112年 - 112-1 國立南科國際實驗高級中學正式教師甄選試題_雙語部:數學專業能力#114450

選擇題 (5)

1. ( ) How many pairs of positive integers (x, y) are there satisfying x² + y²= 6x + 8y? (A) 4 (B) 5 (C) 7 (D) 12

2. ( ) Given f(x) = x³ + 3x² − 2. How many distinct real roots of f(f(x))? (A) 3 (B) 5 (C) 7 (D) 9

3. ( ) Define the notation [t] means “the greatest integer not exceeding t” and {t} means “the fractional part of t”. Which of the following is false? (A) t ＞ 1 (B) 2t＞ 1314 (C) [t] = 657 (D)

4. ( ) Known a, b, c are side of ∆ABC and log_a b is a double root of equation x² − 2x + ?inC + c??C = 0 then which is correct in the followings? (A) acute triangle (B) equilateral triangle (C) obtuse angled triangle (D) isosceles right-angled triangle.

5. ( ) Choose the correct relations of the values of following definite integrals.
(A) I₁＜ I₂, I₃ ＜ I₄ (B) I₁＜ I₂, I₃ ＞ I₄ (C) I₁ ＞ I₂, I₃ ＜ I₄ (D) I₁ ＞ I₂, I₃ ＞ I₄

申論題 (14)

1. Given a polynomial f(x) whose degree is no less than 3. The remainders of f(x) being divided by (x − a)(x − b), (x − b)(x − c), (x − a)(x − c) are −x − 5, 4x, 3x + 3 respectively. Find the remainder of f(x) being divided by (x − a)(x − b)(x − c).

2. How many nonnegative integer solutions are there to the equation x + 2y + 3z = 17?

3. Find the largest possible volume of the tetrahedron PQRS, where P(1,0,0), Q(0,2,0), R(0,0,3) and S lies on the unit sphere centered at the origin.

4. The parametric curve (x, y) = (2 cos θ , 2 sin θ + 3 cos θ), 0 ≤ θ ≤ 2π represents an ellipse. Find the foci that lie in the first quadrant.

5. Let f(x) = x(x − 1)(x − 2) … (x − 2023) and g(x) = f(f(x)). Calculate g ′ (1).

6. Find all a such that both of equations x² + αx + 1996 = 0 and x ² + 1996x + α= 0 has two integer roots.

7. Appling Gram-Schmidt method, it can decomposition vector = (3, −4) and = (4,3), α, β are real number. Please find the 2 × 2 matrix P, such that .

8. If the distribution of discrete random variable X is ,where p is positive real number, please find the variance Var(X) of X.

1. There was a class of 41 people divided into three groups A, B, and C, and the average score of subject natural’s test was 89.7 points; the average score of group A was 89.8 points; the average score of A, B combination was 88.5 points; the average score of A and C combination was 91 points, and how many people were in each of groups A, B and C?

2. Given an angle^θ ，the solution of equation
Please find the solution of equation

3. As figure 1, two 3 by 4 rectangles overlap in such a way that their sides are perpendicular. If the area and perimeter of the shaded region are 22 and 20 respectively, compute length AB.

4. Compute the greatest real number K for which the graphs of (|x| − 5)² + (|y| − 5)²= K and (x − 1)² + (y + 1)² = 37 have exactly two intersection points.

5. Suppose the values of α and β are independently chosen uniformly from the interval (0,2). Provided that the numbers (1, α, β) can form the sides of a triangle, find the probability that the numbers form an acute triangle.

四、證明題：
The sequence {a_n}：1, 1/2, 1/3,….Is it possible to select a subsequence from {an} such that each term in it (except the first two) is the difference of the preceding two?

(1) for 5 terms sequence b₁, b₂, ⋯ , b₅ (2) infinite terms sequence {c_n}.
Explain your answer!

112年 - 112-1 國立南科國際實驗高級中學正式教師甄選試題_雙語部:數學專業能力#114450

試卷資訊

選擇題 (5)

申論題 (14)