Gaokao essentials: original practice · Öffentliche Lernkarten

Frage 1: What subject model do many Chinese provinces use for the Gaokao?

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Antwort

The 3+1+2 model: three core subjects, one choice of physics or history, and two electives.

The core three are Chinese, mathematics, and a foreign language. Students then pick physics or history plus two electives, and each province sets its own paper format.

Frage 2: A ball is dropped from rest and falls 4.9 m. With g = 9.8 m/s^2 and no air resistance, how long is the fall?

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Antwort

1.0 s.

Starting from rest, distance is (1/2)gt^2. Solving 4.9 = 4.9t^2 gives t = 1.0 s.

Frage 3: An arithmetic sequence starts at 2 and increases by 3 each term. What is the 10th term?

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Antwort

29.

The nth term is a1 + (n - 1)d. With a1 = 2, d = 3, and n = 10, the term is 2 + 9 x 3 = 29.

Frage 4: What is the formula for the sum of the first $n$ terms of a geometric sequence with first term $a_1$ and common ratio $r \neq 1$ ?

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$S_n = \frac{a_1(1 - r^n)}{1 - r}$ .

Multiplying $S_n$ by $r$ and subtracting $r S_n$ from $S_n$ cancels intermediate terms, yielding $S_n(1 - r) = a_1(1 - r^n)$ .

Frage 5: In analytic geometry, what is the standard equation and focal length of an ellipse centered at the origin with major axis along the x-axis?

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$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ( $a > b > 0$ ), with focal distance $c = \sqrt{a^2 - b^2}$ and foci at $(\pm c, 0)$ .

The sum of distances from any point on the ellipse to the two foci is constant and equals $2a$ . The eccentricity is $e = c/a \in (0, 1)$ .

Frage 6: What is the dot product of two planar vectors $\vec{u} = (x_1, y_1)$ and $\vec{v} = (x_2, y_2)$ , and what does $\vec{u} \cdot \vec{v} = 0$ imply for non-zero vectors?

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$\vec{u} \cdot \vec{v} = x_1 x_2 + y_1 y_2 = |\vec{u}||\vec{v}|\cos\theta$ ; $\vec{u} \cdot \vec{v} = 0$ indicates that the vectors are orthogonal (perpendicular).

Because $\cos(90^\circ) = 0$ , non-zero vectors with a zero dot product meet at a right angle.

Frage 7: For a differentiable function $f(x)$ , how does the first derivative test determine a local extremum at critical point $x_0$ ( $f'(x_0) = 0$ )?

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Antwort

If $f'(x)$ changes from positive to negative at $x_0$ , $f(x_0)$ is a local maximum; if from negative to positive, a local minimum.

If $f'(x)$ does not change sign around $x_0$ , $x_0$ is an inflection point or saddle rather than a local extremum.

Frage 8: What are the trigonometric angle addition formulas for $\sin(\alpha + \beta)$ and $\cos(\alpha + \beta)$ ?

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$\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$ and $\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta$ .

Setting $\alpha = \beta$ immediately yields the double-angle identities $\sin(2\alpha) = 2\sin\alpha\cos\alpha$ and $\cos(2\alpha) = \cos^2\alpha - \sin^2\alpha$ .

Frage 9: State Newton's second law of motion and explain its relationship to impulse and momentum.

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$\vec{F}_{\text{net}} = m\vec{a} = \frac{d\vec{p}}{dt}$ . Impulse $\vec{J} = \vec{F}_{\text{net}}\Delta t = \Delta \vec{p}$ , equal to change in momentum.

In an isolated system where external forces sum to zero, total momentum is strictly conserved during all collisions.

Frage 10: What is the centripetal acceleration of an object moving at constant speed $v$ in a circle of radius $r$ ?

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$a_c = \frac{v^2}{r} = \omega^2 r$ , directed radially inward toward the center of rotation.

Centripetal acceleration alters the direction of velocity without changing speed. Centripetal force required is $F_c = m a_c = \frac{m v^2}{r}$ .

Frage 11: State the work-energy theorem for a particle of mass $m$ .

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$W_{\text{net}} = \Delta E_k = \frac{1}{2}m v_f^2 - \frac{1}{2}m v_i^2$ .

The net work done by all external forces acting on a body equals the net change in its translational kinetic energy.

Frage 12: What is Coulomb's Law for the electrostatic force between two point charges $q_1$ and $q_2$ separated by distance $r$ ?

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$F = k_e \frac{|q_1 q_2|}{r^2}$ , where $k_e = \frac{1}{4\pi\varepsilon_0} \approx 8.99 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$ .

The force acts along the line joining the charges; like charges repel, whereas opposite charges attract.

Frage 13: What is the magnetic Lorentz force acting on a particle with charge $q$ moving with velocity $\vec{v}$ in a magnetic field $\vec{B}$ ?

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Antwort

$\vec{F} = q(\vec{v} \times \vec{B})$ , with magnitude $F = |q| v B \sin\theta$ where $\theta$ is the angle between $\vec{v}$ and $\vec{B}$ .

Because the force is always perpendicular to velocity ( $\vec{F} \cdot \vec{v} = 0$ ), the magnetic field does no work on a moving charge and does not change its kinetic energy.

Frage 14: What is Faraday's law of electromagnetic induction for a loop of wire enclosing magnetic flux $\Phi_B$ ?

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$\mathcal{E} = -\frac{d\Phi_B}{dt}$ , where the negative sign represents Lenz's law.

The induced electromotive force produces a current whose magnetic field opposes the original change in magnetic flux through the circuit.

Frage 15: State the Ideal Gas Law and identify the standard molar volume of an ideal gas at STP (0°C, 101.3 kPa).

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$PV = nRT$ ; at STP, one mole of any ideal gas occupies approximately $22.4 \text{ L}$ .

$P$ is pressure, $V$ is volume, $n$ is amount in moles, $R \approx 8.314 \text{ J}/(\text{mol}\cdot\text{K})$ is the universal gas constant, and $T$ is absolute temperature in kelvins.

Frage 16: State Le Chatelier's principle regarding chemical equilibrium.

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If a dynamic equilibrium is disturbed by changing conditions (temperature, pressure, or concentration), the position of equilibrium shifts to counteract the change.

Increasing temperature favors the endothermic direction; increasing total pressure shifts equilibrium toward the side with fewer moles of gas.

Frage 17: How is pH defined in an aqueous solution, and what is the relation between pH and pOH at 25°C ( $K_w = 1.0 \times 10^{-14}$ )?

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$\text{pH} = -\log_{10}[\text{H}^+]$ and $\text{pH} + \text{pOH} = 14.0$ at 25°C.

Pure water at 25°C has $[\text{H}^+] = 1.0 \times 10^{-7} \text{ M}$ , corresponding to a neutral pH of 7.0.

Frage 18: What is the Binomial Theorem formula for expanding $(a + b)^n$ for a positive integer $n$ ?

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$(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$ , where $\binom{n}{k} = \frac{n!}{k!(n - k)!}$ .

The general $(k+1)$ -th term in the expansion is $T_{k+1} = \binom{n}{k} a^{n-k} b^k$ , frequently tested in Gaokao algebra questions.

Frage 19: In a galvanic (voltaic) cell, what occurs at the anode and cathode, and in which direction do electrons flow in the external circuit?

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Oxidation occurs at the anode; reduction occurs at the cathode. Electrons flow through the external circuit from anode to cathode.

Anions in the salt bridge migrate toward the anode, while cations migrate toward the cathode to maintain electrical neutrality.

Frage 20: For a real-valued function $f(x)$ , what does it mean for $f(x)$ to be an even function versus an odd function?

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Even: $f(-x) = f(x)$ (symmetric about y-axis); Odd: $f(-x) = -f(x)$ (symmetric about origin).

A standard consequence is that $\int_{-a}^a f(x)\,dx = 0$ for continuous odd functions, and $2\int_0^a f(x)\,dx$ for continuous even functions.