Mathematics

Section 1: General Mathematics

100

What is the probability of randomly pulling a card that is neither red nor a face card from a standard 52-card deck?

0.385

200

Some random guy is making a burger. He has two options for his bun type, three options for meat, five cheese options, seven options for vegetable toppings, and four condiments. He will choose exactly one of each option except for vegetables, where he will choose at least two and no more than four. How many unique burgers can he make?

10,920 unique burgers

300

How long will it take for a deposit to increase in value by 60% if the account offers 4% interest compounded monthly?

11 years and 10 months

400

16 students have to be divided evenly into 4 non-distinct teams. If 4 students want to be on different teams from each other, how many ways of dividing the students are there?

369,600 ways

500

A sum of money is split evenly and deposited into 2 accounts that offer 6% per annum. 1 account pays simple interest. The other pays annual compound interest. After 3 years, the difference in account value is $1500. What is the original sum, rounded to the nearest dollar?

$272,331

Section 2: Right Triangles and Coordinate Geometry

100

What is the distance of (1, –5, 4) from the center of the sphere described by (𝑥 − 3) ² + (𝑦 + 2) ² + (𝑧 − 1) ² = 34?

d = √22

200

Find the area of the quadrilateral with coordinates (–7, 15), (7, 15), (2.5, 19), (–2.5, 19).

A = 38

300

The points (1, 3), (3, a), and (–3, a – 9) are collinear. Find a.

a = 6

400

The point (𝑥, 𝑦) is three times nearer to (–7, 3) than it is to (1, –1). What is 𝑥/𝑦?

𝑥/𝑦 = -5/2

500

Find the equation of the line that divides line 𝐴𝐵, A(4, 2) B(–5, –10), in the ratio 2 : 1, and is perpendicular to the line 5𝑥 + 3𝑦 + 2 = 0.

5y = 3x - 24

Section 2: Polygons and 3D Forms

100

A cone has a radius of 5 inches and a slant height of 13 inches. What is its volume?

V = 314.16 in³

200

A sphere is inscribed in a cube with edge length 7. What is the sphere’s surface area? pi = 3.14.

SA = 153.86 u²

300

A regular polygon’s interior angles are 7 times the measure of its exterior angles. How many sides does the polygon have?

16 sides

400

All but one of a regular convex polygon’s interior angles add to 2005.7°. What is the missing angle?

154.3°

500

A cone with radius 5 and height 12 is painted blue starting from the base to cover 50% of the cone’s total surface area. What percentage of its lateral area is blue?

31%

Section 2: Similarity and Circles

100

What is the area of a circle with the equation 𝑥 ² − 10𝑥 + 𝑦 ² − 6𝑥 = 30?

A = 64pi

200

What is the area of a circle with a diameter from (18, 13) to (–7, 32), pi = 3.14?

A = 774.4

300

2 secants to a circle have internal and external segments of (a; 20) and (12; 16). What is a?

a = 2.4

400

2 tangents to a circle from a common external point enclose an angle of 36°. What are the measures of the intercepted arcs.

144° and 216°

500

Point R lies on a circle. Point Q lies on the same circle and it divides the secant PR in the ratio 1 : 8. What is the length of the tangent from P in terms of line 𝑃𝑄?

3𝑃𝑄

Section 3: Differential Calculus

100

f(x) = 8x⁵ − 3𝑥. By taking repeated derivatives, find the fourth derivative, 𝑓 ′′′′(𝑥).

𝑓 ′′′′(𝑥) = 960x

200

Evaluate lim 𝑡→∞ = 2t−1 / 2t²+t−1 .

lim 𝑡→∞ = 0

300

What is the slope of the line tangent to ℎ(𝑥) = 2𝑥 / 2−√𝑥 at 𝑥 = 2?

ℎ'(𝑥) = 7.536

400

A tank is being filled with water, such that the volume of water in relation to t minutes elapsed is 3𝑉(𝑡) = 6𝑡³ − 12𝑡² − 18𝑡. At what point is V increasing at a rate of 39.5?

3.5 min or 210 seconds

500

Find the quadratic function 𝑔(𝑥) such that 𝑔(𝑥) − 𝑔′(𝑥) = 2𝑥² − 9𝑥 + 5

g(x) = 2𝑥² − 5𝑥