# Free Test Online

## Standardized Tests Math and Science Practice

• Question #1: What is sec(a)sin(a) + csc(a)cos(a) , if a = 45o?
(a) √2
(b) 2
(c) 1
(d) 1/√2
(e) 1/2

Answer: sec(a)sin(a) + csc(a)cos(a) = [1/cos(a)]sin(a) + [1/sin(a)]cos(a) = sin(a)/cos(a) + cos(a)/sin(a) = (√2/2)/(√2/2) + (√2/2)/(√2/2) = 1 + 1 = 2.

• Question #2: In the standard (x,y) coordinate plane, 3 corners of a square are (4,–3), (–4,3), and (–4,-3). What are the coordinates of the square’s fourth corner?
(a) (1/4, -1/3)
(b) (4, 3)
(c) (3, 4)
(d) (3, -4)
(e) (3, -4)

The easiest way to solve this problem is to sketch the 3 corners of the square. We notice that the fourth corner of the square is (4, 3).

• Question #3: Prizes totaling \$60,000 were awarded unequally between 3 contestants. Which of the following choices could be the highest prize?(a) \$5,000
(b) \$10,000
(c) \$15,000
(d) \$25,000
(e) \$65,000

Answer: The highest prize should be higher than one third of \$60,000 and lower than \$60,000. The only answer that satisfies this condition is \$25,000.

• Question #4: What is the area of rectangle ABCD if the length of the side AB is 5?
(a) 40
(b) 50
(c) 25
(d) 25√3
The area of rectangle ABCD is AD·BC = 5√3·5 = 25√3.

• Question #5: What is the circumference of circle O from the figure below if the length of the sides of the ABCD square is 5 inches?(a) 10·¶ inches
(b) 5·¶ inches
(c) ¶ inches
(d) 16 inches
(e) 12; inches

Answer: Angle BOA is 60o and OA = OB, so triangle BOA is equilateral, and the radius of the circle is equal to the side of the square ABCD.
The circumference of the circle is 2·¶5 = 10·¶

• Question #6: Ellipse (x – 3)2/9 + (y – 5)2/5 = 1 is inscribed in the circle O. The transverse of the ellipse is equal to the diameter of the circle O. The equation that describes circle O in the standard (x,y) plane is:
(a) (x + 3)2 + (y + 5)2 = 32
(b) (x – 3)2 + (y – 5)2 = 1
(c) (x – 3)2 + (y – 5)2 = 32
(d) (x – 3)3 + (y – 5)3 = 33
(e) (x – 3)2 + (y – 5)2= 7

Answer: If a>b, the transverse of the ellipse (x – m)2/a2 + (y – n)2/b2 = 1 is equal to 2a.
In our case the transverse diameter that is equal to the diameter of the circle is 2·3 = 6, so the radius of the circle is 3.
The center of the circle is (3, 5) in the (x, y) plane.
A circle with a center in (3, 5) and of radius 3, will be described by the equation:
(x – 3)2 + (y – 5)2 = 32

• Question #7: The marked price of shirts in a store is \$25. During one week, the store sells m shirts at the marked price and n shirts discounted 25% of the marked price. Which of the following is an expression of the average price of the shirts sold?(a) (\$25m + \$18.75n) /(m + n)
(b) (\$25m + \$18.75n) /(2m + 2n)
(c) \$25m + \$18.75n
(d) (\$25 + \$20) /(m + n)
(e) (m + n) / (\$25m + \$20n)

Answer: The price of a discounted shirt is (1 – .25)\$25 = \$18.75.
The total revenue during the week is (\$25m + \$18.75n) and the number of shirts sold is (m + n).
In conclusion the average price of the shirts is (\$25m + \$18.75n)/(m + n).

• Question #8: A box contains 7 blue marbles, 4 red marbles and 6 green marbles. How many additional blue marbles must be added in the box so that the probability of randomly drawing a blue marble is 1/2?

(a) 2
(b) 3
(c) 4
(d) 5
(e) 6

Answer: If the probability of randomly drawing a blue marble is 1/2 after x blue marbles were added, the number of blue marbles must be half of the total numbers of marbles.
1/2 = (x + 7)/(x + 7 + 4 + 6)
x = 3

• Question #9: A number is increased 50% and the resulting number is decreased 50%. The final number is what percent of the original number?(a) 50%
(b) 125%
(c) 75%
(d) 100%
(e) 105%

Answer: If the original number is x, after it is increased 50% the resulting number is 1.5x.
1.5x is decreased 50% so the final number is .5·1.5 = .75·x or 75% of x.

• Question #10: The inequality 2x + 3 > 5x – 6 is equivalent with which of the following inequalities?(a) x > 6
(b) x < 6
(c) x > 3
(d) x < 3
(e) x < 10

Answer:2x + 3 > 5x – 6
9 > 3x
3 > x
x < 3

Examples of ACT geometry questions