Tiêu đề MSTE Solutions.pdf ✅

01 College Math Solutions ▣ 1. If xyz = 8 and y 2 = 12, what is the value of x y if z = 1? [SOLUTION] Divide the two equations, xyz y 2 = 8 12 Simplify and set z = 1, x(1) y = 2 3 x y = 2 3 ▣ 2. When f(x) = x 4 + ax 3 + 7x 2 + bx is divided by x − 2, the remainder is 16. When it is divided by x + 1, the remainder is 10. What is the value of a? [SOLUTION] From the remainder theorem, f(2) = 16 2 4 + a(2 3 ) + 7(2 2 ) + b(2) = 16 8a + 2b = −28 f(−1) = 10 (−1) 4 + a(−1) 3 + 7(−1) 2 + b(−1) = 10 −a − b = 2 From the two equations, a = −4 , b = 2 ▣ 3. Solve for x from x x x x... = 3. [SOLUTION] x x x x... = 3 x 3 = 3 x = √3 3 ▣ 4. In the expansion of (2x − 1 x ) 10 , find the coefficient of the 8th term. [SOLUTION]
Note that any term within the expansion is: 10! k! (10 − k)! (2x) 10−k (− 1 x ) k And the 8th term is at k = 8 − 1 = 7. The coefficient, therefore, is 10! 7! (10 − 7)! (2) 10−7 (−1) 7 = −960 ▣ 5. Given that w varies directly as the product of x and y and inversely as the square of z and that w = 4 when x = 2, y = 6, and z = 3. Find the value of w when x = 1, y = 4, and z = 2. [SOLUTION] The equation of variation is w = k xy z 2 Since k is the constant of variation, then k = wz 2 xy k = ( wz 2 xy ) 1 = ( wz 2 xy ) 2 (4)(3 2 ) (2)(6) = w(2 2 ) (1)(4) w = 3 ▣ 6. Simplify the expression i 1997 + i 1999, where i is the imaginary number. [SOLUTION] Note that: i 1 = i, i 2 = −1, i 3 = −i, i 4 = 1 i 1997 = (i 4 ) 499 ⋅ i = i i 1999 = (i 4 ) 499 ⋅ i 3 = −i Therefore, i + (−i) = 0 ▣ 7. Divide 253 into 4 parts proportional to 2, 5, 7, 9. What is the value of the largest part? [SOLUTION] Since the proportion is 2: 5: 7: 9, then the parts are in the form 2k, 5k, 7k, 9k where k is a real number.
2k + 5k + 7k + 9k = 253 k = 11 Therefore, the largest part is 9k = 9(11) = 99 . ▣ 8. On a Richter scale, the magnitude R of an earthquake of intensity I is given by R = log I I0 where I0 is a certain minimum intensity. If the intensity of an earthquake is 1000I0, find R. [SOLUTION] Substitute I = 1000I0, R = log 1000I0 I0 R = 3 ▣ 9. Find the value of 102x+1 if 10x = 4. [SOLUTION] Manipulating 102x+1 , 102x+1 = 102x ⋅ 10 = (10x ) 2 ⋅ 10 = 4 2 ⋅ 10 = 160 ▣ 10. In a commercial survey involving 1000 people on brand preference, 120 were found to prefer brand X only, 200 prefer brand Y only, 150 prefer brand Z only, 370 prefer either brands X or Y but not Z, 450 prefer brands Y or Z but not X, and 370 prefer either brands Z or X but not Y. How many people have no brand preferences and satisfied with all three brands? [SOLUTION] Shown is the completed Venn Diagram for the problem.
Since there are 1000 in total, (a + b) + 120 + 50 + 200 + 100 + 100 + 150 = 1000 a + b = 280