TestAS Practice Test 2026

About the TestAS exam

The Test for Academic Studies (Germany) (TestAS) is a university entrance examination administered in Germany. It is designed to assess candidates' knowledge and proficiency across 5 key subject areas including Core Test, Humanities, Engineering, Mathematics. This exam is widely recognized as an essential benchmark for students and professionals.

Use short TestAS practice sets to identify topics to revisit. Choose a subject and difficulty, answer questions, then review the explanations. New accounts get 30 question credits; further new questions use paid credits. Retaking previously seen questions is free.

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8 free TestAS practice questions (2026)

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1. A function f(x) is defined as f(x) = x^3 - 6x^2 + 9x. What is the area enclosed by the curve f(x) and the x-axis between x = 0 and x = 3?

   • 4.5
   • 2.25
   • 6.75
   • 8.25
   Show answer & explanation

   Answer: A. 4.5

   The area is given by the definite integral of f(x) from 0 to 3. ∫(x^3 - 6x^2 + 9x) dx from 0 to 3 = [x^4/4 - 2x^3 + (9/2)x^2] from 0 to 3 = (81/4 - 54 + 81/2) - 0 = 4.5.

   Topic: Mathematics · hard

2. A biased coin has a probability of 0.6 of landing heads. If the coin is tossed 3 times, what is the probability of getting exactly 2 heads?

   • 0.432
   • 0.288
   • 0.648
   • 0.36
   Show answer & explanation

   Answer: A. 0.432

   The probability is given by the binomial probability formula: P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of trials, k is the number of successes, and p is the probability of success. In this case, n = 3, k = 2, and p = 0.6. P(X = 2) = (3 choose 2) * (0.6)^2 * (0.4)^1 = 3 * 0.36 * 0.4 = 0.432.

   Topic: Mathematics · hard

3. What is the limit of the sequence a_n = (n^2 + 1) / (2n^2 - n) as n approaches infinity?

   • 0
   • 1
   • 1/2
   • Infinity
   Show answer & explanation

   Answer: C. 1/2

   Divide both the numerator and denominator by the highest power of n, which is n^2. This gives (1 + 1/n^2) / (2 - 1/n). As n approaches infinity, 1/n^2 and 1/n approach 0, so the limit becomes 1/2.

   Topic: Mathematics · hard

4. Find the derivative of the function f(x) = ln(sin(x^2)).

   • 2x * cot(x^2)
   • cot(x^2)
   • 2x * tan(x^2)
   • tan(x^2)
   Show answer & explanation

   Answer: A. 2x * cot(x^2)

   Using the chain rule: f'(x) = (1/sin(x^2)) * cos(x^2) * 2x = 2x * cot(x^2).

   Topic: Mathematics · hard

5. Solve the inequality |2x - 3| < 5.

   • -1 < x < 4
   • -4 < x < 1
   • x < -1 or x > 4
   • x < -4 or x > 1
   Show answer & explanation

   Answer: A. -1 < x < 4

   The inequality |2x - 3| < 5 can be rewritten as -5 < 2x - 3 < 5. Adding 3 to all parts gives -2 < 2x < 8. Dividing by 2 gives -1 < x < 4.

   Topic: Mathematics · hard

6. What is the sum of the infinite geometric series 1 + 1/3 + 1/9 + 1/27 + ... ?

   • 3/2
   • 2/3
   • 1
   • Infinity
   Show answer & explanation

   Answer: A. 3/2

   The sum of an infinite geometric series is given by S = a / (1 - r), where a is the first term and r is the common ratio. In this case, a = 1 and r = 1/3. Therefore, S = 1 / (1 - 1/3) = 1 / (2/3) = 3/2.

   Topic: Mathematics · hard

7. If matrix A = [[2, 1], [3, 4]], what is the determinant of A?

   • 5
   • 11
   • -5
   • -11
   Show answer & explanation

   Answer: A. 5

   The determinant of a 2x2 matrix [[a, b], [c, d]] is ad - bc. In this case, (2 * 4) - (1 * 3) = 8 - 3 = 5.

   Topic: Mathematics · hard

8. A circle has a radius of 5. What is the area of a sector of the circle with a central angle of 72 degrees?

   • 5π
   • 10π
   • 2π
   • 25π
   Show answer & explanation

   Answer: A. 5π

   The area of a sector is given by (θ/360) * πr^2, where θ is the central angle in degrees and r is the radius. In this case, (72/360) * π * 5^2 = (1/5) * 25π = 5π.

   Topic: Mathematics · hard

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TestAS Exam Subjects

Available practice subjects: Core Test, Humanities, Engineering, Mathematics, Natural Sciences. Select a subject for focused preparation. Coverage is not a guarantee of the complete official syllabus.

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