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Junkao Mathematics - independent shared foundations study materials
Junkao Mathematics - independent shared foundations study resources

Junkao Mathematics - independent shared foundations practice questions

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Question 1Functions and differential calculus

What is the domain of definition for the real function f(x) = √(x + 2) / ln(1 - x)?

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

Explanation

For f(x) to be defined, three conditions must hold simultaneously: (1) radicand x + 2 ≥ 0 ⇒ x ≥ -2; (2) logarithm argument 1 - x > 0 ⇒ x < 1; (3) denominator ln(1 - x) ≠ 0 ⇒ 1 - x ≠ 1 ⇒ x ≠ 0. Combining these gives x ∈ [-2, 0) ∪ (0, 1).

Question 2Functions and differential calculus

Which of the following functions is an odd function (奇函数) defined on ℝ?

  1. f(x) = x³ + sin x Correct answer.
  2. f(x) = x² + cos x
  3. f(x) = e^x - 1
  4. f(x) = ln(x² + 1)

Explanation

A function is odd if f(-x) = -f(x) for all x in its domain. For f(x) = x³ + sin x: f(-x) = (-x)³ + sin(-x) = -x³ - sin x = -(x³ + sin x) = -f(x), confirming that it is an odd function.

Question 3Functions and differential calculus

Given the piecewise function f(x) defined by f(x) = 2^x - 1 for x ≤ 1, and f(x) = log₃(x - 1) for x > 1, what is the value of f(f(10))?

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

Explanation

First evaluate the inner function: since 10 > 1, f(10) = log₃(10 - 1) = log₃(9) = 2. Next, evaluate the outer function at x = 2: since 2 > 1, f(2) = log₃(2 - 1) = log₃(1) = 0. Thus, f(f(10)) = 0.

Question 4Functions and differential calculus

Given real numbers a = 2^(0.3), b = 0.3^2, and c = log₂(0.3), which of the following inequalities correctly orders a, b, and c?

  1. c < b < a Correct answer.
  2. c < a < b
  3. b < c < a
  4. a < b < c

Explanation

Comparing each quantity with the reference benchmarks 0 and 1: for a = 2^(0.3), since base 2 > 1 and exponent 0.3 > 0, a > 2^0 = 1; for b = 0.3^2 = 0.09, clearly 0 < b < 1; for c = log₂(0.3), since base 2 > 1 and argument 0.3 < 1, c < log₂(1) = 0. Therefore, c < 0 < b < 1 < a, which gives c < b < a.

Question 5Functions and differential calculus

What is the strictly decreasing interval of the composite function f(x) = log₀.₅(x² - 4x + 3)?

  1. (3, +∞) Correct answer.
  2. (-∞, 1)
  3. (2, +∞)
  4. (-∞, 2)

Explanation

First, the domain requires the inner quadratic u(x) = x² - 4x + 3 > 0 ⇒ (x - 1)(x - 3) > 0 ⇒ x < 1 or x > 3. The outer function y = log₀.₅(u) has base 0.5 < 1, so it is strictly decreasing with respect to u. By the composite function monotonicity rule (同增异减), f(x) is strictly decreasing where u(x) is strictly increasing and within the domain. The quadratic u(x) = (x - 2)² - 1 is strictly increasing on [2, +∞). Intersecting [2, +∞) with the domain x > 3 gives the interval (3, +∞).

Question 6Functions and differential calculus

Let f(x) be an odd function defined on ℝ that satisfies f(x + 2) = -f(x) for all x ∈ ℝ. If f(1) = 3, what is the value of f(2025) + f(2026)?

  1. 3 Correct answer.
  2. 0
  3. -3
  4. 6

Explanation

From f(x + 2) = -f(x), we have f(x + 4) = f((x + 2) + 2) = -f(x + 2) = -(-f(x)) = f(x), so f(x) has period T = 4. Since f(x) is odd and defined at x = 0, f(0) = 0. Then f(2) = f(0 + 2) = -f(0) = 0. Reducing the inputs modulo 4: 2025 = 4 × 506 + 1 ⇒ f(2025) = f(1) = 3; and 2026 = 4 × 506 + 2 ⇒ f(2026) = f(2) = 0. Therefore, f(2025) + f(2026) = 3 + 0 = 3.

Question 7Functions and differential calculus

The graph of the rational function f(x) = (2x - 1) / (x + 1) is symmetric with respect to which of the following points?

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

Explanation

Rewriting the rational fraction in standard hyperbolic form: f(x) = (2(x + 1) - 3) / (x + 1) = 2 - 3/(x + 1). The graph is a translation of y = -3/x shifted left by 1 unit and up by 2 units. Since y = -3/x is centrally symmetric about the origin (0, 0), the transformed curve has center of symmetry (-1, 2).

Question 8Functions and differential calculus

If f(x) = x² ln x, what is the value of its derivative at x = e, denoted f'(e)?

  1. 3e Correct answer.
  2. 2e
  3. e
  4. 3e²

Explanation

Using the product rule for differentiation: f'(x) = (x²)' ln x + x² (ln x)' = 2x ln x + x² (1/x) = 2x ln x + x. Evaluating at x = e: f'(e) = 2e ln(e) + e = 2e(1) + e = 3e.

Question 9Functions and differential calculus

What is the equation of the tangent line to the cubic curve y = x³ - 2x + 1 at the point (1, 0)?

  1. x - y - 1 = 0 Correct answer.
  2. x + y - 1 = 0
  3. 2x - y - 2 = 0
  4. x - 2y - 1 = 0

Explanation

Taking the derivative: y' = 3x² - 2. At the point of tangency x = 1, the slope of the tangent line is k = y'(1) = 3(1)² - 2 = 1. Using the point-slope formula with (x₀, y₀) = (1, 0): y - 0 = 1(x - 1) ⇒ x - y - 1 = 0.

Question 10Functions and differential calculus

What is the equation of the tangent line to the exponential curve y = e^x that passes through the origin (0, 0)?

  1. e·x - y = 0 Correct answer.
  2. x - y = 0
  3. 2ex - y = 0
  4. x - ey = 0

Explanation

Let the point of tangency on the curve be (x₀, e^(x₀)). The derivative is y' = e^x, so the slope of the tangent line is k = e^(x₀). The tangent line equation is y - e^(x₀) = e^(x₀)(x - x₀). Since the line passes through (0, 0), substitute (0, 0): 0 - e^(x₀) = e^(x₀)(0 - x₀) ⇒ -e^(x₀) = -x₀ e^(x₀). Since e^(x₀) > 0, dividing yields x₀ = 1. The slope is k = e¹ = e, so the line is y = e·x, or e·x - y = 0.

Question 11Functions and differential calculus

What is the strictly increasing interval of the function f(x) = x / e^x?

  1. (-∞, 1) Correct answer.
  2. (1, +∞)
  3. (-∞, 0)
  4. (0, 1)

Explanation

Using the quotient rule: f'(x) = (1 · e^x - x · e^x) / (e^x)² = (1 - x) / e^x. Since e^x > 0 for all x ∈ ℝ, the sign of f'(x) is determined entirely by 1 - x. Setting f'(x) > 0 gives 1 - x > 0 ⇒ x < 1. Therefore, f(x) is strictly increasing on (-∞, 1).

Question 12Functions and differential calculus

What is the local minimum value (极小值) of the cubic function f(x) = 2x³ - 9x² + 12x + 1?

  1. 5 Correct answer.
  2. 6
  3. 1
  4. 2

Explanation

Differentiating f(x): f'(x) = 6x² - 18x + 12 = 6(x² - 3x + 2) = 6(x - 1)(x - 2). Setting f'(x) = 0 gives critical points x = 1 and x = 2. Analyzing the sign of f'(x): f'(x) > 0 on (-∞, 1), f'(x) < 0 on (1, 2), and f'(x) > 0 on (2, +∞). Thus, x = 1 is a local maximum and x = 2 is a local minimum. Evaluating at x = 2: f(2) = 2(2)³ - 9(2)² + 12(2) + 1 = 16 - 36 + 24 + 1 = 5.

Question 13Functions and differential calculus

What is the absolute maximum value (最大值) of the function f(x) = x³ - 3x on the closed interval [-2, 2]?

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

Explanation

Differentiating gives f'(x) = 3x² - 3 = 3(x - 1)(x + 1) = 0 ⇒ x = ±1, both inside [-2, 2]. Evaluating f(x) at critical points and interval endpoints: f(-2) = (-2)³ - 3(-2) = -8 + 6 = -2; f(-1) = (-1)³ - 3(-1) = -1 + 3 = 2; f(1) = 1³ - 3(1) = -2; and f(2) = 2³ - 3(2) = 8 - 6 = 2. The absolute maximum value on [-2, 2] is 2.

Question 14Functions and differential calculus

If the cubic function f(x) = x³ + ax² + bx + c attains local extrema at x = 1 and x = 3, what is the value of the ratio a / b?

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

Explanation

The derivative is f'(x) = 3x² + 2ax + b. Since f(x) has extrema at x = 1 and x = 3, x = 1 and x = 3 are the roots of 3x² + 2ax + b = 0. By Vieta's formulas: sum of roots = 1 + 3 = 4 = -2a/3 ⇒ a = -6; product of roots = 1 · 3 = 3 = b/3 ⇒ b = 9. Therefore, the ratio a/b = -6 / 9 = -2/3.

Question 15Functions and differential calculus

For what range of the real parameter m does the cubic equation x³ - 3x - m = 0 have exactly 3 distinct real roots?

  1. (-2, 2) Correct answer.
  2. [-2, 2]
  3. (-∞, -2) ∪ (2, +∞)
  4. (-3, 3)

Explanation

The equation is equivalent to g(x) = x³ - 3x = m. Differentiating g(x): g'(x) = 3x² - 3 = 3(x - 1)(x + 1) = 0 ⇒ x = ±1. The local maximum is g(-1) = (-1)³ - 3(-1) = 2, and the local minimum is g(1) = 1³ - 3(1) = -2. The horizontal line y = m intersects the graph of g(x) at 3 distinct points if and only if m lies strictly between the local minimum and local maximum, i.e., -2 < m < 2.

Question 16Functions and differential calculus

What is the global maximum value of the function f(x) = (ln x) / x on the interval (0, +∞)?

  1. 1/e Correct answer.
  2. e
  3. 1
  4. 1/e²

Explanation

Taking the derivative using the quotient rule: f'(x) = (1 - ln x) / x². Setting f'(x) = 0 gives ln x = 1 ⇒ x = e. For x ∈ (0, e), f'(x) > 0 (function is increasing); for x ∈ (e, +∞), f'(x) < 0 (function is decreasing). Thus, f(x) achieves its global maximum at x = e, with value f(e) = (ln e) / e = 1/e.

Question 17Functions and differential calculus

If the tangent line to the curve y = ax² + ln x at x = 1 is parallel to the straight line y = 3x - 1, what is the value of the constant a?

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

Explanation

Differentiating y = ax² + ln x with respect to x gives y' = 2ax + 1/x. At x = 1, the slope of the tangent line is k = y'(1) = 2a(1) + 1/1 = 2a + 1. Since the tangent line is parallel to y = 3x - 1 (which has slope 3), their slopes must be equal: 2a + 1 = 3 ⇒ 2a = 2 ⇒ a = 1.

Question 18Functions and differential calculus

If the inequality e^x ≥ ax + 1 holds for all real numbers x ∈ ℝ, what is the unique value of the real parameter a?

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

Explanation

At x = 0, e⁰ = 1 and a(0) + 1 = 1, so the exponential curve y = e^x and the line y = ax + 1 intersect at (0, 1). For e^x ≥ ax + 1 to hold for all real x, the line must be tangent to y = e^x at (0, 1) without crossing it. Since (e^x)'|_{x=0} = e⁰ = 1, the tangent line slope is 1, so a = 1. (For a > 1 or a < 1, the line would cross above the curve for some x ≠ 0).

Question 19Functions and differential calculus

Let f(x) be a differentiable function on ℝ satisfying f'(x) > f(x) for all x ∈ ℝ and f(0) = 1. Which of the following relationships must be true regarding f(1)?

  1. f(1) > e Correct answer.
  2. f(1) < e
  3. f(1) = e
  4. f(1) > 1 but f(1) < e

Explanation

Construct the auxiliary function g(x) = f(x) / e^x. Differentiating g(x): g'(x) = (f'(x)e^x - f(x)e^x) / (e^x)² = (f'(x) - f(x)) / e^x. Since f'(x) > f(x) and e^x > 0, g'(x) > 0 for all x ∈ ℝ, meaning g(x) is strictly increasing. Therefore, g(1) > g(0) ⇒ f(1) / e¹ > f(0) / e⁰ = 1 / 1 = 1 ⇒ f(1) > e.

Question 20Functions and differential calculus

If the function f(x) = ln x - a/x has a unique zero-point in the open interval (1, e), what is the range of the parameter a?

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

Explanation

Setting f(x) = 0 gives ln x = a/x ⇒ a = x ln x. Let h(x) = x ln x for x ∈ (1, e). Differentiating h(x): h'(x) = ln x + 1. For x ∈ (1, e), ln x > 0 so h'(x) > 1 > 0, meaning h(x) is strictly increasing on (1, e). Evaluating at the endpoints: h(1) = 1 · ln 1 = 0, and h(e) = e · ln e = e. Since h(x) is strictly continuous and increasing from 0 to e on (1, e), the equation h(x) = a has a unique solution in (1, e) if and only if a ∈ (0, e).

Question 21Trigonometry

If the terminal ray of an angle α in standard position passes through the point P(-3, 4), what is the value of cos α?

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

Explanation

By the standard definition of trigonometric functions for general angles, with x = -3 and y = 4, the radius distance is r = √(x² + y²) = √((-3)² + 4²) = √25 = 5. Then cos α = x / r = -3/5.

Question 22Trigonometry

Given that tan α = 3, what is the value of the trigonometric fraction (2 sin α - cos α) / (sin α + 3 cos α)?

  1. 5/6 Correct answer.
  2. 1/2
  3. 5/4
  4. 2/3

Explanation

Divide every term in both the numerator and denominator by cos α (since cos α ≠ 0 when tan α = 3): (2 sin α - cos α) / (sin α + 3 cos α) = (2 tan α - 1) / (tan α + 3). Substituting tan α = 3 yields (2(3) - 1) / (3 + 3) = (6 - 1) / 6 = 5/6.

Question 23Trigonometry

What is the exact numerical value of sin(7π/6) + cos(2π/3)?

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

Explanation

Applying induced formulas (诱导公式): sin(7π/6) = sin(π + π/6) = -sin(π/6) = -1/2. Similarly, cos(2π/3) = cos(π - π/3) = -cos(π/3) = -1/2. Adding them together: -1/2 + (-1/2) = -1.

Question 24Trigonometry

Given acute angles α, β ∈ (0, π/2) such that sin α = 3/5 and cos β = 5/13, what is the exact value of cos(α - β)?

  1. 56/65 Correct answer.
  2. 16/65
  3. -16/65
  4. 63/65

Explanation

Since α, β ∈ (0, π/2): cos α = √(1 - sin² α) = √(1 - (3/5)²) = 4/5; and sin β = √(1 - cos² β) = √(1 - (5/13)²) = 12/13. Using the cosine difference formula cos(α - β) = cos α cos β + sin α sin β: cos(α - β) = (4/5)(5/13) + (3/5)(12/13) = 20/65 + 36/65 = 56/65.

Question 25Trigonometry

If cos α = -1/3, what is the value of cos 2α?

  1. -7/9 Correct answer.
  2. 7/9
  3. -1/9
  4. 1/9

Explanation

Using the double-angle formula for cosine: cos 2α = 2 cos² α - 1. Substituting cos α = -1/3: cos 2α = 2(-1/3)² - 1 = 2(1/9) - 1 = 2/9 - 9/9 = -7/9.

Question 26Trigonometry

Using the auxiliary angle method (辅助角公式), the expression f(x) = √3 sin x - cos x can be simplified into which of the following standard sinusoidal forms?

  1. 2 sin(x - π/6) Correct answer.
  2. 2 sin(x + π/6)
  3. 2 sin(x - π/3)
  4. 2 cos(x + π/3)

Explanation

Factoring out √(a² + b²) = √((√3)² + (-1)²) = √(3 + 1) = 2 gives: f(x) = 2((√3/2) sin x - (1/2) cos x). Since cos(π/6) = √3/2 and sin(π/6) = 1/2, this becomes 2(sin x cos(π/6) - cos x sin(π/6)) = 2 sin(x - π/6).

Question 27Trigonometry

For the sinusoidal function f(x) = 3 sin(2x + π/4), what are its minimum positive period T and its maximum value M?

  1. T = π, M = 3 Correct answer.
  2. T = 2π, M = 3
  3. T = π, M = 6
  4. T = π/2, M = 3

Explanation

For a sinusoidal function of the form f(x) = A sin(ωx + φ): the amplitude is |A| = 3, so its maximum value is M = 3. The angular frequency is ω = 2, so the minimal positive period is T = 2π / ω = 2π / 2 = π. Therefore, T = π and M = 3.

Question 28Trigonometry

Which of the following lines represents an axis of symmetry for the graph of the function f(x) = sin(2x - π/3) within the interval [0, π/2]?

  1. x = 5π/12 Correct answer.
  2. x = π/6
  3. x = π/3
  4. x = π/12

Explanation

The graph of y = sin(u) has vertical axes of symmetry where the argument reaches an extremum: u = kπ + π/2 for k ∈ ℤ. Setting 2x - π/3 = kπ + π/2 gives 2x = kπ + 5π/6 ⇒ x = kπ/2 + 5π/12. For k = 0, x = 5π/12, which lies in [0, π/2] (since 5π/12 ≈ 1.309 < π/2 ≈ 1.571).

Question 29Trigonometry

To obtain the graph of y = sin(2x + π/3) from the graph of y = sin 2x, which horizontal transformation should be applied?

  1. Shift to the left by π/6 units Correct answer.
  2. Shift to the left by π/3 units
  3. Shift to the right by π/6 units
  4. Shift to the right by π/3 units

Explanation

Factoring the coefficient 2 out of the argument: y = sin(2(x + π/6)). Replacing x with x + π/6 corresponds to shifting the entire graph horizontally to the left by π/6 units.

Question 30Trigonometry

In triangle ABC, the angles are A = 30° and B = 45°, and the side opposite angle A has length a = 2. What is the length of side b opposite angle B?

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

Explanation

Applying the Law of Sines (正弦定理): a / sin A = b / sin B. Substituting the given values: 2 / sin 30° = b / sin 45° ⇒ 2 / (1/2) = b / (√2/2) ⇒ 4 = 2b / √2 ⇒ b = 4(√2/2) = 2√2.

Question 31Trigonometry

In triangle ABC, the three side lengths are a = 3, b = 5, and c = 7. What is the exact measure of angle C?

  1. 120° (2π/3) Correct answer.
  2. 60° (π/3)
  3. 150° (5π/6)
  4. 135° (3π/4)

Explanation

Using the Law of Cosines (余弦定理): cos C = (a² + b² - c²) / (2ab). Substituting the side lengths: cos C = (3² + 5² - 7²) / (2 · 3 · 5) = (9 + 25 - 49) / 30 = -15 / 30 = -1/2. Since C is an interior angle of a triangle (0° < C < 180°), cos C = -1/2 implies C = 120° (2π/3).

Question 32Trigonometry

In triangle ABC, side lengths are b = 4 and c = 6, and the included angle is A = 60°. What is the area of triangle ABC?

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

Explanation

The area formula for a triangle given two sides and the included angle is S = (1/2) bc sin A. Substituting the given values: S = (1/2) · 4 · 6 · sin 60° = 12 · (√3/2) = 6√3.

Question 33Trigonometry

In triangle ABC, if the relation a cos A = b cos B holds, what is the geometric classification of triangle ABC?

  1. Isosceles triangle or right-angled triangle (等腰三角形或直角三角形) Correct answer.
  2. Equilateral triangle only (等边三角形)
  3. Isosceles triangle only (等腰三角形)
  4. Right-angled triangle only (直角三角形)

Explanation

By the Law of Sines, a = 2R sin A and b = 2R sin B. Substituting into the relation gives 2R sin A cos A = 2R sin B cos B ⇒ sin 2A = sin 2B. In a triangle, sin 2A = sin 2B implies either 2A = 2B ⇒ A = B (triangle is isosceles), or 2A + 2B = 180° ⇒ A + B = 90° ⇒ C = 90° (triangle is right-angled). Therefore, triangle ABC is either an isosceles triangle or a right-angled triangle.

Question 34Trigonometry

In triangle ABC, angle A = 60° and the opposite side has length a = √3. What is the maximum possible area of triangle ABC?

  1. 3√3 / 4 Correct answer.
  2. √3 / 2
  3. 3 / 4
  4. √3

Explanation

By the Law of Cosines: a² = b² + c² - 2bc cos A ⇒ 3 = b² + c² - bc. By the AM-GM inequality, b² + c² ≥ 2bc, so 3 = b² + c² - bc ≥ 2bc - bc = bc ⇒ bc ≤ 3, with equality when b = c = √3. The area is S = (1/2) bc sin A = (1/2) bc sin 60° = (√3/4) bc. Since bc ≤ 3, the maximum area is S_max = (√3/4)(3) = 3√3 / 4.

Question 35Trigonometry

In triangle ABC, angle C = 60° and side c = 2. What is the complete range of the perimeter P = a + b + c of triangle ABC?

  1. (4, 6] Correct answer.
  2. (2, 4]
  3. (4, 8]
  4. [4, 6]

Explanation

By the Law of Sines: 2R = c / sin C = 2 / (√3/2) = 4√3 / 3. Then a + b = 2R(sin A + sin B) = (4√3/3)(sin A + sin(120° - A)) = (4√3/3)(√3 sin(A + 30°)) = 4 sin(A + 30°). Since 0° < A < 120°, we have 30° < A + 30° < 150°, so 1/2 < sin(A + 30°) ≤ 1. Thus, a + b ∈ (2, 4], with maximum 4 attained when A = B = 60° (equilateral triangle). Adding c = 2 gives perimeter P = a + b + c ∈ (4, 6].

Question 36Sequences and series

In an arithmetic sequence {a_n}, the first term is a₁ = 3 and the fourth term is a₄ = 12. What is the value of the seventh term a₇?

  1. 21 Correct answer.
  2. 18
  3. 24
  4. 15

Explanation

In an arithmetic sequence, a₄ = a₁ + 3d. Substituting: 12 = 3 + 3d ⇒ 3d = 9 ⇒ d = 3. Then the seventh term is a₇ = a₁ + 6d = 3 + 6(3) = 3 + 18 = 21 (or directly using equal index steps: a₇ = 2a₄ - a₁ = 2(12) - 3 = 21).

Question 37Sequences and series

In an arithmetic sequence {a_n}, the first term is a₁ = 2 and the common difference is d = 3. What is the sum of the first 10 terms, S₁₀?

  1. 155 Correct answer.
  2. 150
  3. 160
  4. 145

Explanation

Using the arithmetic sequence sum formula: S_n = n a₁ + n(n - 1)d / 2. For n = 10: S₁₀ = 10(2) + (10 · 9 · 3) / 2 = 20 + 135 = 155.

Question 38Sequences and series

In an arithmetic sequence {a_n}, if a₃ + a₇ = 18, what is the sum of the first 9 terms, S₉?

  1. 81 Correct answer.
  2. 72
  3. 90
  4. 162

Explanation

By the properties of arithmetic sequences, since 3 + 7 = 1 + 9 = 2 · 5: a₃ + a₇ = 2a₅ = 18 ⇒ a₅ = 9. The sum of the first 9 terms is S₉ = 9(a₁ + a₉) / 2 = 9a₅ = 9 · 9 = 81.

Question 39Sequences and series

In a geometric sequence {a_n} of strictly positive real numbers, the second term is a₂ = 6 and the fourth term is a₄ = 24. What is the value of the sixth term a₆?

  1. 96 Correct answer.
  2. 48
  3. 192
  4. 72

Explanation

In a geometric sequence, a₄ / a₂ = q² = 24 / 6 = 4. Since all terms are positive, the common ratio is q = 2. Then the sixth term is a₆ = a₄ · q² = 24 · 4 = 96.

Question 40Sequences and series

In a geometric sequence {a_n}, the first term is a₁ = 3 and the common ratio is q = 2. What is the sum of the first 6 terms, S₆?

  1. 189 Correct answer.
  2. 192
  3. 93
  4. 381

Explanation

Using the geometric series sum formula: S_n = a₁(1 - q^n) / (1 - q). For n = 6: S₆ = 3(1 - 2⁶) / (1 - 2) = 3(1 - 64) / (-1) = 3(-63) / (-1) = 189.

Question 41Sequences and series

In a geometric sequence {a_n}, if a₃ · a₇ = 16, what is the value of a₅²?

  1. 16 Correct answer.
  2. 4
  3. 256
  4. 8

Explanation

By the geometric mean property of geometric sequences, since 3 + 7 = 5 + 5: a₃ · a₇ = a₅ · a₅ = a₅². Since a₃ · a₇ = 16, it follows immediately that a₅² = 16.

Question 42Sequences and series

If the sum of the first n terms of a sequence {a_n} is given by S_n = 2n² + 3n, what is the formula for the general term a_n?

  1. 4n + 1 Correct answer.
  2. 4n - 1
  3. 4n + 3
  4. 2n + 1

Explanation

For n = 1: a₁ = S₁ = 2(1)² + 3(1) = 5. For n ≥ 2: a_n = S_n - S_{n-1} = (2n² + 3n) - (2(n - 1)² + 3(n - 1)) = 2n² + 3n - (2(n² - 2n + 1) + 3n - 3) = 2n² + 3n - (2n² - n - 1) = 4n + 1. Checking n = 1: 4(1) + 1 = 5 = a₁, so a_n = 4n + 1 for all n ≥ 1.

Question 43Sequences and series

If the sum of the first n terms of a sequence {a_n} is S_n = 3^n - 1, what is the numerical value of the fourth term a₄?

  1. 54 Correct answer.
  2. 80
  3. 27
  4. 162

Explanation

Using the relation a_n = S_n - S_{n-1} for n ≥ 2: a₄ = S₄ - S₃ = (3⁴ - 1) - (3³ - 1) = 3⁴ - 3³ = 81 - 27 = 54.

Question 44Sequences and series

A sequence {a_n} satisfies the first-order linear recurrence relation a_{n+1} = 2a_n + 1 with initial term a₁ = 1. What is the general term formula for a_n?

  1. 2^n - 1 Correct answer.
  2. 2^(n-1)
  3. 2^(n+1) - 3
  4. 2^n + 1

Explanation

Construct an auxiliary geometric sequence by adding 1 to both sides: a_{n+1} + 1 = 2a_n + 2 = 2(a_n + 1). Let b_n = a_n + 1. Then b₁ = a₁ + 1 = 1 + 1 = 2, and b_{n+1} = 2b_n, so {b_n} is a geometric sequence with first term 2 and common ratio 2: b_n = 2 · 2^(n-1) = 2^n. Thus, a_n = b_n - 1 = 2^n - 1.

Question 45Sequences and series

A sequence {a_n} satisfies a₁ = 2 and a_{n+1} - a_n = 3n for all n ≥ 1. What is the value of the tenth term a₁₀?

  1. 137 Correct answer.
  2. 135
  3. 152
  4. 120

Explanation

Using the accumulation method (累加法): for n ≥ 2, a_n = a₁ + ∑_{k=1}^{n-1} (a_{k+1} - a_k) = 2 + 3 ∑_{k=1}^{n-1} k = 2 + 3 · ((n - 1)n / 2). For n = 10: a₁₀ = 2 + 3 · ((9 · 10) / 2) = 2 + 3(45) = 2 + 135 = 137.

Question 46Sequences and series

What is the sum of the first n terms S_n = ∑_{k=1}^n 1 / (k(k + 1)) using the telescoping fractions method (裂项相消法)?

  1. n / (n + 1) Correct answer.
  2. 1 / (n + 1)
  3. (n + 1) / n
  4. n / (2n + 1)

Explanation

Splitting each term into partial fractions: 1 / (k(k + 1)) = 1/k - 1/(k + 1). Summing from k = 1 to n: S_n = (1 - 1/2) + (1/2 - 1/3) + (1/3 - 1/4) + ... + (1/n - 1/(n + 1)). Intermediate terms cancel in pairs, leaving S_n = 1 - 1/(n + 1) = n / (n + 1).

Question 47Sequences and series

Using the grouping summation method (分组求和法), what is the sum of the first n terms S_n of the sequence a_n = 2n + 2^n?

  1. n(n + 1) + 2^(n+1) - 2 Correct answer.
  2. n² + 2^(n+1) - 2
  3. n(n + 1) + 2^n - 1
  4. 2n² + 2^(n+1)

Explanation

Group the terms into an arithmetic part and a geometric part: S_n = ∑_{k=1}^n (2k + 2^k) = 2 ∑_{k=1}^n k + ∑_{k=1}^n 2^k. The arithmetic sum is 2 · (n(n + 1) / 2) = n(n + 1). The geometric sum is 2(1 - 2^n) / (1 - 2) = 2^(n+1) - 2. Adding both parts yields S_n = n(n + 1) + 2^(n+1) - 2.

Question 48Sequences and series

Using the arithmetico-geometric series summation method (错位相减法), what is the sum T_n = ∑_{k=1}^n k · 2^k?

  1. (n - 1)2^(n+1) + 2 Correct answer.
  2. n · 2^(n+1) - 2
  3. (n + 1)2^n - 2
  4. (n - 1)2^n + 2

Explanation

Write out T_n = 1 · 2¹ + 2 · 2² + 3 · 2³ + ... + n · 2^n. Multiply by 2: 2T_n = 1 · 2² + 2 · 2³ + ... + (n - 1) · 2^n + n · 2^(n+1). Subtracting 2T_n from T_n: -T_n = 2¹ + 2² + 2³ + ... + 2^n - n · 2^(n+1). The geometric sum is 2(2^n - 1)/(2 - 1) = 2^(n+1) - 2. Thus -T_n = 2^(n+1) - 2 - n · 2^(n+1) = (1 - n)2^(n+1) - 2. Multiplying by -1 yields T_n = (n - 1)2^(n+1) + 2.

Question 49Sequences and series

In an arithmetic sequence {a_n}, the first term is a₁ = 25 and the common difference is d = -2. For what value of n does the sum of the first n terms S_n achieve its absolute maximum?

  1. 13 Correct answer.
  2. 12
  3. 14
  4. 25

Explanation

The general term is a_n = 25 + (n - 1)(-2) = 27 - 2n. Setting a_n ≥ 0 gives 27 - 2n ≥ 0 ⇒ n ≤ 13.5. Since n is an integer, the terms are strictly positive for n ≤ 13 (with a₁₃ = 27 - 26 = 1 > 0) and strictly negative for n ≥ 14 (with a₁₄ = 27 - 28 = -1 < 0). Adding positive terms increases the sum, while adding negative terms decreases it. Therefore, S_n achieves its unique maximum at n = 13 (where S₁₃ = 13(25 + 1)/2 = 169).

Question 50Sequences and series

What is the infinite sum S = ∑_{n=1}^∞ 1 / ((2n - 1)(2n + 1))?

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

Explanation

Decomposing each term into partial fractions: 1 / ((2n - 1)(2n + 1)) = (1/2)(1/(2n - 1) - 1/(2n + 1)). The partial sum is S_N = (1/2)[(1 - 1/3) + (1/3 - 1/5) + ... + (1/(2N - 1) - 1/(2N + 1))] = (1/2)(1 - 1/(2N + 1)). Taking the limit as N → ∞: S = lim_{N→∞} (1/2)(1 - 1/(2N + 1)) = (1/2)(1 - 0) = 1/2.