MATH OBJECTIVES

Question 1

The sum of the distances from the two foci to any point in what curve is constant?

Answer 1

Ellipse

The definition of an ellipse involves the sum of distances from two foci being constant.

Question 2

Find the area of a square whose side is a2b3.

Answer 2

a4b6

Area of a square is calculated as side squared.

Question 3

A normal to a given plane is.

Answer 3

Perpendicular to the plane

The normal line is defined as perpendicular to the surface at a given point.

Question 4

If a person throws away 3.5 lbs of trash daily, how much trash will the person throw away in one week?

Answer 4

24.5 lbs

Calculation: 3.5 lbs/day * 7 days = 24.5 lbs.

Question 5

What is 20% of 96?

Answer 5

19.20

20% of a number is calculated by multiplying the number by 0.20.

Question 6

What conic section is described by the equation 4x2 – y2 + 8x + 4y = 15?

Answer 6

Hyperbola

Analyzing the equation reveals it fits the form of a hyperbola.

Question 7

Sam scored 96% on his first Calculus quiz; 74% on his second and 85% on his third. What is his quiz average?

Answer 7

85%

Average calculation: (96 + 74 + 85) / 3 = 85%.

Question 8

When two lines are parallel, the slope of one is:

Answer 8

Equal to the other

Parallel lines have identical slopes.

Question 9

If 1 cm = 0.39 in, about how many cm are there in 0.75 in?

Answer 9

1.90 cm

Conversion: 0.75 in * 2.54 cm/in = 1.90 cm.

Question 10

Which of the following is not a multiple of 11?

Answer 10

759

759 does not divide evenly by 11.

Question 11

What is the conic section whose eccentricity is less than 1?

Answer 11

Ellipse

Eccentricity definitions classify conic sections based on their eccentricity values.

Question 12

Nicole had 75 stuffed animals. Her grandmother gave fifteen of them to her. What percentage of the stuffed animals did her grandmother give her?

Answer 12

20%

Calculation: (15/75) * 100% = 20%.

Question 13

What type of conic section is x2 - 4y + 3x + 5 = 0?

Answer 13

Parabola

The equation can be rearranged to fit the standard form of a parabola.

Question 14

The line passing through the focus and perpendicular to the directrix of a parabola is called _____.

Answer 14

axis

The axis of a parabola is a line that reflects its symmetry.

Question 15

Sand is being poured into a conical pile in such a way that the height is always 1/3 of the radius. At what rate is sand being added to the pile when it is 4 ft high and height increasing at 2 in/min?

Answer 15

130,288.13 in/min

Volume and rate calculations involve geometric relationships of a cone.

Question 16

Find the volume generated by revolving the circle x² + y² + 6x + 4y + 12 = 0 about the y-axis.

Answer 16

59.22

Volume is calculated using Pappus’s Theorem.

Question 17

From a sample size of 100, the following description measures were calculated: median = 23; mean = 20; standard deviation = 5; range = 35. What conclusion might you draw?

Answer 17

The distribution is skewed to the right because the median exceeds the mean.

Skewness in distribution is indicated by the relationship of mean and median.

Question 18

What proportion of students have been absent less than 5 days?

Answer 18

0.91

Calculation: 31 out of 34 students were absent less than 5 days.

Question 19

Each of the questions on a quiz is a five-part multiple choice question with exactly only one correct answer. How many questions should the student expect to answer correctly?

Answer 19

3

Expected correct answers calculated as Np = 15 x (1/5) = 3.

Question 20

What is the value of g(f(3))?

Answer 20

-5

This requires substitution based on the provided function tables.

Question 21

Find the volume generated when the area bounded by y = 2x + 3 and y = x2 is revolved about the x-axis.

Answer 21

228

Volume is found using integration techniques for areas of revolution.

Question 22

Solve [y – square root of (x2 + y2)]dx - xdy = 0.

Answer 22

Square root of (x2 + y2) + y = C

This is a differential equation solution involving implicit functions.

Question 23

What is the man’s average between 68 sec and 168 sec?

Answer 23

4 m/s

Average speed is calculated as change in distance over change in time.

Question 24

What is the probability that less than 17 alumni will make a contribution of at least P 50?

Answer 24

0.589

This uses binomial probability calculations.

Question 25

At exactly what time after 5 o’clock will the hour hand and the minute hand be perpendicular for the first time?

Answer 25

5:10 and 54 sec ## Footnote This involves angular relationships of clock hands.

Question 26

Find the equation of the normal line to x2 + y2 = 1 at point (2, 1).

Answer 26

x + y = 1 ## Footnote The normal line is derived from the slope at the point on the curve.

Question 27

What is the smallest positive value for x where y = sin 2x reaches its maximum?

Answer 27

π/4 ## Footnote The maximum of sin occurs at specific intervals based on its periodicity.

Question 28

When the energy/hour required in driving a boat varies as the cube of the velocity, find the most economical rate/hour when going against the current of 4 kph.

Answer 28

6 kph ## Footnote The optimization problem involves calculus to minimize energy consumption.

Question 29

What is the maximum rectangular area that can be fenced in 20 ft using two perpendicular corner sides of an existing wall?

Answer 29

100 square feet ## Footnote The area maximization occurs at specific dimensions derived from calculus.

Question 30

Determine the correct equation for the line with a slope of 7 and y-intercept of – 4.

Answer 30

y = 7x - 4 ## Footnote This is based on the slope-intercept form of a linear equation.

Question 31

With the courier company's charges, what amount would be charged of a parcel weighing 30 kg?

Answer 31

P 460 ## Footnote Cost calculated based on tiered pricing structure.

Question 32

The dimension of a rectangular prism can be expressed as x + 1, x – 2 and x + 4. What is the volume of the prism?

Answer 32

x3 + 3x2 - 6x - 8 ## Footnote Volume is calculated by multiplying the dimensions together.

Question 33

Larry finds the angle of elevation of the top of the tower to be 30 degrees. He walks 85 m nearer the tower and finds its angle of elevation to be 60 degrees. What is the height of the tower?

Answer 33

53.61 m ## Footnote This involves trigonometric relationships to find the height.

Question 34

What is the formula for the volume V of a rectangular prism given dimensions W = x – 2 and H = x + 4?

Answer 34

V = L x W x H = (x + 1)(x – 2)(x + 4) = x^3 + 3x^2 – 6x - 8

Question 35

What is the height of the tower if the angle of elevation to its top changes from 30 degrees to 60 degrees after moving 85 m closer?

Answer 35

73.61 m

Question 36

If log 2 = a, log 3 = b, log 5 = c, what is log(7.5)?

Answer 36

b + c - a

Question 37

From the base of a building, if the angle of elevation to the top of a 4.0 m pole is 18 degrees 50 minutes, what is the height of the building if the angle of depression from the top of the building to the base of the pole is 48 degrees 10 minutes?

Answer 37

13.10 m

Question 38

What is the distance Michael travels to school if he leaves at 7:32 A.M. and arrives 15 minutes later at a speed of 4.5 miles/hr?

Answer 38

1.125 miles

Question 39

Evaluate f(-3) if f(x) = x^2 - 2x + 1.

Answer 39

16

Question 40

Find the sum of the first five terms of the geometric progression if the third term is 144 and the sixth term is 486.

Answer 40

844

Question 41

From the top of a building 100 m high, what is the distance between two cars when the angles of depression are 32 degrees 25 minutes and 58 degrees 33 minutes?

Answer 41

96.31 m

Question 42

Find the equation for line AB given the position vectors of points A and B are 2 + i and 3 – 2i, respectively.

Answer 42

3x + y = 7

Question 43

What is the ratio of the area of a square to the area of an isosceles right triangle formed by a side of the square?

Answer 43

4 : 1

Question 44

In an RL circuit, what is the expression for current I when t = L/R?

Answer 44

0.632(E/R)

Question 45

What is the perimeter of a rectangle if the width is 6 inches less than 3 times the length and the total perimeter is 104 inches?

Answer 45

37.5 inches

Question 46

What is the total value of the area of a square with side length 4 units added to the difference of 11 and 9 divided by 2?

Answer 46

17

Question 47

Where should the fishing port be located to minimize pavement from two towns located 1 km and 2 km from the shore, 6 km apart?

Answer 47

2 km from the point on the shore nearest the second town

Question 48

How long is the feature film if the total movie time is 2 hours and advertisements last 3.8 min, 4.6 min, and 2.9 min?

Answer 48

108.7 min

Question 49

Find the solution to the system of equations x – 2y = 5 and 2x + 5y = 1.

Answer 49

(3, -1)

Question 50

What is the vertex of the parabola given by the equation x^2 = 4y?

Answer 50

(0, 0)

Question 51

What is the value of x if the difference between six times (6x + 1) and three times (x – 1) equals 108?

Answer 51

3

Question 52

What score must Daniel achieve on his last Algebra exam to maintain an average of 95?

Answer 52

97

Question 53

What is the area of the triangle with vertices (1, 1), (3, 2), and (2, 4)?

Answer 53

5/2

Question 54

What is the area of the polygon with vertices 2 + 3i, 3 + i, -2 – 4i, -4, -i, -1 + 2i?

Answer 54

25

Question 55

What is the area bounded by the curves y^2 = 4x and x^2 = 4y?

Answer 55

5.33

Question 56

What is the range of miles Jane can drive her new car if the annual cost is given by C = 0.25m + 1,600 and the cost is between $5,350 and $5,600?

Answer 56

Between 15,000 miles and 16,000 miles

Question 57

What is the equation of the line that passes through the intersection of x – y = 0 and 3x – 2y = 2, forming a triangle of area 9?

Answer 57

x + 2y – 6 = 0

Question 58

How far does a ball travel before coming to rest if it bounces 2/3 of the height from which it falls, starting from 18 feet?

Answer 58

90 feet

Question 59

What is the rate of change of the smaller arc subtended by a chord of a circle of diameter 10 ft decreasing at 1 ft/min when the chord is 8 ft long?

Answer 59

2 ft/min

Question 60

What is the probability that fewer than two reservations will cancel if 20% of the reservations will not be used when four reservations are made?

Answer 60

0.8192

Question 61

What symmetry does a periodic function with zero average value and a Fourier series consisting of only odd cosine terms possess?

Answer 61

odd

Question 62

Evaluate the integral of ln x dx from 1 to e.

Answer 62

1

Question 63

What is the differential equation of the family of parabolas with vertices at the origin and foci on the x-axis?

Answer 63

2x dy - y dx = 0

Question 64

What is the differential equation of the family of parabolas having their vertices at the origin and their foci on the x-axis?

Answer 64

dy/dx – x = 0 ## Footnote The correct differential equation is derived from the properties of parabolas defined in the Cartesian plane.

Question 65

Which of the following equations is an exact differential equation?

Answer 65

(2xy + x)dx + (x^2 + y)dy = 0 ## Footnote Exact differential equations satisfy the condition ∂M/∂y = ∂N/∂x where M and N are the coefficients of dx and dy respectively.

Question 66

Compute log (3-2i)

Answer 66

0.5570 – 0.2554i ## Footnote The logarithm of a complex number is computed using its polar form.

Question 67

What is the maximum value of 3 sin x?

Answer 67

3 ## Footnote The maximum value of sin x is 1, thus 3 sin x achieves a maximum of 3.

Question 68

What is the ratio of the sides of a triangle if the product of the sides is a maximum?

Answer 68

1:1:1 ## Footnote To maximize the product of the sides, they must be equal lengths.

Question 69

Evaluate the limit of x/sqrt(1 + x^2) as x approaches infinity.

Answer 69

1 ## Footnote The limit can be evaluated by dividing numerator and denominator by x.

Question 70

What is the radius of the ball bearing if a cylindrical container of radius 2 cm has a water height increase of 0.6 cm?

Answer 70

1.2 cm ## Footnote The volume of water displaced equals the volume of the sphere.

Question 71

If the coefficient a0 of a Fourier series of a periodic function is zero, it means that the function has:

Answer 71

Odd symmetry ## Footnote A zero average indicates odd symmetry in the function.

Question 72

Solve the equation 5z^2 + 2z + 10 = 0.

Answer 72

1 + i, 1 - 2i ## Footnote The solutions can be found using the quadratic formula.

Question 73

Solve the differential equation y'' – 4y' + 3y = sin x.

Answer 73

y(x) = C₁e^3x + C₂e^x + 1/5 cos x + 1/10 sin x ## Footnote The general solution involves solving the homogeneous and particular solutions.

Question 74

The Fourier series of waveform processing even quarter-wave symmetry contains only _______.

Answer 74

even harmonics ## Footnote Even quarter-wave symmetry implies the absence of odd harmonics.

Question 75

A periodic waveform possessing half-wave symmetry has no _______.

Answer 75

even harmonics ## Footnote Half-wave symmetry excludes even harmonics in the Fourier series representation.

Question 76

Which of the following periodic functions possesses even symmetry?

Answer 76

cos 3t ## Footnote Even functions satisfy f(x) = f(-x).

Question 77

The symbol j represents counterclockwise rotation of a vector through _______ degrees.

Answer 77

90 ## Footnote In complex number representation, j indicates a 90-degree rotation.

Question 78

The conjugate of (-a + jb) is _______.

Answer 78

(a – jb) ## Footnote The conjugate of a complex number is obtained by changing the sign of the imaginary part.

Question 79

When the negative half-cycle of a complex waveform is reversed, it becomes identical to its positive half-cycle. This feature indicates that the complex waveform is composed of:

Answer 79

Even harmonics ## Footnote This symmetry suggests the presence of even harmonics in the waveform.

Question 80

All opposite rays _______.

Answer 80

Extend in the same direction ## Footnote Opposite rays share a common endpoint and extend infinitely in opposite directions.

Question 81

Angles that share a common vertex point cannot _______.

Answer 81

Share a common angle side ## Footnote Common angle sides would imply that the angles are not distinct.

Question 82

If angle EDF and angle HIJ are supplementary angles, and angle SUV and angle EDF are also supplementary angles, then angle HIJ and angle SUV are _______.

Answer 82

Congruent angles ## Footnote Supplementary angles sum to 180 degrees, hence HIJ and SUV are equal.

Question 83

Evaluate sin [arccos (-2/3)].

Answer 83

Square root of 5 ## Footnote This can be solved using the Pythagorean identity.

Question 84

Four is added to the quantity two minus the sum of negative seven and six. What is the result?

Answer 84

21 ## Footnote The calculation follows order of operations.

Question 85

How much will a business have depreciated after 2 years if a computer worth P 21,000 depreciates to 0 in 5 years?

Answer 85

P 8,400 ## Footnote Depreciation is linear over the lifespan.

Question 86

Evaluate lim (2 – x)^tan(πx/2) as x approaches 1.

Answer 86

e ## Footnote Utilizes L'Hôpital's rule due to the indeterminate form.

Question 87

Find the centroid of a semi-circular region of radius a.

Answer 87

4a/3π ## Footnote The centroid of a semicircular area is derived through integration.

Question 88

If y = 2x + sin(2x), find x when y' = 0.

Answer 88

π/2 ## Footnote The derivative is set to zero to find critical points.

Question 89

What is the probability that a student spends less than 6 hours learning a software package if the time spent is normally distributed with a mean of 8 hours and standard deviation of 1.5 hours?

Answer 89

0.09 ## Footnote This probability is calculated using the Z-score for normal distribution.

Question 90

If Mon and Mila can restock an aisle of the supermarket in 1 hour working together, how many aisles will they complete in 2 hours together and 2 hours separately?

Answer 90

4.33 ## Footnote Total aisles are calculated based on their individual rates and combined efforts.

Question 91

How much money did Liza have if she could buy 10 chocolate bars but could only buy 8 after a price increase of 50 centavos?

Answer 91

80 ## Footnote The increase in price led to a reduction in quantity purchased, allowing for calculation of total funds.

Question 92

By what percent would a consumer's gasoline bill change if they reduce their consumption by 10% after a price increase of 10%?

Answer 92

1% ## Footnote The net effect on the bill is calculated through price and quantity adjustments.

Question 93

Evaluate the limit of [(z^2 – 1 – i)/(z^2 - 2z + 2)]^2 as z approaches 1 + i.

Answer 93

-1/4 ## Footnote This limit can be solved through substitution and simplification.

Question 94

Find the volume of a right heptagonal prism with base sides that measure 13 cm, an apothem that measures 6 cm, and a height that measures 2 cm.

Answer 94

546 ## Footnote Volume is calculated using the formula for the area of the base times height.

Question 95

Find the measure of a triangular pyramid’s base side if its volume measures 72√3 cubic meters and its height measures 6 meters.

Answer 95

12 ## Footnote The volume formula for a triangular pyramid is used to solve for the base side length.

Question 96

What is the area of a circle inscribed in a dodecagon with an apothem 13 meters long?

Answer 96

169π meters ## Footnote The area is calculated using the formula A = πr^2, where r is the apothem length.

Question 97

How many miles per gallon does Marci’s car get if it took 12.4 gallons to fill the tank after driving 198.4 miles?

Answer 97

16 miles per gallon ## Footnote Fuel efficiency is calculated by dividing total miles driven by gallons used.

Question 98

If Kelly needs 350 feet of fencing at $3.25 per foot and 6 hours of labor at $15.75 per hour, how much will she owe?

Answer 98

$1,232.00 ## Footnote Total cost is calculated by summing the costs of materials and labor.

Question 99

If the areas of two sections of a garden are 6a + 2 and 5a, what is the difference between the areas of the two sections in terms of a?

Answer 99

a + 2 ## Footnote The difference is found by subtracting the two area expressions.

Question 100

The measures of two complementary angles are in the ratio of 7:8. Find the measure of the smallest angle.

Answer 100

42° ## Footnote Complementary angles sum to 90°, allowing for ratio-based calculations.

Question 101

The sum of three times a greater integer and 5 times a lesser integer is 9. What is the value of the lesser integer?

Answer 101

0 ## Footnote The system of equations is solved to find the integers.