UND MATHEMATICS TRACK MEET Individual Test 1

University of North Dakota Grades 11/12
January 12, 2026
School Team Name
Calculators are allowed. Student Name

• 1.

  Find all real solutions to the equation $2^{x+1}+2^{1-x}=5$.

    (2 pts) 1.

• 2.

  The expression ${\displaystyle \frac{a^2-b^2}{a-b}}$ simplifies to $a+b$ for all $a\ne b$. If $a=3+\sqrt{5}$ and $b=3-\sqrt{5}$, find the exact value of ${\displaystyle \frac{a^3-b^3}{a-b}}$.

    (3 pts) 2.

• 3.

  A rectangle has a diagonal of length 10 cm and one side that is 2 cm longer than the other. Find the dimensions of the rectangle.

    (3 pts) 3.

• 4.

  A surveyor stands on level ground and observes the top of a building. The line of sight to the top of the building makes an angle of elevation of ${35}^{\circ }$. The surveyor then walks 50 meters directly toward the building, where the angle of elevation increases to ${50}^{\circ }$. If the surveyor’s eyes are 1.6 meters above the ground, find the horizontal distance from the first observation point to the building to the nearest meter.

    (3 pts) 4.

• 5.

  Consider the three lines

  $$L_1:2x+3y=12,L_2:y=x-1,L_3:x=2.$$

  Compute the area of the triangle formed by these intersections (three decimal places).

    (3 pts) 5.

• 6.

  A student council has 12 members: 5 seniors, 4 juniors, and 3 sophomores. A committee of 4 students is selected at random. What is the probability that the committee contains exactly two seniors? (three decimal places)

    (3 pts) 6.

• 7.

  A landscape designer plans a triangular flower garden. Two boundary edges measure $110m$ and $75m$, and the angle between them after redesign will be ${63}^{\circ }$. Find the area of the garden to the nearest square meter.

    (3 pts) 7.

TOTAL POINTS