1. The hypotenuse of a right triangle is 34 cm. Find the lengths of the two legs if one leg is 14 cm longer than the other.
A. 15 and 29 cm. B. 16 and 30 cm. C. 17 and 31 cm. D. 18 and 32 cm.
2. The area of a rhombus is 132 sq. m. if its shorter diagonal is 12 m, find the longer diagonal.
A. 20 m B. 22 m C. 36 m D. 28 m
3. One side of the parallelogram is 10 m and its diagonals are 16 m and 24 m respectively, find its area.
A. 158.7 sq. m B. 120 sq. m C. 96 sq. m D. 192 sq. m
4. The diameter of two spheres is in the ratio two is to three and the sum of their volumes is 1260 cu. m. Find the volume of the larger sphere in cu. m.
A. 980 B. 972 C. 960 D. 938
5. The side of the triangle are 5, 7 and 10 respectively. Find the radius of the circumscribed circle.
A. 5.39 m B. 6.40 C. 7.20 D. 4.80
6. The volume of a sphere is 36π cu. m. The surface area of this sphere in sq. m is: A. 25π
B. 42π C.36π D. 54π
7. How many side does a polygon has if the sum of the interior angles is 2520o? A. 14
B. 16 C. 12 D. 10
8. The first term of an arithmetic progression is 3 and the 15th term is 45. Find the sum of the first 15 terms.
B. 360 C. 460 D. 560
9. The first term of an arithmetic progression is -2 and the sum of the first 11 terms is 88. The common difference is:
A. 4 B .3 C. 2 D. 1
10. An ellipse with major axis 8 and a minor axis 6 is revolved about its minor axis. Find the volume of the solid revolution.
A. 201.06 B. 150.80 C. 1608.50 D. 1206.37
11. Find the eccentricity of an ellipse with major axis 8 and minor 6. A. ¾
B.4/2 C. 4/3 D. 0.66
!2. Find the length of the latus rectum of an ellipse with major axis 8 and minor axis 6.
A. 3.5 B. 4.5 C. 4 D. 5
94. Find the smallest number which when you divide by 2, the remainder is 1; when you divide by 3, the remainder is 2; when you divide by 5, the remainder is 4 and which when you divide by 6, the remainder is 5.
A. 39 B. 49 C. 59 D. 69
95. A man bough 20 pcs of assorted calculators for P20, 000. These calculators are of three types namely:
a. programmable at P3,000/pc b. scientific at P1,500/pc c. household type
How many programmable calculators did the man bought? A. 5
B. 2 C. 13 D. 15
96. Two ferryboats ply back and forth across a river with constant but different speeds, turning at the riverbanks without loss time. They leave opposite shores at the same instant, meet for the first time at 900 meter from one shore and meet for the second time 500 meters from the opposite shore. What is the width of the river?
A. 2000 B. 2200 C. 2020 D. 2002
97. How much lead must be added to n alloy which is 50% tin and 25% lead to make an alloy which is 60% tin and 20% lead?
A. 0 B. 1 kg C. 2 kg D. 3 kg
98. A pipe can fill up a tank with drain open in 3 hrs. If the pipe runs with the drain open for 1 hr. and then the drain is closed, it ill take 45 more minutes for the pipe to fill up the tank. If the drain will be closed right at the start of filling how long will it take for the pipe
to fill up the tank? A. 1.1 hrs. B. 1.125 hrs. C. 1.25 hrs. D. 1.3 hrs.
99. Ding can finish the job in 8 hrs. Tito can do it in 5 hrs. If Ding wok for 3 hrs. and then Tito was asked to help him finish it, how long Tito will have to work with Ding?
A. 25 hrs B. 25/13 min C. 1.923 hrs D. 30 hrs
100 Find the volume of the solid generated by revolving the area bounded by x = y2 and x = 2- y 2 about the y – axis.
A. 8 pi/3 B. 16 pi/3 C. 10 pi/3 D. 7 pi/3 1 B 21.C 41.B 61.D 81.A 2. B 22.A 42.C 62.D 82. C 3. A 23.B 43.A 63.B 83. 4. B 24.B 44.B 64.C 84.A 5. A 25.D 45.D 65.A 85.D 6. C 26.A 46.C 66.B 86.A 7. B 27.A 47.A 67.D 87.B 8. B 28.A 48.A 68.C 88.B 9. C 29.C 49.B 69.B 89.A 10 A. 30. 50.C 70.D 90.B
11. D 31.C 51.B 71.A 91.B
12. B 32.B 52.C 72.D 92.A
13. D 33.D 53.A 73.A 93.A
14. D 34.B 54.A 74.A 94.C 15. C 35.B 55.A 75.C 95.B 16. A 36.A 56.C 76.A 96.B 17. D 37.C 57.C 77.A 97.A 18. B 38.B 58.C 78.A 98.B 19. A 39.C 59.B 79.D 99.C 20. D 40.A 60.A 80.C1 00.B MATHEMATICS MODULE 2
1. When the corresponding elements o two rows of determinant are proportional, then the value of the determinants is:
a. Multiplled by the ratio b. Zero
c. Unknown d. One
2. When two rows are interchanged in position, the value of the determinant will be: a. Unchanged
b. Becomes zero c. Multiplied -1 d. Unpredictable
3. If every element of a row (or column) are multiplied by a constant k, then the value of the determinant is:
a. Multiplied by –nk b. k to the n c. Multiplied by k
d. Anyone of the above may be true
4. If the quadratic equation ax2 + bx + c = 0, when b2 is equal than 4ac. then the root are:
a. Equal
b. Real and unequal c. Imaginary
d. Extraneous
5. In the quadratic equation ax2 + bx + c = 0, when b2 is greater than 4ac, then the root are:
a. Equal
b. Real and unequal c. Imaginary d. Extraneous
6. In the quadratic equation ax2 + bx + c = 0, if r
1 + r2 represent the roots, the r1 = r2 is equal to:
a. b/a b. c/a
c. –b/a d. –c/a
7. They are equation whose members are equal only for certain (for possibly) no values of unknown.
a. Conditional equation b. Inequalities
c. fix equation
d. Temporary equation
8. Roots which are equal to zero are called the: a. Trivial roots
b. Identical c. Symmetric d. Rational
9. When all x are replaced by y and all y are replaced by x and the equation remains the same, then the equation is said to be:
a. Equivalent b. Identical c. symmetric d. Rational
10. The number 0.123123123 … is: a. Irrational
b. Surd
c. Transcendental d. Rational
11. To eliminate the surd, we multiply it by its ______________: a. Square
b. Cube c. Reciprocal d. Conjugate
12. The letter D in the Romans Numerals is equivalent to: a. 50
b. 500 c.5000 d.50000
13. A statement which is accepted without proof: a. Postulate
b. Lemma c. Theorem d. corollary
58. There is mo change in the motion of the body unless a resultant force is acting on it. This law is known as:
a. the law of Inertia b. Third law of Newton c. Law of resistance d. Doppler’s principle
59. The energy which body possess by virtue of its positions, configuration or internal mechanism is called ________. a. potential energy b. Kinetic energy c. Mechanical energy d. electrical energy
60. If the mass of the body is expressed in grams and the velocity in cm/sec, the kinetic energy is expressed in:
a. Ergs b. Joules c. Coulombs d. Slugs
61. Energy is given to a body or systems of bodies when work is done upon it. In this process there is merely a transfer of energy from one body to another. In such transfer, no energy is created or destroyed; it merely changes from one to another. This statement is known as:
a. The law of conservation of energy b. Energy transformation
c. Coulomb’s law d. law of power
62. The deformation of elastic body is directly proportional to the applied force, provided that the elastic limit is not exceeded. This theory is known as:
a. Hooke’s law b. Young theory c. Keppler’s law d. Bulk modulus
63. If an external pressure is applied to a confined fluid, the pressure will be increased at every point in the fluid by the amount of external pressure. This theory is known as:
a. Pascal’s law b. Hydraulic law c. Hydrostatic law d. Brangg’s law
64. A body wholly or partially submerged in a fluid experiences an upward force equal to the weight of the fluid displaced. This theory is known as:
a. Archimedes principle b. Boyle’s law
c. Third law of Newton d. Fluid theory
65. If the temperature of a confined gas does not change the product of the pressure and volume is constant. This statement is known as:
a. Boyle’s law b. Young’s law c. Gay lussac law d. Charles law
66. At any two points along a streamline in an ideal fluid in steady flow, the sum of the pressure, the potential energy per unit volume, and the kinetic energy per unit volume has the same value. This concept is known as:
a. Bernoulli’s theorem b. Fluid theory c. Hydraulic theorem d. Pascal theorem Answers: 1. B 21 B 41.A 61.A 2. C 22.B 42.A 62.A 3. C 23.B 43.C 63.A 4. A 24.C 44.B 64.A 5. B 25.B 45.D 65.A 6. C 26.B 46.C 66.A 7. A 27.A 47.B 8. A 28.C 48.B 9. C 29.D 49.A 10.D 30.C 50.D 11.D 31.B 51.D 12.B 32.A 52.A 13.A 33.B 53.D 14.C 34.B 54.C
15.A 35.A 55.A
16.D 36.A 56.C
17.B 37.D 57.B
18.D 38.B 58.A
19.A 39.A 59.A
20 A 40.D 60.A
MATHEMATICS MODULE 1
1. The solid formed by revolving the ellipse about is minor axis is called ________ .
a. spheroid b. oblate spheroid c. prelate spheroid d. ellipsoid
2. When two planes intersect with each other the amount of divergence between the two planes is expressed by measuring the:
a. polyhedral angle b. plane angle c. reflex angle d. dihedral angle
3. If the product of the slope of any two straight lines is negative. One of this are said to be:
a. parallel b. skew
c. perpendicular d. Non-intersecting
4. The logarithm of 1 to any base is: a. zero
b. one c. infinity d. intersecting
5. The maximum displacement of vibration from the equilibrium is called: a. frequency
b. speed c. amplitude d. period
6. If the velocity of the body is doubled, a. The kinetic energy is quadrupled b. the kinetic energy is halved c. the potential energy is halved d. the potential energy is doubled
7. A body traveling to a circle with constants speed: a. Is accelerated
b. Has constant velocity c. Does no move
8. Ivory soap floats in water because: a. All matter has mass
b. The density of ivory soap is unity
c. The specific gravity of ivory soap is greater than that of water d. The specific gravity of ivory soap is greater than that of water
9. When two wave of the same frequency, speed and amplitude traveling in opposite directions are superimposed:
a. Destructive interference always results b. Constructive interference always results c. Standing waves are produced
d. The phase difference is always zero 10. The velocity of a wave is:
a. The product of the frequency and wavelength b. Distance of the crest to the next crest
c. Always equal to 186.00 miles per. sec. d. The ratio of the frequency to wavelength 11. Acceleration is:
a. the same as velocity b. the same as displacement c. the rate of change of velocity d. always zero
12. Mass is the quantitative measure o a. inertia
b. gravity c. weight d. momentum
13. A sequence of number where every term is obtain by adding all the preceding terms of a square number series such as 1, 5, 14, 30, 55, 91 …is called _________. a. Triangular number
b. Tetrahedral number c. Euler’s number
98. Square root of the product of two terms of a geometric progression. a. Median
b. Mean
c. Geometric mean d. Geometric term
99. Which of the following cannot be probability? a. 0.1
b. 0 c. 1
d. 0.232323
100. If a is the number of times that an event will take place and b is the number of times that it will not take place, then the probability that it will take place is:
a. a/b b. b/a c. a/(b+a) d. ab(a+b) Answers: 1. B 21. C 41. A 61. D 81. C 2. D 22. A 42. C 62. C 82. C 3. C 23. A 43. C 63. B 83. D 4. A 24. A 44. A 64. D 84. C
5. A 25. A 45. D 65. A 85. A 6. D 26. B 46. D 66. A 86. B 7. A 27. A 47. D 67. A 87. B 8. D 28. A 48. 68. A 88. A 9. C 29. A 49. A 69. A 89. B 10. A 30. A 50. B 70. B 90. D 11. C 31. A 51. A 71. B 91. D 12. A 32. B 52. A 72. D 92. A 13. D 33. B 53. A 73. C 93. A 14. A 34. C 54. A 74. A 94. B 15. A 35. D 55. B 75. B 95. A 16. A 36. A 56. A 76. C 96. C 17. A 37. D 57. B 77. B 97. C 18. A 38. D 58. A 78. A 98. C 19. A 39. A 59. B 79. B 99. C 20. A 40. B 60. A 80. A 100.C PLANE GEOMETRY Definitions of Terms
Axiom – a statement of truth of which is admitted without proof. Theorem – a statement of truth which must be established by proof.
Corollary – a statement of truth of which follows with little or no proof from a theorem. Postulate – In construction or drawing of lines and figures of which is admitted without proof. Hypothesis – part of a theorem which is assumed to be true.
Conclusion – part of a theorem which is to be proved.
Converse of a Theorem – another theorem wherein the hypothesis and conclusion of the first are reversed, i.e. the hypothesis becomes the conclusion and the conclusion becomes the hypothesis.
CIRCLE
Circle – locus of points which are at the same distant from a point within, called the center. Diameter – a line passing thru the center, terminating at both ends on the circle.
Radius – a line drawn from the center to the circle Arc – a part of the circle.
Chord – a line joining two points on a circle.
Secant – an indefinite line intersecting the circle in two points. Tangent – an indefinite line touching a circle at only one point. Segment – position of a circle between chord and its arc. Sector – position of a circle between two radii and arc.
Inscribed angle – an angle whose vertex is a point in the circle, the sides of which are chords.
Central angle – an angle whose vertex is at the center of the circle the sides of which are radii.
Concentric circles – circle having the same center with unequal radii. Tangent circles – circles tangent to the same line at the same point.
Inscribed circle – (in a polygon) when the sides of the polygon are tangent to it. Circumscribed circle – (about a polygon) when it passes through the vertices of the
polygon.
Circular ring/annulus – area included between two concentric circles of unequal radii.
POLYGON
Polygon – a plane closed by broken lines
Regular polygon – polygon whose angles are equal and all of whose sides are equal Similar polygon – polygon whose corresponding angles are equal and their
corresponding sides are proportional.
Center of a polygon – common center of its inscribed and circumscribed circle. Diagonal of a Polygon – line joining any two non-consecutive vertices.
Apothem (of a regular polygon) – the perpendicular line drawn from the center of the inscribed circle to any one of the sides. It is the radius of the inscribed circle.
Classifications of Polygon
Number of Sides Polygon
3 triangle 4 quadrilateral 5 pentagon 6 hexagon 7 heptagon 8 octagon 9 nonagon 10 decagon 11 undecagon 12 dodecagon
Trapezoid – a quadrilateral with only two sides of which are parallel. Parallelogram – a quadrilateral whose opposite sides are parallel. Rhombus –a parallelogram with equal sides and oblique angles. Rectangle – a parallelogram whose angles are right angles. Square – a rectangle with equal sides.
Isosceles trapezoid – one whose non-parallel sides are equal. Isometric figures – figures whose parameters are equal.
Properties of Plane Figures
1. The exterior angle of a triangle is greater either non- adjacent interior angle and is equal to their sum. α
β φ
φ = α + β
2. The diagonal of a parallelogram divides the parallelogram into two congruent triangles.
∆ ACD ≅∆ ABC
3. The opposite sides and opposite angles of a parallelogram are equal.
AB = CD ∠ADC = ∠ABC AD = BC ∠BAD = BCD
4. The diagonals of a rhombus are perpendicular to each other and bisect the angles through which they pass.
AC⊥BD
∠ACD = ∠ACB = ½ ∠ABC ∠ADB = ∠CDB = ½ ∠ADC
5. The diagonals of a parallelogram bisect each other.
A⋅X = XC = ½ AC
X BX = XD = ½ BD
6. The median of the hypotenuse of a right triangle have equal distances from the three vertices.
Mc AMc = BMc = CMc
7. The line segment which joins the midpoint
C
D E
DE AB
A B DE = ½ AB
8. The median of a trapezoid is parallel to the bases
and equal to one-half of their sum. EF AB
EF = (AB+CD)/2
9. The intersection of the three angle bisectors meet at
a common point is called incenter, which is equidistant from the three sides of a triangle. The inscribed circle is called incenter.
OP = OQ = OR
10. The intersection of the three perpendicular bisectors meet at a common point called circumcenter, which is equidistant from the three vertices of the triangle. The circle whose center is 0 touching the three vertices is called circumcenter.
OB = OA = OC
11. The intersection of the three altitudes of a triangle meet at a common point called orthocenter.
12. The three medians of any triangle meet at a common point which is two-thirds of the distance from each vertex to the midpoint of the opposite side. The point of intersection is called the centroid. 2/3 BR = BM 2/3 CP = CM 2/3 AQ = AM A B C D A B C D A B C D A B C D C A B B C D A F E A B Q C P R o C o B A B P B Q C A R C M A C
13. The altitude upon the hypotenuse of the right triangle is the mean proportional between the segments of the hypotenuse.
CD = √(AD)(DB)
14. A central angle is measured by its intercepted arc.
15. The inscribed angle is measured by one-half of the
intercepted arc. ∠ABC = ½ AC
16. An angle formed by two chords intersecting within a
circle is measured by one half of the sum of the arcs intercepted by it and its vertical angle.
(PA)(PB) = (PC)(PD)
17. The product of one entire secant and its external segment equals the product of the other entire secant and its external segment.
(AD) = √(AB)(AC)
18. The tangent line is a mean proportional between the
entire secant and its external segment.
A1/A2 = (L12) / (L22)
19. The sum of interior angles of any polygon
of n sides is ∠A + ∠B + ∠C… = (n–2)180°
20. Interior angle (θi) and exterior angle (θ0 ) of a regular polygon of n sides is
θI = [(n–2)/n] 180°
θo = 360° / n
Formulas for Plane Figures I. Area of Triangles
I.0 Given sides a, b, and c
(Hero’s formula) a b A = √s(s-a)(s-b)(s-c) A B C D θ C θ B A θ A C B P D C A L1 B D L2 A2 θi A1 F θ0 E D C B A
Where:
s = ½ (a+b+c) c
2. Given base b and altitude h A = ½ bh
Hbb h
b b
3. Given: equilateral triangle of side s A = √ 3 s2
4
s s
s
4. Given: two adjacent sides and the included angle A = ½ bc sin α
A = ½ ac sin β A = ½ ab sin δ
b a
c 5. Given: at least two angles and a side
A = a 2 sin sinβ δ 2 sinα A = b 2 sin sinα δ 2 sin β A = c sin2 α sin β 2 sin δ
6. Given: sides a, b, and c and inscribed in a circle of radius R A = abc
4R
R
7. Given: sides a, b, and c and circumscribed about a circle of radius r A = rs
Where:
s = a + b + c 2
II. Area of Parallelogram
1. Given: base b and altitude h
A = bh b
2. Given: sides a and b and their included angle θ A = ab sin θ
a
θ
III. Area of a Trapezoid
Given: bases a and b and altitude h
A = ½ (a+b) a
h b
IV. Area of cyclic quadrilateral (Bramaguphta’s Formula) Given: sides a, b, c, and d
A =√(s-a)(s-b)(s-c)(s-d) Where: h a r b c a b r c D A a d
s = ½ (a+b+c+d) A+C = 180° B+D = 180° V. Area of Rhombus Given: diagonals d1d2 A = ½ d1d2
VI. Area of Trapezium
1. Given: diagonals d1 and d2 and their included angleφ A = ½ d1d2 sin φ
VII. Area of Regular Polygon
1. Given : n sides, each of length s. A = ½ ns2 cot (π/
n)
2. Given: n sides, and apothem a. A = na2 tan (π/
n)
3. Given: n sides inscribed in a circle of radius R. A = ½ nR2 sin (2π/
n)
VIII. Area (A) and Perimeter (P) of a Circle 1. Given: radius r and diameter d
A = πr2 = π d 2 r 4
P = 2πr = πd
d IX. Area of Annulus
Given: circle of radii r1 and r2
A = π(r12 – r22) r1 Where r1>r2 r2
X. Area of a sector of a circle Given: radius r and θ
s = rθ r
A = ½ r2θ s
XI. Area of a segment of a circle
A = ½ r2 (θ – sinθ) r Where θ is in radius
θ r
XII. Area and Circumference of Ellipse Given: major axis a and minor axis b
A = πab
C = 2π√(a2+b2)/2
a
b
XIII. Area of a parabolic segment Given: base b and height h
A = 2/ 3 bh h b θ B C c b d1 d2 d1 d2 d1 φ A d2 R a a
Supplementary Problems
1. Bisectors of the 3 angles of a triangle meet at a common point called the __________
a. orthocenter b. centroid c. incenter d. circumcenter
2. The perpendicular bisector of the sides of a triangle pass through a common point called the __________
a. orthocenter b. centroid c. incenter d. circumcenter
3. Which of the following is not a property of a circle?
a) through 3 points not in the straight line one circle and only 1 can be drawn
b) a tangent to a circle is perpendicular to the radius at the point of tangency and conversely.
c) an inscribed angle is measured by ½ of the intercepted arc.
d) the arc of 2 circles subtended by equal central angle are equal.
4. Which of the following is not a property of a triangle? a) the sum of the 3 angles of the triangle is equal to two right triangles.
b) the sum of the 2 side of the triangle is less than the 3rd side
c) if the 2 sides of the triangle are unequal, the angles opposite are unequal.
d) the altitude of a triangle meets in a point
5. The radius of the circle inscribed in a polygon is called as a) internal radius b) radius of gyration c) apothem d) hydraulic radius
6. A polygon with 12 sides is called as
a) bidecagon b) dodecagon c) nonagon d) pentedecagon
7. A polyhedron having bases 2 polygons in parallel plane and for lateral faces triangles or trapezoid with 1 side lying on 1 base and the opposite vertex or side lying on the other base of the polyhedron is
a) pyramid b) cone
c) prismatoid d) rectangular parallelepiped
8. An angle greater than a straight angle but less than 2 straight angles is called as a) complement b) supplement
d) complex d) reflex 9. A part of a circle is often called as
a) sector b) cord
b) arc d) segment
10. An angle whose vertex is appoint on the circle and whose sides are cords is known as a) interior angle b) vertical angle
c) acute angle d) inscribed angle
11.Two angles whose sum is 360 degrees are said to be a) supplementary b) complimentary b) elementary d) explementary
12. All circles having the same center but with unequal radii are called as
a. eccentric circles b. concentric circles c. inner circles d. Pythagorean circles
13. A circle is _________ outside the triangle if it is tangent to one side and the other two sides prolonged.
a. inscribed b. escribed c. circumscribed d. tangent
14. A triangle having three sides of unequal length is known as a. equilateral triangle b. scalene triangle
c. isosceles triangle d. equiangular triangle
15. In a proportion of four quantities, the first and the fourth terms are referred to as the
a.means b. extremes c. denominators d. axiom
16. A statement the truth of which is admitted without proof is a. theorem b. corollary
c. postulate d. axiom
17. The part of the theorem which is assumed to be true is the a. corollary b. hypothesis
c. postulate d. conclusion
18. In Geometry,the construction or drawing of lines and figures, the possibility of which is admitted without proof is called the:
a. corollary b. theorem c. postulate d. hypothesis
19. A statement the truth of which follows with little or no proof from the theorem is a. corollary b. axiom
c. postulate d. conclusion
20. A polygon is ______ when no side, when extended, will pass through the interior of the polygon.
a. convex b. equilateral c. isoperimetric d. regular
21. A circle is said to be _______ to a polygon having the same perimeter with that of the circle
a. congruent b.isoperimetric c.proportional d. similar
22. The intersection of the sphere and the plane through the center is the a. great circle b. small circle
23. Points that lie on the same plane are said to be a. collinear b. coplanar
c. dihedral d. parallel
24. What kind of a quadrilateral is always formed by connecting the midpoints of the consecutive sides of a quadrilateral? Ans. parallelogram