I85PREP
⌘ K
Bremen Abitur Math GK study materials
Bremen Abitur Math GK study resources

Bremen Abitur Math GK practice questions

One question at a time. Build your Bremen Abitur Math GK understanding.

First 50 of 190 questions. Each one lists the choices, the correct answer, and an explanation. The checker at the top of this page saves progress in this browser when it is available.

Continue in the practice tool above to load more questions, 50 at a time.

Question 1Calculus

What is the derivative of f(x) = 4x³ − 2x + 7?

  1. f'(x) = x⁴ − x² + 7x
  2. f'(x) = 12x² − 2 Correct answer.
  3. f'(x) = 4x² − 2
  4. f'(x) = 12x² − 2x

Explanation

Differentiate termwise: (4x³)' = 12x², (−2x)' = −2, and the constant 7 vanishes. So f'(x) = 12x² − 2.

Question 2Calculus

What is the derivative of f(x) = e^(2x)?

  1. f'(x) = e^(2x)
  2. f'(x) = 2x·e^(2x−1)
  3. f'(x) = e²
  4. f'(x) = 2e^(2x) Correct answer.

Explanation

Chain rule: d/dx e^(u(x)) = e^u · u' with u = 2x, so u' = 2 and f'(x) = 2e^(2x).

Question 3Calculus

Find the zeros of f(x) = x² − 5x + 6.

  1. x = 1 and x = 6
  2. x = 2 and x = 3 Correct answer.
  3. x = −2 and x = −3
  4. x = 0 and x = 5

Explanation

Factor: x² − 5x + 6 = (x − 2)(x − 3). Alternatively, x = (5 ± √(25 − 24))/2 = (5 ± 1)/2 → 3 and 2.

Question 4Calculus

If f'(x) < 0 on an open interval, then on that interval f is…

  1. constant
  2. always positive
  3. strictly decreasing Correct answer.
  4. strictly increasing

Explanation

A negative first derivative means the function falls as x increases, so f is strictly monotonically decreasing.

Question 5Calculus

A sufficient criterion for a local maximum at an interior point x₀ is…

  1. f''(x₀) = 0 only
  2. f'(x₀) = 0 and f''(x₀) < 0 Correct answer.
  3. f'(x₀) = 0 and f''(x₀) > 0
  4. f(x₀) = 0

Explanation

At a stationary point, f''(x₀) < 0 implies local concavity downward, hence a local maximum (Hochpunkt).

Question 6Calculus

What is f'(x) for f(x) = sin(x) + cos(x)?

  1. cos(x) + sin(x)
  2. cos(x) − sin(x) Correct answer.
  3. sin(x) − cos(x)
  4. −sin(x) − cos(x)

Explanation

(sin x)' = cos x and (cos x)' = −sin x, so f'(x) = cos(x) − sin(x).

Question 7Calculus

For f(x) = x³ − 3x, evaluate f(2).

  1. −2
  2. 8
  3. 2 Correct answer.
  4. 5

Explanation

Substitute x = 2: f(2) = 2³ − 3·2 = 8 − 6 = 2.

Question 8Calculus

Which statement about Stammfunktionen is correct?

  1. Every continuous f has exactly one antiderivative
  2. ∫_a^b f always equals F(b) without subtracting F(a)
  3. Antiderivatives exist only for polynomials
  4. If F' = f, then (F + 7)' = f as well Correct answer.

Explanation

Adding a constant does not change the derivative, so the full family is F + C.

Question 9Calculus

Find the local extrema of f(x) = x³ − 12x + 1. Which statement is correct?

  1. Only one stationary point at x = 0
  2. No stationary points
  3. Local maximum at x = −2 with f(−2) = 17; local minimum at x = 2 with f(2) = −15 Correct answer.
  4. Local minimum at x = −2; local maximum at x = 2

Explanation

f'(x) = 3x² − 12 = 3(x² − 4) = 0 ⇒ x = ±2. f''(x) = 6x, so f''(−2) = −12 < 0 (local max) and f''(2) = 12 > 0 (local min). f(−2) = −8 + 24 + 1 = 17; f(2) = 8 − 24 + 1 = −15.

Question 10Calculus

The area between y = 4 − x² and the x-axis from x = −2 to x = 2 equals…

  1. 16/3
  2. 8
  3. 4
  4. 32/3 Correct answer.

Explanation

∫₋₂² (4 − x²) dx = [4x − x³/3]₋₂² = (8 − 8/3) − (−8 + 8/3) = 16/3 − (−16/3) = 32/3.

Question 11Calculus

Solve f'(x) = 0 for f(x) = x·e^(−x). The only critical point is…

  1. x = −1
  2. x = e
  3. x = 1 Correct answer.
  4. x = 0

Explanation

Product rule: f'(x) = e^(−x) − x·e^(−x) = e^(−x)(1 − x) = 0 ⇒ x = 1.

Question 12Calculus

An inflection point of f(x) = x³ − 3x² + 2 requires f''(x) = 0 with a sign change. Where does this occur?

  1. x = 3
  2. x = 1 Correct answer.
  3. x = 0
  4. x = 2

Explanation

f'(x) = 3x² − 6x; f''(x) = 6x − 6 = 6(x − 1). So f''(x) = 0 at x = 1 with a sign change → Wendepunkt.

Question 13Calculus

For the family f_a(x) = x² + a·x, the vertex (Scheitel) has x-coordinate…

  1. x = −a
  2. x = a
  3. x = −a/2 Correct answer.
  4. x = a/2

Explanation

For ax²+bx+c with a=1, b=a_param: x = −b/(2a) = −a/2. Or f'(x) = 2x + a = 0 ⇒ x = −a/2.

Question 14Calculus

If F is an antiderivative of f and F(0) = 4, F(3) = 10, then ∫ from 0 to 3 of f(x) dx equals…

  1. 10
  2. 6 Correct answer.
  3. 14
  4. 4

Explanation

By the Fundamental Theorem, ∫₀³ f = F(3) − F(0) = 10 − 4 = 6.

Question 15Calculus

For f(x) = −x² + 6x − 5, the maximum value is…

  1. 9
  2. −5
  3. 1
  4. 4 Correct answer.

Explanation

Vertex at x = −6/(2·(−1)) = 3. f(3) = −9 + 18 − 5 = 4.

Question 16Calculus

A particle’s position is s(t) = t³ − 6t² + 9t (metres, t in seconds). Its velocity at t = 2 is…

  1. 9 m/s
  2. −3 m/s Correct answer.
  3. 3 m/s
  4. 0 m/s

Explanation

v(t) = 3t² − 12t + 9; v(2) = 12 − 24 + 9 = −3 m/s.

Question 17Calculus

For f(x) = 2x³ − 9x² + 12x, find f'(1).

  1. 2
  2. 5
  3. 12
  4. 0 Correct answer.

Explanation

f'(x) = 6x² − 18x + 12; f'(1) = 6 − 18 + 12 = 0. Indeed x=1 is a stationary point (f'(x)=6(x²−3x+2)=6(x−1)(x−2)).

Question 18Calculus

The tangent line to f(x) = x² at x = 3 has slope…

  1. 3
  2. 2
  3. 6 Correct answer.
  4. 9

Explanation

f'(x) = 2x, so f'(3) = 6. The tangent is y − 9 = 6(x − 3).

Question 19Calculus

Compute ∫ from −1 to 1 of (x³ + 2) dx.

  1. 1
  2. 4 Correct answer.
  3. 0
  4. 2

Explanation

∫₋₁¹ x³ dx = 0 (odd) and ∫₋₁¹ 2 dx = 4, so the sum is 4.

Question 20Calculus

The area enclosed between y = x and y = x² on [0,1] equals…

  1. 1/2
  2. 1/3
  3. 1/12
  4. 1/6 Correct answer.

Explanation

On [0,1], x ≥ x², so ∫₀¹ (x − x²) dx = [x²/2 − x³/3]₀¹ = 1/2 − 1/3 = 1/6.

Question 21Calculus

For f(x)=x⁴ − 2x² + 1 = (x²−1)², the number of distinct real zeros is…

  1. 3
  2. 4
  3. 2 Correct answer.
  4. 1

Explanation

(x²−1)²=0 ⇒ x=±1, each with multiplicity 2, so two distinct real zeros. Graph touches the axis at both.

Question 22Vectors

Two nonzero vectors are perpendicular if and only if their scalar product is…

  1. −1
  2. equal to the product of their lengths
  3. 0 Correct answer.
  4. 1

Explanation

a · b = |a||b| cos θ; perpendicular means θ=90° and cos 90°=0, so a·b=0.

Question 23Vectors

In Bremen Mathematik GK, which AG topic is explicitly NOT treated computationally in Teil B?

  1. Intersection of a line and a plane
  2. Lagebeziehung of two planes Correct answer.
  3. Scalar products of vectors
  4. Distance from a point to a line

Explanation

GK Konkretisierung: Die Lagebeziehung zweier Ebenen wird nicht rechnerisch betrachtet. Scalar products, point–line distance, and line–plane intersection remain standard.

Question 24Vectors

Find the intersection of the line X = ⟨1,0,0⟩ + t⟨0,1,1⟩ with the plane x + y + z = 4.

  1. (1, 2, 2)
  2. (4, 0, 0)
  3. (1, 0, 0)
  4. (1, 1.5, 1.5) Correct answer.

Explanation

Plug in: 1 + t + t = 4 ⇒ 1 + 2t = 4 ⇒ t = 1.5. Point: ⟨1, 1.5, 1.5⟩.

Question 25Vectors

Are the lines g: X=⟨0,0,0⟩+s⟨1,1,0⟩ and h: X=⟨0,0,1⟩+t⟨1,−1,0⟩ intersecting, parallel, or skew?

  1. Intersecting at the origin
  2. Identical (coincident)
  3. Skew (windschief): not parallel, no intersection Correct answer.
  4. Parallel

Explanation

Directions ⟨1,1,0⟩ and ⟨1,−1,0⟩ are not scalar multiples → not parallel. Setting equal: s=t, s=−t, 0=1 - contradiction. Different planes z=0 vs z=1 → windschief.

Question 26Vectors

Distance from point P(1,2,2) to line X=⟨0,0,0⟩+t⟨1,0,0⟩ (the x-axis) is…

  1. √8 = 2√2 Correct answer.
  2. 1
  3. 2
  4. 3

Explanation

Foot is (1,0,0); distance = |⟨0,2,2⟩| = √8 = 2√2. Formula: |PQ × d|/|d| with Q origin gives same.

Question 27Vectors

Which vector is orthogonal to both ⟨1,0,0⟩ and ⟨0,1,0⟩?

  1. ⟨0,0,1⟩ Correct answer.
  2. ⟨1,1,0⟩
  3. ⟨1,0,1⟩
  4. ⟨0,1,1⟩

Explanation

A vector orthogonal to both i and j must be parallel to k = ⟨0,0,1⟩ (cross product).

Question 28Vectors

The lines X=⟨1,1,1⟩+s⟨2,0,0⟩ and X=⟨3,1,1⟩+t⟨0,1,0⟩ …

  1. intersect at (3,1,1) Correct answer.
  2. are parallel
  3. are identical
  4. are skew and never meet

Explanation

Directions ⟨2,0,0⟩ and ⟨0,1,0⟩ not parallel. Set equal: 1+2s=3, 1=1+t, 1=1 ⇒ s=1, t=0. Point (3,1,1).

Question 29Vectors

Lines g: X=⟨1,0,0⟩+s⟨1,1,1⟩ and h: X=⟨0,1,−1⟩+t⟨2,2,2⟩ are…

  1. parallel (same direction family) and distinct Correct answer.
  2. intersecting at a unique point
  3. perpendicular
  4. identical

Explanation

⟨2,2,2⟩=2⟨1,1,1⟩ → parallel. Is (0,1,−1) on g? 1+s=0, s=1, −s=−1 ⇒ s=−1 from z, but y: 0+s=1 ⇒ s=1 - contradiction. Distinct parallel lines.

Question 30Vectors

A line with direction ⟨1,1,1⟩ is parallel to the plane x − 2y + z = 5 if and only if…

  1. ⟨1,1,1⟩ · ⟨1,−2,1⟩ = 0 Correct answer.
  2. ⟨1,1,1⟩ · ⟨1,−2,1⟩ = 5
  3. the line contains the origin
  4. the normal is ⟨1,1,1⟩

Explanation

Direction ⊥ normal for parallelism to the plane: 1−2+1=0, which holds. The constant 5 does not enter the direction test.

Question 31Probability Statistics

For a Bernoulli trial with success probability p=0.3, the failure probability is…

  1. 0.7 Correct answer.
  2. 0.3
  3. 1.3
  4. 0

Explanation

Success and failure probabilities sum to 1, so q = 1 − p = 1 − 0.3 = 0.7.

Question 32Probability Statistics

A 95%-style rough σ-interval idea at GK: for relative frequency with n=400, 1/√n equals…

  1. 0.05 Correct answer.
  2. 0.025
  3. 0.1
  4. 20

Explanation

1/√400 = 1/20 = 0.05. This magnitude often appears in GK discussions of estimation accuracy (σ ≈ √(p(1−p)/n) ≤ 1/(2√n)).

Question 33Probability Statistics

Which statement correctly reflects Bremen GK Stochastik vs LK emphasis?

  1. GK focuses on binomial models, trees/tables, and estimation ideas; Hypothesentests are an LK Stochastik Alternative (HY), not the GK Schwerpunkt Correct answer.
  2. Every GK paper must include a full Hypothesentest with errors of type I and II
  3. Stochastik is optional and can be skipped in Teil B
  4. Only continuous normal integrals without binomial models appear in GK

Explanation

Bremen SPT assigns Stochastik as a Pflicht Teil-B Aufgabe (15 BE) at GK depth with binomial/conditional/estimation tools. Hypothesentests are listed as an LK Alternative (HY) in the Konkretisierungen - not the GK focus.

Question 34Calculus

Find the natural domain of the real-valued function: f(x) = sqrt(2x - 8).

  1. [4, infinity) Correct answer.
  2. (4, infinity)
  3. (-infinity, 4]
  4. [0, infinity)

Explanation

A square root function is defined over real numbers when its radicand is non-negative: 2x - 8 >= 0 => 2x >= 8 => x >= 4. In interval notation, this is [4, infinity).

Question 35Calculus

Find the range of the quadratic function f(x) = x^2 - 6x + 14 for all real x.

  1. [0, infinity)
  2. (-infinity, 5]
  3. [5, infinity) Correct answer.
  4. [14, infinity)

Explanation

Complete the square: f(x) = (x - 3)^2 - 9 + 14 = (x - 3)^2 + 5. Since (x - 3)^2 >= 0 for all real x, the minimum value occurs at x = 3 with f(3) = 5. Therefore, the range is [5, infinity).

Question 36Calculus

Find the inverse function f^(-1)(x) of the bijective linear function f(x) = (2x + 1) / 3.

  1. (3x + 1) / 2
  2. (3x - 1) / 2 Correct answer.
  3. (2x - 1) / 3
  4. 3 / (2x + 1)

Explanation

Set y = (2x + 1) / 3. Solve for x: 3y = 2x + 1 => 2x = 3y - 1 => x = (3y - 1) / 2. Swapping variables gives f^(-1)(x) = (3x - 1) / 2.

Question 37Calculus

Evaluate the limit: lim_(x -> 3) (x^2 - 9) / (x - 3).

  1. 0
  2. 3
  3. 6 Correct answer.
  4. does not exist

Explanation

Direct substitution yields 0/0 indeterminate form. Factor the numerator: x^2 - 9 = (x - 3)(x + 3). Cancel (x - 3) for x != 3: lim_(x -> 3) (x + 3) = 3 + 3 = 6.

Question 38Calculus

Find the derivative f'(x) of the polynomial function f(x) = 4x^3 - 5x^2 + 7x - 9.

  1. 12x^2 - 10x + 7 Correct answer.
  2. 12x^2 - 5x + 7
  3. 4x^2 - 10x + 7
  4. 12x^3 - 10x^2 + 7x

Explanation

Apply the power rule d/dx[x^n] = n*x^(n-1) to each term: d/dx[4x^3] = 12x^2; d/dx[-5x^2] = -10x; d/dx[7x] = 7; d/dx[-9] = 0. Combining these gives f'(x) = 12x^2 - 10x + 7.

Question 39Calculus

Evaluate the definite integral: int_0^2 (3x^2 - 2x + 1) dx.

  1. 4
  2. 6 Correct answer.
  3. 8
  4. 10

Explanation

Find the antiderivative F(x): int (3x^2 - 2x + 1) dx = x^3 - x^2 + x. Apply the Fundamental Theorem of Calculus from 0 to 2: F(2) = (2)^3 - (2)^2 + 2 = 8 - 4 + 2 = 6. F(0) = 0. Therefore, the integral evaluates to 6 - 0 = 6.

Question 40Calculus

Find the equation of the tangent line to the curve y = x^2 - 4x + 1 at the point where x = 3.

  1. y = 2x - 8 Correct answer.
  2. y = 2x - 6
  3. y = -2x + 4
  4. y = 2x + 8

Explanation

At x = 3, y = (3)^2 - 4(3) + 1 = 9 - 12 + 1 = -2, so the point of tangency is (3, -2). The derivative is y' = 2x - 4. At x = 3, the slope is m = 2(3) - 4 = 2. Using point-slope form: y - (-2) = 2(x - 3) => y + 2 = 2x - 6 => y = 2x - 8.

Question 41Calculus

Find the local maximum value of the polynomial function f(x) = 2x^3 - 3x^2 - 12x + 5.

  1. 5
  2. 18
  3. -15
  4. 12 Correct answer.

Explanation

Compute the first derivative: f'(x) = 6x^2 - 6x - 12 = 6(x^2 - x - 2) = 6(x - 2)(x + 1). Critical points are x = -1 and x = 2. Second derivative: f''(x) = 12x - 6. At x = -1: f''(-1) = -18 < 0, indicating a local maximum. Evaluate: f(-1) = 2(-1)^3 - 3(-1)^2 - 12(-1) + 5 = -2 - 3 + 12 + 5 = 12. (At x = 2, f''(2) = 18 > 0 gives the local minimum f(2) = -15).

Question 42Calculus

Evaluate the limit: lim_(x -> 0) (sin(5x)) / (3x).

  1. 5/3 Correct answer.
  2. 1
  3. 3/5
  4. 0

Explanation

Use the fundamental trigonometric limit lim_(u -> 0) (sin(u) / u) = 1. Rewrite the expression: (sin(5x) / (3x)) = (5/3) * (sin(5x) / (5x)). Taking the limit as x -> 0 gives (5/3) * 1 = 5/3.

Question 43Calculus

Find the derivative value f'(2) for the rational function f(x) = (x^2 + 1) / (x - 1).

  1. -1 Correct answer.
  2. 0
  3. 1
  4. 3

Explanation

Use the quotient rule: f'(x) = [(2x)(x - 1) - (x^2 + 1)(1)] / (x - 1)^2. Simplify the numerator: 2x^2 - 2x - x^2 - 1 = x^2 - 2x - 1. Substitute x = 2: f'(2) = ((2)^2 - 2(2) - 1) / (2 - 1)^2 = (4 - 4 - 1) / 1 = -1.

Question 44Calculus

Evaluate the definite integral: int_0^(pi/2) cos^2(x) dx.

  1. pi/2
  2. pi/4 Correct answer.
  3. 1
  4. pi/8

Explanation

Use the half-angle formula cos^2(x) = (1 + cos(2x)) / 2: int_0^(pi/2) cos^2(x) dx = int_0^(pi/2) (1/2 + (1/2)*cos(2x)) dx = [x/2 + (1/4)*sin(2x)]_0^(pi/2). Evaluating at the limits: (pi/4 + (1/4)*sin(pi)) - (0 + (1/4)*sin(0)) = pi/4 + 0 - 0 = pi/4.

Question 45Calculus

Determine the natural domain of the function: f(x) = ln((x - 2) / (5 - x)).

  1. (-infinity, 2) U (5, infinity)
  2. (2, 5) Correct answer.
  3. [2, 5]
  4. (2, 5]

Explanation

The natural logarithm is defined strictly for positive arguments: (x - 2) / (5 - x) > 0. Multiply numerator and denominator by -1 (or rewrite): (x - 2) / (x - 5) < 0. The critical numbers are x = 2 and x = 5. Testing intervals: for x < 2, (-)/(-) = + > 0; for 2 < x < 5, (+)/(-) = - < 0 (satisfying the inequality); for x > 5, (+)/(+) = + > 0. Thus the domain is (2, 5).

Question 46Calculus

At what point in the first quadrant on the parabola y = x^2 is the tangent line parallel to the line y = 4x - 5?

  1. (1, 1)
  2. (4, 16)
  3. (2, 4) Correct answer.
  4. (2, 3)

Explanation

The slope of the line y = 4x - 5 is m = 4. Two lines are parallel when their slopes are equal, so we set the derivative y' = 2x equal to 4: 2x = 4 => x = 2. The corresponding y-coordinate on the parabola is y = (2)^2 = 4. Thus the point is (2, 4).

Question 47Calculus

Find the absolute minimum value of the function f(x) = x + 4/x on the closed interval [1, 4].

  1. 2
  2. 4 Correct answer.
  3. 5
  4. 8

Explanation

Find critical points by setting the derivative to zero: f'(x) = 1 - 4/x^2 = 0 => x^2 = 4 => x = 2 (since x in [1, 4]). Evaluate f at the critical point and endpoints: f(1) = 1 + 4/1 = 5; f(2) = 2 + 4/2 = 4; f(4) = 4 + 4/4 = 5. The absolute minimum value on [1, 4] is 4.

Question 48Calculus

Calculate the area of the plane region bounded by the parabola y = 4 - x^2 and the line y = 0.

  1. 16/3
  2. 16
  3. 8
  4. 32/3 Correct answer.

Explanation

Find the x-intercepts: 4 - x^2 = 0 => x = -2 and x = 2. Since the parabola is symmetric and non-negative on [-2, 2], the area is A = int_(-2)^2 (4 - x^2) dx = 2 * int_0^2 (4 - x^2) dx = 2 * [4x - x^3/3]_0^2 = 2 * (8 - 8/3) = 2 * (16/3) = 32/3.

Question 49Calculus

Given the function f(x) = e^(2x) * cos(x), evaluate the derivative value f'(0).

  1. 0
  2. 1
  3. 2 Correct answer.
  4. -1

Explanation

Apply the product rule: f'(x) = d/dx[e^(2x)] * cos(x) + e^(2x) * d/dx[cos(x)] = 2*e^(2x)*cos(x) - e^(2x)*sin(x). Evaluating at x = 0: f'(0) = 2*e^0*cos(0) - e^0*sin(0) = 2(1)(1) - (1)(0) = 2.

Question 50Calculus

A rectangle has a fixed perimeter of 24 cm. What is the maximum possible area of this rectangle?

  1. 20 cm^2
  2. 32 cm^2
  3. 36 cm^2 Correct answer.
  4. 48 cm^2

Explanation

Let x and y denote the side lengths: 2x + 2y = 24 => x + y = 12 => y = 12 - x. The area is A(x) = x(12 - x) = 12x - x^2. To find the maximum, take the derivative: A'(x) = 12 - 2x = 0 => x = 6 cm. Then y = 12 - 6 = 6 cm (a square). The maximum area is A(6) = 6 * 6 = 36 cm^2.