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Sample Junkao Mathematics Practice Questions

Try these sample questions to review concepts for the Junkao Mathematics exam. Each question includes a detailed explanation. Start the interactive quiz above for the full 100+ question experience with AI tutoring.

1Given sets A = {x ∈ ℝ | x² - 3x - 4 ≤ 0} and B = {x ∈ ℝ | x > 1}, which of the following sets represents the intersection A ∩ B?
A.(1, 4]
B.[-1, 4]
C.[-1, 1)
D.(1, 4)
Explanation: Factoring the quadratic inequality gives x² - 3x - 4 = (x - 4)(x + 1) ≤ 0, which defines set A as the closed interval [-1, 4]. Intersecting A with set B = (1, +∞) gives A ∩ B = (1, 4].
2Let the universal set be U = ℝ. If set A = {x | x ≥ 2} and set B = {x | x < -1}, what is (∁_U A) ∩ B?
A.{x | x < -1}
B.{x | x < 2}
C.{x | -1 ≤ x < 2}
D.∅
Explanation: The complement of set A in ℝ is ∁_U A = {x | x < 2} = (-∞, 2). Intersecting (-∞, 2) with set B = (-∞, -1) yields (-∞, -1), which is {x | x < -1}.
3For a real number x, proposition p: 'x > 2' is what type of logical condition for proposition q: 'x² > 4'?
A.Sufficient but not necessary condition (充分不必要条件)
B.Necessary but not sufficient condition (必要不充分条件)
C.Sufficient and necessary condition (充要条件)
D.Neither sufficient nor necessary condition (既不充分也不必要条件)
Explanation: If x > 2, then x² > 4 is always true, meaning p ⇒ q (p is sufficient for q). However, x² > 4 means x > 2 or x < -2 (for example, x = -3 gives x² = 9 > 4 but x > 2 is false), so q does not imply p. Thus, p is a sufficient but not necessary condition for q.
4What is the logical negation of the proposition '∀ x ∈ ℝ, x² - x + 1 > 0'?
A.∃ x₀ ∈ ℝ, x₀² - x₀ + 1 ≤ 0
B.∀ x ∈ ℝ, x² - x + 1 ≤ 0
C.∃ x₀ ∈ ℝ, x₀² - x₀ + 1 < 0
D.∀ x ∈ ℝ, x² - x + 1 < 0
Explanation: The negation of a universal statement '∀ x ∈ M, p(x)' is the existential statement '∃ x₀ ∈ M, ¬p(x₀)'. The logical negation of the strict inequality '> 0' is '≤ 0', producing '∃ x₀ ∈ ℝ, x₀² - x₀ + 1 ≤ 0'.
5If the solution set of the quadratic inequality ax² + bx + 2 > 0 is the interval (-1/2, 1/3), what is the value of a + b?
A.-14
B.14
C.-10
D.10
Explanation: Since the solution set of ax² + bx + 2 > 0 is bounded between two finite numbers (-1/2, 1/3), the parabola opens downwards with a < 0, and the roots of ax² + bx + 2 = 0 are x₁ = -1/2 and x₂ = 1/3. By Vieta's formulas: x₁ · x₂ = (-1/2)(1/3) = -1/6 = 2/a ⇒ a = -12. Next, x₁ + x₂ = -1/2 + 1/3 = -1/6 = -b/a = -b/(-12) = b/12 ⇒ b = -2. Therefore, a + b = -12 + (-2) = -14.
6What is the complete solution set of the rational inequality (x - 3) / (x + 1) ≤ 0?
A.(-1, 3]
B.[-1, 3]
C.(-∞, -1) ∪ [3, +∞)
D.(-1, 3)
Explanation: The rational inequality (x - 3)/(x + 1) ≤ 0 is equivalent to (x - 3)(x + 1) ≤ 0 subject to the domain restriction x + 1 ≠ 0. The product inequality yields -1 ≤ x ≤ 3, and excluding x = -1 gives the half-open interval (-1, 3].
7What is the solution set of the absolute value inequality |2x - 1| ≤ 5?
A.[-2, 3]
B.[-3, 2]
C.(-∞, -2] ∪ [3, +∞)
D.[-2, 2]
Explanation: Expanding the absolute value inequality |2x - 1| ≤ 5 gives the double inequality -5 ≤ 2x - 1 ≤ 5. Adding 1 to all sides yields -4 ≤ 2x ≤ 6, and dividing by 2 gives -2 ≤ x ≤ 3, which corresponds to the closed interval [-2, 3].
8For x > 0, what is the minimum value achieved by the function f(x) = x + 9/x?
A.6
B.9
C.3
D.12
Explanation: By the Arithmetic Mean-Geometric Mean (AM-GM) inequality (基本不等式), for any positive real number x, f(x) = x + 9/x ≥ 2√(x · (9/x)) = 2√9 = 6. Equality holds if and only if x = 9/x ⇒ x² = 9 ⇒ x = 3 (since x > 0), so the minimum value is 6.
9Given the system of linear constraints: x + y - 2 ≥ 0, x - y + 2 ≤ 0, and y ≤ 3, what is the maximum value of the objective function z = 2x + y?
A.5
B.3
C.2
D.1
Explanation: The feasible region is a triangle bounded by three vertices: (-1, 3) from the intersection of x + y = 2 and y = 3; (1, 3) from x - y = -2 and y = 3; and (0, 2) from x + y = 2 and x - y = -2. Evaluating the objective function z = 2x + y at these vertices: z(-1, 3) = 2(-1) + 3 = 1; z(0, 2) = 2(0) + 2 = 2; and z(1, 3) = 2(1) + 3 = 5. Therefore, the maximum value is 5.
10If the quadratic inequality x² - 2kx + 4 ≥ 0 holds for all real numbers x ∈ ℝ, what is the complete range of the parameter k?
A.[-2, 2]
B.(-∞, -2] ∪ [2, +∞)
C.[-4, 4]
D.(-2, 2)
Explanation: For the quadratic f(x) = x² - 2kx + 4 (with leading coefficient 1 > 0) to remain non-negative for all x ∈ ℝ, the parabola must lie entirely on or above the x-axis, requiring its discriminant to satisfy Δ ≤ 0. Computing the discriminant: Δ = (-2k)² - 4(1)(4) = 4k² - 16 ≤ 0 ⇒ 4k² ≤ 16 ⇒ k² ≤ 4 ⇒ -2 ≤ k ≤ 2, which is the closed interval [-2, 2].

About the Junkao Mathematics Exam

Mathematics is one of the named cultural subjects in the current soldier military-academy examination. The official paper is in Chinese and requires mathematical working beyond answer recognition. This independent English-language single-answer MCQ bank supports concept and calculation review; it is not an official translation, current-paper replica, or substitute for constructed written solutions.

Exam sponsor: Central Military Commission training and military academy admissions authorities. The requirements and fees below concern the certification or admission exam, separate from our free practice resources.

Assessment

Written mathematics subject; detailed current scope and official preparation materials are supplied through the military examination system.

Time Limit

150 minutes in the published 2023 schedule; verify the current annual notice.

Passing Score

Competitive composite selection; no standalone permanent public pass mark.

Exam / Certification Fees

No public individual fee schedule located for this internal military selection.

Exam sponsor website

Fees, eligibility, and exam policies can change. Confirm them with the exam sponsor before applying or paying.

Our practice resources: topics covered

We aim to reflect publicly available exam outlines and topic information in our study resources. Coverage, format, and difficulty may differ from the actual exam, and we cannot guarantee that every detail is accurate or current. Confirm exam requirements, fees, and policies with the official exam sponsor.

not-published

Algebra and Functions

Equations, inequalities, functions, sequences, and algebraic reasoning.

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Trigonometry and Vectors

Trigonometric relationships, identities, and vector reasoning.

not-published

Geometry

Plane, solid, analytic, and coordinate geometry.

not-published

Probability and Statistics

Counting, probability, data, and statistical reasoning.

not-published

Calculus Foundations

Limits, derivatives, integrals, and applications where included by current authorized scope.

Preparing for the Junkao Mathematics Exam

What You Need to Know

  • Passing score: Competitive composite selection; no standalone permanent public pass mark.
  • Assessment: Written mathematics subject; detailed current scope and official preparation materials are supplied through the military examination system.
  • Time limit: 150 minutes in the published 2023 schedule; verify the current annual notice.
  • Exam / certification fees: No public individual fee schedule located for this internal military selection. Official sources

Using Our Practice Resources

  • Work through all 100 available questions
  • Review every answer and explanation
  • Track weak areas and revisit them
  • Use our AI tutor for tough concepts

Junkao Mathematics: Suggested Study Strategy

1Write complete solutions before checking options.
2Verify domain restrictions and units.
3Use the current authorized Chinese materials for exact scope.

Frequently Asked Questions

Is the official exam an English MCQ test?

No. The official subject is a Chinese-language written mathematics paper; this bank is a study adaptation.

Is a fixed public question count available?

No stable current public source located provides one.

What should supplement this bank?

Timed Chinese-language problems and full written solutions using the current authorized materials.