COMPILATION OF MATH PROBLEMS

ALGEBRA

1. Ten less than four times a certain number is 14. Determine the number. 2. The hypotenuse of a right triangle is 34 cm. Find the length of the two legs if one leg is 14 cm

longer than the other. 3. Find the equation whose roots are the reciprocals of the roots of the equation

2 x2−3x−5=0. 4. The sum of two numbers is 21 and one number is twice the other. Find the numbers. 5. For a particular experiment, you need 5 liters of a 10% solution. You find 7% and 12% solution

on the shelves. How much of the 7% solution should you mix with the appropriate amount of the 12% solution to get 5 liters of a 10% solution.

6. A circle with a radius of 6 has half of its area removed by a cutting border of uniform width. Find the width of the border.

7. Two triangles have equal bases. The altitude of one triangle is 3 units more than its base while the altitude of the other is 3 units less than its base. Find the altitudes if the areas of the triangle differ by 21 square units.

8. The roots of the quadratic equation are 13

and 14

. What is the equation?

9. Find the 30th term of the progression 4, 7, 10, ….. 10. The denominator of a certain fraction is three more than twice the numerator. If 7 is added to

both terms of the fraction, the resulting fraction is 35

. Find the original fraction.

11. A piece of wire is shaped to enclose a square whose area is 169 sq.cm. It is then reshaped to enclose a rectangle whose length is 15 cm. Find the area of the rectangle.

12. Find the sum of 6, -2, 2/3, ….. 13. In the expansion of (x+4 y)12, find the numerical coefficient of the 5th term. 14. Find the sum of the first 10 terms of the progression 2, 4, 8, 16,…. 15. Find the ratio of an infinite geometric progression if the sum is 2 and the first term is ½. 16. Find the 30th term of the sequence 4, 7, 10, …. 17. A man rows downstream at the rate of 5 mph and upstream at the rate of 2mph. How far

downstream should he go if he is to return in 7/4 hours after leaving? 18. Find the mean proportional of 4 and 36. 19. Mary is 24 years old. Mary is twice as old as Ana was when Mary was as old as Ana is now. How

old is Ana now? 20. Determine x so that x, 2x + 7, 10x – 7 will be a geometric sequence. 21. Mike, Loui and Joy can mow the lawn in 4, 6 and 7 hours, respectively. What fraction of the yard

can they mow in 1 hour if they work together?

22. Given: f(x) = (x + 3) (X – 4) + 4. When f(x) is divided by (x – k), the remainder is k. Find k. 23. The sum of the digits of a two-digit number is 11. If the digits are reversed, the resulting

number is seven more than twice the original number. What is the original number? 24. The time required for the examinees to solve the same problem differ by two minutes.

Together, they can solve 32 problems in one hour. How long will it take for the slower problem solver to solve a problem?

25. Find the sum of the roots of 5 x2−10 x+2=0. 26. In a box there are 25 coins consisting of quarters, nickels and dimes with a total amount of

$2.75. If the nickels were dimes, the dimes were quarters and the quarters were nickels, the total amount would be $3.75. How many quarters are there?

27. A man travels in a motorized banca at the rate of 15 kph from his barrio to the poblacion and come back to his barrio at the rate of 12 kph. If his total time of travel back and forth is 3 hours, find the distance from the barrio to the poblacion.

28. One leg of a right triangle is 20 cm and the hypotenuse is 10 cm longer than the other leg. Find the length of the hypotenuse.

29. Three times the sine of a certain angle is twice of the square of the cosine of the same angle. Find the angle.

30. A man is 41 years old and his son is 9. In how many years will the father be three times as old as his son?

31. A tank is filled with an intake pipe that fills it in 2 hours and an outlet pipe that empty it in 6 hours. If both pipes are left open, how long will it take to fill the empty tank?

32. A piece of wire of length 50 m is cut into two parts. Each part is then bent to form a square. It is found that the total area of the square is 100 m2. Find the difference in length of the sides of the two squares.

33. A purse contains $11.65 in quarters and dimes. If the total number of coins is 70, find how many dimes are there?

34. How many liters of water must be added to 35 liters of 89% HCl solution to reduce its strength to 75%?

35. Find the value of m that will make 4 x2−4mx+4m+5a perfect square trinomial. 36. Ana is 5 yrs. older than Beth. In 5 yrs., the product of their ages is 1.5 times the product of their

present ages. How old is Beth now? 37. Find the coefficient of the term involving b4 in the expansion of (a2 – 2b)10. 38. The seating section in a Coliseum has 30 seats in the first row, 32 seats in the second row, 34

seats in the third row, and so on, until the tenth row is reached, after which there are ten rows each containing 50 seats. Find the total number of seats in the section.

39. Pedro started running at a speed of 10 kph. Five minutes later, Mario started running in the same direction and catches up with Pedro in 20 minutes. What is the speed of Mario?

40. The sum of two numbers is 35 and their product is 15. Find the sum of their reciprocals.

41. The ten’s digit of a certain two digit number exceeds the unit’s digit by four and is one less than twice the unit’s digit. Find the number.

42. One pipe can fill a tank in 6 hours and another pipe can fill the same tank in 3 hours. A drain pipe can empty the tank in 24 hours. With all three pipes open, how long will it take to fill in the tank?

43. A piece of paper is 0.05 inch thick. Each time the paper is folded into half, the thickness is doubled. If the paper was folded twelve times, how thick in feet the folded paper be?

44. A man invested part of Php 20,000 at 18% and the rest at 16%. The annual income from 16% investment was Php 620 less than three times the annual income from 18% investment. How much did he invest at 18%?

45. If 3x=9 y and 27 y=81z, find x/z. 46. It takes an airplane one hour and forty five minutes to travel 500 miles against the wind and

covers the same distance in one hour and fifteen minutes with the wind. What is the speed of the airplane?

47. An airplane travels from points A and B with a distance of 1500 kms and a wind along its flight line. If it takes the airplane 2 hours from A to B with the tailwind and 2.5 hours from B to A with the headwind, what is the velocity?

TRIGONOMETRY

1. If sin A = 35

and A is in quadrant II while cos B = 725

and B is in quadrant I, find sin (A + B).

2. Solve for x in the equation: Arctan 2x + Arctan x = (π2

) −45°

3. If 84 °−0.4 x=arctan (cot 0.25 x ) . Find x.

4. Simplify: 4 cosy siny (1−2sin2 y )

5. Evaluate: cos [arctan 158 −arctan 724 ].

6. What is the measure in degrees of 2.25 revolutions counterclockwise? 7. Find the value of x in the equation cscx+cot x=3.

8. If sec2 A=5

2, what is the quantity 1−sin2 A?

9. Of what quadrant is A, if sec A is positive and csc A is negative? 10. The angle of a sector is 30° and the radius is 15 cm. What is the area of the sector? 11. A man finds the angle of elevation of the top of the tower to be 30°. He walks 85 m nearer the

tower and finds its angle of elevation to be 60°. What is the height of the tower?

12. Evaluate: Arctan[2 cos (arc sin(√32

¿]

13. Points A and B are 1000 m apart are plotted on a straight highway running east and west. From A, the bearing of tower C is N32°W and from B the bearing of C is N26°E. Approximate the shortest distance of the tower from the highway.

14. A rotating wheel has a radius of 2 feet and 6 inches. A point on the rim of the wheel moves 30 feet in 2 sec. Find the angular velocity of the wheel.

15. Solve for A: cos2 A=1−cos2 A 16. Evaluate: sin (270° - A) 17. If sec2 A = 5/2, find 1 – sin2A.

18. Simplify : (cos A )4−(sin A )4 19. Assuming that the earth is a sphere whose radius is 6400 km. Find the distance along a 3°arc at

the equator of the earth’s surface. 20. A central angle of 45° subtends an arc of 12 cm. What is the radius of the circle? 21. If tan 4A = cot 6A, then what is the value of angle A? 22. A railroad is to be laid-off in a circular path. What should be the radius if the track is to change

direction by 30° at a distance of 157.08 m?

23. Solve for x: arctan x + arctan (1/3) = π4

24. You are given one coin with 5-cm diameter and a large supply of coins with diameter of 2 cm. What is the maximum number of the smaller coins that may be arranged tangentially around the larger without any overlap?

25. Determine the period of the curve y = sin (1/2) x.

26. Given: x=cosB tan B−sin B

cosB. Solve for x if B = 30°.

27. A flywheel of radius 14 inches is rotating at the rate of 1000 rpm. How fast does a point on the rim travels in ft/sec?

28. Solve angle A of an oblique triangle with vertices ABC, if a = 25, b = 16 and C = 94° 6´ 29. If sin A = 2.5x and cos A = 5.5 x, find the value of A. 30. Triangle ABC is a right triangle with right angle at C. CD is perpendicular to AB. BC = 4 and

CD = 1. Find the area of the triangle ABC. 31. A ladder 5 m long leans against the wall of an apartment house forming an angle of 50° 32´with

the ground. How high up the wall does it reach? 32. If cot 2A cot 68° = 1, then what is tan A?

33. Simplify the expression: sinB+cosB tanB

cosB.

34. If A is in the III quadrant and cos A = -15/17, find the value of cos (A/2).

SOLID MENSURATION

1. The sum of the interior angles of a polygon is 540°. Find the number of sides. 2. A regular octagon is inscribed in a circle of radius 10. Find the area of the octagon.3. The volume of a sphere is 36π cu.m. What is its surface area? 4. One side of a regular octagon is 2. Find the area of the region inside the octagon. 5. The distance between the centers of the three circles which are mutually tangent to each other

externally are 10, 12 and 14 units. What is the area of the largest circle?

6. If the sides of a parallelogram and an included angle are 6, 10 and 100° respectively, find the length of the shorter diagonal.

7. A trapezoid has an area of 36 m2 and an altitude of 2 m. Its two bases have ratio of 4:5. What are the lengths of the bases?

8. The sides of a right triangle are 8, 15 and 17 units. If each side is doubled, how many square units will the area of the new triangle?

9. Find the measure of each interior angle in degrees of a regular dodecagon. 10. If an equilateral triangle is circumscribed about a circle of radius 10 cm, determine the side of

the triangle. 11. A metal washer 1-inch in diameter is pierced by ½ inch hole. What is the volume of the washer

if it is 1/8 inch thick? 12. What polygon has 27 diagonals? 13. The volume of the two spheres is in the ratio 27:343 and the sum of their radii is 10. Find the

radius of the smaller sphere. 14. A regular hexagonal pyramid has a slant height of 4 cm and the length of each side of the base is

6 cm. Find the lateral area. 15. What is the area of an isosceles right triangle if its perimeter is 6.6824? 16. What is the distance in cm between two vertices of a cube which are farthest from each other, if

an edge measures 8 cm? 17. The area of the rhombus is 132 sq.m. If its shorter diagonal is 12 m, find the longer diagonal.

18. One of the diagonals of a rhombus is 25 units and its area is 75 u2. Determine the length of the

sides. 19. Find the area of a parabola having a span of 30 m and a height of 20 m. 20. A regular dodecagon is inscribed in a circle of radius 24. Find the perimeter of the dodecagon.

21. The lateral area of the right circular water tank is 92 cm2 and its volume is 342 m3. Determine its

radius. 22. A cone and a cylinder have the same height and the same volume. Find the ratio of the radius of

the cone to the radius of the cylinder. 23. It is desired that the volume of the sphere be tripled. By how many times will the radius be

increased? 24. What is the area of a parabola with a base of 15 cm and a height of 20 cm? 25. The circumference of a great circle of a sphere is 18π. Find the volume of the sphere.

ANALYTIC GEOMETRY

1. Find the equation of the directrix of the parabola y2=16 x. 2. The midpoint of the line segment between P1 and P2 (-2, 4) is P (2, -1). Find P1

3. Given an ellipse: x2

36+ y

2

32=1. Determine the distance between foci.

4. Convert the θ = π3

to the Cartesian equation.

5. Find the coordinates of the point P(2, 4) with respect to the translated axis with origin at (1, 3).

6. The segment from (-1, 4) to (2, -2) is extended three times its own length. Find the terminal point.

7. Determine B such that 3x + 2y – 7 = 0 is perpendicular to 2x – By + 2 = 0. 8. Find the value of k for which the equation x2+ y2+4 x−2 y−k=0 represents a point circle.

9. What is the diameter of a circle described by 9 x2+9 y2=16? 10. The major axis of the elliptical path in which the earth moves around the sun is approximately

186,000,000 miles and the eccentricity of the ellipse is 1/60. Determine the apogee of the earth.

11. Point P(x, y) moves with a distance from point (0, 1) one-half of its distance from the line y = 4. What is the equation of the locus?

12. A line passes through point (2, 2). Find the equation of the line if the length of the line segment intercepted by the coordinate axis is √5.

13. Find the area of the triangle which the line 2x – 3y + 6 = 0 forms with the coordinate axes.

14. Determine the coordinates of the point which is three-fifths of the way from the point (2, -5) to the point (-3, 5).

15. Two vertices of a triangle are (2, 4) and (-2, 3) and the area is 2 square units. Find the locus of the third vertex.

16. If the points (-2, 3), (x, y), and (-3, 5) lie on a straight line, find the equation of the line.

17. Find the inclination of the line passing through (-5, 3) and (10, 7). 18. Find the distance of the directrix from the center of an ellipse if its major axis is 10 and its minor

axis is 8. 19. A point moves so that its distance from the point (2, -1) is equal to its distance from the x-axis.

What is the equation of the locus? 20. The parabolic antenna has an equation of y2+8 x=0. Determine the length of the latus

rectum. 21. Find the equation of the ellipse with center at (4, 2), major axis horizontal and of length 8 and

with minor axis of length 6. 22. Find the area of the hexagon ABCDEF formed by joining the points A(1, 4), B(0, -3), C(2, 3), D(-1,

2), E(-2, -1) and F(3, 0). 23. The directrix of a parabola is the line y = 5 and its focus is at the point (4, -3). What is the length

of its latus rectum? 24. Find the eccentricity of an ellipse when the length of the latus rectum is 2/3 of the length of the

major axis.

25. Find the equation of the parabola whose axis is parallel to the x-axis and passes through the points (3, 1), (0, 0) and (8, -4).

26. A point P (x, 2) is equidistant from the points (-2, 9) and (4, -7). What is the value of x?

27. Find the equation of a line whose x-intercept is 2 and y-intercept is -2. 28. If the length of the latus rectum of an ellipse is three-fourth of the length of its minor axis, find

its eccentricity.

Calculus 1

1. 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 0.02 cu.m?

2. Find the altitude of a cylinder of maximum volume which can be inscribed in a right circular cone of radius r and height h.

3. Find the approximate change in the volume of a cube of side “x” inches caused by increasing its side by 1%.

4. Three sides of a trapezoid are each 8-cm long. How long is the fourth side when the area of the trapezoid has the greatest value?

5. A statue 3 m high is standing on a base of 4 m high. If an observer’s eye is 1.5 m above the ground, how far should he stand from the base in order that the angle subtended by the statue is a maximum.

6. A balloon is rising vertically over a point A on the ground at the rate of 15 ft/sec. A point B on the ground level with and 30 ft from A. When the balloon is 40 ft from A, at what rate is its distance from B changing?

7. Find the point in the parabola y2=4 x at which the rate of change of the ordinate and abscissa are equal.

8. Find the slope of x2 y=8 at the point (2, 2).

9. Find the equation of the normal to x2+ y2=1 at the point (2, 1). 10. The depth of water in a cylindrical tank 4 m in diameter is increasing at the rate of 0.70 m/min.

Find the rate at which the water is flowing into the tank. 11. Find the coordinates of the vertex of the parabola y=x2−4 x+1 if the slope of the tangent at

the vertex is zero. 12. Find the minimum distance from the point (4, 2) to the parabola y2=8 x . 13. Two posts, one 8 m and the other 12 m high are 15 cm apart. If the posts are supported by a

cable running from the top of the first post to a stake on the ground and then back to the top of the second post, find the distance from the lower post to the stake to use minimum amount of wire?

14. What is the area of the largest rectangle that can be inscribed in a semi-circle of radius 10?

15. Find the approximate increase in the volume of the sphere if the radius increases from 2 to 2.05 in one second.

16. Differentiate: ln(ln y) + ln y = ln x

17. The volume of the sphere is increasing at the rate of 6 cm3/hr. At what rate is its surface area increasing when the radius is 50 cm?

18. If the sum of two numbers is 4, find the minimum value of the sum of their cubes. 19. Water is running out a conical funnel at the rate of 1 cu.inch per second. If the radius of the

base of the funnel is 4 inches and the altitude is 8 inches, find the rate at which the water level is dropping when it is 2 inches from the top.

20. Divide the number 120 into two parts such that the product of one and the square of the other is a maximum.

21. If y = 2x + sin 2x, find x if y’ = 0.

22. What is the equation of the tangent to the curve y=x+5x

at the point (1, 3)?

23. If y=arc tan ¿¿, find y’ at x = 1/e. 24. Find the change in y = 2x – 3 if x changes from 3.3. to 3.5. 25. The radius of a sphere is r inches at time t seconds. Find the radius when the rates of increase

of the surface area and the radius are numerically equal. 26. Using the two existing corner sides of an existing wall, what is the maximum rectangular area

that can be fenced by a fencing material 30 ft. long? 27. A man on a wharf 3.6 m above sea level is pulling a rope tied to a raft at 0.60 m/sec. How fast is

the raft approaching the wharf when there are 6 m of rope out? 28. The sum of four positive integers is 32. Find the greatest possible product of these four

numbers.

29. Find the maximum point of y=x+1x

30. Find the height of a right circular cylinder of maximum volume which can be inscribed in a sphere of radius 10 cm.

CALCULUS 2

1. Find the area bounded by the curve defined by the equation x2=8 y and its latus rectum.

2. Evaluate: ∫0

π6

cos83 A dA

3. Find the area bounded by the parabolas x2−2 y=0 and x2+2 y−8=0.

4. Evaluate:∫√1−cos xdx

5. Find the area bounded by the curve y=9−x2 and the x-axis.

6. Evaluate: ∫0

13x

e xdx

7. Evaluate: ∫ tan2 xdx

CALCULUS 3

1. Find the value of (1 + i)5. 2. Simplify the expression: i1997+i1999 3. What is the quotient when 4 + 8i is divided by i3? 4. If ( x+ yi ) (2−4 i)=14−8 i, find x. 5. Find the angle between the planes 3x – y + z – 5 = 0 and x +2y +2z + 2 = 0 6. Find the area of the geometric figure whose vertices are at (3, 0, 0), (3, 3, 0), (0, 0, 4) and

(0, 3, 4). 7. Find the length of the vector (2, 4, 4)

TERMINOLOGIES:

1. A function F(x) is called ________ of f(x) if F’(x) = f(x). Ans: antiderivative2. In an ellipse, a chord which contains a focus and is in a line perperdicular to the major axis is a

______________. Ans: Latus rectum3. If all y-terms have even exponents, the curve is symmetric with respect to the _____. Ans: x-axis4. Convergent series is a sequence of decreasing numbers or when the succeeding term is _____

than the preceding term. Ans: lesser5. The graph of r = a + b cos θ is a ____________. Ans: limacon6. 4 x2−256=0 is the equation of __________. Ans: Parallel lines7. Each of the faces of a regular hexahedron is a _________. Ans: square8. It is a sequence of numbers such that successive terms differ by a constant. Ans: Arithmetic9. Equations relating x and y that cannot readily solved explicitly for y as a function of x or for x as a

function of y. Such function may nonetheless determine y as a function of x or vice versa. Such a function is called ____________. Ans: Implicit Function

10. If the roots of an equation are zero, then they are classified as _______. Ans: Trivial Solution11. If a = b, then b = a. This illustrates what axiom in Algebra? Ans: Symmetric axiom12. The integral of any quotient whose numerator is the differential of the denominator is the

________. Ans: logarithm13. It is a polyhedron of which two faces are equal polygons in parallel planes and the other faces

are parallelograms. Ans: Prism14. When f”(x)is negative, the curve of y = f(x) is concave ________. Ans: downward 15. In polar coordinate system, the length of the ray segment from a fixed origin is known as

_________. Ans: radius vector16. If the first derivative of the function is constant, then the function is _________. Ans: linear17. A locus of a point which moves so that it is always equidistant from a fixed point to a fixed line is

a __________. Ans: parabola18. A line which a curve approaches indefinitely near as its tracing point passes off to the infinity is

called the _________. Ans: Asymptote19. __________ is the concept of finding the derivative of composite functions. Ans: Chain Rule

20. It is a part of a circle bounded by a chord and an arc. Ans: Segment21. In Algebra, the operation of root extraction is called as ___________. Ans: evolution22. The chords of an ellipse, which pass through the center, are known as _______. Ans; diameters23. A horizontal line has a slope of _____. Ans: zero24. The line passing through the focus and is perpendicular to the directrix of a parabola is called

the _________. Ans: Axis of the parabola25. At maximum point the value of y” is ________. Ans: negative26. The altitude of the sides of a triangle intersect at the point known as ______. Ans: orthocenter27. A sequence of numbers where the succeeding term is greater than the preceding term

is_______. Ans: Divergent series28. A line, which is perpendicular to the x-axis, has a slope equal to ________. Ans: infinity29. Convergent series is a sequence of decreasing numbers or when the succeeding term is ______

than the preceding term. Ans: lesser30. The area of the region bounded by two concentric circles is called ______. Ans: annulus31. Point of derivatives which do not exists (and so equals zero) are called _____.

Ans: maximum and minimum points32. It can be defined as the set of all points in the plane whose distances from two fixed points is a

constant is called __________. Ans: Ellipse 33. If the equation is unchanged by the substitution of y for x, it curve is symmetric with respect to

the ________. Ans: origin34. The apothem of a polygon is the _______ of its inscribed circle. Ans: radius