Algebra Workshop 2013 Student Copy

7/25/2019 Algebra Workshop 2013 Student Copy

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C T lligraphy

100 hours workshop to tame CAT

Day 7Al ebra


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1. During a recent span of time, eleven days had some rain. A morning rain was always followed by a

clear afternoon, and an afternoon rain was always preceded by a clear morning. In all, nine

mornings and twelve afternoons were clear. How many days had no rain at all?

a. 4 b. 5 c. 6 d. 9

2. Ten boxes each contain 9 balls. The balls in one box each weigh 0.9 kg; the rest all weigh 1 kg. In

how many least number of weighing you can determine the box with the light balls?

a. 2 b. 1 c. 3 d. 9

3. When P(x) = x4-2x3-3x2+8x-4 is divided by which factor is the remainder the greatest?

a. x-1 b. x+1 c. x-2 d. x-3

4. How many integral values of a are there such that the quadratic expression (x + a) (x + 1991) +

1 can be factored as a product (x + b) (x + c), where b and c are integers?

a. Two b. One c. None d. Zero

5. Let f(x + 2) + f(5x + 6) = 2x 1 for all real x. Find the value of f(1).

a. -2 b. -1 c. -5/3 d. -3/2

6. Find the value: .....6701067010

a. 6 b. 63 c. 36 d. 43

7. Which of the following are even functions of x?

I.

2

16)( xxf

II. xxxf

)( III. 168)( 2

xxxf IV. 325)( 24

xxxxfa. I only b. I, II only c. III only d. III, IV only

8. If 18)( 34578 DxCxBxAxxxf , where DCBA ,,, are positive values, then )(xf could

have:

I. 4 positive, 4 non-real roots III. 4 negative, 4 non-real roots

II. 2 positive, 4 negative, 2 non-real roots IV. 8 non-real roots

a. I, II only b. I, IV only c. II, III only d. II, IV only

9. Brijesh and Manish are trying to solve a quadratic equation. Brijesh writes down the equation but

makes a mistake when copying down the constant term and gets that this equations roots are

3,6. Manish also tries to write down the equation but makes a mistake when copying down the x termand obtains the roots 1, 10. Assuming that both students solved their respective equationscorrectly, what were the original equations roots?a. 1, 3 b. 2 , 5 c. 0, 9 d. 6, 10

10. Find the least value of xy + 2xz + 3yz for positive numbers x, y, z, satisfying xyz = 48.

a. 0 b. 24 c. 72 d. 1

11. The value of805402010

15

is

a. )25(5 b. )22(5 c. )21(5 d. )23(5


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12. 4 )21217(

a. 12 b. )12(2 4/1 c. 122 d. None of these

13. If the roots of 0)()()( 2 baxacxcb are equal then ca

a. 2b b. b2 c. 3b d. b

14. A multiple choice test has 30 questions and 5 choices for each question. If a student answers all 30

questions and the score is [number right - (number wrong/4)] then which of the following is a

possible score?

a. 10 b. 5.25 c. 7.75 d. 8.75

15. If, )( are roots of the equation 02 cbxx where )0( bc then

a. 0 b. ||0 c. 0 d. ||0

16. If f(x) = x3 4x + p, and f(0) and f(1) are of opposite signs, then which of the following is

necessarily true?a. 1 < p < 2 b. 0 < p < 3 c. 2 < p < 1 d. 3 < p < 0

17. If for real values ofx, 0232 xx and 0432 xx , thena. 11 x

b. 41 x

c. 11 x or 42 x

d. 42 x

18. If ab , then the equation 01))(( bxax , has

a. Both roots in [a b]

b. Both roots in ( , a)

c. Both roots in (b,)

d. One root in ( , a) and other in (b, +)

19. The sum of the reciprocals of the solutions of the equation 027450845 245 xxxx is:

a. 50

3b. 15 c.

5

3d.

5

3

20. For which of the following function does f(a+b) = f(a) + f(b)?

a. f(x) = x2

b. f(x) = 4x+1 c. f(x) = 1/x d. f(x) = 7x

21. An infinite geometric series has sum 2005. A new series, obtained by squaring each term of the

original series, has 10 times the sum of the original series. The common ratio of the original series

is

a. 1995/2015 b. 2000/2015 c. 2005/2015 d. None of these

22. Find the minimum value of the function f(x) =

a. 1 b. 0 c. d. 2

23. What is the minimum value of 3131)(

xxxxxf

?

a. -34 b. -16 c. 16 d. 34

92

922

2

xx

xx


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24. Find all ordered pairs (a, b) which satisfy 5y2+2xy-80=0 such that (a, b) are both integers.

a. 6 b. 8 c. 12 d. 16

25. There are 5 Arithmetic Progressions of 9 terms each. The common property of all these APs is that

they all are composed of positive integers and have the same middle term, X. If it is known that

the sum of all the numbers of the APs is 4500, then find out the value of X.

a. 50 b. 100 c. 200 d. 45

26. The first term of an arithmetic progression (AP) is -1 and the 8th term of a geometric progression

(GP) is 80. The 4th terms of both progressions are equal. If the 6 th term of the GP is equal to the

sum of the 6th and 7th terms of the AP, find the smallest possible integral value of the 9 th term of

the GP.

a. 80 b. -80 c. 160 d. -160

27. If a, b and c are real numbers and a+b+c = 16, ca=b2a, 2c=2.4a and abc < 0.

Evaluate 9a -6b +9c

a. 121 b. 199 c. 221 d. None of these

28. Let x, y, and z be distinct real numbers that sum to 0. Find the maximum possible value of

222 zyx

zxyzxy

a. 3 b. 2 c. -1 d. -1/2

29. If a1, a2, , an are positive real numbers whose product is a fixed number c, then the minimum

value of a1+ a2+ + an-1 + 2an is

a. n(2c)1/n b. (n+1) c1/n c. 2nc1/n d. (n+1) (2c) 1/n

30. The area bounded by the curves y = |x| 1 and y = |x| + 1 is

a. 1 b. 2 c. 22 d. 4

31. Consider the expansion (2 + x + x3)5 = a0 + a1x + a2x2 + + a15x

15. What is the value of a0 +

a2 + a4 + + a14?


32. What is the remainder when x100 -2x51 + 1 is divided by x2-1?

a. 2x + 2 b. 2x 2 c. -2x + 2 d. 2

33. The expression (1 + x) (1 + x2) (1 + x4) (1 + x8) (1 + x16) (1 + x32) (1 + x64), equals (where x 1,

& x < 1)

a.

1281

1

x

x

b.

1271

1

x

x

c.

1 2 ..... 621

1

x

x

d. None of these

34. Let a, b, c, d be four real numbers such that a + b + c + d = 8, ab + ac + ad + bc + bd + cd =

12. Find the greatest possible value of d.

a. 4 b. 32+4 c. 32+2 d. 0

35. For a positive integer n let f(n) be the value of24 4 1

2 1 2 1

n n

n n

. Calculate f(1) + f(2) + + f(40)

a. 729 b. 366 c. 364 d. 0


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36. Find the minimum value of

66

6

33

3

1 12

1 1

x xx x

x xx x

for x > 0.

a. 0 b. 1 c. 3 d. 6

37. Determine F(2010) if for all real x and y, F(x)F(y) F(xy) = x + y.

a. 1 b. 2010 d. 2011 d. 0

38. The equation x2 + ax + (b + 2) = 0 has real roots. What is the minimum value of a2 + b2?

a. 2 b. 4 c. 0 d. 8

39. Let P(x) be a polynomial such that P(x) = x19 201x18 + 201x17 -...- 201x2 + 201x.

Calculate P(200).

a. 1 b. 201 c. 200 d. 0

40. For the real numbers a, b and c, it is known that , and a + b + c = 1. Find

the value of the expression, M = 1 1 1

1 1 1a ab b bc c ca .

a. 0 b. 1 c. 2 d. 3

41. Find the infinite sum: ...1 1 2 3 5 8 13 21

4 8 16 32 64 128 256 512

a. 2/3 b. 3/2 c. 1 d. 5/4

42. The houses in a street are spaced so that each house of one lane is directly opposite to a house ofother lane. The houses are numbered 1, 2, 3, and so on up one side, continuing the order back

down the other side. Number 39 is opposite to 66. How many houses are there?

a. 103 b. 102 c. 104 d. 105

43. One number is removed from the set of integers from 1 to n. The average of the remaining

numbers is 40.75. Which of the following is true about the removed number?

a. The number lies between 50 and 60

b. The number lies between 60 and 70

c. The number lies between 100 and 110

d. The number lies between 20 and 30

44. If f(x)=x3+x2-x+19 has a relative minimum at x=a and the relative maximum at x=b, find ab.

a. -1 b. 3 c. 1/3 d. 1

45. A group of n students is sitting equally spaced around a circular table, and each student is labeled

with a number from 1 through n (corresponding to the clockwise order in which they are sitting). If

student 7 is sitting directly across from student 81, what is the value of n?

a. 147 b. 148 c. 149 d. 150

46. The three roots of f(x)=3x3-10x2+Kx-8 are a, b and c. If c is integer and a+b=ab, find the value of

ab.a. 4/3 b. 8/3 c. 2 d. Cannot be determined

1 1 11

ab bc ca


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47. A single-elimination tournament is played with 128 teams. In such a tournament, all undefeated

teams are paired together, and only the winning teams advance to the next round. This process is

repeated until only one undefeated team remains. Which of the following expressions correctly

represents the total number of games played in such a tournament?

a. 127 b. 128 c. 64 d. 63

48. Let f(x) be a cubic polynomial with integral coefficients. If f(1)=-1, f(2)=f(3)=-5, f(4)=5, andf(5)=31, find the value of f(6)?

a. 48 b. 67 c. 79 d. 81

49. If 2

3x 31 A B

x 2x 15 x 3 x 5

, find AB.

a. -93 b. -15 c. -10 d. 93

50. If 6 2 5 A B , find A + AB + B.

a. 5 b. 6 c. 11 d. 6+145

51. Given a cubic function, M, such that when M(x) = 0, the solution set is {0, 2, -3}. Find the

solution set when M (x + 5) = 0.

a. {-5} b. {-3, -2, 1} c. {-3, -2, 0} d. {-8, -5, -3}

Direction 52 -54: {x}=x-[x] where [x] denotes the greatest integer function.

52. Find the value of {} +{-}

a. 6-2 b. 0 c. 2 -6 d. 1

53. Find the inequality that gives the range of {x}

a. -1 {x} 1 b. -1a

56. ?565656

a. -8 b. 7 c. 15/2 d. 8

57. Find the sum of the solutions to 103log5log 293 xxxa. -5 b. 0 c. 2 d. None of these

58. Find the sum of the real roots of .1577 33 2 xxxy

a. 7 b. 15 c. 151 d. 153

59. The minimum value of the expression (x+y)(y+z), where x, y and z are positive real numbers

satisfying xyz(x + y + z) = 1.a. 1 b. 2 c. 0 d. 4


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60. The roots of f(x) = x3-15x2+71x-105 form an arithmetic progression. What is the common

difference for this progression?

a. 0.5 b. 1 c. 1.5 d. 2

61. If f(x) is a quadratic function with f(2)=4, f(-3)=-1and f(-4)=5, evaluate f(0).

a. -23 b. -17 c. -13 d. -5

62. A function f from the integers to the integers is defined as follows: f(n) = n + 3 if n is odd

if n is even

Suppose k is odd and f(f(f(k))) = 27. What is the sum of the digits of k?

a. 3 b. 6 c. 9 d. 12

63. What is the smallest value of the expression x2 + 2xy + 2y2 10y + 3?

a. -22 b. -14 c. -12 d. 3

64. )))).4(((,

1,1

13,5

3,52

)(2

ffffevaluate

xifx

xif

xifx

xf

a. 1 b. 5 c. 24 d. 575

65. If f(3x-5) = x4- x3 -3x +7, then what is the sum of the coefficients of f(x)?

a. 4 b. 6 c. 9 d. 25

66. If f(4x+8) = 2x3- 2x2 -3x +5, then what is the constant term of f(x)?

a. -13 b. 5 c. 6 d. 23

67. The number of common terms of the two sequences 17, 21, 25, ., 417 and 16, 21, 26, 466.a. 21 b. 29 c. 20 d. 2

68. The equation 1 13 5 34x x has

a. no solution b. one solution c. two solutions d. more than two solution

69. Let a, b, and c be positive real numbers. Determine the largest total number of real roots that the

following three polynomials may have among them: ax2 + bx + c, bx2 + cx + a, and cx2 + ax + b.

a. 2 b. 4 c. 6 d. 0

70. 2 112

F nF n for n = 1, 2, 3 and F(1) = 2. Then F(101) equals:


71. Find the quadratic equation whose roots are half of the reciprocal of the roots of the equation2 0.ax bx c

a. 24 2 0ax bx c

b. 24 2 0cx bx a

c. 22 0cx bx a

d.

2

2 0ax bx a


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72. The number of ordered pairs of integers (x, y) satisfying the equation 2 26 4x x y is

a. 2 b. 4 c. 6 d. 8

73. The number of points at which the curve 6 3 2y x x cuts the x-axis is

a. 1 b. 2 c. 4 d. 6

74. 2 2 2, , ,p qy rz q rz px r px qy then the value of

x y z

p x q y r z

is

a. pqr b. pqr

xyzc. 0 d. 1

75. If the minimum value of 24 3x px is realized for x = a, where a > 0, which of the following

statements is necessarily true?

a. 8a + p < 0 b. 6a + p > 0 c. 11a + 2p 0 d. 15a + 2p


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84. If the domain for the function f(x) =cxx 2

12

is (-,), which of the following best describes all

possible values of c ?

a. c>1 b. c=1 c. c1 d. c0,

find the minimum value of |f(x)|

a. 1/2 b. 4/15 c. 7/15 d. 3/5

88. Find the number of elements of the set A = {2, 3, 6, 7, 10, 11, 14, 15, . , 2006, 2007, 2010,2011}.

a. 1004 b. 1006 c. 1008 d. 1010

89. This equation describes the relationship between two quadratic functions g(x) = 2f(x)+2. Which ofthe following could represent the graphs of the two functions?

a. b. c. d.

90. The minimum value of the quantity (a2+3a+1)(b2+3b+1)(c2+3c+1)/abc is:

a. 113/23 b. 125 c. 25 d. 27

91. Among following options which one is factor of x2 - y2 - z2 + 2yz + x + y z

a. x y + z + 1 b. x + y + z c. x + y z + 1 d. x y z + 1

92. Find the coefficient of x.y.z4 in the expansion of (x+3y+2z)6 with like terms combined

a. 160 b. 480 c. 1440 d. 4320

93. Find the value ofabcdabcdabcdabcd dabcdabcdabc loglogloglog

1111

a. 0 b. 1 c. 3 d. 4


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94. If x , y and z are positive real numbers and x3+y3+z3 = 4, what is the maximum value of(1+x3)(1+y3)(1+z3)-(yz)3-(xz)3-(xy)3

a. 199/27 b. 32 c. 125 d. 64

95. Find the area of the region enclosed by 2( ) 64f x x and ( ) 0g x

a. 4 b.16 c.32 d. 64

96. The roots of f(x) = x3-15x2+71x-105 form an arithmetic progression. What is the commondifference for this progression?

a. 1/2 b. 1 c. 3/2 d. 2

97. If ;)(24

4

x

x

xf find the value of

1999

1998

1999

3

1999

2

1999

1ffff ......

a. 1998 b. 1999 c. 998 d. 999

98. How many integral points are within the graph of |x-2|+|y+6|8

a. 144 b. 145 c. 180 d. 181

99. What is the coefficient of the x3. 4 term in the expansion of (2 +3)9?

a. 38,420 b. 64,560 c. 76,840 d. 90,720

100. Given f(n) = f(n + 1)f(n 1), and f(1) = 1; f(2) = 2, find the summation of the series

2013321 2222 ffff log......logloglog

a. 1 b. 2 c. 2012 d. 2013

101. Find the 100th term of the series 21,32,54,87,1211,1716,2322,....

a. 49004899 b. 49524951 c. 52205219 d. 52105209

102. Find the sum of the first 125 terms of the series 1,2,1,3,2,1,4,3,2,1,5,4,3,2,1,6,5,4,3,2,1,.......

a. 460 b. 738 c. 750 d. 680

103. If a1=2,a2=3 and an+2+an=2an+1+1,for every positive integer n, then a51 is

a. 1277 b. 1276 c. 1127 d. 1126

104. Let {sn} be a unique sequence of positive numbers satisfying the following properties:s1 = 64800 = 30s2 and sn sn+1= sn-1. Which of the following numbers is a perfect square?

a. 36S3 b. 50/S7 c. 12S4 d. 5S5

105. p is the root of the equation x2 + ax + b = 0, 0 < b a 1. Which of the following bestdescribes p?

a. p|1

b. |p| 3/2

c. |p| 4/3

d. 1/2|p|3/2


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Logarithm Practice Problem

1. Find the zero(s) of the graph ofy=2log2(x+3).

a. 3

b. 22c. 6

d. 9

2. 3x+1 + 22y-1 = 144. Find the sum x+y.

a. 7/2

b. 5/2

c. 2

d. 1

3. If logbc=d then 3logcb+2 equals:

a. 3d+2

b. D3+2

c.2

3

d

d

d.d

d23

4. Solve for x if log2x=3-log2(x+2)

a. 1

b. 2

c. 3

d. 4

5. 2log3x logx2 +log(x+1) when simplified

equals log w. Find w.

a.x

x

2

)1(3

b. 1+7x-x2

c.2

)1(9

x

x

d.1

9

x

6. Find the value of log28- log54+ log2516-

log2(1/16)

a. 16

b. 4

c. -4

d. -5

7. The sum of solutions to: 9x-2(3x+1)+8=0 is

a. log32

b. log34

c. log36d. log38

8. Find the real values of log2X+ log4X+

log16X=7

a. 2

b. 4c. 8

d. 16

9. Which of the following are equivalent?

I. y=43x-2

II. y=2(23x-2)

III. y=23x-1

a. I,II

b. I,III

c. II,III

d. I,II,III

10. y=(log23)( log34).. (log3132) then

a. 4


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15. 128(x+1) divided by 4(2x+3) equals 32x, find the

value of (x/2).

a. 2

b. 17/6

c. -13/6

d. -13/4

16. Find the greatest value of x that solves

xx 9log9log =10/3

a. 27

b. 8

c. 729

d. 33

17. Find the value of 32log25log 52 -

4log2

a. -3

b. -11

c. 0d. 2

18. Find the solution to 3x.92x-1= 3(1/2)81-1

a. -3/10

b. -1/2

c. 1/6

d. 2/5

19. Find the value of x that satisfies x3log =(-

2+ 100log2 )( 2log3 )

a. 210log3b. 20

c. -3

d. 5

20. axlog =2, bxlog =-3, cxlog =5. Find the

value of )(log2

3

c

acabx

a. 18/5

b. -13.5

c. -2710d. 54/25

21. If x10log - y10log =1 and x y10log =100,

find 2x-3y.

a. 160

b. 170

c. 180

d. 200

22. 43m-1=1+82(k+3) find mk.

a. 27

b. 21c. 1/8

d. 3

23. Solve:

)2(log5 x = 2(3 2log5 - 1.5* )2(log5 x )a. -2 + 22

b. 22

c. 2+22

d. 2

24. If xxxaaa

42 logloglog =c fid x in terms

of c.

a. 7c/4

b. 47c

a

c. ca 74

d. 47

c

Keys:

1 b 16 c

2 a 17 b

3 d 18 a

4 b 19 d

5 d 20 b

6 d 21 b

7 d 22 c

8 d 23 c

9 c 24 b

10 b

11 d

12 a

13 c

14 b

15 d

Documents

Algebra Workshop 2013 Student Copy