1
Sample question. (20) Simplify:
27
4
18
4
Ans: 3
2
V1.1 (40) Find the triple (x, y, z) so that
x + 2y - z = 4
x - 4y + z = 2
x +2y
=0
Ans: (2,-1,-4)
V1.2. (30) The universe is {1, 2, 3, 4, 5, 6, 7, 8, 9}. A = {1, 2, 3, 4, 5}. B = {2, 4, 6, 8}.
Find: ( A ' B ') '
Ans: {2,4}
V1.3 (30) We define a binary operation by
x * y  2 x  y . Solve: 4*(x*4) = 6.
V1.4 (30) Give the equations of all asymptotes to the graph: y 
Ans: x = 3
2 x2  5
x2  x  2
Ans: x = -2, x = 1, y = 2
V1.5 (15) Give, in interval notation, the range of the function f ( x)  x3  3x  5
Ans: (, )
V1.6 (40) Find the coefficient of a 2 b 2 in the expansion of (a  b)4
Ans: 6
V1.7. (30) Find the y-coordinate of the vertex of the parabola y  x 2  4 x  2
Ans: -2
V1.8 (30) f ( x)  2 x  3 g ( x)  x  1
Ans: 4x +3
V1.9 (30) Calculate

k  20
k 1
Find: f ( g ( f ( g ( x))))
(2k  1)
Ans: 400
V1.10 (30) (-2,0) and (2,0) are the end points of the hypotenuse of a right triangle. Give, in
standard form, the equation of the locus of points for the third vertex.: Ans: x 2  y 2  4
V1.11 (30) I roll two dice. What is the probability their product is 4?
Ans: 1/12
V1.12 (45) Find all solutions in [0,  ) of tan  cot 
Ans:
 3
4
,
4
2
V2.1. (40) How many different ways are there to arrange the letters {a, b, c, d, e} in order so
there are no two consecutive consonants. Ans: 12
V2.2. (40) Solve for x: (
2x 2
2x
)  2(
) 1  0
3x  2
3x  2
V2.3 (30) What is the range of the function
V2.4 (30) Solve for x:
V2.5 (30) Simplify
x  3  x  10
Ans: 2
f ( x)  x 4  4 x 2 ?
Ans: [4, )
 7
Ans: 19
log 2 27
log 2 9
Ans: 3/2
V2.6 (40) If (3a  2b)4 is multiplied out, what is the coefficient of a 2 b 2 ?
Ans: 216
V2.7 (30) Solve for x: log3 ( x  1)  log3 ( x  1)  3
Ans: 14/13
V2.8 (40) cos 36o 
5 1
. Write sin180 in radical form.
4
x2
V2.9 (20) What is the area inside the ellipse:  y 2  1 ?
4
Ans:
5 1
or
4
3 5
8
Ans 2
V2.10 (30) If I roll two dice, what is the probability the product of the numbers is even?
Ans: 3/4
V2.11 (30) sin   3 / 5,  / 2     . What is tan 2 ?
Ans -24/7
V2.12. (30) Simplify 4log2 7
Ans: 49
3
V3.1 (40) The numbers x, y, and z are in ratio 2 to 3 to 4. Their product is 3. What is x?
Ans: 1
V3.2 (40) If we multiply out the polynomial (x-1)(x-2)(x-3)(x-4) then what is the coefficient of
x3? Ans: -10
V3.3 (40)
2x  1
x  3x  2
2
2
V3.4 (20) an  an 1
3

A
x2

B
. What is A?
x 1
a0  8 . What is a3 ?
Ans: 3
Ans: 27
V3.5 (40) If i 2  1 then find in simplified form (1  i)8
Ans 16
V3.6 (30) Find the determinant:
2 1 1
1 4 2
Ans: 0
6 3 3
V3.7 (30) Perform the base five division: 4004 five /13 five  ???? five
V3.8. (30) Find the sum of the binomial coefficients:
Ans: 223 or 223 five
 4  4  4  4  4
     
 0  1   2   3   4 
V3.9. (30) Solve, giving your answer in interval form, | |x - 1| - 2| < 3
Ans: 16
Ans: (-4,6)
V3.10. (40) I draw 3 cards at random from an ordinary deck. What is the probability (in lowest
terms) that all three are red? Ans: 2/17
V3.11 (30) A triangle has angles 300 , 450 , and 1050 . The second longest side has length
What is the length of the shortest side?
Ans: 1
x10
x  2 x
V3.12. (20) Find the limit: lim
Ans: 0
2.
4
V4.1 (30) f ( x)  3x  8
g ( x)  2 x  1
Find
f 1 ( g 1 (3))
Ans: -2
V4.2. (30) If we multiply out the polynomial (x+1)(x+2)(x+3)(x+4) then what is the coefficient
of x? Ans: 50
V4.3 (40) Two sides of a triangle have lengths 2 and 3. The angle between them is  / 3 . What
is the length of the third side?
Ans 7
V4.4. (40)
b? Ans: 1
3  2 2 can be written in the form a  b 2 where a and b are integers. What is
V4.5 (30) i 2  1 . Write ( 3  i ) 6 in simplest form
Ans: -64
 4  4  4  4  4
V4.6 (30) Find:              
 0  1   2   3   4 
Ans: 0
V4.7 (40) Give, in interval notation, the range of the function f ( )  sin   cos  ?
Ans [ 2, 2]
V4.8 (30) Three of the roots of f ( x)  x 4  ax3  bx 2  cx  12 are 1, -1, and 2. What is the
fourth root? Ans: -6
1 1 
V4.9 (40) Find the inverse matrix of: 

1 2 
 2  1
Ans: 

 1 1 
V4.10. (30) Give the mathematicians A , B, C, and D in correct historical order.
A. Hilbert
B. Gauss
C. Euclid
D. Fermat
Ans: C D B A
V4.11 (40) f ( x ) 
3x  2
x 1
What is f '(2) ?
Ans: -5
V4.12 (30) What is the name for the set of functions which, for all a, satisfy, lim f ( x)  f (a) ?
x a
Ans: continuous or continuous functions or everywhere continuous functions