Mathematics Practice Questions with Solutions

This page contains a set of math practice questions with solutions covering various topics such as algebra, trigonometry, geometry, and calculus. The questions are designed to help students improve their math skills and prepare for exams. Each question comes with a detailed solution to help students understand the concepts better. The topics covered include equations, functions, logarithms, geometry, and more. The questions are suitable for students in high school and college. Course codes, names, and universities are not mentioned.

Multiple Choices
Question 1
3−4 x ≤−7∨3−4 x ≥ 7
For 3−4 x ≤−7
−4 x ≤−10
x ≤ 10
4
x ≤ 5
2
3−4 x ≥7
−4 x ≤ 4
x ≤−1
Combining the two , we have : x ≤−1∨x ≥ 5
2
Interval Notation : (−∞ ,−1 ] ∪ [ 5
2 , ∞ )(D)
Question 2
D ¿ f ( x )=−6 x3 + x+ 8
Question 3
3 log( x +2)+log 1−log z
¿ log ( x +2 )3+ log 1
z
¿ log ( x+2 ) 3
z (C)
Question 4
C ¿ ( −∞ , 1 ) ∧(4 , ∞)
Question 5

D ¿ f ( x )=e x+2
Question 6
A ¿ f ( x )=3−cos x
Short Answer
Question 7
a) d= √ ( x1−x2 )2 + ( y1− y2 )2 = √ (−8+2 )2 + ( 5−7 )2=√ ¿
¿ 2 √10
b) Center of the ˚¿ ( x1 + x2
2 , y1 + y2
2 )=(−10
2 , 12
2 )=(−5,6)
c) Equation of a ˚is givenby : ( x−h ) 2 + ( y−k ) 2=r2 ,
Where ( h , k ) is thecenter of the ˚¿
( x +5 ) 2 + ( y−6 ) 2= ( 2 √ 10 ) 2=4∗10=40
x2+ 10 x + y2−12 y=−21
Question 8
log3
1
81
Let x=log3
1
81
3x= 1
81
( 1
3 )
−x
= ( 1
3 )
4
−x=4
x=−4
log3
1
81 =−4
Question 9
Commission Earned=5237.20−1650=3587.20
Thus 7.6 %=3587.20

100 %=?
100
7.6 ∗3587.20
¿ $ 47200
Question 10
T ( t ) =35+ 43 e− ( 0.058 t )
T ( t ) =35+ 43 e− ( 0.058∗12 ) =35+21.4=56.4 ° F
Question 11
f ( x )= 3
8 − 5
6 x
x= 3
8 − 5
6 y
y=−24 x +9
20
Inverse of the function is : y=−24 x +9
20
Question 12
a) Ar =360 °− Ac=360−300=60 °
b) 30 °= π
6
300 °=?
300
30 ∗π
6 = 10 π
6 = 5 π
3 radians
Question 13
a) Period :π
b) Phase Shift : π
2
Question 14
− √2 cos x−1=0

x= 3 π
4 , 5 π
4 .
Thus x=135 ° ,225 °
2 sin x−1= π
6 , 5 π
6 ,
Thus x=30 ° ,150 °
x=135 ° ,225 ° , 30° , 150°
Question 15
a) 0
b) π
3
Question 16
a) Direction of the Parabola: Up
b) Vertex: (5,-3)
c) Focus: (5,-1)
Question 17
a) Domain: (−∞,−3 ) ∪(−3 , ∞)
b) Vertical Asymptotes: x=−3
c) Horizontal Asymptotes: y=−1
d) Graph C
Short Answer: Work Required
Question 18
m= ( y2− y1 )
x2−x1
=−9−3
7−1 =−12
6 =−2
y−3
x−1 =−2
y−3=−2 x+ 2
y=−2 x +5
Question 19

4 x2 =1+12 x
4 x2 −12 x−1=0
x1,2=−b ± √ b2−4 ac
2 a
x= 3+ √10
2 ∨x= 3− √ 10
2
Question 20
a) ( f Ο g ) ( x )=2 ( x −4 )2−9
2 [ x2 −8 x+18 ]=2 x2−16 x +36
( f Ο g ) ( x )=2 x2−16 x +36
b) ( f Ο g ) ( x )=2 x2−16 x +36=2 (−2 )2−16 (−2 ) +36
8+32+36
¿ 76
Question 21
a) h=−16 t2 +72 t+18
At Maximumheight ,
dh
dt =0
dh
dt =−32 t+72=0
−32 t =−72
t=2.25 seconds
b) h=−16 t2 +72 t+18
h=−16∗2.252+72∗2.25+18
¿ 99 feet
Question 22
x+11
x +6 + 60
x2−36 =0
x+11
x +6 + 60
x2−36 = x +11
x +6 ( x +6 ) ( x−6 ) + 60
x2−36 ( x+6 ) ( x−6 )
¿ 0∗( x+6 ) (x−6)
( x +11 ) ( x−6 ) +60=0

x=1 , x=−6
Sincethe equation isundefined at x=−6 , x=1
Question 23
a) sin θ=2
7
( sin θ )2= 4
49
( cos θ )2=1− ( sin θ )2 =1− 4
49 = 45
49
cos θ= √ 45
49 = √ 45
7 = 3 √ 5
7
b) sin 2 θ=2 sin θ cos θ=
2∗2
7 ∗ √45
7 = 4 √45
49 =12 √ 5
49
Question 24
( sin x +cos x ) 2=sin2 x+ 2sin x cos x+ cos2 x
But sin2 x +cos2 x=1∧2sin x cos x=sin 2 x
Thus sin2 x+2sin x cos x+ cos2 x=1+ sin 2 x
Question 25
tan x= Opp
Adj
Thus tan 41.7 °= Height of thetree
55
Height of the Tree=55∗tan 41.7
¿ 49 feet
Question 26
Sine Rule : sin A
a = sin B
b = sin C
c
C=180−48−62=70 °
sin 70
35 =sin 48
a

a= 35∗sin 48
sin 70 =27.68
Question 27
a ¿ :⃗ u .⃗ v=a1 b1 +a2 b2
¿ 9∗4 + (−3 )∗12=36−36=0
b ¿ : Let the angle⃗ u∧⃗ v=x
cos x= (⃗ u⃗ . v )
||⃗u ||.∨¿⃗ v∨¿ = 0
( √92+ (−3 )2 )∗ ( √122+42 ) =0
cos x=¿0
x=90 °
Question 28
a) The major axis is horizontal
b) For an ellipse withmajor axis∥¿ the x−axis ,the foci are defined as
( h+c , k ) , ( h−c , k ) , where c= √a2−b2
is the distance ¿ the center ( h , k ) ¿ a focus .
With the given Equation, Ellipse withcenter ( h , k )= (−1,16 ) ,∧semi−axis
a=8 , b=4
(−1+c , 16 ) , (−1−c ,16 )
c= √ 82−42
¿ 4 √3
( −1+ 4 √ 3 ,16 ) , ( −1−4 √ 3 , 16 )