david-essner-preliminary-2003
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Rumson Fair Haven Reg H *
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Course
101
Subject
Mathematics
Date
Nov 24, 2024
Type
Pages
4
Uploaded by CoachRiverTiger30
David Essner Exam 22 2002-2003 1
. If one class of 30 students averaged 80 on an exam, a second class of 40 students averaged 60 on the exam and a third class of 20 students averaged 50 on the exam then the combined average of all 3 classes is nearest the integer (a) 62 (b) 63 (c) 64 (d) 65 (e) 66 2.
Given that a
*
b
= b
a
a
+
then x
*(
x
*
x
) equals (a) 1
2
2
+
x
x
(b) 1
2
+
x
x
(c) 1
+
x
x
(d) 1
2
+
x
x
(e) 2
2
+
x
x
3
. Given that the values b,c are among the set of integers {1,2,3,4,5,6} then there are how many equations of the form x
2
+ bx
+ c
= 0 such that all roots are real and rational?. (a) 4 (b) 7 (c) 8 (d) 10 (e) 12 4
. In the game of basketball John made 90% of his free throws and Bill made 80% of his free throws. If they shot the same number of free throws and John missed x
free throws then Bill missed how many free throws? (a) 9x/8 (b) 8
x
/9 (c) 10
x
(d) 2
x
(e) x
+ 10 5
. A person invests $1,000 at a fixed rate of interest compounded 4 times per year. If after 5 years the value of the investment is $1,500, then after 10 years the value of the investment is (a) $2,000 (b) $2,125 (c)$2,250 (d) $2,375 (e) cannot be determined from the given information 6
. Each year one of the three schools Central, Western and Northeastern is equally likely to be selected to host a math competition. What is the probability that over a three year period each of the three schools is selected exactly once? (a) 1/3 (b) 4/27 (c) 2/9 (d) 5/27 (e) 1/2 7
. Given a triangle whose sides are of length 3,4,5, if h
is the length of the altitude to the longest side then h
equals (a) 5/2 (b) 8/3 (c) 9/4 (d) 7/3 (e) 12/5 8
. For how many positive integers n
are n
, n
+ 2 and n
+ 4 all prime numbers? (a) none (b) 1 (c) 2 (d) more than 2 but a finite number (e) an infinite number 9
. If P
(
x
) = ax
3
+ bx
2
+ cx
+ d
is a real number polynomial function, P
(1) = P
(2) = P
(-1) = 0 and P
(-2) = -24 then P
(3) = (a) 16 (b) 12 (c) 8 (d) 48 (e) 72
10
. For each non-empty subset T
of {1,2,3,4,5} let S
T
be the sum of all numbers in T
. The sum of all S
T
is (a) 180 (b) 190 (c) 200 (d) 220 (e) 240 11
Between years 1990 and 2000 at a certain university the number of boys increased by 10%, the number of girls by 40% and the total number of students by 30%. The ratio of boys to girls in 1990 was (a) 2 to 3 (b) 1 to 3 (c) 1 to 4 (d) 3 to 4 (e) 1 to 2 12
. For which values of x
does the parabola y
= 5
x
2
+ x
– 3 lie above the parabola y = 2x
2
+ 6x – 1? (a) x
< -1/2 or x
> 7/2 (b) x
< -1/4 or x > 5/2 (c) x
< -2/3 or x
> 3 (d) x
< -1/3 or x
> 2 (e) x
< -3/5 or x
> 7/3 13
. If n
is the smallest integer such that 616
n
is a perfect square, then the sum of the digits of n
is (a) 8 (b) 10 (c) 13 (d) 17 (e) 25 14
. Measured by weight a given salt solution of 100 pounds is 90% water. If after evaporation the solution is by weight 60% water then the weight of the remaining solution in pounds is (a) 70 (b) 54 (c) 40 (d) 36 (e) 25 15. If m,n
are positive integers and m
+ n
2
= 2
24
41
+
then m + n
= (a) 3 (b) 5 (c) 7 (d) 9 (e) 11 16
. There are how many different (non-congruent) triangles with sides of integer length and perimeter 16? (a) 3 (b) 4 (c) 5 (d) 6 (e) 7 17
. A system of equations ax
+ by
= c
and dx
+ ey
= f
has solution x
= 2, y
= 1 when c
= 6 and f
= 8, and has solution x
= 1, y
= 2 when c
= 6 and f
= 4. If c
= 8 and f = 12 then x + y equals (a) 2 (b) 3 (c) 4 (d) 5 (e) 6 18
. Tom drives from town A
to town B
in 6 hours and Bill drives from town B
to town A
in 8 hours. If they both start at the same time and drive at a constant rate , then what is the number of hours after the starting time until they meet? (a) 7/2 (b) 10/3 (c) 16/5 (d) 19/5 (e) 24/7 19
. If a,b
,
c
are positive real numbers and log
4
a
=log
6
b
= log
9
(
a
+ b
) then b/a
= (a) 3/2 (b) 2/3 (c) (
3 - 1)/2 (d) ( 1 + 5 )/2 (e) 15
/2
20
. Circle C
1
has radius 2 and Circle C
2
has radius 3, and the distance between the centers of C
1
and C
2
is 7. If two lines, one tangent to both circles and the other passing through the center of both circles, intersect at a point P
which lies between the centers of C
1
and C
2
, then the distance between P
and the center of C
1
is (a) 9/4 (b) 7/3 (c) 8/3 (d) 13/5 (e) 5/2 21
. In a league of 8 teams each team played each other team 10 times. The number of wins of the 8 teams formed an arithmetic sequence. What is the least possible number of games won by the champion? (a) 42 (b) 45 (c) 48 (d) 50 (e) 54 22
. In the coordinate plane the point (
a
,0) has distance 2 from the line y
= 2
x
; if a
> o then a
equals (a) 5/2 (b) 7/2 (c) 6
(d) 2
2
(e) 5
23
. For what value of r
is the line through the points (2,0) and (0,4) tangent to the circle x
2
+ y
2
= r
2
? (a) 2 (b) 5/2 (c) 4/
5 (d) 1 + 5 (e) 7
/2 24
. Given that A
= 2
5/8
, B
= 3
1/3
and
C
= 4
1/4
then (a) A
> B
> C (b) A
> C
> B
(c) C
> B
> A
(d) C
> A
> B
(e) B > A > C 25
. If 0 < x
< .01 then 2
2
1
2
1
2
−
−
+
x
x
is (a)
between 0 and 1 (b) between 1 and 2 ( c) between 2 and 1,000 (d) greater than 1000 (e) less than 0 26
. Let A,B,C
be vertices of an equilateral triangle, and let D,E
be points on the side AB
such that segments AD, DE
, and EB
each have length 1. Then tan ∠
CDE
equals (a) 3 (b) 3
2
(c) 2
3 (d) 3
3
(e) 3
3
/2 27
. By a ≡ b
mod c
is meant that (b - a) is divisible by c. If 41
≡
n mod 72 and k ≡
n
mod 18, where 0 ≤
k < 18, then k
equals (a) 13 (b) 11 (c) 9 (d) 8 (e) 5 28
. If 7
100
is divided by 100 then the remainder is (a) 1 (b) 7 (c) 14 (d) 43 (e) 49
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29.
Given a regular decagon (10 sided polygon), there are how many diagonals (lines joining vertices and lying inside the decagon)? (a) 30 (b) 35 (c) 40 (d) 45 (e) 90 30
. If m,n are integers and 2m – n = 5 then m – 3n (a)
can be any integer (b) is a multiple of 3 (c) is an even integer (d) is a multiple of 5 (e) is none of (a)-(d)