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Math Level 2 SAT Subject Test

Richard Ku, Howard P. Dodge

Chapter 2

Model Test

Educators


Problem 1

The slope of a line perpendicular to the line whose equation is $\frac{x}{3}-\frac{y}{4}=1$ is
(A) $-3$
(B) $^{-\frac{4}{3}}$
(C) $^{-\frac{3}{4}}$
(D) $\frac{1}{4}$
(E) $\frac{4}{3}$

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Problem 2

What is the range of the data set $8,12,12,15,18 ?$
(A) 10
(B) 12
(C) 13
(D) 15
(E) 18

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Problem 3

If $f(x)=\frac{x-7}{x^{2}-49}$ , for what value(s) of $x$ does the graph of $y$
$=f(x)$ have a vertical asymptote?
(A) $-7$
(B) 0
(C) $-7,0,7$
(D) $-7,7$
(E) $-7,7$
(E) 7

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Problem 4

If $f(x)=\sqrt{2 x+3}$ and $g(x)=x^{2}+1,$ then $f(g(2))=$
(A) 2.24
(B) 3.00
(C) 3.61
(D) 6.00
(E) 6.16

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Problem 5

$\left(-\frac{1}{16}\right)^{2 / 3}=$
(A) $-0.25$
(B) $-0.16$
(C) 0.16
(D) 6.35
(E) The value is not a real number.

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Problem 6

The circumference of circle $x^{2}+y^{2}-10 y-36=0$ is
(A) 38
(B) 49
(C) 54
(D) 125
(E) 192

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Problem 7

Twenty-five percent of a group of unrelated students are
only children. The students are asked one at a time
whether they are only children. What is the probability
that the 5 th student asked is the first only child?
(A) 0.08098
(B) 0.08
(C) 0.24
(D) 0.25
(E) 0.50

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Problem 8

If $f(x)=2$ for all real numbers $x,$ then $f(x+2)=$
(A) 0
(B) 2
(C) 4
(D) $x$
(E) The value cannot be determined.

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Problem 9

The volume of the region between two concentric spheres of radii 2 and 5 is
(A) 28
(B) 66
(C) 113
(D) 368
(E) 490

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Problem 10

If $a, b,$ and $c$ are real numbers and if $^{5} b^{3} c^{8}=\frac{9 a^{3} c^{8}}{b^{-3}}$
then $a$ could equal
(A) $\frac{1}{9}$
(B) $\frac{1}{3}$
(B) 3
(D) 3
(E) 9$b^{6}$

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Problem 11

In right triangle $A B C, A B=10, B C=8, A C=6 .$ The sine
of $\angle A$ is
(A) $\frac{3}{5}$
(B) $\frac{3}{4}$
(B) $\frac{5}{4}$
(E) $\frac{5}{4}$
(E) $\frac{4}{3}$

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Problem 12

If $16^{x}=4$ and $5^{x+y}=625,$ then $y=$
(A) 1
(B) 2
(C) $\frac{7}{2}$
(D) 5
(E) $\frac{25}{2}$

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Problem 13

If the parameter is eliminated from the equations $x=t^{2}+$
1 and $y=2 t,$ then the relation between $x$ and $y$ is
(A) $y=x-1$
(B) $y=1-x$
(B) $y^{2}=x-1$
(D) $y^{2}=(x-1)^{2}$
(E) $y^{2}=4 x-4$

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Problem 14

Let $f(x)$ be a polynomial function: $f(x)=x^{5}+\cdots .$ If $f(1)$
$=0$ and $f(2)=0,$ then $f(x)$ is divisible by
(A) $x-3$
(B) $x^{2}-2$
(C) $x^{2}+2$
(D) $x^{2}-3 x+2$
(E) $x^{2}+3 x+2$

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Problem 15

If $x-y=2, y-z=4,$ and $x-y-z=-3,$ then $y=$
(A) 1
(B) 5
(C) 9
(D) 11
(E) 13

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Problem 16

If $z>0, a=z \cos \theta,$ and $b=z \sin \theta,$ then $\sqrt{a^{2}+b^{2}}=$
(A) 1
(B) $z$
(C) 2$z$
(D) $z \cos \theta \sin \theta$
(E) $z(\cos \theta+\sin \theta$

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Problem 17

If the vertices of a triangle are $(u, 0),(v, 8),$ and $(0,0),$
then the area of the triangle is
(A) 4$|u|$
(B) 2$|v|$
(C) $|u v|$
(D) 2$|u v|$
(E) $\frac{1}{2}|u v|$

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Problem 18

If $f(x)=\left\{\begin{array}{cl}{\frac{5}{x-2},} & {\text { when } x \neq 2} \\ {k,} & {\text { when } x=2 \text { what must the value of } k \text { be }}\end{array}\right.$ in order for $f(x)$ to be a continuous function?
(A) $-2$
(B) 0
(C) 2
(D) 5
(E) No value of $k$ will make $f(x)$ a continuous function.

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Problem 19

What is the probability that a prime number is less than
$7,$ given that it it less than 13$?$
(A) $\frac{1}{3}$
(B) $\frac{2}{5}$
(C) $\frac{1}{2}$
(D) $\frac{3}{5}$
(E)$\frac{3}{4}}$

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Problem 20

The ellipse $4 x^{2}+8 y^{2}=64$ and the circle $x^{2}+y^{2}=9$
intersect at points where the $y$ -coordinate is
(A) $\pm \sqrt{2}$
(B) $\pm \sqrt{5}$
(C) $\pm \sqrt{6}$
(D) $\pm \sqrt{6}$
(E) $\pm \sqrt{7}$
(E) $\pm 10.00$

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Problem 21

Each term of a sequence, after the first, is inversely proportional to the term preceding it. If the first two terms are 2 and $6,$ what is the twelfth term?
(A) 2
(B) 2
(C) 46
(D) 46
(D) 2$\cdot 3^{11}$
(E) The twelfth term cannot be determined.

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Problem 22

A company offers you the use of its computer for a fee.
Plan A costs $\$ 6$ to join and then $\$ 9$ per hour to use the computer. Plan $\mathrm{B}$ costs $\$ 25$ to join and then $\$ 2.25$ per hour to use the computer. After how many minutes of use would the cost of plan A be the same as the cost of plan B?
(A) $18,052$
(B) 173
(C) 169
(D) 165
(E) 157

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Problem 23

If the probability that the Giants will win the NFC championship is $p$ and if the probability that the Raiders will win the AFC championship is $q$ , what is the
probability that only one of these teams will win its respective championship?
(A) $p q$
(B) $p+q-2 p q$
(C) $|p-q|$
(D) $1-p q$
(E) $2 p q-p-q$

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Problem 24

If a geometric sequence begins with the terms $\frac{1}{3}, 1, \cdots,$ what is the sum of the first 10 terms?
(A) $9841^{\frac{1}{3}}$
(B) 6561
(C) 3281
(D) $3280^{\frac{1}{3}}$
(D) 333
(E) 6

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Problem 25

The value of $\frac{453 !}{450 ! 3 !}$ is
(A) greater than $10^{100}$
(B) between $10^{10}$ and $10^{100}$
(C) between $10^{5}$ and $10^{10}$
(D) between 10 and $10^{5}$
(E) less than 10

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Problem 26

If $A$ is the angle formed by the line $2 y=3 x+7$ and the $x$
-axis, then $\angle A$ equals
(A) $-45^{\circ}$
(B) $0^{\circ}$
(C) $56^{\circ}$
(D) $72^{\circ}$
(E) $215^{\circ}$

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Problem 27

A U.S. dollar equals 0.716 European euros, and a
Japanese yen equals 0.00776 European euros. How
many U.S. dollars equal a Japanese yen?
(A) 0.0056
(B) 0.011
(C) 0.71
(D) 94.2
(E) 179.98

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Problem 28

If $(x-4)^{2}+4(y-3)^{2}=16$ is graphed, the sum of the
distances from any fixed point on the curve to the two
foci is
(A) 4
(B) 4
(B) 8
(D) 12
(D) 16
(E) 32

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Problem 29

In the equation $x^{2}+k x+54=0,$ one root is twice the
other root. The value(s) of $k$ is (are)
(A) $-5.2$
(B) 15.6
(C) 22.0
(D) $\pm 5.2$
(E) $\pm 15.6$

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Problem 30

The remainder obtained when $3 x^{4}+7 x^{3}+8 x^{2}-2 x-3$ is
divided by $x+1$ is
(A) $-3$
(B) 0
(C) 3
(D) 5
(E) 13

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Problem 31

If $f(x)=e^{x}$ and $g(x)=f(x)+f^{-1}(x),$ what does $g(2)$
equal?
(A) 5.1
(B) 7.4
(C) 7.5
(D) 8.1
(E) 8.3

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Problem 32

If $x_{0}=3$ and $x_{n+1}=\sqrt{4+x_{n}}$ , then $x_{3}=$
(A) 2.65
(B) 2.58
(C) 2.56
(D) 2.55
(E) 2.54

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Problem 33

For what values of $k$ does the graph of
$\frac{(x-2 k)^{2}}{1}-\frac{(y-3 k)^{2}}{3}=1$
(A) only 0
(B) only 1
(C) $\pm 1$
(D) $\pm \sqrt{5}$
(E) no value

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Problem 34

If $\frac{1-\cos \theta}{\sin \theta}=\frac{\sqrt{3}}{3},$ then $\theta=$
(A) $15^{\circ}$
(B) $30^{\circ}$
(C) $45^{\circ}$
(D) $60^{\circ}$
(E) $75^{\circ}$

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Problem 35

If $x^{2}+3 x+2<0$ and $f(x)=x^{2}-3 x+2,$ then
(A) $0<f(x)<6$
(B) $f(x) \geq \frac{3}{2}$
(C) $f(x)>12$
(D) $f(x)>0$
(E) $6<f(x)<12$

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Problem 36

If $f(x)=|x|+[x],$ the value of $f(-2.5)+f(1.5)$ is
(A) $-2$
(B) 1
(C) 1.5
(D) 2
(E) 3

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Problem 37

If $(\sec x)(\tan x)<0,$ which of the following must be
true?
I. $\tan x<0$
II. $\csc x \cot x<0$
III. $x$ is in the third or fourth quadrant
(A) I only
(B) II only
(C) III only
(D) $I I$ and III
(E) $I$ and II

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Problem 38

At the end of a meeting all participants shook hands
with each other. Twenty-eight handshakes were
exchanged. How many people were at the meeting?
(A) 7
(B) 8
(C) 14
(D) 28
(E) 56

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Problem 39

Suppose the graph of $f(x)=2 x^{2}$ is translated 3 units
down and 2 units right. If the resulting graph represents
the graph of $g(x),$ what is the value of $g(-1.2) ?$
(A) $-1.72$
(B) $-0.12$
(C) 2.88
(D) 17.48
(E) 37.28
$$
\begin{array}{|c|c|c|c|c|}\hline x & {-5} & {-3} & {-1} & {1} \\ \hline y & {0} & {4} & {-3} & {0} \\ \hline\end{array}
$$

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Problem 40

Four points on the graph of a polynomial $P$ are shown in
the table above. If $P$ is a polynomial of degree $3,$ then
$P(x)$ could equal
(A) $(x-5)(x-2)(x+1)$
(B) $(x-5)(x+2)(x+1)$
(C) $(x+5)(x-2)(x-1)$
(D) $(x+5)(x+2)(x-1)$
(E) $(x+5)(x+2)(x-1)$
(E) $(x+5)(x+2)(x+1)$

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Problem 41

If $f(x)=a x+b,$ which of the following make(s) $f(x)=f$
1 $f(x)$ ?
I. $a=-1, b=$ any real number
II. $a=1, b=0$
III. $a=$ any real number, $b=0$
(A) only I
(B) only II
(C) only III
(D) only I and II
(E) only I and III

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Problem 42

In the figure above, $\angle A=110^{\circ}, a=\sqrt{6}$ and $b=2 .$ What
is the value of $\angle C ?$
(A) $50^{\circ}$
(B) $25^{\circ}$
(C) $20^{\circ}$
(D) $15^{\circ}$
(E) $10^{\circ}$

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Problem 43

If vector $\vec{v}=(1, \sqrt{3})$ and vector $\vec{u}=(3,-2),$ find the value
of $|3 \vec{v}-\vec{u}|$
(A) 5.4
(B) 6
(C) 7
(D) 7.2
(E) 52

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Problem 44

If $f(x)=\sqrt{x^{2}-1}$ and $^{g(x)=\frac{10}{x+2}}$ , then $g(f(3))=$
(A) 0.2
(B) 1.7
(D) 3.1
(E) 3.5
(E) 8.7

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Problem 45

In $\triangle A B C$ above, $a=2 x, b=3 x+2, c=\sqrt{12},$ and $\angle C=$
$60^{\circ} .$ Find $x$
(A) 0.50
(B) 0.64
(C) 0.77
(D) 1.64
(E) 1.78

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Problem 46

If $\log _{a} 5=x$ and $\log _{a} 7=y,$ then $\log _{a} \sqrt{1.4}=$
(A) $\frac{1}{2} x y$
(B) $\frac{1}{2} x-y$
(C) $\frac{1}{2}(x+y)$
(D) $\frac{1}{2}(y-x)$
(E) $\frac{y}{2 x}$

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Problem 47

If $f(x)=3 x^{2}+4 x+5,$ what must the value of $k$ equal so
that the graph of $f(x-k)$ will be symmetric to the $y-$
axis?
(A) $-4$
(A) $-4$
(B) $-\frac{4}{3}$
(B) $-\frac{2}{3}$
(B) $\frac{2}{3}$
(E) $\frac{4}{3}$

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Problem 48

If $f(x)=\cos x$ and $g(x)=2 x+1,$ which of the following
are even functions?
I. $f(x) \cdot g(x)$
II. $f(x) )$
III. $f(g(x))$
(A) only I
(B) only II
(B) only III
(D) only I and II
(E) only II and III

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Problem 49

A cylinder whose radius is 3 is inscribed in a
sphere of radius $5 .$ What is the difference between the
volume of the sphere and the volume of the cylinder?
(A) 88
(B) 297
(C) 354
(D) 448
(E) 1345

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Problem 50

Under which conditions is $\overline{x-y}$ negative?
(A) $0<y<x$
(B) $x<y<0$
(C) $x<0<y$
(D) $y<x<0$
(E) none of the above

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