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Math Level 2 SAT Subject Test

Richard Ku, Howard P. Dodge

Chapter 1

Diagnostics Test

Educators


Problem 1

A linear function, $f$ has a slope of $-2 . f(1)=2$ and $f(2)$ $=q .$ Find $q$
(A) 0
(B) $\frac{3}{2}$
(C) $\frac{5}{2}$
(D) 3
(E) 4

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Problem 2

A function is said to be even if $f(x)=f(-x) .$ Which of the following is not an even function?
(A) $y=|x|$
(B) $y=\sec x$
(C) $y=\log x^{2}$
(D) $y=x^{2}+\sin x$
(E) $y=3 x^{4}-2 x^{2}+17$

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Problem 3

What is the radius of a sphere, with center at the origin, that passes through point $(2,3,4) ?$
(A) 3
(B) 3.31
(C) 3.32
(D) 5.38
(E) 5.39

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Problem 4

If a point $(x, y)$ is in the second quadrant, which of the following must be true?
I. $x<y$
II. $x+y>0$
III. $\frac{x}{y}<0$
(A) only I
(B) only II
(C) only III
(D) only I and II
(E) only I and III

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Problem 5

If $f(x)=x^{2}-a x,$ then $f(a)=$
(A) $a$
(B) $a^{2}-a$
(C) 0
(D) 1
(E) $a-1$

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Problem 6

The average of your first three test grades is 78 . What grade must you get on your fourth and final test to make your average 80$?$
(A) 80
(B) 82
(C) 84
(D) 86
(E) 88

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Problem 7

$\log _{7} 9=$
(A) 0.89
(B) 0.95
(C) 1.13
(D) 1.21
(E) 7.61

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Problem 8

If $\log _{2} m=x$ and $\log _{2} n=y,$ then $m n=$
(A) $2^{x+y}$
(B) $2^{x y}$
(C) $4^{x y}$
(D) $4^{x+y}$
(E) cannot be determined

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Problem 9

How many integers are there in the solution set of $| x-$ $2 | \leq 5 ?$
(A) 0
(B) 7
(C) 9
(D) 11
(E) an infinite number

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Problem 10

If $f(x)=\sqrt{x^{2}},$ then $f(x)$ can also be expressed as
(A) $x$
(B) $-x$
(C) $\pm x$
(D) $|x|$
(E) $f(x)$ cannot be determined because $x$ is unknown.

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Problem 11

The graph of $\left(x^{2}-1\right) y=x^{2}-4$ has
(A) one horizontal and one vertical asymptote
(B) two vertical but no horizontal asymptotes
(C) one horizontal and two vertical asymptotes
(D) two horizontal and two vertical asymptotes
(E) neither a horizontal nor a vertical asymptote

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Problem 12

$\lim _{x \rightarrow \infty}\left(\frac{3 x^{2}+4 x-5}{6 x^{2}+3 x+1}\right)=$
(A) $-5$
(B) $\frac{1}{5}$
(C) $\frac{1}{2}$
(D) 1
(E) This expression is undefined.

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Problem 13

A linear function has an $x$ -intercept of $\sqrt{3}$ and a $y-$ intercept of $\sqrt{5}$ . The graph of the function has a slope of
(A) $-1.29$
(B) $-0.77$
(C) 0.77
(D) 1.29
(E) 2.24

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Problem 14

If $f(x)=2 x-1,$ find the value of $x$ that makes $f(f(x))$ $\quad=9$
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6

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Problem 15

The plane $2 x+3 y-4 z=5$ intersects the $x$ -axis at $(a, 0,0),$ the $y$ -axis at $(0, b, 0),$ and the $z$ -axis at $(0,0, c) .$ The value of $a+b+c$ is
(A) 1
(B) $\frac{35}{12}$
(C) 5
(D) $\frac{65}{12}$
(E) 9

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Problem 16

Given the set of data $1,1,2,2,2,3,3,4,$ which one of the following statements is true?
(A) mean $\leq$ median $\leq$ mode
(B) median $\leq$ mean $\leq$ mode
(C) median $\leq$ mode $\leq$ mean
(D) mode $\leq$ mean $\leq$ median
(E) mode $\leq$ mean $\leq$ median the median cannot be calculated.

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Problem 17

If $\frac{x-3 y}{x}=7$ , what is the value of $\overline{y} ?$
(A) $^{-\frac{8}{3}}$
(B) $-2$
(C) $^{-\frac{1}{2}}$
(D) $\frac{3}{8}$
(E) 2

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Problem 18

Find all values of $x$ that make $\left(\begin{array}{ccc}{2} & {-1} & {4} \\ {3} & {0} & {5} \\ {4} & {1} & {6}\end{array}\right)=\left(\begin{array}{ll}{x} & {4} \\ {5} & {x}\end{array}\right)$
(A) 0
(B) $\pm 1.43$
(C) $\pm 3$
(D) $\pm 4.47$
(E) 5.34

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Problem 19

Suppose $f(x)=\frac{1}{2} x^{2}-8$ for $-4 \leq x \leq 4,$ then the maximum value of the graph of $|f(x)|$ is
(A) $-8$
(B) 0
(C) 2
(D) 4
(E) 8

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Problem 20

If tan $\quad \theta=\frac{2}{3},$ then $\sin \theta=$
(A) $\pm 0.55$
(B) $\pm 0.4$
(C) 0.55
(D) 0.83
(E) 0.89

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Problem 21

If $a$ and $b$ are the domain of a function and $f(b) < f(a),$ which of the following must be true?
(A) $a < b$
(B) $b < a$
(C) $a=b$
(D) $a \neq b$
(E) $a=0$ or $b=0$

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Problem 22

Which of the following is perpendicular to the line $y$ $\quad=-3 x+7 ?$
(A) $y=\frac{1}{-3 x+7}$
(B) $y=7 x-3$
(C) $^{y}=\frac{1}{3} x+5$
(D) $^{y}=-\frac{1}{3} x+7$
(E) $y=3 x-7$

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Problem 23

The statistics below provide a summary of IQ scores of 100 children.
Mean: 100
Median: 102
Standard Deviation: 10
First Quartile: 84
Third Quartile: 110
About 50 of the children in this sample have IQ scores that are
(A) less than 84
(B) less than 110
(C) between 84 and 110
(D) between 64 and 130
(E) more than 100

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Problem 24

If $f(x)=\frac{1}{\sec x}$ then
(A) $f(x)=f(-x)$
(B) $f\left(\frac{1}{x}\right)=-f(x)$
(C) $f(-x)=-f(x)$
(D) $f(x)=f\left(\frac{1}{x}\right)$
(E) $f(x)=\frac{1}{f(x)}$

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Problem 25

The polar coordinates of a point $P$ are $\left(2,240^{\circ}\right) .$ The Cartesian (rectangular) coordinates of $P$ are
(A) $(-1,-\sqrt{3})$
(B) $(-1, \sqrt{3})$
(C) $(-\sqrt{3},-1)$
(D) $(-\sqrt{3}, 1)$
(E) $(1,-\sqrt{3})$

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Problem 26

The height of a cone is equal to the radius of its base. The radius of a sphere is equal to the radius of the base of the cone. The ratio of the volume of the cone to the volume of the sphere is
(A) $\frac{1}{12}$
(B) $\frac{1}{4}$
(C) $\frac{1}{3}$
(D) $\frac{1}{3}$

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Problem 27

In how many distinguishable ways can the seven letters in the word MINIMUM be arranged, if all the letters are used each time?
(A) 7
(B) 42
(C) 420
(D) 840
(E) 5040

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Problem 28

Which of the following lines are asymptotes of the graph of $^{y=\frac{x}{x+1}}$
I. $x=1$
II. $x=-1$
III. $y=1$
(A) I only
(B) II only
(C) III only
(D) I and II
(E) II and III

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Problem 29

What is the probability of getting at least three heads when flipping four coins?
(A) $\frac{3}{16}$
(B) $\frac{1}{4}$
(C) $\frac{5}{16}$
(D) $\frac{7}{16}$
(E) $\frac{3}{4}$

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Problem 30

The positive zero of $y=3 x^{2}-4 x-5$ is, to the nearest tenth, equal to
(A) 0.8
(B) $0.7+1.1 i$
(C) 0.7
(D) 2.1
(E) 2.2

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Problem 31

In the figure above, $S$ is the set of all points in the shaded region. Which of the following represents the set consisting of all points $(2 x, y),$ where $(x, y)$ is a point in $S ?$
Graph cannot copy

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Problem 32

If a square prism is inscribed in a right circular cylinder of radius 3 and height $6,$ the volume inside the cylinder but outside the prism is
(A) 2.14
(B) 3.14
(C) 61.6
(D) 115.6
(E) 169.6

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Problem 33

What is the length of the major axis of the ellipse whose equation is $10 x^{2}+20 y^{2}=200 ?$
(A) 3.16
(B) 4.47
(C) 6.32
(D) 8.94
(E) 14.14

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Problem 34

The fifth term of an arithmetic sequence is $26,$ and the eighth term is $41 .$ What is the first term?
(A) 3
(B) 4
(C) 5
(D) 6
(E) 7

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Problem 35

What is the measure of one of the larger angles of the parallelogram that has vertices at $(-2,-2),$ $(0,1),(5,1),$ and $(3,-2) ?$
(A) $117.2^{\circ}$
(B) $123.7^{\circ}$
(C) $124.9^{\circ}$
(D) $125.3^{\circ}$
(E) $131.0^{\circ}$

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Problem 36

If $f(x)=\frac{k}{x}$ for all nonzero real numbers, for what value of $k$ does $f(f(x))=x ?$
(A) only 1
(B) only 0
(C) all real numbers
(D) all real numbers except 0
(E) no real numbers

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Problem 37

$F(x)=\left\{\begin{array}{c}{\frac{3 x^{2}-3}{x-1}, \text { when } x \neq 1} \\ {k, \text { when } x=1}\end{array}\right.$ For what value(s) of $k$ is $F$ a continuous function?
(A) 1
(B) 2
(C) 3
(D) 6
(E) no value of $k$

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Problem 38

If $f(x)=2 x^{2}-4$ and $g(x)=2^{x},$ the value of $g(f(1))$ is
(A) $-4$
(B) 0
(C) $\frac{1}{4}$
(D) 1
(E) 4

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Problem 39

If $f(x)=3 \sqrt{5 x},$ what is the value of $f^{-1}(15) ?$
(A) 0.65
(B) 0.90
(C) 5.00
(D) 7.5
(E) 25.98

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Problem 40

Which of the following could be the equation of one cycle of the graph in the figure above?
I. $y=\sin 4 x$
II. $y=\cos \left(4 x-\frac{\pi}{2}\right)$
III. $y=-\sin (4 x+\pi)$
(A) only I
(B) only I and II
(C) only II and III
(D) only II
(E) I, II, and III

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Problem 41

If $2 \sin ^{2} x-3=3 \cos x$ and $90^{\circ} < x < 270^{\circ},$ the number of values that satisfy the equation is
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

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Problem 42

If $A=\tan ^{-1}$ $\left(-\frac{3}{4}\right)$ and $A+B=315^{\circ},$ then $B=$
(A) $278.13^{\circ}$
(B) $351.87^{\circ}$
(C) $-8.13^{\circ}$
(D) $171.87^{\circ}$
(E) $233.13^{\circ}$

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Problem 43

Observers at locations due north and due south of a rocket launchpad sight a rocket at a height of 10 kilometers. Assume that the curvature of Earth is negligible and that the rocket's trajectory at that time is perpendicular to the ground. How far apart are the two observers if their angles of elevation to the rocket are $80.5^{\circ}$ and $68.0^{\circ} ?$
(A) 0.85 $\mathrm{km}$
(B) 4.27 $\mathrm{km}$
(C) 5.71 $\mathrm{km}$
(D) 20.92 $\mathrm{km}$
(E) 84.50 $\mathrm{km}$

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Problem 44

The vertex angle of an isosceles triangle is $35^{\circ} .$ The length of the base is 10 centimeters. How many centimeters are in the perimeter?
(A) 16.6
(B) 17.4
(C) 20.2
(D) 43.3
(E) 44.9

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Problem 45

If the graph below represents the function $f(x),$ which of the following could represent the equation of the inverse of $f ?$
(A) $x=y^{2}-8 y-1$
(B) $x=y^{2}+11$
(C) $x=(y-4)^{2}-3$
(D) $x=(y+4)^{2}-3$
(E) $x=(y+4)^{2}+3$

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Problem 46

If $k > 4$ is a constant, how would you translate the graph of $y=x^{2}$ to get the graph of $y=x^{2}+4 x+k ?$
(A) left 2 units and up $k$ units
(B) right 2 units and up $(k-4)$ units
(C) left 2 units and up $(k-4)$ units
(D) right 2 units and down $(k-4)$ units
(E) left 2 units and down $(k-4)$ units

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Problem 47

If $f(x)=\log _{b} x$ and $f(2)=0.231,$ the value of $b$ is
(A) 0.3
(B) 1.3
(C) 13.2
(D) 20.1
(E) 32.5

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Problem 48

If $f_{n+1}=f_{n-1}+2 f_{n}$ for $n=2,3,4, \ldots,$ and $f_{1}=1$ and $f_{2}=1, \text { then } f_{5}=$
(A) 7
(B) 11
(C) 17
(D) 21
(E) 41

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Problem 49

Suppose cos $\theta=u$ in $^{0<\theta<\frac{\pi}{2}}$ . Then tan $\theta=$
(A) 1
(B) $\frac{1}{\sqrt{1-u^{2}}}$
(C) $\sqrt{1-u^{2}}$
(D) $\sqrt{1-u^{2}}$
(E) $\frac{\sqrt{1-u^{2}}}{u}$

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Problem 50

A certain component of an electronic device has a probability of 0.1 of failing. If there are 6 such components in a circuit, what is the probability that at least one fails?
(A) 0.60
(B) 0.47
(C) 0.167
(D) 0.000006
(E) 0.000001

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