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Math 201-015-50 Final Exam—December 2010: Mathematics Problem Solving, Exams of Algebra

University of AllahabadAlgebra

The final exam for math 201-015-50 from december 2010. The exam covers various topics in mathematics including simplifying expressions, factoring polynomials, performing operations, rationalizing numerators, long division, solving equations, finding intercepts, and graphing functions. It also includes problems on logarithms, trigonometry, and calculus.

Typology: Exams

2012/2013

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Math 201-015-50 Final Exam—December 2010
[marks]
Page 1 of 4
1. Simplify each expression. Your final answer must contain positive exponents only.[4]
(a) x3y−1(x−1y)2
x3y
(b) n3m33
√nm−3
√nm6
2. Factor the following polynomials as much as possible.[6]
(a) 5xy + 25x+y2+ 5y
(b) 12x2+ 15x+ 3
(c) x3+ 8y3
3. Perform the indicated operations and simplify the results.[6]
(a) x2−6x−16
x2−4x−32 ÷4x2−16
x+ 4
(b)
x
x−2+3
x−4
x+ 2
4. Rationalize the numerator and simplify. √2−√y
2
[2]
5. Use long division to find the quotient and the remainder: 2x3+ 2x2+ 3x+ 3
x−2
[2]
6. Solve each of the following for x.
(a) −2(3x+ 5) >10x−4[2]
(b) 3x2−x= 10
[2]
(c) x5−4x4−21x3= 0
[2]
7. Solve each of the following for x.
(a) 3x−9
x2−4x−5=1
x−5−4
x+ 1
[3]
(b) x+√x−4 = 4[3]
8. Find the equation of the vertical line through the point (−2,7).[1]
9. Calculate the distance between the points (−3,7) and (4,9).[1]
10. For the line given by 4x−3y+ 24 = 0, find the xand y-intercepts.[1]
11. Find the midpoint of the line segment with endpoints (−7,15) and (−2,4).[1]
12. The line Lis given by the equation 3y+ 2x= 6. Find the equation of the line perpendicular to Lthat[2]
passes through the x-intercept of L.
13. Give the exact value of the expression. (No Decimals.)[2]
e(−4/3) ln 27
14. Use the change of base formula to calculate log23 to two decimal places.[1]
15. Solve for x. Give your answer to two decimal places.[2]
7(4x) = 21000
16. Solve for x. Give exact simplified values.[3]
ln(3x+ 1) = ln 5 −ln(x+ 1)
17. Write the following expression as a single logarithm:[2]
2 log(x+ 3) + log(x−4) −log(x−1)
pf3
pf4

Partial preview of the text

Download Math 201-015-50 Final Exam—December 2010: Mathematics Problem Solving and more Exams Algebra in PDF only on Docsity!

[marks]

[4] 1. Simplify each expression. Your final answer must contain positive exponents only.

(a)

x^3 y−^1 (x−^1 y)^2 x^3 y

(b)

n^3 m^3

nm−^3 √ nm^6

[6] 2. Factor the following polynomials as much as possible.

(a) 5 xy + 25x + y^2 + 5y (b) 12 x^2 + 15x + 3 (c) x^3 + 8y^3

[6] 3. Perform the indicated operations and simplify the results.

(a) x^2 − 6 x − 16 x^2 − 4 x − 32

÷

4 x^2 − 16 x + 4

(b)

x x− 2 +^

3 x− 4 x + 2

  1. Rationalize the numerator and simplify.

y 2

[2]

  1. Use long division to find the quotient and the remainder:

2 x^3 + 2x^2 + 3x + 3 x − 2

[2]

  1. Solve each of the following for x.

[2] (a) −2(3x + 5) > 10 x − 4

[2]^ (b) 3x^2 −^ x^ = 10

[2]^ (c)^ x^5 −^4 x^4 −^21 x^3 = 0

  1. Solve each of the following for x.

(a)

3 x − 9 x^2 − 4 x − 5

x − 5

x + 1

[3]

(b) x +

[3] x − 4 = 4

[1] 8. Find the equation of the vertical line through the point (− 2 , 7).

[1] 9. Calculate the distance between the points (− 3 , 7) and (4, 9).

[1] 10. For the line given by 4x − 3 y + 24 = 0, find the x and y-intercepts.

[1] 11. Find the midpoint of the line segment with endpoints (− 7 , 15) and (− 2 , 4).

[2] 12. The line L is given by the equation 3y + 2x = 6. Find the equation of the line perpendicular to L that

passes through the x-intercept of L.

[2] 13. Give the exact value of the expression. (No Decimals.)

e(−^4 /3) ln 27

[1] 14. Use the change of base formula to calculate log 2 3 to two decimal places.

[2] 15. Solve for x. Give your answer to two decimal places.

7(4x) = 21000

[3] 16. Solve for x. Give exact simplified values.

ln(3x + 1) = ln 5 − ln(x + 1)

[2] 17. Write the following expression as a single logarithm:

2 log(x + 3) + log(x − 4) − log(x − 1)

[marks]

[3] 18. Write as a sum/difference of mulitiples of logarithms. Each logarithm should be as simple as possible.

ln

ex

(x + 1)^2 z^2

[5] 19. Sketch the function f (x) = − 1 − 2 ex. Then state the domain, range, the coordinates of the y- and x-

intercepts (if any), and the equation of the asymptote.

  1. Find all y-intercepts, x-intercepts, vertical asymptotes, and horizontal asymptotes of the following functions:

(a) f (x) =

3 x + 1 2 − x

[2]

(b) f (x) = x^2 x^3 − 8

[2]

  1. Given f (x) =

x + 1

and g(x) =

x − 1, find the following:

[1] (a) f (g(x))

[2]^ (b)^ f^ −^1 (x) , the inverse of^ f^.

  1. Find the domain and range of f (x) =

[2] 2 − x.

  1. For the function f (x) = x^2 + 1, find and simplify

f (x + h) − f (x) h

[2].

  1. Graph the following function: f (x) =

2 x if x < 2 6 − x if x ≥ 2

[2]

[6] 25. Consider the function: y = x^2 + 4x + 5.

(a) Complete the square. (b) Find the coordinates of the the y-intercept, x-intercept(s) if any, vertex, axis of symmetry, and the domain and the range. Then sketch the graph.

[1] 26. If $5000 is invested at a rate of 3% interested compounded annually, what is the investment worth

after 9 years? (Round your answer to the nearest cent.)

  1. Calculate to at least 4 decimal places.

[1] (a) sec(101◦)

[1] (b) cos(3)

[1] 28. Convert 12◦^ to radians.

  1. Convert

3 π 5

[1] radians to degrees.

[2] 30. Find two angles θ between 0◦^ and 360◦^ with tan(θ) = 5. (Give your answer to two decimal places.)

  1. Simplify the following expressions:

(a) sin A cos A tan A sec A csc A cot A

[2]

(b)

1 − cos B sin B

sin B 1 + cos B

[2]

[2] 32. Prove: csc A − cos A cot A = sin A.

[2] 33. Prove: cot^2 C(sec^2 C − 1) = 1.

[2] 34. Find the amplitude and period of y = 5 sin(2x). Then sketch the function.

[marks]

20(a). y-int: (0, 1 /2) x-int: (− 1 / 3 , 0) VA: x = 2 HA: y = − 3

20(b). y-int: (0, 0) x-int: (0, 0) VA: x = 2 HA: y = 0

21(a).

x − 1 + 1

21(b). f −^1 (x) = (^) x^1 − 1 22. Domain: (−∞, 2], Range: [0, ∞) 23. 2x + h

x

y

25(a). y = (x + 2)^2 + 1

25(b). y-int: (0, 5) x-int: None Vertex: (− 2 , 1) Axis of Sym: x = − 2 Domain: R Range: [1, ∞)

x

y

  1. $6, 523. 87 27(a). − 5. 2408 27(b). − 0. 9900 28. 0.2094 (Radians) 29. 108◦
    1. 69 ◦, 258. 69 ◦^ 31(a). sin^4 A 31(b). 0
  3. csc A − cos A cot A =

sin A

− cos A

cos A sin A

1 − cos^2 A sin A

sin^2 A sin A

= sin A

  1. cot^2 C(sec^2 C − 1) = cot^2 C tan^2 C =

tan^2 C

· tan^2 C = 1

Amp = 5 Per= 180◦ 1

− 90 ◦^90 ◦^180 ◦^270 ◦^360 ◦

x

y

    1. 52 ◦^ 36. 38. 79 ◦^ 37. 60◦^ 38. 19m