CSAT (Suneung) essentials: original practice · 公開単語カード

問題 1: How are CSAT (Suneung) results reported, and how do universities use them?

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解答

As standards-based grades (stanine 1-9) and percentiles, combined with school records and other admissions materials.

The CSAT does not publish one total score. Each section becomes a standards-based grade and a percentile, and universities weigh those results with school records.

問題 2: For f(x) = x^3 - 3x, at which x-values is the tangent line horizontal?

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解答

At x = -1 and x = 1.

The derivative is f'(x) = 3x^2 - 3. Setting it to zero gives x^2 = 1, so the slope is zero at x = -1 and x = 1.

問題 3: A passage says people who sleep longer earn higher scores, then concludes that sleep causes higher scores. What is the flaw?

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解答

It treats correlation as causation. Another factor may drive both.

The report shows only that the two measures move together. Without a controlled comparison, a third factor such as health or study habits could explain both.

問題 4: What is the Fundamental Theorem of Calculus, Part 1, for $g(x) = \int_a^x f(t)\,dt$ where $f$ is continuous?

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解答

$g'(x) = \frac{d}{dx}\left[\int_a^x f(t)\,dt\right] = f(x)$ .

Differentiation and integration are inverse operations. Furthermore, $\int_a^b f(x)\,dx = F(b) - F(a)$ where $F'(x) = f(x)$ .

問題 5: In CSAT Mathematics, how is the summation formula $\sum_{k=1}^n k^2$ calculated?

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解答

$\sum_{k=1}^n k^2 = \frac{n(n + 1)(2n + 1)}{6}$ .

Similarly, $\sum_{k=1}^n k = \frac{n(n + 1)}{2}$ and $\sum_{k=1}^n k^3 = \left[\frac{n(n + 1)}{2}\right]^2$ . These identities frequently appear in sequence problems.

問題 6: What is the formula for the number of combinations $\binom{n}{r}$ (or $_nC_r$ ) of choosing $r$ items from $n$ distinct objects without regard to order?

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解答

$\binom{n}{r} = \frac{n!}{r!(n - r)!}$ .

In contrast, permutations $_nP_r = \frac{n!}{(n - r)!}$ account for ordering. A key property is symmetry: $\binom{n}{r} = \binom{n}{n - r}$ .

問題 7: How is a continuous random variable $X \sim N(\mu, \sigma^2)$ standardized into a standard normal variable $Z$ ?

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解答

$Z = \frac{X - \mu}{\sigma}$ , where $Z \sim N(0, 1)$ .

Standardization allows calculating probabilities such as $P(a \le X \le b)$ by converting values to standard deviations from the mean using standard $Z$ -tables.

問題 8: What is the condition for a function $f(x)$ to be continuous at $x = c$ ?

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解答

1. $f(c)$ is defined; 2. $\lim_{x \to c} f(x)$ exists; 3. $\lim_{x \to c} f(x) = f(c)$ .

Both one-sided limits $\lim_{x \to c^-} f(x)$ and $\lim_{x \to c^+} f(x)$ must exist and equal each other as well as $f(c)$ .

問題 9: What are the product rule and quotient rule for differentiating functions $u(x)$ and $v(x)$ ?

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解答

Product: $(uv)' = u'v + uv'$ ; Quotient: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$ ( $v \neq 0$ ).

These rules derive directly from the definition of the derivative and are foundational across CSAT differential calculus questions.

問題 10: What are the period and amplitude of the trigonometric function $g(x) = A \sin(Bx + C) + D$ with $A, B > 0$ ?

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解答

Amplitude $= A$ ; Period $= \frac{2\pi}{B}$ ; Vertical shift $= D$ ; Phase shift $= -\frac{C}{B}$ .

The graph oscillates between maximum $D + A$ and minimum $D - A$ , completing one full oscillation over an interval of $\frac{2\pi}{B}$ .

問題 11: State the law of conservation of mechanical energy for an isolated system under conservative forces.

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解答

$E_{\text{mech}} = E_k + E_p = \text{constant}$ , so $\frac{1}{2}mv_1^2 + mgh_1 = \frac{1}{2}mv_2^2 + mgh_2$ .

When non-conservative forces such as friction or drag do work, the total mechanical energy changes by $W_{\text{nc}} = \Delta E_{\text{mech}}$ .

問題 12: State Snell's Law for light passing from medium 1 (refractive index $n_1$ ) into medium 2 (refractive index $n_2$ ).

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解答

$n_1 \sin\theta_1 = n_2 \sin\theta_2$ , where angles $\theta$ are measured relative to the normal line.

When moving into an optically denser medium ( $n_2 > n_1$ ), light bends toward the normal line ( $\theta_2 < \theta_1$ ).

問題 13: In CSAT English reading comprehension, what is the role of contrast transitional adverbs such as 'however', 'conversely', and 'nevertheless'?

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解答

They signal a shift in perspective, qualify a preceding assertion, or introduce the author's primary counter-thesis.

Questions evaluating paragraph insertion, blank cloze, or main idea often pivot directly at these contrast transitions.

問題 14: What is an 'ad hominem' logical fallacy frequently analyzed in critical reasoning questions?

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解答

Attacking the person making an argument rather than addressing the substance, evidence, or logic of the argument itself.

Discrediting a speaker's character or motives does not invalidate the factual validity of their premises.

問題 15: What is the Intermediate Value Theorem (IVT) and how is it used to locate roots of continuous functions?

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解答

If $f$ is continuous on $[a, b]$ and $k$ is between $f(a)$ and $f(b)$ , there exists at least one $c \in (a, b)$ such that $f(c) = k$ .

If $f(a)$ and $f(b)$ have opposite signs ( $f(a) \cdot f(b) < 0$ ), IVT guarantees at least one root $c \in (a, b)$ where $f(c) = 0$ .

問題 16: What is the formula for the distance $d$ from a point $(x_0, y_0)$ to the line $Ax + By + C = 0$ ?

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解答

$d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}}$ .

This formula is derived from vector projection and is a standard tool in CSAT geometry problems involving tangent lines to circles.

問題 17: State the law of cosines for $\triangle ABC$ to find side $c$ when sides $a, b$ and included angle $C$ are known.

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解答

$c^2 = a^2 + b^2 - 2ab\cos C$ .

When $C = 90^\circ$ , $\cos 90^\circ = 0$ , reducing the law of cosines to the Pythagorean theorem $c^2 = a^2 + b^2$ .

問題 18: In probability theory, what is Bayes' Theorem for finding the posterior probability $P(A|B)$ ?

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解答

$P(A|B) = \frac{P(B|A)P(A)}{P(B)}$ .

By law of total probability, the denominator can be expressed as $P(B) = P(B|A)P(A) + P(B|A^c)P(A^c)$ .

問題 19: What is Hooke's Law for an ideal spring, and what is its stored elastic potential energy?

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解答

Force $F = -kx$ ; Potential energy $E_p = \frac{1}{2}kx^2$ , where $k$ is the spring constant and $x$ is displacement from equilibrium.

The negative sign signifies that the spring exerts a restoring force directed opposite to displacement.

問題 20: What does the Mean Value Theorem (MVT) state for a function $f(x)$ continuous on $[a, b]$ and differentiable on $(a, b)$ ?

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解答

There exists at least one $c \in (a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}$ .

Geometrically, there is at least one point where the instantaneous tangent line is parallel to the secant line between the endpoints.