北京市高二英语上学期高考模拟试卷

阅读下面短文，从每题所给的A、B、C、D四个选项中选出最佳选项。 Dr. Sarah Chen, a renowned mathematician, often tells her students that the beauty of mathematics lies in its ___21___. She recalls her own early struggles with calculus, which ___22___ led her to develop a unique teaching method. “Many students fear math because they see it as a ___23___ subject,” she says. “But I believe it’s about ___24___ and creativity.” Her approach involves using real-world problems to ___25___ abstract concepts. For instance, she uses the example of a roller coaster’s path to explain derivatives. “The ___26___ of the track at any point is like the slope of the tangent line,” she explains. This method helps students ___27___ the practical applications of math. Dr. Chen also emphasizes the importance of ___28___. “Making mistakes is a ___29___ part of learning,” she insists. “Each error is a stepping stone to ___30___.” Her students often find that this mindset ___31___ their anxiety and boosts their confidence. Beyond the classroom, Dr. Chen runs a popular online forum where students can ___32___ problems and share insights. “The goal is to create a ___33___ where learning is collaborative,” she says. Her efforts have ___34___ many students to pursue careers in STEM fields. She remains a(n) ___35___ to all who believe that with the right guidance, anyone can master mathematics.

阅读下面短文，在空白处填入1个适当的单词或括号内单词的正确形式。 The concept of zero, which seems so simple today, was a revolutionary idea in the history of mathematics. Its invention is ___36___ (credit) to ancient Indian mathematicians. Before zero, counting systems were ___37___ (limit) and cumbersome. The introduction of zero as a placeholder made arithmetic ___38___ (significance) easier. Zero also plays a crucial role in algebra. It is the identity element for addition, meaning any number ___39___ (add) to zero remains unchanged. Furthermore, zero is the foundation ___40___ the concept of negative numbers, ___41___ (allow) for a complete number line. In calculus, the concept of a limit approaches zero. The derivative of a function at a point is defined as the limit of the difference quotient as the change in x approaches ___42___. This idea is fundamental to understanding rates of change. Despite its utility, zero presented philosophical challenges. Early Greek mathematicians struggled with the idea of “nothingness” ___43___ (represent) by a symbol. Today, zero is not just a number; it is a ___44___ (symbol) of the power of abstract thought. Its journey from a dot in ancient manuscripts to a cornerstone of modern science is a testament ___45___ human ingenuity.

Given the function f(x) = x³ - 3x + 1. (1) Find the intervals where f(x) is increasing and decreasing. (2) Find the local maximum and minimum values of f(x).

In triangle ABC, the sides opposite to angles A, B, C are a, b, c respectively. Given that a = 2√3, b = 2, and angle A = 60°. (1) Find angle B. (2) Find the area of triangle ABC.

A sequence {aₙ} is an arithmetic sequence with a common difference d. It is known that a₃ = 7 and a₇ = 19. (1) Find the general term formula for {aₙ}. (2) Find the sum of the first 10 terms, S₁₀.

Read the following passage and answer the questions. The concept of mathematical proof is central to mathematics. Unlike in science, where theories are tested by experiments, mathematical truths are established through logical deduction. A proof is a sequence of logical steps that starts from axioms (self-evident truths) or previously proven theorems and leads to a new conclusion. One famous proof is Euclid’s proof of the infinitude of prime numbers. He assumed there was a largest prime number, say P. Then he considered the number Q = (2 × 3 × 5 × ... × P) + 1. This number Q is either prime itself or divisible by some prime. However, Q is not divisible by any prime up to P, because it leaves a remainder of 1 when divided by any of them. Therefore, Q must be divisible by a prime larger than P, contradicting the assumption that P is the largest prime. Hence, there are infinitely many primes. This proof is elegant because it uses contradiction. It shows that assuming the opposite of what we want to prove leads to an impossibility, thereby confirming the original statement. 1. According to the passage, how are mathematical truths established? 2. What is the key idea in Euclid’s proof about prime numbers? 3. What logical method does Euclid’s proof use?

Explain why the function f(x) = 1/x is not defined at x = 0, and describe its behavior as x approaches 0 from the positive side and the negative side.

A company models its daily profit P(x) (in thousands of dollars) from selling x units (in hundreds) of a product using the quadratic function P(x) = -2x² + 12x - 10, where x > 0. (1) How many units must be sold to maximize profit? What is the maximum profit? (2) For what range of units sold is the profit positive?