SAT Math Practice Test — 50 Questions
Total Time: 60 Minutes Calculator: Permitted (some questions may be done faster without)
Question 1
The measures of the angles of triangle QRS are m∠Q = 2x + 4, m∠R = 4x − 12, and m∠S = 3x + 8. QR = y + 9, RS = 2y − 7, and QS = 3y − 13. The perimeter of △QRS is:
(A) 11 (B) 20 (C) 44 (D) 55 (E) 68
Question 2
Given g(x) = (3x + 2)/(5x − 1), find g(−3/4):
(A) −1/2 (B) 1/19 (C) 1/11 (D) 7/16 (E) 1/2
Question 3
Simplify: ³√(81x⁷y¹⁰)
(A) 9x³y⁵ ³√x (B) 9x²y³ ³√(xy)
(C) 3x²y³ ³√(9xy) (D) 3x²y³ ³√(3xy)
(E) 3x³y⁵ ³√x
Question 4
The vertices of triangle GHK have coordinates G(−3,4), H(1,−3), and K(2,7). The equation of the altitude to HK is:
(A) 10x + y = −26 (B) 10x + y = 7 (C) 10x + y = 27
(D) x + 10y = 37 (E) x + 10y = 72
Question 5
Diagram (rotating a semicircle about its vertical axis):
|
|
┌───┼───┐
│ │ │
│ │ │
│ │ │
├───┼───┤
│ │ │
│ │ │
└───┼───┘
|
| ← vertical axis of rotation
A semicircle with radius 6 cm is rotated about its vertical diameter. The total surface area of the solid formed is:
(A) 144π (B) 108π (C) 72π (D) 36π (E) 9π + 12
Question 6
If f(x) = 4x² − 1 and g(x) = 8x + 7, find g(f(2)):
(A) 15 (B) 23 (C) 127 (D) 345 (E) 2115
Question 7
If p and q are positive integers with pq = 36, then p/q cannot be:
(A) 1/4 (B) 4/9 (C) 1 (D) 2 (E) 9
Question 8
Simplify:
$$ \frac{\frac{2}{3} + \frac{1}{x-4}}{1 - \frac{2}{3x-12}} $$
(A) (2x)/(3x−2) (B) (2x−8)/(3x−2)
(C) (2x−8)/(3x−12) (D) (2x−7)/(3x−12)
(E) (2x−5)/(3x−14)
Question 9
If $\left(\frac{1}{125}\right)^{a^2+4ab} = \left(\sqrt[3]{625}\right)^{3a^2-10ab}$ and a and b are nonzero, find a/b:
(A) 4/21 (B) 2 (C) 76/21 (D) 4 (E) 76/3
Question 10
Diagram (isosceles triangle with parallel segments):
K
/|\
/ | \
/ | \
/ | \
L----N----M
/ | \
/ | \
H-------P-------J
HJ = 8
K is 10 cm from base HJ
KL = 0.4 · KH
NL ⟂ HJ and MP ⟂ HJ
Find the area of triangle LNH.
(A) 4 (B) 4.8 (C) 6 (D) 7.2 (E) 16
Question 11
The equation of the perpendicular bisector of the segment joining A(−9,2) and B(3,−4) is:
(A) y − 1 = −½(x − 3) (B) y + 1 = −½(x + 3)
(C) y + 1 = 2(x + 3) (D) y + 3 = 2(x + 1)
(E) y − 1 = 2(x − 3)
Question 12
Diagram (tangent and secant to a circle):
T
\
\ tangent
\
B
/|
/ |
/ |
/ |
/ |
C-----A
\ /
\ /
O
TC = 6, CA = 10
AB is a diameter
TB is tangent at B
Find CB.
(A) 2√6 (B) 4√6 (C) 2√15 (D) 10 (E) 2√33
Question 13
Define $p \oplus q = \frac{p^q}{q-p}$. Find $(5 \oplus 3) - (3 \oplus 5)$:
(A) −184 (B) −59 (C) 0 (D) 59 (E) 184
Question 14
Grades: mean = 68, median = 64, standard deviation = 12. The teacher adds 7 points to every score. Which statements are true?
I. New mean = 75 II. New median = 71 III. New standard deviation = 7
(A) I only (B) III only
(C) I and II only (D) I, II, and III
(E) None
Question 15
Diagram (midpoint triangle nest):
A
/ \
/ \
/ \
D-------E
/ \ / \
/ \ / \
/ \ / \
F-------G-------H
\ / \ /
\ / \ /
\ / \ /
I-------J
\ /
\ /
\ /
K
Largest triangle ABC has area 128. The smallest triangle (DEF) is formed by joining midpoints repeatedly. Find its area.
(A) 1/8 (B) 1/4 (C) 1/2 (D) 1 (E) 4
Question 16
Graph of y = f(x) shown (points (0,3), (3,0), (5,0)). Which graph represents g(x) = 2f(x−2) + 1?
(Options not shown; this is a visual transformation question.)
Question 17
Bacteria population (in thousands) is modeled by:
$$ b(t) = \frac{380 e^{2.31t}}{175 + e^{3.21t}} $$
where t = days. According to the model, the maximum population the culture can support is:
(A) 2.17 (B) 205 (C) 380 (D) 760 (E) ∞
Question 18
A sphere with diameter 50 cm is cut by a plane 14 cm from its center. What is the area of the circular cross-section?
(A) 49π (B) 196π (C) 429π (D) 576π (E) 2304π
Question 19
Given g(x) = (3x−1)/(2x+9), find g(g(x)):
(A) (9x−4)/(6x+7) (B) (7x−12)/(24x+79)
(C) (x−10)/(21x+80) (D) (7x+6)/(24x+79)
(E) (9x²−6x+1)/(4x²+36x+81)
Question 20
Area of △QED = 750. QE = 48, QD = 52. To the nearest degree, what is the measure of the largest possible angle of △QED?
(A) 76° (B) 77° (C) 78° (D) 143° (E) 145°
Question 21
Given $\log_3(a) = c$ and $\log_3(b) = 2c$, then $a =$
(A) 3c (B) c+3 (C) b² (D) √b (E) b/2
Question 22
Diagram (isosceles trapezoid with parallel segments):
W_________X_________Y
|\ | /|
| \ | / |
| \ | / |
| \ | / |
| \ | / |
| \ | / |
| \ | / |
| \ | / |
| \|/ |
H---------E---------Z
\ | /
\ | /
\ | /
\ | /
\ | /
\ | /
\ | /
\ | /
\|/
T
WH ∥ XZ ∥ TY
m∠TWH = 120°, m∠HWE = 30°
WH = 30
XZ passes through E (intersection of diagonals)
Find the ratio XZ : TY.
(A) 1:2 (B) 2:3 (C) 3:4 (D) 4:5 (E) 5:6
Question 23
Sides of a triangle are 25, 29, and 34. To the nearest tenth of a degree, the largest angle is:
(A) 77.6° (B) 77.7° (C) 87.6° (D) 87.7° (E) 102.3°
Question 24
One root of a quadratic with integer coefficients is $-\frac{4}{5} + \frac{3\sqrt{2}}{8}i$. Which equation has that root?
(A) 800x² + 1280x + 737 = 0
(B) 800x² − 1280x + 737 = 0
(C) 800x² + 1280x + 287 = 0
(D) 800x² − 1280x + 287 = 0
(E) 800x² + 1280x − 287 = 0
Question 25
Parametric equations: $x = \cos(2t) + 1$, $y = 3\sin(t) + 2$. They correspond to a subset of which graph?
(A) (x−1)²/1 + (y−2)²/9 = 1
(B) (x−1)²/1 − (y−2)²/9 = 1
(C) x−2 = (2/9)(y−2)²
(D) x−2 = −(2/9)(y−2)²
(E) x−2 = −(2/3)(y−2)
Question 26
A committee of 5 is chosen from 8 Democrats and 6 Republicans. What is the probability that Democrats have more members than Republicans?
(A) 686/2002 (B) 1316/2002 (C) 1876/2002
(D) 2688/24024 (E) 21336/24024
Question 27
Vectors $u = [-5,4]$ and $v = [3,-1]$. Find $|2u - 3v|$.
(A) [-19,11] (B) √502 (C) [19,11]
(D) 2√41 − 3√10 (E) 2√65
Question 28
Triangle ABC: A(−11,4), B(−3,8), C(3,−10). Find the circumcenter.
(A) [−2,−3] (B) [−3,−2] (C) [3,2]
(D) [2,3] (E) [−1,−1]
Question 29
Each side of the base of a square pyramid is reduced by 20%. By what percent must the height be increased to keep volume the same?
(A) 20% (B) 40% (C) 46.875% (D) 56.25% (E) 71.875%
Question 30
Simplify:
$$ \frac{20,\text{cis}(19\pi/18)}{5,\text{cis}(2\pi/9)} $$
(A) −2√3 + 2i (B) −2√3 − 2i (C) 2√3 + 2i
(D) −2 + 2i√3 (E) 2 − 2i√3
Question 31
Given $\log_b(a) = x$ and $\log_b(c) = y$, find $\log_{a^2}\left(\sqrt[3]{b^5 c^4}\right)$:
(A) 5/3 + y⁴ (B) (5+4y)/(6x) (C) (20y)/(3x)
(D) 2x + 4y (E) 2x + (20y)/3
Question 32
A hyperbola has asymptotes $y-1 = \pm \frac{3}{4}(x+3)$ and a focus at (7,1). Its equation is:
(A) (x+3)²/16 − (y−1)²/9 = 1
(B) (y−1)²/9 − (x+3)²/16 = 1
(C) (x+3)²/64 − (y−1)²/36 = 1
(D) (y−1)²/36 − (x+3)²/64 = 1
(E) (x+3)²/4 − (y−1)²/3 = 1
Question 33
An inverted cone (vertex down) has height 12 and base radius 8. Water fills it. When half-filled (by volume), what is the height of the water?
(A) 6 (B) 6∛4 (C) 8∛6
(D) 9∛4 (E) 9∛6
Question 34
Solve $\sin(t) = \cos(2t)$ for $-4\pi \le t \le -2\pi$.
(Answer choices omitted; this is a free-response style in original.)
Question 35
Solve $12x - 3|x-11| < 2$:
(A) 1/5 < x < 5
(B) 1/5 < x < 2 and 2 < x < 5
(C) x < 1/5 or x > 5
(D) x < 5
(E) x > 1/5
Question 36
Evaluate:
$$ \sum_{k=0}^{\infty} 12\left(\frac{2}{3}\right)^k - \sum_{k=0}^{\infty} 18\left(-\frac{1}{2}\right)^k $$
(A) −5 (B) 5 (C) 24 (D) 27 (E) 30
Question 37
Solve the system:
$$ \frac{3}{x} - \frac{4}{y} + \frac{2}{z} = 3,\quad \frac{2}{x} - \frac{8}{y} - \frac{1}{z} = -8,\quad \frac{4}{x} - \frac{6}{y} - \frac{3}{z} = 1 $$
Find $\frac{1}{x - y + z}$.
(A) 2/25 (B) 30/31 (C) 31/30 (D) 19/2 (E) 25/2
Question 38
For $f(x) = 3 - 2\cos\left(\frac{3\pi}{5}x - \frac{3\pi}{10}\right)$, which statements are true?
I. Amplitude = 2 II. Shifted right by 2 III. $f(5/2) = f(31/6)$
(A) I only (B) II only
(C) III only (D) I and II only
(E) I and III only
Question 39
All of the following solve $z^5 = 32i$ EXCEPT:
(A) 2 cis(π/2) (B) 2 cis(9π/10)
(C) 2 cis(11π/10) (D) 2 cis(13π/10)
(E) 2 cis(17π/10)
Question 40
Sequence: $a_1 = 4,; a_2 = -2,; a_n = 2a_{n-2} - 3a_{n-1}$. Find the smallest n such that $|a_n| > 1{,}000{,}000$.
(A) 11 (B) 12 (C) 13 (D) 14 (E) 15
Question 41
For $f(x) = \frac{(2x-3)(x+2)(2x-1)}{4x^2-9}$, which are true?
I. $f(x) = 9/4$ has two solutions. II. $f(x) = 7/6$ has two solutions. III. The range is all real numbers.
(A) III only (B) I and II only
(C) II and III only (D) I and III only
(E) I, II, and III
Question 42
$g(x) = 9\log_8(x-3) - 5$. Find $g^{-1}(13)$.
(A) 3 (B) 6 (C) 61 (D) 67 (E) 259
Question 43
Diagram (isosceles triangle with centroid):
Q
/|\
/ | \
/ | \
/ | \
/ | \
R-----T-----S
QR = QS = 60, RS = 30. T is the centroid. Find the distance from T to side QR.
(A) 2√15 (B) (5/2)√15 (C) 3√15
(D) (7/2)√15 (E) 5√15
Question 44
Diagram (quadrilateral with perpendicular diagonals):
D
/ \
/ \
/ \
/ \
A---------C
|\ /|
| \ / |
| \ / |
| \ / |
| B |
AD = DC = 8, AC = BC = 6, m∠ADC = 60°. Diagonals AC and BD are perpendicular. Find the area of ABCD.
(A) 4√5 + 8√3 (B) 16√3 (C) 32√3
(D) 8√5 + 16√3 (E) 48
Question 45
Simplify:
$$ \cos\left(2\csc^{-1}\left(\frac{x+4}{5}\right)\right) $$
(A) (x²+8x−16)/(x+4) (B) (x²+8x−16)/(x+4)²
(C) (x²+8x−34)/(x+4) (D) (x²+8x−34)/(x+4)²
(E) (−16−8x−x²)/(x+4)²
Question 46
For $(a+b)^n - (a-b)^n$, which statements are true?
I. If n is even, it has n/2 terms. II. If n is odd, it has (n+1)/2 terms. III. The exponent on the last term is always n.
(A) I only (B) II only
(C) I and II only (D) I and III only
(E) II and III only
Question 47
In △VWX, $\sin(X) = 8/17$ and $\cos(W) = -3/5$. Find $\cos(V/2)$.
(A) −77/85 (B) −2/√85 (C) 2/√85 (D) 9/√85 (E) 77/85
Question 48
Intersection of the hyperbola $\frac{(x+1)^2}{8} - \frac{(y-1)^2}{9} = 1$ and the ellipse $\frac{(x+1)^2}{32} + \frac{(y-1)^2}{18} = 1$ gives which points?
(A) (3,4), (3,−4), (−5,4), (−5,−4)
(B) (3,2), (3,−2), (−5,2), (−5,−2)
(C) (3,4), (3,−2), (−5,4), (−5,−2)
(D) (−3,4), (−3,−2), (5,4), (5,−2)
(E) (3,−4), (3,2), (−5,−4), (−5,2)
Question 49
Graph of $8x^6 + 72x^5 + bx^4 + cx^3 - 687x^2 - 2160x - 1700 = 0$ shows roots at x = −2.5 (double), x = −2, and x = 2. The equation has two complex roots. Find their product.
(A) −4 (B) 17/2 (C) 9 (D) −687/2 (E) 425/2
Question 50
If $\sin A = -8/17$ (QIV) and $\cos B = -24/25$ (QIII), find $\cos(2A + B)$.
(A) −5544/7225 (B) −2184/7225 (C) −3696/7225
(D) 2184/7225 (E) 5544/7225
Answer Key
Q | Ans | Q | Ans | Q | Ans | Q | Ans | Q | Ans |
|---|---|---|---|---|---|---|---|---|---|
1 | D | 11 | C | 21 | D | 31 | B | 41 | D |
2 | B | 12 | C | 22 | B | 32 | C | 42 | D |
3 | D | 13 | A | 23 | B | 33 | B | 43 | C |
4 | B | 14 | C | 24 | A | 34 | C | 44 | D |
5 | B | 15 | C | 25 | D | 35 | B | 45 | D |
6 | C | 16 | B | 26 | B | 36 | C | 46 | C |
7 | D | 17 | C | 27 | B | 37 | B | 47 | D |
8 | E | 18 | C | 28 | B | 38 | E | 48 | C |
9 | A | 19 | B | 29 | D | 39 | C | 49 | B |
10 | D | 20 | D | 30 | A | 40 | D | 50 | A |
Essential SAT Math Formulas
Geometry & Measurement
Formula | Description |
|---|---|
$A = \frac12 bh$ | Area of triangle |
$A = \pi r^2$ | Area of circle |
$C = 2\pi r$ | Circumference |
$V = \frac13 \pi r^2 h$ | Volume of cone |
$V = \frac13 B h$ | Volume of pyramid |
$a^2 + b^2 = c^2$ | Pythagorean theorem |
$A = \frac12 d_1 d_2$ | Area of rhombus/kite |
Trigonometry
Identity | Formula |
|---|---|
Pythagorean | $\sin^2\theta + \cos^2\theta = 1$ |
Double-angle | $\cos(2\theta) = \cos^2\theta - \sin^2\theta$ |
Double-angle | $\cos(2\theta) = 1 - 2\sin^2\theta$ |
Law of Cosines | $c^2 = a^2 + b^2 - 2ab\cos C$ |
Half-angle | $\cos(\theta/2) = \pm\sqrt{\frac{1+\cos\theta}{2}}$ |
Algebra & Sequences
Concept | Formula |
|---|---|
Quadratic formula | $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ |
Infinite geometric series | $S = \frac{a_1}{1-r},; \lvert r\rvert<1$ |
Change of base | $\log_a b = \frac{\log_c b}{\log_c a}$ |
Sum & product of roots | sum = $-b/a$, product = $c/a$ |
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