MATH OBJECTIVES Flashcards
The sum of the distances from the two foci to any point in what curve is constant?
Ellipse
The definition of an ellipse involves the sum of distances from two foci being constant.
Find the area of a square whose side is a2b3.
a4b6
Area of a square is calculated as side squared.
A normal to a given plane is.
Perpendicular to the plane
The normal line is defined as perpendicular to the surface at a given point.
If a person throws away 3.5 lbs of trash daily, how much trash will the person throw away in one week?
24.5 lbs
Calculation: 3.5 lbs/day * 7 days = 24.5 lbs.
What is 20% of 96?
19.20
20% of a number is calculated by multiplying the number by 0.20.
What conic section is described by the equation 4x2 – y2 + 8x + 4y = 15?
Hyperbola
Analyzing the equation reveals it fits the form of a hyperbola.
Sam scored 96% on his first Calculus quiz; 74% on his second and 85% on his third. What is his quiz average?
85%
Average calculation: (96 + 74 + 85) / 3 = 85%.
When two lines are parallel, the slope of one is:
Equal to the other
Parallel lines have identical slopes.
If 1 cm = 0.39 in, about how many cm are there in 0.75 in?
1.90 cm
Conversion: 0.75 in * 2.54 cm/in = 1.90 cm.
Which of the following is not a multiple of 11?
759
759 does not divide evenly by 11.
What is the conic section whose eccentricity is less than 1?
Ellipse
Eccentricity definitions classify conic sections based on their eccentricity values.
Nicole had 75 stuffed animals. Her grandmother gave fifteen of them to her. What percentage of the stuffed animals did her grandmother give her?
20%
Calculation: (15/75) * 100% = 20%.
What type of conic section is x2 - 4y + 3x + 5 = 0?
Parabola
The equation can be rearranged to fit the standard form of a parabola.
The line passing through the focus and perpendicular to the directrix of a parabola is called _____.
axis
The axis of a parabola is a line that reflects its symmetry.
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?
130,288.13 in/min
Volume and rate calculations involve geometric relationships of a cone.
Find the volume generated by revolving the circle x² + y² + 6x + 4y + 12 = 0 about the y-axis.
59.22
Volume is calculated using Pappus’s Theorem.
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?
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.
What proportion of students have been absent less than 5 days?
0.91
Calculation: 31 out of 34 students were absent less than 5 days.
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?
3
Expected correct answers calculated as Np = 15 x (1/5) = 3.
What is the value of g(f(3))?
-5
This requires substitution based on the provided function tables.
Find the volume generated when the area bounded by y = 2x + 3 and y = x2 is revolved about the x-axis.
228
Volume is found using integration techniques for areas of revolution.
Solve [y – square root of (x2 + y2)]dx - xdy = 0.
Square root of (x2 + y2) + y = C
This is a differential equation solution involving implicit functions.
What is the man’s average between 68 sec and 168 sec?
4 m/s
Average speed is calculated as change in distance over change in time.
What is the probability that less than 17 alumni will make a contribution of at least P 50?
0.589
This uses binomial probability calculations.
At exactly what time after 5 o’clock will the hour hand and the minute hand be perpendicular for the first time?
5:10 and 54 sec
This involves angular relationships of clock hands.
Find the equation of the normal line to x2 + y2 = 1 at point (2, 1).
x + y = 1
The normal line is derived from the slope at the point on the curve.
What is the smallest positive value for x where y = sin 2x reaches its maximum?
π/4
The maximum of sin occurs at specific intervals based on its periodicity.
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.
6 kph
The optimization problem involves calculus to minimize energy consumption.
What is the maximum rectangular area that can be fenced in 20 ft using two perpendicular corner sides of an existing wall?
100 square feet
The area maximization occurs at specific dimensions derived from calculus.
Determine the correct equation for the line with a slope of 7 and y-intercept of – 4.
y = 7x - 4
This is based on the slope-intercept form of a linear equation.
With the courier company’s charges, what amount would be charged of a parcel weighing 30 kg?
P 460
Cost calculated based on tiered pricing structure.
The dimension of a rectangular prism can be expressed as x + 1, x – 2 and x + 4. What is the volume of the prism?
x3 + 3x2 - 6x - 8
Volume is calculated by multiplying the dimensions together.
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?
53.61 m
This involves trigonometric relationships to find the height.
What is the formula for the volume V of a rectangular prism given dimensions W = x – 2 and H = x + 4?
V = L x W x H = (x + 1)(x – 2)(x + 4) = x^3 + 3x^2 – 6x - 8
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?
73.61 m
If log 2 = a, log 3 = b, log 5 = c, what is log(7.5)?
b + c - a
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?
13.10 m
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?
1.125 miles
Evaluate f(-3) if f(x) = x^2 - 2x + 1.
16
Find the sum of the first five terms of the geometric progression if the third term is 144 and the sixth term is 486.
844
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?
96.31 m
Find the equation for line AB given the position vectors of points A and B are 2 + i and 3 – 2i, respectively.
3x + y = 7
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?
4 : 1
In an RL circuit, what is the expression for current I when t = L/R?
0.632(E/R)
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?
37.5 inches
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?
17
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?
2 km from the point on the shore nearest the second town
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?
108.7 min
Find the solution to the system of equations x – 2y = 5 and 2x + 5y = 1.
(3, -1)
What is the vertex of the parabola given by the equation x^2 = 4y?
(0, 0)
What is the value of x if the difference between six times (6x + 1) and three times (x – 1) equals 108?
3
What score must Daniel achieve on his last Algebra exam to maintain an average of 95?
97
What is the area of the triangle with vertices (1, 1), (3, 2), and (2, 4)?
5/2
What is the area of the polygon with vertices 2 + 3i, 3 + i, -2 – 4i, -4, -i, -1 + 2i?
25
What is the area bounded by the curves y^2 = 4x and x^2 = 4y?
5.33
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?
Between 15,000 miles and 16,000 miles
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?
x + 2y – 6 = 0
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?
90 feet
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?
2 ft/min
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?
0.8192
What symmetry does a periodic function with zero average value and a Fourier series consisting of only odd cosine terms possess?
odd
Evaluate the integral of ln x dx from 1 to e.
1
What is the differential equation of the family of parabolas with vertices at the origin and foci on the x-axis?
2x dy - y dx = 0
What is the differential equation of the family of parabolas having their vertices at the origin and their foci on the x-axis?
dy/dx – x = 0
The correct differential equation is derived from the properties of parabolas defined in the Cartesian plane.
Which of the following equations is an exact differential equation?
(2xy + x)dx + (x^2 + y)dy = 0
Exact differential equations satisfy the condition ∂M/∂y = ∂N/∂x where M and N are the coefficients of dx and dy respectively.
Compute log (3-2i)
0.5570 – 0.2554i
The logarithm of a complex number is computed using its polar form.
What is the maximum value of 3 sin x?
3
The maximum value of sin x is 1, thus 3 sin x achieves a maximum of 3.
What is the ratio of the sides of a triangle if the product of the sides is a maximum?
1:1:1
To maximize the product of the sides, they must be equal lengths.
Evaluate the limit of x/sqrt(1 + x^2) as x approaches infinity.
1
The limit can be evaluated by dividing numerator and denominator by x.
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?
1.2 cm
The volume of water displaced equals the volume of the sphere.
If the coefficient a0 of a Fourier series of a periodic function is zero, it means that the function has:
Odd symmetry
A zero average indicates odd symmetry in the function.
Solve the equation 5z^2 + 2z + 10 = 0.
1 + i, 1 - 2i
The solutions can be found using the quadratic formula.
Solve the differential equation y’’ – 4y’ + 3y = sin x.
y(x) = C₁e^3x + C₂e^x + 1/5 cos x + 1/10 sin x
The general solution involves solving the homogeneous and particular solutions.
The Fourier series of waveform processing even quarter-wave symmetry contains only _______.
even harmonics
Even quarter-wave symmetry implies the absence of odd harmonics.
A periodic waveform possessing half-wave symmetry has no _______.
even harmonics
Half-wave symmetry excludes even harmonics in the Fourier series representation.
Which of the following periodic functions possesses even symmetry?
cos 3t
Even functions satisfy f(x) = f(-x).
The symbol j represents counterclockwise rotation of a vector through _______ degrees.
90
In complex number representation, j indicates a 90-degree rotation.
The conjugate of (-a + jb) is _______.
(a – jb)
The conjugate of a complex number is obtained by changing the sign of the imaginary part.
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:
Even harmonics
This symmetry suggests the presence of even harmonics in the waveform.
All opposite rays _______.
Extend in the same direction
Opposite rays share a common endpoint and extend infinitely in opposite directions.
Angles that share a common vertex point cannot _______.
Share a common angle side
Common angle sides would imply that the angles are not distinct.
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 _______.
Congruent angles
Supplementary angles sum to 180 degrees, hence HIJ and SUV are equal.
Evaluate sin [arccos (-2/3)].
Square root of 5
This can be solved using the Pythagorean identity.
Four is added to the quantity two minus the sum of negative seven and six. What is the result?
21
The calculation follows order of operations.
How much will a business have depreciated after 2 years if a computer worth P 21,000 depreciates to 0 in 5 years?
P 8,400
Depreciation is linear over the lifespan.
Evaluate lim (2 – x)^tan(πx/2) as x approaches 1.
e
Utilizes L’Hôpital’s rule due to the indeterminate form.
Find the centroid of a semi-circular region of radius a.
4a/3π
The centroid of a semicircular area is derived through integration.
If y = 2x + sin(2x), find x when y’ = 0.
π/2
The derivative is set to zero to find critical points.
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?
0.09
This probability is calculated using the Z-score for normal distribution.
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?
4.33
Total aisles are calculated based on their individual rates and combined efforts.
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?
80
The increase in price led to a reduction in quantity purchased, allowing for calculation of total funds.
By what percent would a consumer’s gasoline bill change if they reduce their consumption by 10% after a price increase of 10%?
1%
The net effect on the bill is calculated through price and quantity adjustments.
Evaluate the limit of [(z^2 – 1 – i)/(z^2 - 2z + 2)]^2 as z approaches 1 + i.
-1/4
This limit can be solved through substitution and simplification.
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.
546
Volume is calculated using the formula for the area of the base times height.
Find the measure of a triangular pyramid’s base side if its volume measures 72√3 cubic meters and its height measures 6 meters.
12
The volume formula for a triangular pyramid is used to solve for the base side length.
What is the area of a circle inscribed in a dodecagon with an apothem 13 meters long?
169π meters
The area is calculated using the formula A = πr^2, where r is the apothem length.
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?
16 miles per gallon
Fuel efficiency is calculated by dividing total miles driven by gallons used.
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?
$1,232.00
Total cost is calculated by summing the costs of materials and labor.
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?
a + 2
The difference is found by subtracting the two area expressions.
The measures of two complementary angles are in the ratio of 7:8. Find the measure of the smallest angle.
42°
Complementary angles sum to 90°, allowing for ratio-based calculations.
The sum of three times a greater integer and 5 times a lesser integer is 9. What is the value of the lesser integer?
0
The system of equations is solved to find the integers.
