Problem Analysis #calculus_27

Motif 3 · Run 1

Python ● execution succeeded
Correct
41.24s
13,850 toks

Infron · Motif 3 · 46문항 × 3회 · Python TAR. 131,072 output tokens · 기본 1,800초 · 6+1 turns. ACC는 전체 138회, Pass@1은 각 문항의 첫 유효 시도, Pass@3는 3회 내 정답으로 계산합니다.

Problem Statement

매개변수 t로 나타내어진 곡선 x=e4t(1+sin⁡2πt)x=e^{4t}(1+\sin^2\pi t), y=e4t(1−3cos⁡2πt)y=e^{4t}(1-3\cos^2\pi t)를 C라 하자. 곡선 C가 직선 y=3x−5ey=3x-5e와 만나는 점을 P라 할 때, 곡선 C 위의 점 P에서의 접선의 기울기는?

① 3π−4π+4\frac{3\pi-4}{\pi+4} ② 3π−2π+6\frac{3\pi-2}{\pi+6} ③ 3ππ+8\frac{3\pi}{\pi+8} ④ 3π+2π+10\frac{3\pi+2}{\pi+10} ⑤ 3π+4π+12\frac{3\pi+4}{\pi+12}

최종 답은 선택지 번호 1~5 중 하나를 \boxed{} 안에 쓰세요.

Ground Truth
2
Recorded model response / solution
Reasoning Process

Recorded Python call:

import sympy as sp

# define symbols
t = sp.symbols('t', real=True)
pi = sp.pi

# define x and y
x = sp.exp(4*t) * (1 + sp.sin(pi*t)**2)
y = sp.exp(4*t) * (1 - 3*sp.cos(pi*t)**2)

# condition: y = 3x - 5e
eq = sp.Eq(y, 3*x - 5*sp.exp(1))
# solve for t
sol = sp.solve(eq, t)
sol
Tool Output
Execution succeeded
Reasoning Process

Recorded Python call:

t_val = sp.Rational(1,4)
# compute dy/dx = (dy/dt)/(dx/dt) at t=1/4
dx_dt = sp.diff(x, t)
dy_dt = sp.diff(y, t)
slope = (dy_dt/dx_dt).subs(t, t_val).simplify()
slope
Tool Output
Execution failed name 'sp' is not defined
Reasoning Process

Recorded Python call:

import sympy as sp
t = sp.symbols('t', real=True)
pi = sp.pi
x = sp.exp(4*t) * (1 + sp.sin(pi*t)**2)
y = sp.exp(4*t) * (1 - 3*sp.cos(pi*t)**2)
# Condition check at t=1/4
t_val = sp.Rational(1,4)
x_val = x.subs(t, t_val).simplify()
y_val = y.subs(t, t_val).simplify()
x_val, y_val, (y_val - (3*x_val - 5*sp.exp(1))).simplify()
Tool Output
Execution succeeded
Reasoning Process

Recorded Python call:

import sympy as sp
t = sp.symbols('t', real=True)
pi = sp.pi
x = sp.exp(4*t) * (1 + sp.sin(pi*t)**2)
y = sp.exp(4*t) * (1 - 3*sp.cos(pi*t)**2)
t_val = sp.Rational(1,4)
x_val = x.subs(t, t_val).simplify()
y_val = y.subs(t, t_val).simplify()
print("x =", x_val)
print("y =", y_val)
print("line check:", (y_val - (3*x_val - 5*sp.exp(1))).simplify())

dx_dt = sp.diff(x, t)
dy_dt = sp.diff(y, t)
slope = (dy_dt/dx_dt).subs(t, t_val).simplify()
print("slope =", slope)
Tool Output
Execution succeeded x = 3*E/2 y = -E/2 line check: 0 slope = (-2 + 3*pi)/(pi + 6)
Reasoning Process

풀이

주어진 곡선은

x(t)=e4t(1+sin⁡2πt),y(t)=e4t(1−3cos⁡2πt)x(t)=e^{4t}(1+\sin^2\pi t),\qquad y(t)=e^{4t}(1-3\cos^2\pi t)

이고, 직선은 y=3x−5ey=3x-5e이다. 교점 PP는 다음을 만족한다.

e4t(1−3cos⁡2πt)=3e4t(1+sin⁡2πt)−5ee^{4t}(1-3\cos^2\pi t)=3e^{4t}(1+\sin^2\pi t)-5e e4t[1−3cos⁡2πt−3−3sin⁡2πt]=−5ee^{4t}\bigl[1-3\cos^2\pi t-3-3\sin^2\pi t\bigr]=-5e e4t[−2−3(cos⁡2πt+sin⁡2πt)]=−5ee^{4t}\bigl[-2-3(\cos^2\pi t+\sin^2\pi t)\bigr]=-5e −5e4t=−5e⟹e4t=e⟹t=14-5e^{4t}=-5e \quad\Longrightarrow\quad e^{4t}=e \quad\Longrightarrow\quad t=\frac14

따라서 점 PP는 t=14t=\frac14에 대응한다.

접선의 기울기 dydx=dy/dtdx/dt\frac{dy}{dx}=\frac{dy/dt}{dx/dt}를 계산한다.

dxdt=e4t[4(1+sin⁡2πt)+πsin⁡(2πt)],\frac{dx}{dt}=e^{4t}\bigl[4(1+\sin^2\pi t)+\pi\sin(2\pi t)\bigr], dydt=e4t[4(1−3cos⁡2πt)+3πsin⁡(2πt)].\frac{dy}{dt}=e^{4t}\bigl[4(1-3\cos^2\pi t)+3\pi\sin(2\pi t)\bigr].

t=14t=\frac14에서 sin⁡π4=cos⁡π4=22\sin\frac\pi4=\cos\frac\pi4=\frac{\sqrt2}{2}, sin⁡π2=1\sin\frac\pi2=1, sin⁡2π4=cos⁡2π4=12\sin^2\frac\pi4=\cos^2\frac\pi4=\frac12이므로

dydx∣t=14=4(1−32)+3π4(1+12)+π=−2+3π6+π=3π−2π+6.\frac{dy}{dx}\bigg|_{t=\frac14}= \frac{4(1-\frac32)+3\pi}{4(1+\frac12)+\pi} =\frac{-2+3\pi}{6+\pi} =\frac{3\pi-2}{\pi+6}.

따라서 정답은 ②이다.

2\boxed{2}
Final Model Output
2

Comparative Analysis (Pcalculus_27) (Average across runs)

Averages use reported measurements only. Missing usage is not treated as zero.

Avg Time Taken

Union Alpha
90.3
Motif 3
63.3
Solar Pro 4
41.2
Grok 4.7
36.2
MiMo V2.6 Pro
27.1
Muse Spark 1.3
12.5
DeepSeek V4.1 Flash
10.8
Solar Mini 4 Preview
9.6

Avg Token Usage

Motif 3
17,230
Grok 4.7
13,359
Solar Mini 4 Preview
7,445
DeepSeek V4.1 Flash
5,702
Union Alpha
5,645
Muse Spark 1.3
5,510
Solar Pro 4
4,984
MiMo V2.6 Pro
3,867
    CSAT 2026 · Problem calculus_27 · Motif 3 | EntropyMath