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 – 1.3644i 
 B. 0.8562 + 1.3644i     D. 1.3644 – 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 Euler’s Theorem in which m is a prime 
 number p or ?? ?? -1
= 1 ( mod ?? ) . 
 A. Fibonacci Theorem              C. Euler’s Prime Theorem 
 B. Fermat’s Little Theorem              D. Lerma’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. Dirac’s Theorem    C. Euler’s Theorem 
 B. Vertex Removal Argument  D. Hamilton’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 
 the unshaded region. 
 
 
 
 
 
 
 
 
 
 
 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 – 1.3644i 
 B. 0.8562 + 1.3644i     D. 1.3644 – 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 Euler’s Theorem in which m is a prime 
 number p or ?? ?? -1
= 1 ( mod ?? ) . 
 A. Fibonacci Theorem              C. Euler’s Prime Theorem 
 B. Fermat’s Little Theorem              D. Lerma’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. Dirac’s Theorem    C. Euler’s Theorem 
 B. Vertex Removal Argument  D. Hamilton’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 
 the unshaded region. 
 
 
 
 
 
 
 
 
 
 
 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 – 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) – 
 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 pilot’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. Simpsons’ 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. Euler’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, … . . ?? , ?????? 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 – 1.3644i 
 B. 0.8562 + 1.3644i     D. 1.3644 – 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 Euler’s Theorem in which m is a prime 
 number p or ?? ?? -1
= 1 ( mod ?? ) . 
 A. Fibonacci Theorem              C. Euler’s Prime Theorem 
 B. Fermat’s Little Theorem              D. Lerma’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. Dirac’s Theorem    C. Euler’s Theorem 
 B. Vertex Removal Argument  D. Hamilton’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 
 the unshaded region. 
 
 
 
 
 
 
 
 
 
 
 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 – 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) – 
 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 pilot’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. Simpsons’ 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. Euler’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, … . . ?? , ?????? 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. Riccati’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,….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. Skew–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. – 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. – 3 i      C. 3 i – 4 j 
 B. 3 i     D. 3 i – 6 k 
 
78 Find the orthogonal trajectories of the family of hyperbolas x
2
 – 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 – 1.3644i 
 B. 0.8562 + 1.3644i     D. 1.3644 – 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 Euler’s Theorem in which m is a prime 
 number p or ?? ?? -1
= 1 ( mod ?? ) . 
 A. Fibonacci Theorem              C. Euler’s Prime Theorem 
 B. Fermat’s Little Theorem              D. Lerma’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. Dirac’s Theorem    C. Euler’s Theorem 
 B. Vertex Removal Argument  D. Hamilton’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 
 the unshaded region. 
 
 
 
 
 
 
 
 
 
 
 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 – 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) – 
 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 pilot’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. Simpsons’ 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. Euler’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, … . . ?? , ?????? 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. Riccati’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,….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. Skew–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. – 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. – 3 i      C. 3 i – 4 j 
 B. 3 i     D. 3 i – 6 k 
 
78 Find the orthogonal trajectories of the family of hyperbolas x
2
 – 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
 – 5 at any point (x, y).  
A. 2x
3
 – 15x – 3y + 2 = 0   
B. 3x
3
 – 5x – 2y – 1 = 0  
C. 2x
3
 + 5x – 3y – 21 = 0  
D. 5x
3
 – 3x – 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 – 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 
about the x-axis. 
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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