1. No category

Sample Math Placement Test
1.
Divide [(-18) – (6)(-2)+34 – 15] by (1-11) and
simplify.
2.
Find the missing numerator for
3.
Arrange
4.
5.
6.
7.
3
x
.
=
7 42
7 5 6
,
,
from smallest to largest.
19 13 17
2 1
+
Simplify and reduce to lowest terms: 3 2
1 2
3 −
2 3
Change 0.4825 to a fraction in lowest terms.
Divide 36.174 by 52.19 and round to two
decimal places.
Convert 6.73 kL to L.
8.
Multiply: (3a – 2b)(2a + b)
7
9. Write 16 % as a fraction in lowest terms.
8
10. Given T = 3 (T2 – T1), solve for T1.
25. Which of the following are true?
a) -(x – 1) = -x + 1
a −b
a
b)
= −1 +
b
b
m
c)
=0
0
d)
x2 − 9 = x −3
e)
(q + 2) 2 = q 2 + 4
26. State the values(s) of t for which the expression
t 2 − 7t + 12
t2 −9
⎛ 2 p 2 q − 2
27. Evaluate ⎜
⎜ r −3
⎝
13. Simplify:
128x − 5 2x
(
3
29. Evaluate
xy ) 6
x2 − a2
10
17. Solve 3 log (2x – 3) = 0 for x.
18. Find α if cot α = -4.5247 and 0° ≤ α ≤ 360°.
19. Find the value of α if the terminal side of angle α
(an angle in standard position) passes through the
point (1.41, 3.51).
20. Given the right triangle ABC, find side b if
∠C = 90.0°, ∠B = 76.0°, and side a = 14.5 cm.
21. Change
3
t − 3r 1 / 3
radians to degrees.
22. Evaluate csc
31. Simplify:
a+ x
2
[log t − 5 log n + 2 log y] as a single logarithm.
3
π
for p=2 , q=-2 and r=1.
for r=8 and t=4.
⎛ y 1 / 3 r 2 / 3
30. Simplify: ⎜
⎜ m 3 r 4 / 3
⎝
15. Change log b x = y to exponential form.
16.Write
3r − 2t 1 / 2
x
14. Simplify:
−3
⎞
⎟
⎟
⎠
28. Write –7x2y0 z-5 without zero or negative
exponents.
11. Write 0.00934 in scientific notation.
12. Simplify:
is undefined.
5π
and round to 4 decimal places.
4
23. Indicate the slope (m) and y-intercept (b) for the
equation 4x – 3y = 12.
24. Solve the following system of equations:
2x – y + 2z = 6
x+y+z=6
3x + 2y – z = 4
32. Simplify:
⎞
⎟
⎟
⎠
3
m2 − 9
5m − 15
÷ 2
2m − 12 m − 2m − 24
x
2
x − 4x + 3
−
3
2x
3
+
4
3
x − 3x 2
33. Solve for k: -3 (k – 1) = -2 (k – 4)
34. Solve for x:
2
5
5
− =
x − 2 x 2x
E R+r
=
e
r
36. An employer gave an employee a raise of 15%.
The new salary is $36,000 per year. What was the
employee's salary before the raise?
35. Solve for r:
37. A rectangle has a perimeter of 300 cm. The length is 10
more than 1 1/3 times the width. Find the length and
width.
38. How much water must be added to a litre of 90%
pure alcohol to reduce it to 80% pure alcohol?
39. For the right triangle ABC, ∠C = 90.0°,
c = 8.20 mm, and a = 6.50 mm. Find ∠B.
40. A guy wire whose grounded end is 11.5 m from the
pole it supports makes an angle of 64.3° with the ground.
Find the length of the guy wire.
Sample MPT solutions !!" ! ! !! !!! !!"
1.
2.
3.
4.
5.
6.
7.
8.
9.
!
=
!
!
,
!
!"
!
!"
= !!" = −6 !!"
⇒ 3 42 = 7𝑥 ⇒ 𝑥 = 18 !
,
!"#$ !"!# !"#$
⇒ !" !" !"
! !
!
! !
! !
! !
! !
!!"!!"!!"!!"
=
!!!!
!
!
!"
!
=
,
,
!"## !"## !"##
7
6
6
17
= ×
4825
10000
=
7
17
⇒ !
,
!
,
!
!" !" !"
193
400
0.4825 =
=
36.174 ÷ 52.19 = 0.693121288 = 0.69 1000 L
6.73 kL × 1 kL = 6370 L 3a − 2b 2a + b = 6a! + 3ab − 4ab − 2 = 6a! − ab − 2b! 7
16 8 %
=
135
8
135
%
8
= 100
27
= 135
= 160
800
3T2 −T
3
10.
T = 3 T! − T! ⇒ T = 3T! − 3T! ⇒ 3T! = 3T! − T ⇒ T! =
11.
0.00934 = 1000 = 9.34× 1000 = 9.34×10!! 128𝑥 − 5 2𝑥 = 64 2𝑥 − 5 2𝑥 = 8 2𝑥 − 5 2𝑥 = 3 2𝑥 12.
13.
9.34
( ! !" )!
16.
17.
( ! !" )!
!
10
𝑥 ! − 𝑎! 𝑎+𝑥 ⇒
14.
15.
!
=
1
!
=
!" !
!
=
!!!!
!
= 𝑥𝑦 ! 10
𝑥 + 𝑎 (𝑥 − 𝑎) 𝑎+𝑥 ⇒
(𝑥 − 𝑎) 10 ⇒
10(𝑥 − 𝑎) log ! 𝑥 = 𝑦 ⇒ 𝑏 ! = 𝑥 2
2
10
4
t 2 3y4 3
23
10 3
43
logt
−
5log
n
+
2
log
y
=
logt
−
log
n
+
log
y
=
logt
−
log
n
+
log
y
=
log
[
]
3
3
3
3
n10 3
0
⇒ log(2 x − 3) = 0
3
⇒ x=2
3 log(2 x − 3) = 0 ⇒ log(2 x − 3) =
⇒ 1 = 2x − 3 ⇒ 4 = 2x
⇒ 10 0 = 2 x − 3 cot α = −4.5247 First find the reference angle, α ref
18.
1
1
⎛ 1 ⎞
= − 4.5247 ⇒ tan α ref =
⇒ α ref = tan −1 ⎜
⎟ ⇒ α ref = 12.46 tan α ref
4.5247
⎝ 4.5247 ⎠
Since cot α is negative, then α lies in the second and fourth quadrants. Solution 1: α = 180° − α ref ⇒ α = 167.54° ; Solution 2 : α = 360° − α ref ⇒ α = 347.54° 19.
The right triangle setup yields tan α =
20.
tan 76° =
b
14.5
!
21.
!
radians =
csc
22.
!!
!
=
!
!
×
!
5𝜋
!"#
4
3.51
⎛ 3.51 ⎞
⇒ α = tan −1 ⎜
⎟ ⇒ α = 68.1° 1.41
⎝ 1.41 ⎠
⇒
b = 14.5 tan 76° ⇒
!"#°
=
=
!
!"#°
!
!!.!"!#
!
b = 52.8cm = 60° = −1.4142 !
4𝑥 − 3𝑦 = 12 ⇒ 𝑦 = ! 𝑥 − 4 This is 𝑦 = 𝑚𝑥 + 𝑏 form. 23.
!
Slope (𝑚) = ! and y-­‐intercept (b) = −4 Equation 1: Equation 2: Equation 3: 24.
2x – y + 2z = 6 x + y + z = 6 3x + 2y – z = 4 Step 1: Add equation 1 to equation 2 2x – y + 2z = 6 (+) x + y + z = 6 3x + 3z = 12 Equation 4 Step 2: Multiply equation 1 by 2 and add to equation 3 2x – y + 2z = 6 ⇒ 25.
4x – 2y + 4z = 12 (+) 3x + 2y – z = 4 7x + 3z = 16 Equation 5 Step 3: Subtract equation 4 from equation 5 3x + 3z = 12 (-­‐)7x + 3z = 16 -­‐4x = -­‐4 x=1 Step 4: Substitute x=1 into equation 4; z=3 Step 5: Substitute x=1 and z=3 into equation 2; y=2 (a) True. (b) True. (c) False. Division by zero is undefined. (d) False. x ! − 9 = (x − 3)(x + 3) ≠ x − 3. (e) False. (q + 2)! = q + 2 q + 2 ≠ q! + 4 26.
! ! !!!!!"
! ! !!
! ! !!!!!"
=
is undefined when the denominator is zero. !!! !!!
𝑡 + 3 𝑡 − 3 = 0 ⇒ 𝑡 = 3 𝑜𝑟 𝑡 = −3 27.
!!
!!! ! !!
! !!
!!! !!! ! !
=
!!
=
!!
!! ! ! ! !
28.
1
𝑧5
−7𝑥 ! 𝑦 ! 𝑧 !! = −7𝑥 ! 1
29.
!!!!! ! !
! ! !!
31.
!(!)!! !
=
!
!!! !
=
! ! ! !" !
!!!!"
! ! !!!!!"
!
!"
=
! (!)!
!
(!)(!")(!)
= !
!"!!
!!!
!! !
!! ! !
= =
! ! !!
!!!!"
=
!"
!!
!
!! ! !
= −10 ! ! !!!!!"
𝑥
!!!!"
= 𝑚 − 3 (𝑚 + 3) (𝑚 − 6)(𝑚 + 4)
(𝑚 + 3)(𝑚 + 4)
𝑥
= 2(𝑚 − 6)
5(𝑚 − 3)
10
!
− !
! 𝑥3
+
!
𝑥3 !!𝑥2
⇒ !
!!! !!!
− !
! 𝑥3
+
!
𝑥2
!!!
⇒ 𝑥 2𝑥3 − 3 𝑥 − 3 𝑥 − 1 + 4 2𝑥 𝑥 − 1
2𝑥4 − 3 𝑥2 − 4𝑥 + 3 + 8𝑥 𝑥 − 1
⇒
2𝑥3 𝑥 − 3 𝑥 − 1
2𝑥3 𝑥 − 3 𝑥 − 1
⇒
2𝑥4 − 3𝑥2 + 12𝑥 − 9 + 8𝑥2 − 8𝑥 2𝑥4 + 5𝑥2 + 4𝑥 − 9
⇒
2𝑥3 𝑥 − 3 𝑥 − 1
2𝑥3 𝑥 − 3 𝑥 − 1
−3 𝑘 − 1 = −2 𝑘 − 4 ⇒ −3𝑘 + 3 = −2𝑘 + 8 ⇒ −1𝑘 = 5 ⇒ 𝑘 = −5 34.
=
!!
−7𝑥2
𝑧5
=
!! !!! !
=
÷
!!!!"
𝑥2 !!!!!
33.
!!! !
!! ! ! !
!
!
!! !!! !
30.
32.
!!!! !
=
!!!! ! !
(!!)!
=
!
!!!
!
!
!
!!
− = ⇒ !!!(!)(!!!)
(!!!)(!)
= !
!!
⇒
!!!!!!!"
(!!!)(!)
= !
!!
−3𝑥 + 10 2𝑥 = 5 𝑥 − 2 𝑥 ⇒ −3𝑥 + 10 2 = 5 𝑥 − 2 ⇒ −6𝑥 + 20 = 5𝑥 − 10 ⇒ −11𝑥 = −30 ⇒ 𝑥 = 35.
!
!
= !!!
!
⇒ 𝐸
𝑟
=
𝑒
𝑅+𝑟
30
8
= 2 11
11
⇒ 𝐸𝑟 = 𝑒𝑅 + 𝑒𝑟 ⇒ 𝐸𝑟 − 𝑒𝑟 = 𝑒𝑅 𝑒𝑅
⇒ 𝑟 𝐸 − 𝑒 = 𝑒𝑅 ⇒ 𝑟 = 𝐸−𝑒 36.
37.
36 000
(1.15)(x) = 36 000 ⇒ 𝑥 = 1.15 ⇒ 𝑥 = $31 304.45 = original salary Equation 1: 2 L + 2 W = 300 !
Equation 2: L = ! W + 10 Substitute equation 2 into equation 1: 4
8
2 W + 10 + 2 𝑊 = 300 ⇒ W + 20 + 2 𝑊 = 300 3
3
8
6
⇒ W + W = 280 ⇒ 14 W = 840 ⇒ W = 60 cm 3
3
Substituting W=60 back into equation 1: 38.
2 L + 2 (60) = 300 ⇒ 2L = 180 ⇒ L = 90 cm (x)(0%) + (1)(90%) = (x + 1)(80%) ⇒ 0 + .90 = 0.80x + 0.8 ⇒ 0.1 = 0.80x ⇒ 0.125 L = x 39.
cos 𝐵 = 40.
6.50
⇒ 𝐵 = cos−1 6.50
⇒ ∠𝐵 = 37.6o 8.20
8.20
cos 64.3! = 11.5
⇒ 𝑥 = 26.5 𝑚 𝑥