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mathematics
basic technical mathematics

Questions and Answers of Basic Technical Mathematics

Perform the indicated multiplication. [ 4 -211- -1 0 26
For matrices A and B, find A − 2B. A || 32 3 -1 4 0-2 2 B = 145 -1 -2 3
Find the inverse of each of the given matrices by the method of Example 1 of this section.Data from Example 1Find the inverse of the matrixFirst, we interchange the elements on the principal diagonal
Determine each of the following as being either true or false. If it is false, explain the reason why. 2 3-1 0 2 6-2 0 2
Determine each of the following as being either true or false. If it is false, explain the reason why.If A = -1 1 -3 4 then A-1 -3 -4 -1
Evaluate the literal symbols. 2x х+2 x - у 20 Z · ] - [ $ a y + z 6 -2 4 b c
Determine each of the following as being either true or false. If it is false, explain the reason why. 13 BEA] 02 -2 3-1 5 10 -4 -2
Solve the given systems of equations by Gaussian elimination. If there is an unlimited number of solutions, find two of them.x + 2y = 43x − y = 5
Solve the given systems of equations by using the inverse of the coefficient matrix.2x − 3y = −6 (3)−2x + 4y = 11
In Example 1, change the element −7 to −5 and then find the inverse using the same method.Data from Example 1Find the inverse of the matrixFirst, we interchange the elements on the principal
For the mirror shown in Fig. 15.13, the reciprocal of the focal distance f equals the sum of the reciprocals of the object distance p and image distance q (in in.). Find p, if q = p + 4 and f = (p +
A computer science student is to write a computer program that will print out the values of n for which x + r is a factor of xn + rn . Write a paragraph that states which are the values of n and
Three electric capacitors are connected in series. The capacitance of the second is 1μF more than the first, and the third is 2μF more than the second. The capacitance of the combination is
The entrance to a garden area is a parabolic portal that can be described by y = 4 − x2 (in m). Find the largest area of a rectangular gate that can be installed by graphing the function for area
The radius of one ball bearing is 1.0 mm greater than the radius of a second ball bearing. If the sum of their volumes is 100 mm3, find the radius of each.
A rectangular door has a diagonal brace that is 0.900 ft longer than the height of the door. If the area of the door is 24.3 ft 2, find its dimensions.
A grain storage bin has a square base, each side of which is 5.5 m longer than the height of the bin. If the bin holds 160 m3 of grain, find its dimensions.
The height of a cylindrical oil tank is 3.2 m more than the radius. If the volume of the tank is 680 m3, what are the radius and the height of the tank?
An architect is designing a window in the shape of a segment of a circle. An approximate formula for the area is where A is the area, w is the width, and h is the height of the segment. If the
In finding the electric current in a certain circuit, it is necessary to factor the denominator ofIs (a) (s − 2)(b) (s + 5) a factor? 2s s3+5s² + 4s + 201
A cubical tablet for purifying water is wrapped in a sheet of foil 0.500 mm thick. The total volume of the tablet and foil is 33.1% greater than the volume of the tablet alone. Find the length of the
In order to find the diameter d (in cm) of a helical spring subject to given forces, it is necessary to solve the equation 64d3 − 144d2 + 108d − 27 = 0. Solve for d.
A company determined that the number s (in thousands) of computer chips that it could supply at a price p of less than $5 is given by s = 4p2 − 25, whereas the demand d (in thousands) for the chips
A computer analysis of the number of crimes committed each month in a certain city for the first 10 months of a year showed that n = x3 − 9x2 + 15x + 600. Here, n is the number of monthly crimes
In finding the volume V (in cm3) of a certain gas in equilibrium with a liquid, it is necessary to solve the equation V3 − 6V2 + 12V = 8. Use synthetic division to determine if V = 2 cm3.
The edge of a cube is 10 cm greater than the radius of a sphere. If the volumes of the figures are equal, what is this volume?
In the theory of the motion of a sphere moving through a fluid, the function f(r) = 4r3 − 3ar2 − a3 is used. Is (a) r = a or(b) r = 2a a zero of f(r)?
A silo is to be constructed in the shape of a cylinder with a hemisphere as its top. Because of design constraints, the total height is to be 40.0 ft. Find the radius that would be required in order
The length of a rectangular box is 3 cm longer than its width. If the volume as a function of the width is f(w) = 2w3 + 5w2 − 3w, find the height if the box.
Where does the graph of the function f(x) = 2x4 − 7x3 + 11x2 − 28x + 12 cross the x-axis?
If f(x) = 3x3 − 5ax2 − 3a2x + 5a3, find f(x) ÷ (x + a).
Solve the following system algebraically: x2 = y + 3; xy = 2
Do the functions f(x) and f(−x) have the same zeros? Explain.
If f(x) = 3x4 − 18x3 − 2x2 + 13x − 6, and f(x) = g(x)(x − 6), find g(x).
If f(x) = −g(x), do the functions have the same zeros? Explain
Form a polynomial equation of degree 3 with integer coefficients and having roots of j and 5.
Use synthetic division: (2x3 − 7x2 + 10x − 6) ÷ [x − (1 + j)].
Explain how to find k if x − 3 is a factor of f(x) = kx4 − 15x2 − 5x − 12. What is k?
Use synthetic division: (x3 − 3x2 + x − 3) ÷ (x + j).
Explain how to find k if x + 2 is a factor of f(x) = 3x3 + kx2 − 8x − 8. What is k?
By division, show that 2x − 1 is a factor of f(x) = 4x3 + 8x2 − x − 2. May we therefore conclude that f(1) = 0? Explain.
What are the possible number of real zeros (double roots count as two, etc.) for a polynomial with real coefficients and of degree 5?
Using synthetic division, divide ax2 + bx + c by x + 1.
Graph the function f(x) = 8x4 − 22x3 − 11x2 + 52x − 12, and use the graph as an assist in factoring the function.
If f(x) = 2x3 + 3x2 − 19x − 4, and f(x) = (x + 4) g(x), find g(x).
Graph the function f(x) = 3x4 − 7x3 − 26x2 + 16x + 32, and use the graph as an assist in factoring the function.
Use synthetic division to determine whether or not the given numbers are zeros of the given functions.2x3 + 13x2 + 10x − 4; 1/2
An equation f(x) = 0 involves only odd powers of x with positive coefficients. Explain why this equation has no real root except x = 0.
Solve the given equation.2x4 + 5x3 − 14x2 − 23x + 30 = 0
If a, b, and c are positive integers, find the combinations of the possible positive, negative, and nonreal complex roots if f(x) = ax3 − bx2 + c = 0
Solve the given equation.4t4 − 17t2 + 14t − 3 = 0
Elliptic curve cryptography uses equations of the form y2 = x3 + ax + b and a type of point addition where the “sum” R of points P and Q is found by extending a line through P and Q, determining
Use synthetic division to determine whether or not the given numbers are zeros of the given functions.r4 + 5r3 − 18r − 8; −4
Each of three revolving doors has a perimeter of 6.60 m and revolves through a volume of 9.50 m3 in one revolution about their common vertical side. What are the door’s dimensions?
Solve the given equation.6y3 + 19y2 + 2y = 3
Use synthetic division to determine whether or not the given numbers are zeros of the given functions.x4 − 5x3 − 15x2 + 5x + 14; 7
Solve the given equation.6x3 − x2 − 12x = 5
Use the factor theorem and synthetic division to determine whether or not the second expression is a factor of the first.3x4 − 2x3 + x2 + 15x + 4; 3x + 4
A rectangular safe is to be made of steel of uniform thickness, including the door. The inside dimensions are 1.20 m, 1.20 m, and 2.00 m. If the volume of steel is 1.25 m3, find its thickness.
Solve the given equation.2x3 − 3x2 − 11x + 6 = 0
Use the factor theorem and synthetic division to determine whether or not the second expression is a factor of the first.4x4 + 2x3 − 8x2 + 3x + 12; 2x + 3
The radii of four different-sized ball bearings differ by 1.00 mm in radius from one size to the next. If the volume of the largest equals the volumes of the other three combined, find the radii.
Solve the given equation.6r3 − 9r2 + 3 = 0
Use the factor theorem and synthetic division to determine whether or not the second expression is a factor of the first.6x4 + 5x3 − x2 + 6x − 2; 3x − 1
The angle θ (in degrees) of a robot arm with the horizontal as a function of time t (in s) is given by θ = 15 + 20t2 − 4t3 for 0 ≤ t ≤ 5s. Find t for θ = 40°.
Solve the given equation.x3 − 8x2 + 20x = 16
Use the factor theorem and synthetic division to determine whether or not the second expression is a factor of the first.2Z4 − Z3 − 4Z2 + 1; 2Z − 1
A rectangular tray is made from a square piece of sheet metal 10.0 cm on a side by cutting equal squares from each corner, bending up the sides, and then welding them together. How long is the side
Solve the given equation.x3 + x2 − 10x + 8 = 0
Use the factor theorem and synthetic division to determine whether or not the second expression is a factor of the first.3x3 − 5x2 + x + 1; x + 1/3
The specific gravity s of a sphere of radius r that sinks to a depth h in water is given byFind the depth to which a spherical buoy of radius 4.0 cm sinks if s = 0.50. S 3rh²h³ 4r3
The pressure difference p (in kPa) at a distance x (in km) from one end of an oil pipeline is given by p = x5 − 3x4 − x2 + 7x. If the pipeline is 4 km long, where is p = 0?
Find all the roots of the given equations, using synthetic division and the given roots.2x6 − x5 + 8x² − 4x = 0 (r1 = 1/2, r2 = 1 + j)
Use the factor theorem and synthetic division to determine whether or not the second expression is a factor of the first.4x3 − 9x2 + 2x − 2; x − 1/4
Cubic Bezier curves are commonly used to control the timing of animations. A certain “ease in” curve is given by x = 0.1t3 − 1.2t2 + 2.1t, y = −2t 3 + 3t 2 for 0 ≤ t ≤ 1, where x
Find all the roots of the given equations, using synthetic division and the given roots.V5 + 4V4 + 5V3 − V2− 4V − 5 = 0(r1 = 1, r2 = −2 + j)
Use the factor theorem and synthetic division to determine whether or not the second expression is a factor of the first.t5 − 3t4 − t2 − 6; t − 3
Find all the roots of the given equations, using synthetic division and the given roots.24x5 + 10x4 + 7x² − 6x + 1 = 0 (r1 = −1, r2=1/4, r3 = 1/3)
Use the factor theorem and synthetic division to determine whether or not the second expression is a factor of the first.2x5 − x3 + 3x2 − 4 : x + 1
The deflection y of a beam at a horizontal distance x from one end is given by y = k(x4 − 2Lx3 − L3x), where L is the length of the beam and k is a constant. For what values of x is the
Find all the roots of the given equations, using synthetic division and the given roots.s5 + 3s4 − s3 − 11s2 − 12s − 4 = 0 (−1 is a triple root)
Find all the roots of the given equations, using synthetic division and the given roots.x4 + 2x³ − 4x − 4 = 0 (r1 = −1 + j)
Perform the indicated divisions by synthetic division.(6t4 + 5t3 − 10t + 4) ÷ (3t − 2)
Form a polynomial equation of the smallest possible degree and with integer coefficients, having a double root of 3, and a root of j.
In finding one of the dimensions d (in in.) of the support columns of a building, the equation 3d3 + 5d2 − 400d − 18,000 = 0 is found. What is this dimension?
Find all the roots of the given equations, using synthetic division and the given roots.4x4 + 4x³ + x² + 4x − 3 = 0 (r1 = j)
Perform the indicated divisions by synthetic division.(2x4 + x3 + 3x2 − 1) ÷ (2x − 1)
Equations of the form y2 = x3 + ax + b are called elliptic curves and are used in cryptography. If y = 3 for the curve y2 = x3 − 4x + 6, use synthetic division to show that one possible value of x
The angular acceleration α (in rad/s2) of the wheel of a car is given by α = −0.2t3 + t2 , where t is the time (in s). For what values of t is α = 2.0/rad/s2?
Find all the roots of the given equations, using synthetic division and the given roots.15x4 + 4x³ + 56x² + 16x − 16 = 0 (r1 = 2/5, r2 = −2/3)
Perform the indicated divisions by synthetic division.(20x4 + 11x3 − 89x2 + 60x − 77) ÷ (x + 2.75)
By checking only the equation and the coefficients, determine the smallest and largest possible rational roots of the equation 2x4 + x2 + 22x + 26 = 0.
Find all the roots of the given equations, using synthetic division and the given roots.4p4 − p² − 18p + 9 = 0 (r1 = 1/2, r2 = 3/2)
Perform the indicated divisions by synthetic division.(x7 − 128) ÷ (x − 2)
Find k such that x − 2 is a factor of 2x3 + kx2 − kx − 2.
Find all the roots of the given equations, using synthetic division and the given roots.2x4 − 2x³ − 10x² − 2x − 12 = 0 (r1 = 3, r2 = −2)
By checking only the equation and the coefficients, determine the smallest and largest possible rational roots of the equation 2x4 + x2 − 22x + 8 = 0.
Perform the indicated divisions by synthetic division.(x5 + 4x4 − 8) ÷ (x + 1)
How can the graph of a fourth-degree polynomial equation have its only x-intercepts as 0, 1, and 2?
Where does the graph of the function f(s) = 2s4 − s3 − 5s2 + 7s − 6 cross the s-axis?

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