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# Math Enhancement Exam, Engineering Math Class 1 Notes | EduRev

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## Class 1 : Math Enhancement Exam, Engineering Math Class 1 Notes | EduRev

``` Page 1

RF REVIEW CENTER ENHANCEMENT EXAM MATHEMATICS

RF Review Center : MANILA-CEBU-BAGUIO (0932-175-1218) Page 1

Instruction: Select the correct answer for each of the following
questions. Mark only one answer for each item by shading the box
corresponding to the letter of your choice on the answer sheet provided.

1. The total number of relations from the set A to set B is equal to:
a. n(A)n(B)    c. n(A)+n(B)
b. 2
n(A)n(B)

d. 2
n(A)+n(B)

2. Find the join of A = |
1 0
0 1
| and B = |
0 0
1 1
|.
A. |
1 0
1 1
|     C. |
0 0
0 1
|
B. |
0 1
0 1
|    D. |
1 0
1 0
|
3. The graph shown below is known as:

A. Injection Graph   C. Bigraph
B. Digraph     D. Arrow graph

4. What is the characteristic and mantissa of the logarithm of
0.0000065766 to the base 10?
A. -5, 0.818001   C. -6, 0.818001
B. -5, 0.181999   D. -6, 0.181999

5. Which of the following is the logarithm of -7.182?
A. 1.3644 + 0.8562i    C. 0.8562 â€“ 1.3644i
B. 0.8562 + 1.3644i     D. 1.3644 â€“ 0.8562i

6. If p(a) is true for some specific element a in the universe of
discourse, then ?x p(x) is true;

A. Existential Generalization
B. Existential Instantiation
C. Universal Generalization
D. Universal Instantiation

7. ________ minimum time needed to execute the algorithm among
all inputs of a given size n.
A. Average-case time   C. Execution time
B. Worst-case time   D. Best-case time

8. Determine the period of the function ?? ( ?? ) = cos
2
(
?? 5
?? ) + sin
3
(
?? 3
?? ).
A. 15    C. 45
B. 30     D. 60

9. _______ are certain simple arguments known to be valid and used
to make a proof step by step.
A. Rules of Affirmation   C. Rules of Proposition
B. Rules of Detachment   D. Rules of interference

10. The particular case of Eulerâ€™s Theorem in which m is a prime
number p or ?? ?? -1
= 1 ( mod ?? ) .
A. Fibonacci Theorem              C. Eulerâ€™s Prime Theorem
B. Fermatâ€™s Little Theorem              D. Lermaâ€™s Theorem

11. If G is a simple graph with n vertices with n = 3 such that the degree
of every vertex in G is at least n/2, then G has a Hamilton circuit.
A. Diracâ€™s Theorem    C. Eulerâ€™s Theorem
B. Vertex Removal Argument  D. Hamiltonâ€™s Theorem

12. Find the convolution between cos(t) and sin(t).
A. ½ t sin(t)     C. ½ t
2
sin(t)
B. ½ t cos(t)    D. ½ t
2
cos(t)

13. Find the larger angle and longer diagonal of a parallelogram with
two sides identified by vectors from the origin to the points (3,5) and
(8,0).
A. 116.855°, 11.704    C. 128.445°, 13.782
B. 120.964°, 12.083     D. 135.727°, 14.068

14. Evaluate: ? ?? x
n
ln?? ???? (x > 0, n ? -1).

A. (ln?? -
?? ?? +1
)(
?? ?? ?? +1
?? +1
) + ??
B. (ln?? +
????
?? +1
)(
?? ?? ?? +1
?? +1
) + ??
C. (ln?? -
?? ?? +1
)(
?? ?? ?? +1
?? +1
) + ??
D. (ln?? +
????
?? +1
)(
?? ?? ?? +1
?? +1
) + ??

15. Approximate the area of the graph ?? =
1
?? +1
formed within the closed
interval [2, 3] using Simpson's Rule with n=4
A. 0.2876831     C. 0.2876813
B. 0.2878682    D. 0.2867682

16. The side of the largest square has length 1. Find the total area of

A. 1/3     C. 4/3
B. 2/3    D. 5/3

17. Derive a set notation from the Venn diagram shown.

A. (AUC)\(AUB)    C. C\(AUC)
B. A\(AUC)     D. (AUB)\ (AUC)

18. Which of the following numbers is a derangement of the number
12345?
A. 14325     C. 21453
B. 13254     D. 21543

19. Define a sequence by b 1=2 and

?? ?? +?? =
?? + ?? ?? ?? - ?? ??   ?????? ?? = ??

What is the value of b2006?
A. -3       C. -1/2
B. 2      D. 1/3

20. The formula shown below is used in finding the approximate root of
a function using __________

A. Bisection Method   C. Regula Falsi
B. Newton Raphson Method  D. Secant Method

21. Given a rectangular coordinate (x,y,z) as (1,4,-7). Find its equivalent
value in spherical coordinate (r, ?, ø).
A. (8.124, 76°, 150°)    C. (9.361, 76°, 120°)
B. (8.124, 85°, 120°)   D. (9.361, 85°, 150°)

22. Let U={a,b,c,d,e,f,g}, A={a,b,c,d,e}, B={a,c,e,g} and C={b,e,f,g}.
Find (?? \?? ?? )
A. {a,c,e}     C. {b,e,f,g}
B. {a,c,d}     D. {b,d,f,g}

23. A function f satisfies (?? ?? ) = ?? (?? ) + ?? ?? ?? ?? ?? = ?? and f(2)=8.
Find f(65536).
A. 14     C.26
B. 20     D. 32

24. A total of 1232 students have taken a course in Spanish, 879 have
taken a course in French, and 114 have taken a course in Russian.
Further, 103 have taken courses in both Spanish and French, 23
have taken courses in both Spanish and Russian, and 14 have
taken courses in both French and Russian. If 2092 students have
taken at least one of Spanish, French, and Russian, how many
students have taken a course in all three languages?

Page 2

RF REVIEW CENTER ENHANCEMENT EXAM MATHEMATICS

RF Review Center : MANILA-CEBU-BAGUIO (0932-175-1218) Page 1

Instruction: Select the correct answer for each of the following
questions. Mark only one answer for each item by shading the box
corresponding to the letter of your choice on the answer sheet provided.

1. The total number of relations from the set A to set B is equal to:
a. n(A)n(B)    c. n(A)+n(B)
b. 2
n(A)n(B)

d. 2
n(A)+n(B)

2. Find the join of A = |
1 0
0 1
| and B = |
0 0
1 1
|.
A. |
1 0
1 1
|     C. |
0 0
0 1
|
B. |
0 1
0 1
|    D. |
1 0
1 0
|
3. The graph shown below is known as:

A. Injection Graph   C. Bigraph
B. Digraph     D. Arrow graph

4. What is the characteristic and mantissa of the logarithm of
0.0000065766 to the base 10?
A. -5, 0.818001   C. -6, 0.818001
B. -5, 0.181999   D. -6, 0.181999

5. Which of the following is the logarithm of -7.182?
A. 1.3644 + 0.8562i    C. 0.8562 â€“ 1.3644i
B. 0.8562 + 1.3644i     D. 1.3644 â€“ 0.8562i

6. If p(a) is true for some specific element a in the universe of
discourse, then ?x p(x) is true;

A. Existential Generalization
B. Existential Instantiation
C. Universal Generalization
D. Universal Instantiation

7. ________ minimum time needed to execute the algorithm among
all inputs of a given size n.
A. Average-case time   C. Execution time
B. Worst-case time   D. Best-case time

8. Determine the period of the function ?? ( ?? ) = cos
2
(
?? 5
?? ) + sin
3
(
?? 3
?? ).
A. 15    C. 45
B. 30     D. 60

9. _______ are certain simple arguments known to be valid and used
to make a proof step by step.
A. Rules of Affirmation   C. Rules of Proposition
B. Rules of Detachment   D. Rules of interference

10. The particular case of Eulerâ€™s Theorem in which m is a prime
number p or ?? ?? -1
= 1 ( mod ?? ) .
A. Fibonacci Theorem              C. Eulerâ€™s Prime Theorem
B. Fermatâ€™s Little Theorem              D. Lermaâ€™s Theorem

11. If G is a simple graph with n vertices with n = 3 such that the degree
of every vertex in G is at least n/2, then G has a Hamilton circuit.
A. Diracâ€™s Theorem    C. Eulerâ€™s Theorem
B. Vertex Removal Argument  D. Hamiltonâ€™s Theorem

12. Find the convolution between cos(t) and sin(t).
A. ½ t sin(t)     C. ½ t
2
sin(t)
B. ½ t cos(t)    D. ½ t
2
cos(t)

13. Find the larger angle and longer diagonal of a parallelogram with
two sides identified by vectors from the origin to the points (3,5) and
(8,0).
A. 116.855°, 11.704    C. 128.445°, 13.782
B. 120.964°, 12.083     D. 135.727°, 14.068

14. Evaluate: ? ?? x
n
ln?? ???? (x > 0, n ? -1).

A. (ln?? -
?? ?? +1
)(
?? ?? ?? +1
?? +1
) + ??
B. (ln?? +
????
?? +1
)(
?? ?? ?? +1
?? +1
) + ??
C. (ln?? -
?? ?? +1
)(
?? ?? ?? +1
?? +1
) + ??
D. (ln?? +
????
?? +1
)(
?? ?? ?? +1
?? +1
) + ??

15. Approximate the area of the graph ?? =
1
?? +1
formed within the closed
interval [2, 3] using Simpson's Rule with n=4
A. 0.2876831     C. 0.2876813
B. 0.2878682    D. 0.2867682

16. The side of the largest square has length 1. Find the total area of

A. 1/3     C. 4/3
B. 2/3    D. 5/3

17. Derive a set notation from the Venn diagram shown.

A. (AUC)\(AUB)    C. C\(AUC)
B. A\(AUC)     D. (AUB)\ (AUC)

18. Which of the following numbers is a derangement of the number
12345?
A. 14325     C. 21453
B. 13254     D. 21543

19. Define a sequence by b 1=2 and

?? ?? +?? =
?? + ?? ?? ?? - ?? ??   ?????? ?? = ??

What is the value of b2006?
A. -3       C. -1/2
B. 2      D. 1/3

20. The formula shown below is used in finding the approximate root of
a function using __________

A. Bisection Method   C. Regula Falsi
B. Newton Raphson Method  D. Secant Method

21. Given a rectangular coordinate (x,y,z) as (1,4,-7). Find its equivalent
value in spherical coordinate (r, ?, ø).
A. (8.124, 76°, 150°)    C. (9.361, 76°, 120°)
B. (8.124, 85°, 120°)   D. (9.361, 85°, 150°)

22. Let U={a,b,c,d,e,f,g}, A={a,b,c,d,e}, B={a,c,e,g} and C={b,e,f,g}.
Find (?? \?? ?? )
A. {a,c,e}     C. {b,e,f,g}
B. {a,c,d}     D. {b,d,f,g}

23. A function f satisfies (?? ?? ) = ?? (?? ) + ?? ?? ?? ?? ?? = ?? and f(2)=8.
Find f(65536).
A. 14     C.26
B. 20     D. 32

24. A total of 1232 students have taken a course in Spanish, 879 have
taken a course in French, and 114 have taken a course in Russian.
Further, 103 have taken courses in both Spanish and French, 23
have taken courses in both Spanish and Russian, and 14 have
taken courses in both French and Russian. If 2092 students have
taken at least one of Spanish, French, and Russian, how many
students have taken a course in all three languages?

RF REVIEW CENTER ENHANCEMENT EXAM MATHEMATICS

RF Review Center : MANILA-CEBU-BAGUIO (0932-175-1218) Page 2

A. 6      C. 9
B. 8      D. 7

25. What is the gradient of the function y
2
+ 3x â€“ 4y + 6 = 0 at a given
point (2,-1)?
A. 0.25    C. 0.75
B. 0.5     D. 1

26. What is the approximate value of x 3 in the function f(x) = cos(x) â€“
x
3
given x 0 = 0.5.
A. 0.90967269   C. 0.86547714
B. 0.86726382    D. 0.86547403

27. The property of a set given as ?? ? = ?? :
A. Idempotent Law   C. Involution Law
B. Nilpotent Law   D. Identity Law

28. The functions f(t) = 2t
2
and g(t) = t
4
is ___________ with each other.
A. Linearly dependent   C. Symmetric
B. Linearly independent   D. Antisymmetric

29. A ____________ is a graphical representation of a partially ordered
set in which each element is represented by a dot (node or vertex
of the diagram) as shown in the figure below:

A. Hilbert Diagram   C. Heibniz Diagram
B. Hasse Diagram    D. Hollerith Diagram

30. Find the laplacian of the function ?? ( ?? , ?? ) = 3 + ?????? (
?? 2
) ?????? (
?? 2
).
A.
3
2
???? +
1
2
?????? (
?? 2
)?????? (
?? 2
)
B.
1
2
?????? (
?? 2
)?????? (
?? 2
)
C. -
3
2
???? +
1
2
?????? (
?? 2
)?????? (
?? 2
)
D. -
1
2
?????? (
?? 2
)?????? (
?? 2
)

31. Find the sum of the infinite series:

1
3
+
2
3
2
+
1
3
3
+
2
3
4
+
1
3
5
+
2
3
6
+ ?

A. 5/9    C. 3/4
B. 2/3    D. 5/8

32. Two graphs G1 = (V 1,E 1), G2 = (V 2,E 2), are called ________ if there
is a bijection f : V 1 ? V 2 and a bijection g : E 1 ? E 2 such that an edge
e is adjacent to vertices v and w if and only if g(e) is adjacent to f(v)
and f(w).
A. Isomorphic    C. Orthomorphic
B. Symmetric    D. Isometric

33. Two sets are equal if and only if they have the same elements, i.e.:
A = B = ?x (x?A ? x?B)

A. Principle of Equivalence              C. Principle of Symmetry
B. Principle of Extension               D. Principle of Equality

34. Evaluate and determine |A|.

?? = [[
3 4 4
6 6 1
3 4 8
] [
4 2 -1
1 1 2
3 1 -1
]
?? - 3 [
4 1 5
6 1 3
-2 1 4
]]
A. 3412    C. 4312
B. 2212    D. 2172

35. The probability that a student pilot passes a written test for his
private pilotâ€™s license is 0.7. Find the probability that a person
passes the test before the 4
th
try.
A. 0.973     C. 0.922
B. 0.812    D. 0.954

36. Which bracketing method is faster than the others in finding the
roots of an expression?
A. Simpsonsâ€™ Rule   C. Bisection Method
B. Newton Raphson    D. False Position

37. The conditions that a function must satisfy so that it can be
expanded into a fourier series.
A. Dirichlet conditions    C. Fourier conditions
B. Lejuene conditions   D. Eulerâ€™s conditions

38. What is the cofactor of the element X in the matrix?

?? = [
2 3 4 8
5 -7 -6 3
7
5
1
2
2
?? 3
-7
]

A. 402    C. 111
B. -402     D. -111

39. All are first-order differential equations except _______.
A. Bernoulli    C. Clairaut
B. Riccati    D. Euler

40. If the sum of the squares of 10 numbers is 645 and their standard
deviation is 2.87, what is the arithmetic mean?
A. 6.5    C. 8.5
B. 7.5     D. 9.5

41. What is the approximate value of the root of the function,
?? ( ?? ) = ?? 3
- ?? - 2 after 8
th
iterations using Bisection method?
A. 1.5234375    C. 1.5205078
B. 1.5214844    D. 1.5195313

42. The integral of an odd function is ________.
A. Infinite    C. one
B. zero     D. it depends

43. Find the Jacobian transformation of the following functions:
x = r sin? cosø
y = r sin? sinø
z = r cos?

A. r
2
C. r
2
cos?
B. r
2
sin?     D. r
2
sin ?cos?

44. Find the meet of X = |
1 0
0 1
| and Y = |
0 0
1 1
|.
A. |
1 0
1 1
|    C. |
0 0
0 1
|
B. |
0 1
0 1
|    D. |
1 0
1 0
|

45. For the following tabulation, compute the correlation coefficient
between x and y.
X 80 84 88 92 98 104
Y 4 8 10 8 12 14

A. 0.94    C. 0.88
B. 0.92     D. 0.84

46. What is the Jacobian transformation of the functions: x = ?sin? and
y = ?cos??
A. ?      C. sin?
B. cos?    D. ?
2
sin?cos?

47. If ? ?? ( ?? ?? ) h
?? ?? ?? =1
has a limit as h
?? ? 0 ( ?? = 1,2, â€¦ . . ?? , ?????? h ?? ? 8) , no
matter how the region is subdivided, then this is called the ________ of
f(x).
A. Riemann Integral    C. Partial Integral
B. Parseval Integral   D. Bessel Integral

48. A tetrahedron ABCD, edge AB has length 3 cm. The area of the face
ABC is 15 cm
2
and the area of face ABD is 12 cm
2
. These two faces
meet each other at a 30° angle. Find the volume of the tetrahedron
in cm
3
. ABC is the base of the tetrahedron while D is its apex.
A. 15    C. 25
B. 20     D. 30

49. The circuit of the given figure realizes the Boolean function:

Page 3

RF REVIEW CENTER ENHANCEMENT EXAM MATHEMATICS

RF Review Center : MANILA-CEBU-BAGUIO (0932-175-1218) Page 1

Instruction: Select the correct answer for each of the following
questions. Mark only one answer for each item by shading the box
corresponding to the letter of your choice on the answer sheet provided.

1. The total number of relations from the set A to set B is equal to:
a. n(A)n(B)    c. n(A)+n(B)
b. 2
n(A)n(B)

d. 2
n(A)+n(B)

2. Find the join of A = |
1 0
0 1
| and B = |
0 0
1 1
|.
A. |
1 0
1 1
|     C. |
0 0
0 1
|
B. |
0 1
0 1
|    D. |
1 0
1 0
|
3. The graph shown below is known as:

A. Injection Graph   C. Bigraph
B. Digraph     D. Arrow graph

4. What is the characteristic and mantissa of the logarithm of
0.0000065766 to the base 10?
A. -5, 0.818001   C. -6, 0.818001
B. -5, 0.181999   D. -6, 0.181999

5. Which of the following is the logarithm of -7.182?
A. 1.3644 + 0.8562i    C. 0.8562 â€“ 1.3644i
B. 0.8562 + 1.3644i     D. 1.3644 â€“ 0.8562i

6. If p(a) is true for some specific element a in the universe of
discourse, then ?x p(x) is true;

A. Existential Generalization
B. Existential Instantiation
C. Universal Generalization
D. Universal Instantiation

7. ________ minimum time needed to execute the algorithm among
all inputs of a given size n.
A. Average-case time   C. Execution time
B. Worst-case time   D. Best-case time

8. Determine the period of the function ?? ( ?? ) = cos
2
(
?? 5
?? ) + sin
3
(
?? 3
?? ).
A. 15    C. 45
B. 30     D. 60

9. _______ are certain simple arguments known to be valid and used
to make a proof step by step.
A. Rules of Affirmation   C. Rules of Proposition
B. Rules of Detachment   D. Rules of interference

10. The particular case of Eulerâ€™s Theorem in which m is a prime
number p or ?? ?? -1
= 1 ( mod ?? ) .
A. Fibonacci Theorem              C. Eulerâ€™s Prime Theorem
B. Fermatâ€™s Little Theorem              D. Lermaâ€™s Theorem

11. If G is a simple graph with n vertices with n = 3 such that the degree
of every vertex in G is at least n/2, then G has a Hamilton circuit.
A. Diracâ€™s Theorem    C. Eulerâ€™s Theorem
B. Vertex Removal Argument  D. Hamiltonâ€™s Theorem

12. Find the convolution between cos(t) and sin(t).
A. ½ t sin(t)     C. ½ t
2
sin(t)
B. ½ t cos(t)    D. ½ t
2
cos(t)

13. Find the larger angle and longer diagonal of a parallelogram with
two sides identified by vectors from the origin to the points (3,5) and
(8,0).
A. 116.855°, 11.704    C. 128.445°, 13.782
B. 120.964°, 12.083     D. 135.727°, 14.068

14. Evaluate: ? ?? x
n
ln?? ???? (x > 0, n ? -1).

A. (ln?? -
?? ?? +1
)(
?? ?? ?? +1
?? +1
) + ??
B. (ln?? +
????
?? +1
)(
?? ?? ?? +1
?? +1
) + ??
C. (ln?? -
?? ?? +1
)(
?? ?? ?? +1
?? +1
) + ??
D. (ln?? +
????
?? +1
)(
?? ?? ?? +1
?? +1
) + ??

15. Approximate the area of the graph ?? =
1
?? +1
formed within the closed
interval [2, 3] using Simpson's Rule with n=4
A. 0.2876831     C. 0.2876813
B. 0.2878682    D. 0.2867682

16. The side of the largest square has length 1. Find the total area of

A. 1/3     C. 4/3
B. 2/3    D. 5/3

17. Derive a set notation from the Venn diagram shown.

A. (AUC)\(AUB)    C. C\(AUC)
B. A\(AUC)     D. (AUB)\ (AUC)

18. Which of the following numbers is a derangement of the number
12345?
A. 14325     C. 21453
B. 13254     D. 21543

19. Define a sequence by b 1=2 and

?? ?? +?? =
?? + ?? ?? ?? - ?? ??   ?????? ?? = ??

What is the value of b2006?
A. -3       C. -1/2
B. 2      D. 1/3

20. The formula shown below is used in finding the approximate root of
a function using __________

A. Bisection Method   C. Regula Falsi
B. Newton Raphson Method  D. Secant Method

21. Given a rectangular coordinate (x,y,z) as (1,4,-7). Find its equivalent
value in spherical coordinate (r, ?, ø).
A. (8.124, 76°, 150°)    C. (9.361, 76°, 120°)
B. (8.124, 85°, 120°)   D. (9.361, 85°, 150°)

22. Let U={a,b,c,d,e,f,g}, A={a,b,c,d,e}, B={a,c,e,g} and C={b,e,f,g}.
Find (?? \?? ?? )
A. {a,c,e}     C. {b,e,f,g}
B. {a,c,d}     D. {b,d,f,g}

23. A function f satisfies (?? ?? ) = ?? (?? ) + ?? ?? ?? ?? ?? = ?? and f(2)=8.
Find f(65536).
A. 14     C.26
B. 20     D. 32

24. A total of 1232 students have taken a course in Spanish, 879 have
taken a course in French, and 114 have taken a course in Russian.
Further, 103 have taken courses in both Spanish and French, 23
have taken courses in both Spanish and Russian, and 14 have
taken courses in both French and Russian. If 2092 students have
taken at least one of Spanish, French, and Russian, how many
students have taken a course in all three languages?

RF REVIEW CENTER ENHANCEMENT EXAM MATHEMATICS

RF Review Center : MANILA-CEBU-BAGUIO (0932-175-1218) Page 2

A. 6      C. 9
B. 8      D. 7

25. What is the gradient of the function y
2
+ 3x â€“ 4y + 6 = 0 at a given
point (2,-1)?
A. 0.25    C. 0.75
B. 0.5     D. 1

26. What is the approximate value of x 3 in the function f(x) = cos(x) â€“
x
3
given x 0 = 0.5.
A. 0.90967269   C. 0.86547714
B. 0.86726382    D. 0.86547403

27. The property of a set given as ?? ? = ?? :
A. Idempotent Law   C. Involution Law
B. Nilpotent Law   D. Identity Law

28. The functions f(t) = 2t
2
and g(t) = t
4
is ___________ with each other.
A. Linearly dependent   C. Symmetric
B. Linearly independent   D. Antisymmetric

29. A ____________ is a graphical representation of a partially ordered
set in which each element is represented by a dot (node or vertex
of the diagram) as shown in the figure below:

A. Hilbert Diagram   C. Heibniz Diagram
B. Hasse Diagram    D. Hollerith Diagram

30. Find the laplacian of the function ?? ( ?? , ?? ) = 3 + ?????? (
?? 2
) ?????? (
?? 2
).
A.
3
2
???? +
1
2
?????? (
?? 2
)?????? (
?? 2
)
B.
1
2
?????? (
?? 2
)?????? (
?? 2
)
C. -
3
2
???? +
1
2
?????? (
?? 2
)?????? (
?? 2
)
D. -
1
2
?????? (
?? 2
)?????? (
?? 2
)

31. Find the sum of the infinite series:

1
3
+
2
3
2
+
1
3
3
+
2
3
4
+
1
3
5
+
2
3
6
+ ?

A. 5/9    C. 3/4
B. 2/3    D. 5/8

32. Two graphs G1 = (V 1,E 1), G2 = (V 2,E 2), are called ________ if there
is a bijection f : V 1 ? V 2 and a bijection g : E 1 ? E 2 such that an edge
e is adjacent to vertices v and w if and only if g(e) is adjacent to f(v)
and f(w).
A. Isomorphic    C. Orthomorphic
B. Symmetric    D. Isometric

33. Two sets are equal if and only if they have the same elements, i.e.:
A = B = ?x (x?A ? x?B)

A. Principle of Equivalence              C. Principle of Symmetry
B. Principle of Extension               D. Principle of Equality

34. Evaluate and determine |A|.

?? = [[
3 4 4
6 6 1
3 4 8
] [
4 2 -1
1 1 2
3 1 -1
]
?? - 3 [
4 1 5
6 1 3
-2 1 4
]]
A. 3412    C. 4312
B. 2212    D. 2172

35. The probability that a student pilot passes a written test for his
private pilotâ€™s license is 0.7. Find the probability that a person
passes the test before the 4
th
try.
A. 0.973     C. 0.922
B. 0.812    D. 0.954

36. Which bracketing method is faster than the others in finding the
roots of an expression?
A. Simpsonsâ€™ Rule   C. Bisection Method
B. Newton Raphson    D. False Position

37. The conditions that a function must satisfy so that it can be
expanded into a fourier series.
A. Dirichlet conditions    C. Fourier conditions
B. Lejuene conditions   D. Eulerâ€™s conditions

38. What is the cofactor of the element X in the matrix?

?? = [
2 3 4 8
5 -7 -6 3
7
5
1
2
2
?? 3
-7
]

A. 402    C. 111
B. -402     D. -111

39. All are first-order differential equations except _______.
A. Bernoulli    C. Clairaut
B. Riccati    D. Euler

40. If the sum of the squares of 10 numbers is 645 and their standard
deviation is 2.87, what is the arithmetic mean?
A. 6.5    C. 8.5
B. 7.5     D. 9.5

41. What is the approximate value of the root of the function,
?? ( ?? ) = ?? 3
- ?? - 2 after 8
th
iterations using Bisection method?
A. 1.5234375    C. 1.5205078
B. 1.5214844    D. 1.5195313

42. The integral of an odd function is ________.
A. Infinite    C. one
B. zero     D. it depends

43. Find the Jacobian transformation of the following functions:
x = r sin? cosø
y = r sin? sinø
z = r cos?

A. r
2
C. r
2
cos?
B. r
2
sin?     D. r
2
sin ?cos?

44. Find the meet of X = |
1 0
0 1
| and Y = |
0 0
1 1
|.
A. |
1 0
1 1
|    C. |
0 0
0 1
|
B. |
0 1
0 1
|    D. |
1 0
1 0
|

45. For the following tabulation, compute the correlation coefficient
between x and y.
X 80 84 88 92 98 104
Y 4 8 10 8 12 14

A. 0.94    C. 0.88
B. 0.92     D. 0.84

46. What is the Jacobian transformation of the functions: x = ?sin? and
y = ?cos??
A. ?      C. sin?
B. cos?    D. ?
2
sin?cos?

47. If ? ?? ( ?? ?? ) h
?? ?? ?? =1
has a limit as h
?? ? 0 ( ?? = 1,2, â€¦ . . ?? , ?????? h ?? ? 8) , no
matter how the region is subdivided, then this is called the ________ of
f(x).
A. Riemann Integral    C. Partial Integral
B. Parseval Integral   D. Bessel Integral

48. A tetrahedron ABCD, edge AB has length 3 cm. The area of the face
ABC is 15 cm
2
and the area of face ABD is 12 cm
2
. These two faces
meet each other at a 30° angle. Find the volume of the tetrahedron
in cm
3
. ABC is the base of the tetrahedron while D is its apex.
A. 15    C. 25
B. 20     D. 30

49. The circuit of the given figure realizes the Boolean function:

RF REVIEW CENTER ENHANCEMENT EXAM MATHEMATICS

RF Review Center : MANILA-CEBU-BAGUIO (0932-175-1218) Page 3

A. Y = ( A
¯
+ B
¯
) C + DE
¯ ¯ ¯ ¯

B. Y = A
¯
+ B
¯
+ C
¯
+ D
¯
+ E
¯

C. Y = AB + C + DE
D. Y = AB + C( D + E)

50. The equation of the form
????
????
= P( x) y
2
+ Q( x) y + R( x) is _________
differential equation.
A. Bernoulli    C. Cauchy
B. Riccatiâ€™s     D. Euler

51. Find the wronskian of the f(t) = cos(t) and g(t) = sin(t)
A. 0     C. -1
B. 1     D. ?

52. Determine the Wronskian of two solutions to the following
differential equation.

?? 4
?? '
- 2?? 3
?? '
- ?? 8
?? = 0

A. 2ct    C. 2ct
3
B. ct
2
D. ct
4

53. Lot ABCDEFA is a closed traverse in the form of a regular hexagon
with each side equal to 100m The bearing of AB is N 25°. What is
the bearing of CD?
A. S 35° E     C. S 30° E
B. S 45° E    D. S 40° E

54. Find A in 2(covercosine A) + haversine A = 3.112.
A. 24.08°     C. 28.05°
B. 26.76°    D. 30.09°

55. The vertices of a triangle have their polar coordinates at (0, 0°), (6,
30°) and (9, 70°). The perimeter of the triangle is:
A. 34.45    C. 25.67
B. 20.85     D. 21.34

56. Five people are sitting at a table in a restaurant. Two of them order
coffee and the other three order tea. The waiter forgot who ordered
what and puts the drinks in a random order for the five persons.
Determine the probability that each person gets the correct drink.
A. 0.10     C. 0.16
B. 0. 13    D. 0.19

57. Pete tosses n + 1 fair coins and John tosses n fair coins. What is
the probability that Pete gets more heads than John?
A. 1/4    C. 1/3
B. ½     D. 1/5

58. Twelve married couples participate in a tournament. The group of
24 people is randomly split into eight teams of three people each,
where all possible splits are equally likely. What is the probability
that none of the teams has a married couple?
A. 0.6553    C. 0.4388
B. 0.5612    D. 0.3447

59. A random number is repeatedly drawn from 1, 2, 3, 4, 5, 6, 7, 8, 9,
and 10. What is the probability that not all of the numbers 1,
2,â€¦.10 show up after 50 draws?
A. 0.0512    C. 0.0502
B. 0. 0507    D. 0.0497

60. In the spherical triangle, A = 120°, B = 135° and c = 30°. What is
C?
A. 79.82°     C. 67.33°
B. 46.78°    D. 98.23°

61. An "n x n" complex matrix A is _____ if and only if A = A
T
A. Unitary    C. Singular
B. Hermitian      D. Skewâ€“Hermitian

62. What is the modulus of 9 - 3i?
A. 5v2    C. 2v3
B. 6v2    D. 3v10

63. Solve ?? ????
????
= ?? + ?? cos
2
?? ??
A. tan
?? ?? = log|???? |     C. tan
?? ?? = log|???? |
B. tan
?? ?? = log|???? |   D. tan
?? ?? = log|???? |

64. If ?? ( ?? ) is the Lapalce Transform of function ?? ( ?? ) , then Laplace
transform of ? ?? ( ?? ) ????
?? 0
is

A.
1
?? ?? ( ?? )   C. ???? ( ?? )- ?? ( 0)
B.
1
?? ?? ( ?? )- ?? ( 0)  D. ? ?? ( ?? ) ????

65. With initial condition ?? ( 1) = 0.5, the solution of the differential
equation ?? ????
????
+ ?? = ?? , is
A. ?? = ?? - 1/2    C. ?? = ?? 2
/2
B. ?? = ?? 2
- 1/2   D. ?? = ?? /2

66. If ?? = [
1 2
3 -5
] then its adjoint is
A. [
-1 -2
3 5
]  C. [
-5 -2
-3 1
]

B. [
5 -2
-3 1
]  D. [
-5 -2
3 -1
]

67. The principal value of log(
1
?? 4
) is
A. ????  C. ???? /4
B. ???? /2  D. ???? /8

68. The real part of the principal value of 4
4-?? is
A. 256cos ( ln4)    C. 16cos ( ln4)
B. 64cos ( ln4)  D. 4 cos ( ln4)

69. One of the eigen vectors of the matrix ?? = [
2 2
1 3
] is
A. [
2
-1
]    C. [
4
1
]
B. [
2
1
]  D. [
1
-1
]

70. For a matrix ?? = [
3/5 4/5
?? 3/5
], the transpose of the matrix is equal
to the inverse of the matrix, ?? ?? = ?? -1
. The value of x is given by
A. -4/5    C. 3/5
B. -3/5    D. 4/5

71. The particular integral of ?? '''
( ?? )- 8?? ( ?? ) = 1 is
A. 1/8    C. â€“ 1/8
B. 1/4     D. 1/10

72 The particular integral of
?? 2
?? ?? ?? 2
+ 9?? = cos3?? is
A.
cos ?? 8
C.
cos 3?? 9

B.
cos ?? 16
D. None of these

73. Find the integrating factor of the differential equation cos?? ????
????
+
?? ?????? ?? = 1.
A. ?????? ??    C. ?????? ??
B. ?????? ??      D. ?????? ??

74. Under certain conditions, cane sugar in water is converted into
dextrose at a rate proportional to the amount that is unconverted at
any time. If, of 75 kg at time t = 0, 8kg are converted during the first
30 minutes, find the amount converted in 2 hours.
A. 72.73 kg     C. 27.23 kg
B. 23.27 kg     D. 32.72 kg

75. A tank contains 400 liters of brine holding 100 kg of salt in solution.
Water containing 125 g of salt per liter flows into the tank at the rate
of 12 liters per minute, and the mixture, kept uniform by stirring,
flows out at the same rate. Find the amount of salt at the end of 90
minutes.
A. 53.36 kg      C. 53.63 kg
B. 56.33 kg     D. 65.33 kg

76. Find the Laplace transform of the function
1
v?? .
A.
?? v??  C.
?? ??
B.
v?? v??      D.
v?? ?? 2

77. The curl of vector V( x,y,z) = 2x
2
i +3z
2
j + y
3
k at x = y = z = 1 is:
A. â€“ 3 i      C. 3 i â€“ 4 j
B. 3 i     D. 3 i â€“ 6 k

78 Find the orthogonal trajectories of the family of hyperbolas x
2
â€“ y
2

= ay.
A.?? 2
+ 3???? = ??  C.  ?? 3
+ 2?? ?? 2
= ??
B. ?? 3
+ 3?? ?? 2
= ??    D.  ?? 3
+ 2?? 2
?? = ??

Page 4

RF REVIEW CENTER ENHANCEMENT EXAM MATHEMATICS

RF Review Center : MANILA-CEBU-BAGUIO (0932-175-1218) Page 1

Instruction: Select the correct answer for each of the following
questions. Mark only one answer for each item by shading the box
corresponding to the letter of your choice on the answer sheet provided.

1. The total number of relations from the set A to set B is equal to:
a. n(A)n(B)    c. n(A)+n(B)
b. 2
n(A)n(B)

d. 2
n(A)+n(B)

2. Find the join of A = |
1 0
0 1
| and B = |
0 0
1 1
|.
A. |
1 0
1 1
|     C. |
0 0
0 1
|
B. |
0 1
0 1
|    D. |
1 0
1 0
|
3. The graph shown below is known as:

A. Injection Graph   C. Bigraph
B. Digraph     D. Arrow graph

4. What is the characteristic and mantissa of the logarithm of
0.0000065766 to the base 10?
A. -5, 0.818001   C. -6, 0.818001
B. -5, 0.181999   D. -6, 0.181999

5. Which of the following is the logarithm of -7.182?
A. 1.3644 + 0.8562i    C. 0.8562 â€“ 1.3644i
B. 0.8562 + 1.3644i     D. 1.3644 â€“ 0.8562i

6. If p(a) is true for some specific element a in the universe of
discourse, then ?x p(x) is true;

A. Existential Generalization
B. Existential Instantiation
C. Universal Generalization
D. Universal Instantiation

7. ________ minimum time needed to execute the algorithm among
all inputs of a given size n.
A. Average-case time   C. Execution time
B. Worst-case time   D. Best-case time

8. Determine the period of the function ?? ( ?? ) = cos
2
(
?? 5
?? ) + sin
3
(
?? 3
?? ).
A. 15    C. 45
B. 30     D. 60

9. _______ are certain simple arguments known to be valid and used
to make a proof step by step.
A. Rules of Affirmation   C. Rules of Proposition
B. Rules of Detachment   D. Rules of interference

10. The particular case of Eulerâ€™s Theorem in which m is a prime
number p or ?? ?? -1
= 1 ( mod ?? ) .
A. Fibonacci Theorem              C. Eulerâ€™s Prime Theorem
B. Fermatâ€™s Little Theorem              D. Lermaâ€™s Theorem

11. If G is a simple graph with n vertices with n = 3 such that the degree
of every vertex in G is at least n/2, then G has a Hamilton circuit.
A. Diracâ€™s Theorem    C. Eulerâ€™s Theorem
B. Vertex Removal Argument  D. Hamiltonâ€™s Theorem

12. Find the convolution between cos(t) and sin(t).
A. ½ t sin(t)     C. ½ t
2
sin(t)
B. ½ t cos(t)    D. ½ t
2
cos(t)

13. Find the larger angle and longer diagonal of a parallelogram with
two sides identified by vectors from the origin to the points (3,5) and
(8,0).
A. 116.855°, 11.704    C. 128.445°, 13.782
B. 120.964°, 12.083     D. 135.727°, 14.068

14. Evaluate: ? ?? x
n
ln?? ???? (x > 0, n ? -1).

A. (ln?? -
?? ?? +1
)(
?? ?? ?? +1
?? +1
) + ??
B. (ln?? +
????
?? +1
)(
?? ?? ?? +1
?? +1
) + ??
C. (ln?? -
?? ?? +1
)(
?? ?? ?? +1
?? +1
) + ??
D. (ln?? +
????
?? +1
)(
?? ?? ?? +1
?? +1
) + ??

15. Approximate the area of the graph ?? =
1
?? +1
formed within the closed
interval [2, 3] using Simpson's Rule with n=4
A. 0.2876831     C. 0.2876813
B. 0.2878682    D. 0.2867682

16. The side of the largest square has length 1. Find the total area of

A. 1/3     C. 4/3
B. 2/3    D. 5/3

17. Derive a set notation from the Venn diagram shown.

A. (AUC)\(AUB)    C. C\(AUC)
B. A\(AUC)     D. (AUB)\ (AUC)

18. Which of the following numbers is a derangement of the number
12345?
A. 14325     C. 21453
B. 13254     D. 21543

19. Define a sequence by b 1=2 and

?? ?? +?? =
?? + ?? ?? ?? - ?? ??   ?????? ?? = ??

What is the value of b2006?
A. -3       C. -1/2
B. 2      D. 1/3

20. The formula shown below is used in finding the approximate root of
a function using __________

A. Bisection Method   C. Regula Falsi
B. Newton Raphson Method  D. Secant Method

21. Given a rectangular coordinate (x,y,z) as (1,4,-7). Find its equivalent
value in spherical coordinate (r, ?, ø).
A. (8.124, 76°, 150°)    C. (9.361, 76°, 120°)
B. (8.124, 85°, 120°)   D. (9.361, 85°, 150°)

22. Let U={a,b,c,d,e,f,g}, A={a,b,c,d,e}, B={a,c,e,g} and C={b,e,f,g}.
Find (?? \?? ?? )
A. {a,c,e}     C. {b,e,f,g}
B. {a,c,d}     D. {b,d,f,g}

23. A function f satisfies (?? ?? ) = ?? (?? ) + ?? ?? ?? ?? ?? = ?? and f(2)=8.
Find f(65536).
A. 14     C.26
B. 20     D. 32

24. A total of 1232 students have taken a course in Spanish, 879 have
taken a course in French, and 114 have taken a course in Russian.
Further, 103 have taken courses in both Spanish and French, 23
have taken courses in both Spanish and Russian, and 14 have
taken courses in both French and Russian. If 2092 students have
taken at least one of Spanish, French, and Russian, how many
students have taken a course in all three languages?

RF REVIEW CENTER ENHANCEMENT EXAM MATHEMATICS

RF Review Center : MANILA-CEBU-BAGUIO (0932-175-1218) Page 2

A. 6      C. 9
B. 8      D. 7

25. What is the gradient of the function y
2
+ 3x â€“ 4y + 6 = 0 at a given
point (2,-1)?
A. 0.25    C. 0.75
B. 0.5     D. 1

26. What is the approximate value of x 3 in the function f(x) = cos(x) â€“
x
3
given x 0 = 0.5.
A. 0.90967269   C. 0.86547714
B. 0.86726382    D. 0.86547403

27. The property of a set given as ?? ? = ?? :
A. Idempotent Law   C. Involution Law
B. Nilpotent Law   D. Identity Law

28. The functions f(t) = 2t
2
and g(t) = t
4
is ___________ with each other.
A. Linearly dependent   C. Symmetric
B. Linearly independent   D. Antisymmetric

29. A ____________ is a graphical representation of a partially ordered
set in which each element is represented by a dot (node or vertex
of the diagram) as shown in the figure below:

A. Hilbert Diagram   C. Heibniz Diagram
B. Hasse Diagram    D. Hollerith Diagram

30. Find the laplacian of the function ?? ( ?? , ?? ) = 3 + ?????? (
?? 2
) ?????? (
?? 2
).
A.
3
2
???? +
1
2
?????? (
?? 2
)?????? (
?? 2
)
B.
1
2
?????? (
?? 2
)?????? (
?? 2
)
C. -
3
2
???? +
1
2
?????? (
?? 2
)?????? (
?? 2
)
D. -
1
2
?????? (
?? 2
)?????? (
?? 2
)

31. Find the sum of the infinite series:

1
3
+
2
3
2
+
1
3
3
+
2
3
4
+
1
3
5
+
2
3
6
+ ?

A. 5/9    C. 3/4
B. 2/3    D. 5/8

32. Two graphs G1 = (V 1,E 1), G2 = (V 2,E 2), are called ________ if there
is a bijection f : V 1 ? V 2 and a bijection g : E 1 ? E 2 such that an edge
e is adjacent to vertices v and w if and only if g(e) is adjacent to f(v)
and f(w).
A. Isomorphic    C. Orthomorphic
B. Symmetric    D. Isometric

33. Two sets are equal if and only if they have the same elements, i.e.:
A = B = ?x (x?A ? x?B)

A. Principle of Equivalence              C. Principle of Symmetry
B. Principle of Extension               D. Principle of Equality

34. Evaluate and determine |A|.

?? = [[
3 4 4
6 6 1
3 4 8
] [
4 2 -1
1 1 2
3 1 -1
]
?? - 3 [
4 1 5
6 1 3
-2 1 4
]]
A. 3412    C. 4312
B. 2212    D. 2172

35. The probability that a student pilot passes a written test for his
private pilotâ€™s license is 0.7. Find the probability that a person
passes the test before the 4
th
try.
A. 0.973     C. 0.922
B. 0.812    D. 0.954

36. Which bracketing method is faster than the others in finding the
roots of an expression?
A. Simpsonsâ€™ Rule   C. Bisection Method
B. Newton Raphson    D. False Position

37. The conditions that a function must satisfy so that it can be
expanded into a fourier series.
A. Dirichlet conditions    C. Fourier conditions
B. Lejuene conditions   D. Eulerâ€™s conditions

38. What is the cofactor of the element X in the matrix?

?? = [
2 3 4 8
5 -7 -6 3
7
5
1
2
2
?? 3
-7
]

A. 402    C. 111
B. -402     D. -111

39. All are first-order differential equations except _______.
A. Bernoulli    C. Clairaut
B. Riccati    D. Euler

40. If the sum of the squares of 10 numbers is 645 and their standard
deviation is 2.87, what is the arithmetic mean?
A. 6.5    C. 8.5
B. 7.5     D. 9.5

41. What is the approximate value of the root of the function,
?? ( ?? ) = ?? 3
- ?? - 2 after 8
th
iterations using Bisection method?
A. 1.5234375    C. 1.5205078
B. 1.5214844    D. 1.5195313

42. The integral of an odd function is ________.
A. Infinite    C. one
B. zero     D. it depends

43. Find the Jacobian transformation of the following functions:
x = r sin? cosø
y = r sin? sinø
z = r cos?

A. r
2
C. r
2
cos?
B. r
2
sin?     D. r
2
sin ?cos?

44. Find the meet of X = |
1 0
0 1
| and Y = |
0 0
1 1
|.
A. |
1 0
1 1
|    C. |
0 0
0 1
|
B. |
0 1
0 1
|    D. |
1 0
1 0
|

45. For the following tabulation, compute the correlation coefficient
between x and y.
X 80 84 88 92 98 104
Y 4 8 10 8 12 14

A. 0.94    C. 0.88
B. 0.92     D. 0.84

46. What is the Jacobian transformation of the functions: x = ?sin? and
y = ?cos??
A. ?      C. sin?
B. cos?    D. ?
2
sin?cos?

47. If ? ?? ( ?? ?? ) h
?? ?? ?? =1
has a limit as h
?? ? 0 ( ?? = 1,2, â€¦ . . ?? , ?????? h ?? ? 8) , no
matter how the region is subdivided, then this is called the ________ of
f(x).
A. Riemann Integral    C. Partial Integral
B. Parseval Integral   D. Bessel Integral

48. A tetrahedron ABCD, edge AB has length 3 cm. The area of the face
ABC is 15 cm
2
and the area of face ABD is 12 cm
2
. These two faces
meet each other at a 30° angle. Find the volume of the tetrahedron
in cm
3
. ABC is the base of the tetrahedron while D is its apex.
A. 15    C. 25
B. 20     D. 30

49. The circuit of the given figure realizes the Boolean function:

RF REVIEW CENTER ENHANCEMENT EXAM MATHEMATICS

RF Review Center : MANILA-CEBU-BAGUIO (0932-175-1218) Page 3

A. Y = ( A
¯
+ B
¯
) C + DE
¯ ¯ ¯ ¯

B. Y = A
¯
+ B
¯
+ C
¯
+ D
¯
+ E
¯

C. Y = AB + C + DE
D. Y = AB + C( D + E)

50. The equation of the form
????
????
= P( x) y
2
+ Q( x) y + R( x) is _________
differential equation.
A. Bernoulli    C. Cauchy
B. Riccatiâ€™s     D. Euler

51. Find the wronskian of the f(t) = cos(t) and g(t) = sin(t)
A. 0     C. -1
B. 1     D. ?

52. Determine the Wronskian of two solutions to the following
differential equation.

?? 4
?? '
- 2?? 3
?? '
- ?? 8
?? = 0

A. 2ct    C. 2ct
3
B. ct
2
D. ct
4

53. Lot ABCDEFA is a closed traverse in the form of a regular hexagon
with each side equal to 100m The bearing of AB is N 25°. What is
the bearing of CD?
A. S 35° E     C. S 30° E
B. S 45° E    D. S 40° E

54. Find A in 2(covercosine A) + haversine A = 3.112.
A. 24.08°     C. 28.05°
B. 26.76°    D. 30.09°

55. The vertices of a triangle have their polar coordinates at (0, 0°), (6,
30°) and (9, 70°). The perimeter of the triangle is:
A. 34.45    C. 25.67
B. 20.85     D. 21.34

56. Five people are sitting at a table in a restaurant. Two of them order
coffee and the other three order tea. The waiter forgot who ordered
what and puts the drinks in a random order for the five persons.
Determine the probability that each person gets the correct drink.
A. 0.10     C. 0.16
B. 0. 13    D. 0.19

57. Pete tosses n + 1 fair coins and John tosses n fair coins. What is
the probability that Pete gets more heads than John?
A. 1/4    C. 1/3
B. ½     D. 1/5

58. Twelve married couples participate in a tournament. The group of
24 people is randomly split into eight teams of three people each,
where all possible splits are equally likely. What is the probability
that none of the teams has a married couple?
A. 0.6553    C. 0.4388
B. 0.5612    D. 0.3447

59. A random number is repeatedly drawn from 1, 2, 3, 4, 5, 6, 7, 8, 9,
and 10. What is the probability that not all of the numbers 1,
2,â€¦.10 show up after 50 draws?
A. 0.0512    C. 0.0502
B. 0. 0507    D. 0.0497

60. In the spherical triangle, A = 120°, B = 135° and c = 30°. What is
C?
A. 79.82°     C. 67.33°
B. 46.78°    D. 98.23°

61. An "n x n" complex matrix A is _____ if and only if A = A
T
A. Unitary    C. Singular
B. Hermitian      D. Skewâ€“Hermitian

62. What is the modulus of 9 - 3i?
A. 5v2    C. 2v3
B. 6v2    D. 3v10

63. Solve ?? ????
????
= ?? + ?? cos
2
?? ??
A. tan
?? ?? = log|???? |     C. tan
?? ?? = log|???? |
B. tan
?? ?? = log|???? |   D. tan
?? ?? = log|???? |

64. If ?? ( ?? ) is the Lapalce Transform of function ?? ( ?? ) , then Laplace
transform of ? ?? ( ?? ) ????
?? 0
is

A.
1
?? ?? ( ?? )   C. ???? ( ?? )- ?? ( 0)
B.
1
?? ?? ( ?? )- ?? ( 0)  D. ? ?? ( ?? ) ????

65. With initial condition ?? ( 1) = 0.5, the solution of the differential
equation ?? ????
????
+ ?? = ?? , is
A. ?? = ?? - 1/2    C. ?? = ?? 2
/2
B. ?? = ?? 2
- 1/2   D. ?? = ?? /2

66. If ?? = [
1 2
3 -5
] then its adjoint is
A. [
-1 -2
3 5
]  C. [
-5 -2
-3 1
]

B. [
5 -2
-3 1
]  D. [
-5 -2
3 -1
]

67. The principal value of log(
1
?? 4
) is
A. ????  C. ???? /4
B. ???? /2  D. ???? /8

68. The real part of the principal value of 4
4-?? is
A. 256cos ( ln4)    C. 16cos ( ln4)
B. 64cos ( ln4)  D. 4 cos ( ln4)

69. One of the eigen vectors of the matrix ?? = [
2 2
1 3
] is
A. [
2
-1
]    C. [
4
1
]
B. [
2
1
]  D. [
1
-1
]

70. For a matrix ?? = [
3/5 4/5
?? 3/5
], the transpose of the matrix is equal
to the inverse of the matrix, ?? ?? = ?? -1
. The value of x is given by
A. -4/5    C. 3/5
B. -3/5    D. 4/5

71. The particular integral of ?? '''
( ?? )- 8?? ( ?? ) = 1 is
A. 1/8    C. â€“ 1/8
B. 1/4     D. 1/10

72 The particular integral of
?? 2
?? ?? ?? 2
+ 9?? = cos3?? is
A.
cos ?? 8
C.
cos 3?? 9

B.
cos ?? 16
D. None of these

73. Find the integrating factor of the differential equation cos?? ????
????
+
?? ?????? ?? = 1.
A. ?????? ??    C. ?????? ??
B. ?????? ??      D. ?????? ??

74. Under certain conditions, cane sugar in water is converted into
dextrose at a rate proportional to the amount that is unconverted at
any time. If, of 75 kg at time t = 0, 8kg are converted during the first
30 minutes, find the amount converted in 2 hours.
A. 72.73 kg     C. 27.23 kg
B. 23.27 kg     D. 32.72 kg

75. A tank contains 400 liters of brine holding 100 kg of salt in solution.
Water containing 125 g of salt per liter flows into the tank at the rate
of 12 liters per minute, and the mixture, kept uniform by stirring,
flows out at the same rate. Find the amount of salt at the end of 90
minutes.
A. 53.36 kg      C. 53.63 kg
B. 56.33 kg     D. 65.33 kg

76. Find the Laplace transform of the function
1
v?? .
A.
?? v??  C.
?? ??
B.
v?? v??      D.
v?? ?? 2

77. The curl of vector V( x,y,z) = 2x
2
i +3z
2
j + y
3
k at x = y = z = 1 is:
A. â€“ 3 i      C. 3 i â€“ 4 j
B. 3 i     D. 3 i â€“ 6 k

78 Find the orthogonal trajectories of the family of hyperbolas x
2
â€“ y
2

= ay.
A.?? 2
+ 3???? = ??  C.  ?? 3
+ 2?? ?? 2
= ??
B. ?? 3
+ 3?? ?? 2
= ??    D.  ?? 3
+ 2?? 2
?? = ??

RF REVIEW CENTER ENHANCEMENT EXAM MATHEMATICS

RF Review Center : MANILA-CEBU-BAGUIO (0932-175-1218) Page 4

79.  Which of the following is the Laplace transform of vt?
A.
vp
vs
C.
vp
2s
3/2

B.
vp
2vs
D.
vp
2s

80.  The following differential equation has
3 (
?? 2
?? ????
2
) + 4 (
????
????
)
3
+ ?? 2
+ 2 = ??

A. degree = 2, order = 1                C. degree = 4, order = 3
B. degree = 1, order = 2                 D. degree = 2, order = 3

81. A trough filled with liquid is 2 m long and has a cross section of an
isosceles trapezoid 30 cm wide of 50 cm. If the through leaks
water at the rate of 2000 cm3/min, how fast is the water level
decreasing when the water is 20 cm deep.
A. 13/25     C. 5/21
B. 1/46     D. 11/14

82. The height of a projectile thrown vertically at any given time is
define by the equation h(t) = -16t2 + 256t. What is the maximum
height reach by the projectile?
A. 1567 ft     C. 1247 ft
B. 1920 ft     D. 1024 ft

83. Find the equation of the curve passing through the point (3, 2) and
having slope 2x
2
â€“ 5 at any point (x, y).
A. 2x
3
â€“ 15x â€“ 3y + 2 = 0
B. 3x
3
â€“ 5x â€“ 2y â€“ 1 = 0
C. 2x
3
+ 5x â€“ 3y â€“ 21 = 0
D. 5x
3
â€“ 3x â€“ 3y + 1 = 0

84. Find the centroid of the region bounded by y = x
2
, y = 0, and x = 1.
A. (1/4, 2/3)    C. (3/4, 3/10)
B. (2/3, 5/4)    D. (3/5, 5/10)

85. A man walks across a bridge at the rate of 5 ft/s as a boat directly
beneath him at 10 ft/s. If the bridge is 10 feet above the boat, how
fast are the man and the boat separating 1 second later?
A. 8 ft/s   C. 8.33 ft/s
B. 8.25 ft/s   D. 8.67 ft/s

86. Find the arc length of the given curve y = 3x â€“ 2 from x=0 to x=1.
A. sqrt 3   C. sqrt 10
B. sqrt 5   D. 2sqrt 2

87. Evaluate ?
?? 3
+1
?? +2
????
A. ?? 3
+ ?? 2
+ 4?? + 7 ???? ( ?? + 2)+ ??
B. ?? 3
-
?? 2
2
+ 4?? - ???? ( ?? + 2)+ ??
C.
?? 3
3
- ?? 2
+ 4?? - 7 ???? ( ?? + 2)+ ??
D.
?? 3
3
- ?? 2
+ 4?? - ???? ( ?? + 2)+ ??

88. Divide 60 into 3 parts so that the product of the three parts will be
the maximum. Find the product.
A. 6,000    C. 4,000
B. 8,000     D. 12,000

89. Find the centroid of the region bounded by y = x
2
, y = 0 and x = 1.
A. (1/4, 2/3)   C. (3/4, 3/10)
B. (2/3, 3/4)   D. (3/5, 1/2)
90. Find the volume of the solid obtained by revolving 4x
2
+ 9y
2
= 36
A. 8p   C.  12p
B. 18p   D.  16p

91. What is the allowable error in measuring the edge of the cube that
is intended to hold 8 cu. m., if the error of the computed volume is
not to exceed 0.03 cu. m?
A. 0.002 m   C. 0.0025 m
B. 0.003 m   D. 0.001 m

92. Two cities A and B are 8 km and 12 km, respectively, north of a river
which runs due east. City B being 15 km east of A. a pumping station
is to be constructed (along the river) to supply water for the two
cities. Where should the station be located so that the amount of
pipe is a minimum?
A. 3 km east of A  C. 9 km east of A
B. 4 km east of A  D. 6 km east of A

93. Which of the following integrals gives the length of the graph of
?? = tan?? between ?? = ?? and ?? = ?? , where 0 < ?? < ?? <
?? 2
?
A. ? v?? + ?????? ?? ????
?? ??
B. ? v1 + ?????? 2
?? ????
?? ??
C. ? v1 + ?????? 2
?? ????
?? ??
D. ? v1 + ?????? 4
?? ????
?? ??

94. Given the area in the first quadrant by x
2
= 8y, the line x = 4 and the
x-axis. What is the volume generated by revolving this area about
the y-axis.
A. 53.26 cubic units  C. 51.26 cubic units
B. 52.26 cubic units  D. 50.26 cubic units

95. Locate the centroid of the plane area bounded by the equation y
2
=
4x, x = 1 and the x-axis on the first quadrant.
A. (3/4, 3/5)   C. (2/3, 3/5)
B. (3/5, 3/4)     D. (3/5, 2/3)

96. Find the length of the arc of the parabola x
2
= 4y from x = -2 to x =
2.
A. 4.2 units   C. 4.9 units
B. 4.6 units     D. 5.2 units

97. Find the slope of x
2
y = 8 at the point (2, 2)
A. 2    C. -1/2
B. -1   D. -2

98. "If my computations are correct and I pay the electric bill, then i will
run out of money. If i don't pay the electric bill, the power will be
turned off. Therefore, if I don't run out of money and power is still
on, then my computations are incorrect"
Convert this argument into logical notation using the variables
c,b,r,p for propositions of computations, electric bills, out of money
and power respectively.
A if (c^b) -> r and ~b -> ~p, then (~r^p) -> ~c
B if (c
v
b) -> r and ~b -> ~p, then (r^p) -> c
C. if (c^b) -> r and ~p -> ~b, then (~r
v
p) -> ~c
D. if(c
v
b) -> r and ~b -> ~p, then (~r^p) -> ~c

99. Find the area of the curve r
2
= a
2
cos 2?.
A. a sq. units   C. a
2
sq. units
B. 2a sq. units   D. a
3
sq. units

100. Find the moment of inertia of the area bounded by the parabola
?? 2
= 4?? , x-axis and the line ?? = 1, with respect to the x-axis.
A. 1.067      C. 0.968
B. 1.244    D. 0.878

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