possible roots.
13. Determine the nature of roots of f(x) = x^6 – 4x^5 + 5x^4 – 3x^3 +
2x^2 + x.
BINOMIAL THEOREM
34. Determine the 5th term of the expansion (2x^3 + 3y^2)^9
35. Find the term involving (x^4)(y^2) in the expansion (x^2 + y^2 – 1)^5
PARTIAL FRACTION
3𝑥 2 +32𝑥−51
14.
(𝑥−1)(𝑥−2)(𝑥−3)
=
5𝑥 2 −25𝑥+8
15.
(3𝑥+2)(𝑥−3)2
𝐴
𝐵
+ 𝑥−3
(3𝑥+2)
𝐴
𝑥−1
+
𝐵
𝐶
+ 𝑥+3
𝑥−2
36. Find the sum of the coefficients and exponents in the expansion of
(3x^2 + y – 2)^4
REMAINDER THEOREM
=
+
𝐶
(𝑥−3)2
37. Find the quotient and remainder when 4y^3 + 18y^2 + 8y – 4 is
divided by 2y + 3?
g. rasco, ree, rmp
Page 1 of 4
GARCIA COLLEGE OF TECHNOLOGY
Kalibo, Aklan
Electrical Engineering Department
EE 412 (ELECTRICAL ENGINEERING CORRELATION 1)
MATHEMATICS SUBJECTS
38. What is the remainder when 3x4 + 2x3 – 5x2 + x + 7 is divided by x –
3?
39. Let f(x) = x^5 + ax^4 – 3x^3 + bx – 4. If f(x) is divided by (x + 7), the
remainder is -3805, when divided by (x + 1), the remainder is -1. What is
the value of a and b?
a. 1
b. 2
c. -2
d. 0
12. Determine the value of C
a. 1
c. -2
b. 2
d. 0
13. Determine the value of D
a. 1
c. -2
b. 2
d. 0
Problem Sets:
1. Solve for solution set: 3x^2 + 2x + 2 &lt; 2x^2 + x + 4.
a. -2 &lt; x &lt; 1
c. -2 &lt; x &lt; -1
b. 1 &lt; x &lt; 2
d. 0 &lt; x &lt; 1
2. (x + 5) (x -1)(x-2) &lt; 0
a. x &lt; -5 U 1 &lt; x &lt; 2
b. x &gt; -5  1 &lt; x &lt; 2
c x &lt; -5  1 &lt; x &lt; 2
d. x &lt; 5  1 &lt; x &lt; 2
3. Which of the following is the smallest upper bound for the equation
f(x) = x^3 + 4x^2 – 3x + 5 = 0
a. -1
c. 1
b. 7
d. -3
A polynomial of smallest degree, with integral coefficient, with roots, 3 –
i and 2 – 3i, has a form of x^4 - ax^3 +47x^2 - cx + d = 0
4. Which of the following gives the value of a.
a. 8
c. 10
b. 5
d. -9
5. Which of the following gives the value of c.
a. 85
c. 118
b. -55
d. -94
6. Which of the following gives the value of d.
a. 124
c. 130
b. 47
d. -97
Resolve into partial fraction
3𝑥 2 − 10𝑥 + 16
𝐴
𝐵𝑥 + 𝐶
=
+
(𝑥 − 3)(𝑥 2 + 𝑥 + 1) 𝑥 − 3 𝑥 2 + 𝑥 + 1
7. Determine the value of A
a. -4
c. 4
b. 1
d. -6
8. Determine the value of B
a. -6
c. 4
b. 2
d. 6
9. Determine the value of C
a. 8
c. 0
b. -5
d. -10
Resolve into partial fraction
4𝑥 4 − 7𝑥 3 + 5𝑥 2 − 𝑥 + 1
𝐴
𝐵𝑥 + 𝐶
𝐷𝑥 + 𝐸
=
+ 2
+ 2
2
2
(2𝑥 − 1)(𝑥 − 𝑥 + 1)
2𝑥 − 1 𝑥 − 𝑥 + 1 (𝑥 − 𝑥 + 1)2
10. Determine the value of A
a. 1
c. -2
b. 2
d. 0
11. Determine the value of B
14. Determine the value of E
a. 1
c. -2
b. 2
d. 0
15. Which of the following is not a possible nature of the roots of f(x) =
x^4 + 2x^3 + 3x^2 + 4x + 5
a. 4 negative numbers
b. 2 negative and 2 complex numbers
c. 4 complex number
d. 1 positive, 1 negative, and 2 complex
16. Determine the domain of x for which y will be real:
𝑦 = √2𝑥 2 − 7𝑥 + 3
a. x  1/2 U x P3
c. x P 1/2 U x 3
b. x  -1/2 U x P3
d. x P -4 U x  -2
17. Determine the domain of x for which y will be real:
𝑥𝑦 2 + 3𝑥𝑦 + 3𝑥 − 4𝑦 − 4 = 0
a. x  -5 U x P2
c x  5 U x P-2
b. x  -5  x P2
d. x P -5 U x  2
18. Given the discriminant of the quadratic equation b^2 – 4ac, which of
the following is FALSE?
a. when equal to zero, the quadratic equation has 2 repeated real roots
b. when equal to a positive square, the quadratic equation has 2 distinct
real roots
c. when equal to negative, the quadratic equation has 2 imaginary roots
d. none of the above
19. Solve the following: absolute value of (x-3).
a. 2  x  4
c. 2  x
b. 2 &lt; x &lt; 4
d. x  4
20. What is the sum of the coefficients of the expansion (2x-1)^20?
a. 3486784401
c. 0
b. 3486784400
d. 2
21. What is the sum of the numerical coefficient of the expansion of
(2a^3 – 7b)^4?
a. 6561
c. 625
b. 1
d. 5934
22. Given the term (x^2 + 2y + 3z)^9. Determine the number of
terms of the expansion.
a. 3
c. 5
b. 251
d. 27
23. In a survey on the chewing gum preferences of baseball players,
it was found that 22 like fruit, 25 like spearmint, 39 like grape, 9 like
spearmint and fruit, 17 like fruit and grape, 20 like spearmint and
grape, 6 like all flavors, 4 like none. How many players were
surveyed?
a. 46
c. 34
b. 48
d. 50
24. If n(A-B) = 23, n(AB)=68 and n(B)=20, solve n(B).
a. 43
c. 45
b. 45
d. 49
g. rasco, ree, rmp
Page 2 of 4
GARCIA COLLEGE OF TECHNOLOGY
Kalibo, Aklan
Electrical Engineering Department
EE 412 (ELECTRICAL ENGINEERING CORRELATION 1)
MATHEMATICS SUBJECTS
25. A type of inequality that is true only for some intervals of x.
a. algebraic
c. absolute
b. conditional
d. recursive
26. A type of inequality that is true for all real numbers.
a. algebraic
c. absolute
b. conditional
d. recursive
27. Analysis of the nature of and algebraic solutions of algebraic
equation of polynomial.
a. theory of equations
b. fundamental theorem of algebra
c. remainder theorem
d. rational roots theorem
28. If a polynomial f(x) is divided by x-k until the remainder is a
constant, then this remainder is f(k).
a. rational zeroes theorem
b. factor theorem
c. remainder theorem
d. quadratic surd
29. A number of the form &plusmn; sq rt (a), where a is a positive rational
number which is not the square of another rational number.
a. pure quadratic surd
b. factor theorem
c. rational zeroes theorem
d. mixed quadratic surd
30. It states that every equation which can be put in the form with
zero on one side of the equal sign and a polynomial of degree
greater than or equal to one with real or complex coefficients on the
other has at least one root which is real or complex number.
a. remainder theorem
b. rational zeroes theorem
c. factor theorem
d. fundamental theorem of algebra
31. Which of the following gives the value of k so that x-3 is a factor
of x^4 – (k^2)(x^2) – kx -81.
a. -7/3
c. 3/4
b. -1/3
d. -1/3
32. Which of the following is a possibility for the roots of
x^4 + 2x^3 + 3x^2 + 4x + 5 = 0.
a. 4 positive real number
b. 3 positive real root and 1 negative real root
c. 2 negative real number and 2 positive real number
d. 4 negative real number
33. Form a quartic equation with real coefficients having 1-2i and 3 +
i as roots
a. (x^2-4x+2)(x^2-x+10)
b. (x^2+5x+2)(x^2+x-5)
c. (2x^2+5x-3)(3x^2+x-5)
d. (x^2-x+1)(x^2-x+10)
34. Solve the inequality [(x+7)/(x+3)] 0.
a. (-infinity, -7) union (-3, positive infinity)
b. (-infinity, -7] union [3, positive infinity)
c. (-infinity, -7] union (-3, positive infinity)
d. (-infinity, -7) intersection [-3, positive infinity)
35. Solve the inequality -2x^2 &lt; -11x + 5
a. (-infinity, 1/2) intersection [5, positive infinity)
b. (-infinity, 1/2) union (5, positive infinity)
c. (-infinity, 1/2) union (5, positive infinity)
d. (-infinity, 1/2) union [5, positive infinity)
36. Solve the inequality -5  0.5(4-3x) &lt; 1.
a. 2/3  x  14/3
c. 2/3 &gt; x &gt; 14/3
b. 2/3 &lt; x  14/3
d. 2/3 &lt; x  14/3
For the following equation:
f (x) = sq rt (3x – 1)
37. Find the domain
a. [1/3, infinity)
b. (1/3, infinity)
c. [0, infinity)
d. [-1/3, infinity)
38. Find the range
a. (negative infinity, positive infinity)
b. (negative infinity, 1/3)
c. [1/3, positive infinity)
d. negative infinity, -1/3)
39. Solve the inequality 1/(x-2) &gt; 0
a. [2, positive infinity)
b. (-2, positive infinity)
c. (negative infinity, positive infinity)
d. (2, positive infinity)
For the partial fraction:
(x^4 – x^3 + 14x^2 – 2x + 22) / [(x+1)(x^2+4)(x^2 -2x +5)] = A/(x+1) +
(Bx + C)/(x^2 +4) + (Dx +E)/(x^2 -2x +5)
40. Which of the following gives the value of A
a. 1
c. -2
b. 3
d. 0
41. Which of the following gives the value of C
a. 1
c. -2
b. 3
d. 0
42. Which of the following gives the value of D
a. 1
c. -2
b. 3
d. 0
43. Which of the following gives the value of E
a. 1
c. -2
b. 3
d. 0
44. Which of the following is an upper bound for the roots of x^4 –
x^3 -2x^2 -4x -24 = 0.
a. -1
c. 2
b. 1
d. 3
45. Find the 11th term of (x +2/sq rt of y)^15
a. 3075082 x^5/y^5
b. 3075072 x^5/y^4
c. 3075072 x^6/y^5
d. 3075072 x^5/y^5
46. Find the term containing x^9 in the expansion of (x^3 + 1/x)^11.
a. 123x^9
c. 233x^9
b. 550x^9
d. 462x^9
47. Find the sum of the coefficients (x^2 + 3y -2z^3)^10
a. 1678999
c. 60466176
b. 1024
d. 124444
g. rasco, ree, rmp
Page 3 of 4
GARCIA COLLEGE OF TECHNOLOGY
Kalibo, Aklan
Electrical Engineering Department
EE 412 (ELECTRICAL ENGINEERING CORRELATION 1)
MATHEMATICS SUBJECTS
48. Ninety people at a Superbowl party were surveyed to see what
they ate while watching the game. The following data were
collected: 48 had nachos, 39 had wings, 35 had a potato skins, 20
had both wings and potato skins, 19 had both potato skins and
nachos, 22 had both wings and nachos, 10 had nachos, wings and
potato skins. How many ad nothing?
a. 19
c. 15
b. 12
d. 14
49. Suppose Walter’s online music store conducts a customer survey
to determine the preferences of its customers. Customers are asked
what type of music they like. They may choose from the following
categories: Pop (P), Jazz (J), Classical (C), and none of the above (N).
Of 100 customers, some of the results are as follows: 44 like
Classical, 27 like all three, 15 like only Pop, 10 like Jazz and Classical,
but not Pop. How many like Classical but not Jazz?
a. 12
c. 11
b. 7
d. 10
50. From the equation 5x^2 + (3k-2)x -4k -1 = 0, determine the value
of k so that the sum and product of the roots are equal.
a. 2
c. -1
b. -3
d. 2
g. rasco, ree, rmp
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