Algebra
If f(x)=2x²-3x+1, find f(-2).
15
A rectangle has a length of 8 units and a width of 5 units. Find its perimeter.
26 units
The derivative tells you the instantaneous rate of change of a function. Find the derivative of f(x)=3x²+2x.
f'(x)=6x+2
Five students need to line up for a picture. The first position can be filled by any of the 5 students, the second by any of the remaining 4, and so on. How many different lineups are possible?
5×4×3×2×1=120 different lineups
List the multiples of 18 and 24 until you find the first number that appears in both lists. What is the least common multiple of 18 and 24?
72
Solve for x: 3(x-2)+2=2x+7.
x = 11
A triangle has a base of 10 units and a height of 6 units. Calculate its area.
30 square units
Evaluate the limit: lim as x approaches 3 of (x²-9)/(x-3).
6
A bag contains 5 red marbles, 4 blue marbles, and 3 green marbles. If you randomly select one marble, what is the probability that it is blue?
4/12 = 1/3
When any number is divided by 6, the remainder tells you how far it is from a multiple of 6. Since 7=6+1, find the remainder when 7¹⁰⁰ is divided by 6. Hint: What is the remainder when 7 is divided by 6? What happens when you multiply numbers with remainder 1?
1
A quadratic function is given by f(x)=x²-6x+11. Find its minimum value.
f(x)=(x-3)²+2. The minimum value is 2, occurring at x=3.
A circle has a radius of 4 units. Find its circumference.
8π units.
A definite integral can be evaluated using an antiderivative. Use the rule ∫x^n dx=x^(n+1)/(n+1). Evaluate the integral from 0 to 3 of (2x+1) dx. Find an antiderivative first, then subtract its value at 0 from its value at 3.
12
A standard deck has 52 cards, including 4 aces. You draw two cards without replacing the first card. What is the probability that both cards are aces?
(4/52)×(3/51)=1/221
Look at the sequence: 1, 4, 9, 16, 25, 36, ?. Find the difference between each pair of consecutive numbers. Use the pattern in those differences to determine the next number.
49. The differences are 3, 5, 7, 9, 11, so the next difference is 13.
Solve for x: x²-5x+6=0.
x = 2 or x = 3
A polygon has 6 sides. You can divide it into triangles by drawing diagonals from one vertex. Each triangle has an angle sum of 180°. How many triangles can you create, and what is the polygon's total interior angle sum?
4 triangles; 4(180°) = 720°
A function is given by f(x)=x³-3x²-9x+5. Critical points occur where the derivative equals zero. (1) Find the derivative. (2) Set it equal to zero and solve. (3) Use the sign of the derivative before and after each critical point to determine whether it is a local maximum or minimum.
f'(x)=3x²-6x-9=3(x-3)(x+1). Critical points: x=-1 and x=3. At x=-1 there is a local maximum; at x=3 there is a local minimum.
A fair six-sided die is rolled once. The possible outcomes are 1, 2, 3, 4, 5, and 6. You win $12 if you roll a 6, and you win $0 otherwise. Calculate the expected value of your winnings using E(X)=ΣP(x)·x.
E(X)=(1/6)(12)+(5/6)(0)=$2
Find the smallest positive integer that satisfies both conditions: (1) When divided by 5, it leaves a remainder of 2. (2) When divided by 7, it leaves a remainder of 3. Hint: Start with numbers that leave a remainder of 2 when divided by 5: 2, 7, 12, 17, 22, 27, ... Which is the first one that leaves a remainder of 3 when divided by 7?
17
Two functions are defined as f(x)=2x-3 and g(x)=x²+1. Evaluate g(f(2))+f(g(2)).
9
A square has side lengths of 10 units. A circle is drawn inside the square so that it touches all four sides. (1) What is the radius of the circle? (2) Find the area of the square. (3) Find the area inside the square but outside the circle.
Radius = 5. Square area = 100. Circle area = 25π. Area inside square but outside circle = 100 - 25π square units
Evaluate the integral from 0 to 1 of xe^(x²) dx. Hint: Use substitution. Let u=x². Then du=2x dx. Rewrite the integral in terms of u, change the limits, and evaluate.
(e-1)/2
A fair coin is flipped 5 times. Each flip has two possible outcomes: heads or tails. How many different sequences of 5 flips are possible? How many sequences contain exactly 3 heads? Use your results to find the probability of getting exactly 3 heads.
There are 2⁵=32 total sequences. There are 10 sequences with exactly 3 heads. Probability = 10/32 = 5/16.
A two-digit number has digits that add up to 11. When its digits are reversed, the new number is 27 greater than the original number. Find the original number.
47. The digits add to 11, and reversing it gives 74, which is 27 greater than 47.