/
AI Flashcards
Save to my account
Sign up
AI Flashcards
Mathematics Practice Problems
Study
1
Question
Evaluate the expression using order of operations: 5(1 + 4)^2 − 10.
Page 2
Answer
First evaluate inside parentheses: 1 + 4 = 5. Then exponent: 5^2 = 25. Multiply: 5 × 25 = 125. Finally subtract: 125 − 10 = 115. Answer: 115.
2
Question
What is an equilateral triangle?
Page 4
Answer
An equilateral triangle is a triangle with all three sides equal in length. Consequently, all three interior angles are equal and each measures 60∘
3
Question
What is an isosceles triangle?
Page 4
Answer
An isosceles triangle is a triangle with at least two sides of equal length. The base angles opposite those equal sides are congruent (equal in measure).
4
Question
Find the area of a square with side length $s=7$ units.
Page 4
Answer
Area of a square is $A=s^2$. Substitute $s=7$: $A=7^2=49$. Answer: 49 square units.
5
Question
Find the perimeter of a rectangle with length $L=12$ and width $W=5$.
Page 4
Answer
Perimeter of a rectangle is $P=2(L+W)$. Substitute: $P=2(12+5)=2(17)=34$. Answer: 34 units.
6
Question
Find the perimeter of a square with side length $a=9$.
Page 4
Answer
Perimeter of a square is $P=4a$. Substitute $a=9$: $P=4(9)=36$. Answer: 36 units.
7
Question
Find the area of a parallelogram with base b=8 and height $h=5$.
Page 4
Answer
Area of a parallelogram is $A=b\times h$. Substitute: $A=8\times 5=40$. Answer: 40 square units.
8
Question
What is the formula for the surface area of a rectangular prism with dimensions L, W, H?
Page 2
Answer
Surface area is $SA=2(LW+LH+WH)$. This sums the areas of opposite faces and doubles them. Answer: $2(LW+LH+WH)$.
9
Question
What is the formula for the volume of a rectangular prism with dimensions L, W, H?
Page 2
Answer
Volume is $V=L\times W\times H$. Answer: $LWH$.
10
Question
A wheelchair ramp rises 2 ft over a horizontal distance of 24 ft. What is the slope of the ramp?
Page 2
Answer
Slope $m=\frac{\text{rise}}{\text{run}}=\frac{2}{24}=\frac{1}{12}$. Answer: $\frac{1}{12}$.
11
Question
Find the slope between points \(5,10\) and \(20,30\).
Page 2
Answer
Slope $m=\frac{\text{rise}}{\text{run}}=\frac{2}{24}=\frac{1}{12}$.
12
Question
Using point-slope form with point (5,10) and slope $m=\frac{4}{3}$, write the equation of the line in point-slope form.
Page 2
Answer
Point-slope form: $y-y_1=m(x-x_1)$. Substitute: $y-10=\frac{4}{3}(x-5)$. Answer: $y-10=\frac{4}{3}(x-5)$.
13
Question
If a line passes through (0,-3) and has slope 2, what is the equation of the line in slope-intercept form?
Page 2
Answer
Slope-intercept form: $y=mx+b$. Given $m=2$ and point $(0,-3)$ gives $b=-3$. So $y=2x-3$. Answer: $y=2x-3$.
14
Question
Find the midpoint between the points (2,6) and (8,10)
Page 2
Answer
Midpoint formula: $(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$. Compute: $(\frac{2+8}{2},\frac{6+10}{2})=(5,8)$. Answer: $(5,8)$.
15
Question
A ladder leans against a wall with base 6 ft from the wall and the top touches 8 ft high. Find the length of the ladder (hypotenuse).
Page 2
Answer
Use Pythagorean theorem: $c=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10$. Answer: 10 ft.
16
Question
Find the distance between points (-3,4) and (5,1)
Page 2
Answer
Distance formula: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. Compute: $\sqrt{(5-(-3))^2+(1-4)^2}=\sqrt{8^2+(-3)^2}=\sqrt{64+9}=\sqrt{73}$. Answer: $\sqrt{73}$.
17
Question
In a right triangle, \(\sin(\theta)=\tfrac{3}{5}\). Find \(\cos(\theta)\).
Page 3
Answer
In a right triangle, \(\sin\theta=\frac{\text{opp}}{\text{hyp}}=\frac{3}{5}\). Opp = 3, hyp = 5, so adjacent = \(\sqrt{5^2-3^2}=\sqrt{25-9}=\sqrt{16}=4\). Thus \(\cos\theta=\frac{4}{5}\). Answer: \(\frac{4}{5}\).
18
Question
A population of 1,000 doubles every 5 years. What will the population be after 15 years?
Page 3
Answer
Doubling every 5 years means in 15 years there are \(15/5=3\) doubling periods. Population = \(1000\times 2^3=1000\times 8=8000\). Answer: 8,000.
19
Question
Which of the following is a geometric sequence: 3, 6, 9, 12 ; 2, 6, 18, 54 ; 4, 7, 10, 13 ; 10, 20, 30, 40? Identify the geometric sequence.
Page 3
Answer
A geometric sequence has a constant ratio between consecutive terms. Check ratios: 6/3=2, 9/6=1.5 (not constant). For 2,6,18,54 ratios: 6/2=3, 18/6=3, 54/18=3 (constant). So sequence 2,6,18,54 is geometric. Answer: 2, 6, 18, 54.
20
Question
If \(f(x)=2x-1\) and \(g(x)=x^2\), find \((f\circ g)(2)\) (that is, \(f(g(2))\)).
Page 2
Answer
First compute \(g(2)=2^2=4\). Then \(f(g(2))=f(4)=2(4)-1=8-1=7\). Answer: 7.
21
Question
If \(h(x)=|x-3|\), find \(h(7)\) and \(h(1)\).
Page 2
Answer
Compute: \(h(7)=|7-3|=|4|=4\). \(h(1)=|1-3|=|-2|=2\). Answer: \(h(7)=4\), \(h(1)=2\).
22
Question
If \(f(x)=3x+2\), solve \(f(x)=11\) for \(x\).
Page 2
Answer
Set \(3x+2=11\). Subtract 2: \(3x=9\). Divide by 3: \(x=3\). Answer: 3.
23
Question
If \(g(x)=(x-1)(x+1)\), find \(g(-2)\).
Page 2
Answer
Compute: \(g(-2)=(-2-1)(-2+1)=(-3)(-1)=3\). Answer: 3.
24
Question
If \(p(x)=\dfrac{1}{x}\), find \(p(0.5)\).
Page 2
Answer
Compute \(p(0.5)=\dfrac{1}{0.5}=2\). Answer: 2.
25
Question
Given the table: when \(x=1, f(x)=2\); when \(x=2, f(x)=5\); when \(x=3, f(x)=10\). Find \(f(2)\).
Answer
From the table \(f(2)=5\). Answer: 5.
26
Question
Solve the proportion \(\dfrac{7}{x+3}=\dfrac{8}{x+5}\) for \(x\).
Page 2
Answer
Cross-multiply: \(7(x+5)=8(x+3)\). Expand: \(7x+35=8x+24\). Subtract 7x: \(35=x+24\). Subtract 24: \(x=11\). Answer: 11.
27
Question
In similar triangles, corresponding sides are proportional. In the figure, triangle ABC has side AB = 26.25, AC (vertical) = 14, BC (diagonal) = 29.75, and triangle XYZ has one corresponding side YX (horizontal) = ?, and YZ (diagonal) = 17 (corresponds to BC). Find the perimeter of triangle XYZ.
Page 2
Answer
Similarity ratio = \(\dfrac{YZ}{BC}=\dfrac{17}{29.75}=\dfrac{17}{29.75}=0.571428...=\dfrac{4}{7}\) (since 29.75=\(\tfrac{7}{4}\times17\)). Multiply each corresponding side: X Y (corresponds to AB): \(AB\times\frac{4}{7}=26.25\times\frac{4}{7}=15\). Vertical side (corresponds to AC=14): \(14\times\frac{4}{7}=8\). Diagonal is 17. Perimeter = \(15+8+17=40\). Answer: 40.
28
Question
Compute the area of a circle with radius \(r=6\). (Use \(\pi\) in the answer.)
Answer
Area of a circle: \(A=\pi r^2\). Substitute \(r=6\): \(A=\pi(6^2)=36\pi\). Answer: \(36\pi\) square units.
29
Question
Compute the circumference of a circle with diameter \(d=10\).
Answer
Circumference formula: \(C=\pi d\) or \(C=2\pi r\). With \(d=10\): \(C=10\pi\). Answer: \(10\pi\).
30
Question
Volume of a sphere with radius \(r=3\). (Use \(\pi\) in the answer.)
Answer
Volume of a sphere: \(V=\dfrac{4}{3}\pi r^3\). Substitute \(r=3\): \(V=\dfrac{4}{3}\pi(27)=36\pi\). Answer: \(36\pi\) cubic units.