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mathematics
basic technical mathematics

Questions and Answers of Basic Technical Mathematics

Evaluate the given determinants by expansion by minors. -2 6 6-1 1-2 2 -4 3 5 -3 -5 4 -2-3 1-2
Use the determinants and evaluate each using a calculator -2 6 1 -2 -5 2 3 5 -4 65 -3 6-1 T 4 1-2-3
Use the determinants and evaluate each using a calculator. 7 2 1 -5 3 0 4 -3 -6
Use the matrix N.Show that N2 = −I. [ N = -1 10 0
Use the determinants and evaluate each using a calculator 14 0-3 2 5 -2 -2 -4 1 -1 6 3-4 32 3 1
For the matrixfind (a) N2 , (b) N3, (c) N4 .What is N20? Explain. N 11 1 1
Iffind the A. B-1 || 12 3 4 and AB || -1 3 02
Use the matrix N.Show that N−1 = −N. [ N = -1 10 0
Forverify the associative law of multiplication, A(BC) = (AB)C. A = 12 34 , B 2-3 3 5 | c = [ 01 24
For any real number n, show that. 1+n n 1- n-n 2 = I.
Use matrices A and B.Show that (A + B)(A − B) ≠ A2 − B2 . A = 1 1-2 3 0 B = -3 1 2 - 1
Use matrices A and B.Show that (A + B)2 ≠ A2 + 2AB + B2 A = 1 1-2 3 0 B = -3 1 2 - 1
Use matrices A and B.Show that the inverse of 2A is A−1/2. A = 1 1-2 3 0 B = -3 1 2 - 1
Use matrices A and B.Show that the inverse of B/2 is 2B−1. A = 1 1-2 3 0 B = -3 1 2 - 1
To find the electric currents (in A) indicated in Fig. 16.24, it is necessary to solve the following equations.Find IA, IB, and IC. IA+IB + IC = 0 = = -4 51A - 21 B 21B - 1C = 0 Ic
A beam is supported as shown in Fig. 16.23. Find the force F and the tension T by solving the following system of equations:Fig. 16.23. 0.500F 0.866T = 0.866F+0.500T = 350
A beam is supported as shown in Fig. 16.23. Find the force F and the tension T by solving the following system of equations:Fig. 16.23. 0.500F 0.866T = 0.866F+0.500T = 350
To find the electric currents (in A) indicated in Fig. 16.24, it is necessary to solve the following equations.Find IA, IB, and IC. IA+IB + IC = 0 = = -4 51A - 21 B 21B - 1C = 0 Ic
By mass, three alloys have the following percentages of lead, zinc, and copper.How many grams of each of alloys A, B, and C must be mixed to get 100 g of an alloy that is 44% lead, 38% zinc, and 18%
Two electric resistors, R1 and R2, are tested with currents and voltages such that the following equations are found:2R₁ + 3R₂ = 26 3R₁ + 2R₂ = 24Find the resistances R1 and R2 (inΩ).
A company produces two products, each of which is processed in two departments. Considering the worker time available, the numbers x and y of each product produced each week can be found by solving
Set up a matrix representing the information given in Exercise 91. A given shipment contains 500 g of alloy A, 800 g of alloy B, and 700 g of alloy C. Set up a matrix for this information. By
The matrix equationmay be used to represent the system of equations relating the currents and resistances of the circuit in Fig. 16.25. Find this system of equations by performing the indicated
Two electric resistors, R1 and R2, are tested with currents and voltages such that the following equations are found:2R₁ + 3R₂ = 26 3R₁ + 2R₂ = 24Find the resistances R1 and R2 (inΩ).
A company produces two products, each of which is processed in two departments. Considering the worker time available, the numbers x and y of each product produced each week can be found by solving
In Example 2, change x + 3 to 3 − x and then draw the graph of the resulting inequality.Data from Example 2Draw a sketch of the graph of the inequality y < x + 3. First, we draw the function y =
A crime suspect passes an intersection in a car traveling at 110 mi/h. The police pass the intersection 3.0 min later in a car traveling at 135 mi/h. How long is it before the police overtake the
A contractor needs a backhoe and a generator for two different jobs. Renting the backhoe for 5.0 h and the generator for 6.0 h costs $425 for one job. On the other job, renting the backhoe for 2.0 h
In Example 2, change the > to <, solve the resulting inequality, and graph the solution.Data from Example 2Solve the inequality |2x − 1| > 5.By using Eq. (17.1), we have 2x − 1 < −5
In Example 1, change the last constraint to 2x + y ≤ 8. Then graph the feasible points and find the maximum value of F.Data from Example 1Find the maximum value of F, where F = 2x + 3y and x and y
On a 750-mi trip from Salt Lake City to San Francisco that took a total of 5.5 h, a person took a limousine to the airport, then a plane, and finally a car to reach the final destination. The
In Example 2, change 3 to 25 and solve the resulting inequality.Data from Example 2Solve the following inequality: 3 − 2x ≥ 15.We have the following solution: inequality reversed 3- 2x ≥ 15 -
An automobile maker has two assembly plants at which cars with either 4, 6, or 8 cylinders and with either standard or automatic transmission are assembled. The annual production at the first plant
In Example 2, change the − sign before the 3 to + and change 2x to 4x, then solve the resulting inequality, and graph the solution.Data from Example 2Solve the inequality x2 − 3 > 2x.We first
In Example 4, change x2 − 4 to 4 − x2 and then draw the graph of the resulting inequality.Data from Example 4Although the graph of y = x2 − 4 is not a straight line, the method of solution is
A person prepared a meal of the following items, each having the given number of grams of protein, carbohydrates, and fat, respectively. Beef stew: 25, 21, 22; coleslaw: 3, 10, 10; (light) ice cream:
In Example 4, change the 1/4 to 7/4 and then solve and display the resulting inequality.Data from Example 4Solve the inequalityNote that the sense of the inequality was reversed when we divided by
A hardware company has 60 different retail stores in which 1500 different products are sold. Write a paragraph or two explaining why matrices provide an efficient method of inventory control, and
In Example 4(b), change ≤ to > and then graph the resulting inequality.Data from Example 4(b)To graph x ≤ 1, we follow the same basic procedure as in part (a), except that we use a solid
In Example 2, in the first paragraph, change the > to < and then complete the meaning of the resulting inequality as in the first sentence. Rewrite the meaning as in the second line.Data from
In Example 2, change the last constraint to 3y − x ≤ 4. Then graph the feasible points and find the maximum and minimum values of F.Data from Example 2Find the maximum and minimum values of the
Determine each of the following as being either true or false. If it is false, explain why.x < −3 or x > 1 may also be written as 1 < x < −3.
State conditions on x and y in terms of inequalities if the point (x, y) is in the second quadrant.
In Example 6, change the + in the middle member to − and then solve the resulting inequality. Graph the solution.Data from example 6Solve: −1 < 2x + 3 < 6.We have the following solution.The
In Example 5, change the exponent on (x − 2) from 2 to 3, then solve the resulting inequality, and graph the solution.Data from Example 5Solve the inequalitySince this function is a fraction, the
In Example 7, change the inequality to −2 > −4 and then perform the two operations shown in color.Data from Example 7Using property 3 on the inequality 4 > −2, we have the following
Determine each of the following as being either true or false. If it is false, explain why.The solution of the inequality −3x > 6 is x > −2.
Solve the given inequalities algebraically and graph each solution.−x/2 ≥ 3
Draw a sketch of the graph of the given inequality.y > x − 1
Graphing the constraints of a linear programming problem shows the consecutive vertices of the region of feasible points to be (0, 0), (12, 0), (10, 7), (0, 5), and (0, 0). What are the maximum and
Solve the given inequalities. Graph each solution.|x − 4| < 1
Determine each of the following as being either true or false. If it is false, explain why.The solution of the inequality x 2 − 2x − 8 > 0 is x < −2 or x > 4.
Solve the given inequalities algebraically and graph each solution.3x + 1 < −5
Solve the given inequalities. Graph each solution. It is suggested that you also graph the function on a calculator as a check.x2 − 16 < 0
Prove that secθ − tanθ/cscθ = cosθ
Use the half-angle formulas to evaluate the given functions.cos 3π/8
Evaluate the given functions with the following information: sin α = 4 / 5 (α in first quadrant) and cos β = −12 / 13 (β in second quadrant).sin(α + β)
Find cos 1/2x if sin x = −3/5 and 270° < x < 360°.
Find tan 120° by using the functions of 60°.
Write down the meaning of each of the given equations. See Example 1.y = 2 sin−1 xData from Example 1(a) y = cos−1 x is read as “y is the angle whose cosine is x.” In this case, x = cos y.(b)
Multiply and simplify.cos x(tan x − sec x)
Solve the given trigonometric equations analytically (using identities when necessary for exact values when possible) for values of x for 0 ≤ x < 2π.1 − 2 cos x = 0
Find sin 120° by using 120° = 90° + 30°.
Use the half-angle formulas to evaluate the given functions.sin 11π/12
Evaluate the given functions with the following information: sin α = 4 5 (α in first quadrant) and cos β = −12 13 (β in second quadrant).tan(β − α)
The intensity of a certain type of polarized light is given by I = I0 sin 2θ cos2θ. Solve for θ.
Find cos 60° by using the functions of 30°.
Simplify the given expressions by giving the results in terms of one-half the given angle. Then use a calculator to verify the result. - cos 236° 2
Write down the meaning of each of the given equations. See Example 1.y = 3 tan−1 2xData from Example 1(a) y = cos−1 x is read as “y is the angle whose cosine is x.” In this case, x = cos
Simplify the given expressions by giving the results in terms of one-half the given angle. Then use a calculator to verify the result. 1 + cos98° 2
Multiply and simplify.csc y(sin y + 3cos y)
Solve the given trigonometric equations analytically (using identities when necessary for exact values when possible) for values of x for 0 ≤ x < 2π.4 tan x + 2 = 3(1 + tan x)
Find cos 30° by using 30° = 90° − 60°.
Evaluate the given functions with the following information: sin α = 4 5 (α in first quadrant) and cos β = −12 13 (β in second quadrant).cos(α + β)
In Example 1, change (−4, 1) to (4,−1).Data from Example 1Find the equation of the line that passes through (−4,1) with a slope of −1/2. See Fig. 21.16. Substituting in Eq. (21.6), we find
Solve graphically: x − 2cos x = 5.
In Example 1, change (y + 2)2 to (y + 1)2.Data from Example 1The equation (x − 1)2 + (y + 2)2 = 16 represents a circle with center at (1,−2) and a radius of 4. We determine these values by
Use a calculator to verify the values found by using the double-angle formulas.Find sin 100° directly and by using functions of 50°.
Write down the meaning of each of the given equations. See Example 1.y = 5 cos−1 (2x − 1)Data from Example 1(a) y = cos−1 x is read as “y is the angle whose cosine is x.” In this case, x =
Multiply and simplify.cosθ cotθ(secθ − 2 tanθ)
Solve the given trigonometric equations analytically (using identities when necessary for exact values when possible) for values of x for 0 ≤ x < 2π.4 − sec2 x = 0
Find sin 315° by using 315° = 360° − 45°.
Evaluate the given functions with the following information: sin α = 4 5 (α in first quadrant) and cos β = −12 13 (β in second quadrant).sin(α − β)
Find the exact value of x: cos−1 x = −tan−1(−1).
Using graphs displayed on a calculator, verify the identity in Exercise 43.Data from Exercises 43Show that sin 2x/sin x = 2cos x.
Describe the curve represented by each equation. Identify the type of curve and its center (or vertex if it is a parabola). Sketch each curve.(y − 2)2 = 4(x + 1)
Determine the center and the radius of each circle.(x − 3)2 + (y + 4)2 = 49
Determine each of the following as being either true or false. If it is false, explain why.The foci of the hyperbola 9x2 −16y2 =144 are (−5, 0) and (5, 0).
Plot the polar curve r = 3 + cos θ.
Find the equation of each of the circles from the given information.The origin and (−6, 8) are the ends of a diameter
Concentric with the circle x2 + y2 + 2x − 8y + 8 = 0 and passes through (−2, 3)
Evaluate the given expressions. cos(sin-12 + cos -¹³3)
When designing a solar-energy collector, it is necessary to account for the latitude and longitude of the location, the angle of the sun, and the angle of the collector. In doing this, the equation
In Example 3, change the − before the 2sinθ to +.Data from Example 3Plot the graph of r = 1 − 2sinθ. Choosing values of θ and then finding the corresponding values of r, we find the following
In Example 5, change the + before 8y to −.Data from Example 5Find the center and radius of the circle. x2 + y2 − 6x + 8y − 24 = 0We can find this information if we write the given equation in
In Example 3(a), change 4y to −6y.Data from Example 3(a)The parabola x2 = 4y fits the form of Eq. (21.16). Therefore, its axis is along the y-axis and its vertex is at the origin. From the
Identify each of the equations as representing either a circle, a parabola, an ellipse, a hyperbola, or none of these.x2 − y − 5 = 0
Find the polar equation of the curve whose rectangular equation is x2 = 2x − y2.
In Example 7, change sin to cos.Data from Example 7Find the rectangular equation of the rose r = 4sin 2θ. Using the trigonometric identity sin2θ = 2sinθcosθ and Eqs. (21.41) and (21.42) leads to
Find the coordinates of the vertices and foci of the given ellipses. Sketch each curve. 16x² + 9y² 25 = 1

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