Problem Analysis #common_17

Motif 3 · Run 1

Python ● execution succeeded
Correct
17.91s
3,577 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

함수 f(x)=4x3−2xf(x)=4x^3-2x의 한 부정적분 F(x)F(x)에 대하여 F(0)=4F(0)=4일 때, F(2)F(2)의 값을 구하시오.

최종 정수 답을 \boxed{} 안에 쓰세요.

Ground Truth
16
Recorded model response / solution
Reasoning Process

Recorded Python call:

import sympy as sp
x = sp.symbols('x')
f = 4*x**3 - 2*x
F_gen = sp.integrate(f, x) + sp.Symbol('C')
print("F(x) = ", F_gen)
C = sp.Symbol('C')
eq = sp.Eq(F_gen.subs(x, 0), 4)
sol = sp.solve(eq, C)
print("C =", sol[0])
F_specific = F_gen.subs(C, sol[0])
print("F(x) =", F_specific)
F_2 = F_specific.subs(x, 2)
print("F(2) =", F_2)
Tool Output
Execution succeeded F(x) = C + x**4 - x**2 C = 4 F(x) = x**4 - x**2 + 4 F(2) = 16
Reasoning Process

The antiderivative is found by integrating f(x)=4x3−2xf(x) = 4x^3 - 2x:

F(x)=∫(4x3−2x) dx=x4−x2+C.F(x) = \int (4x^3 - 2x)\,dx = x^4 - x^2 + C.

Using the condition F(0)=4F(0) = 4, we substitute x=0x = 0 to get 0−0+C=40 - 0 + C = 4, so C=4C = 4. Thus,

F(x)=x4−x2+4.F(x) = x^4 - x^2 + 4.

Evaluating at x=2x = 2:

F(2)=24−22+4=16−4+4=16.F(2) = 2^4 - 2^2 + 4 = 16 - 4 + 4 = 16.

The Python computation confirms this result.

16\boxed{16}

Final Model Output
16

Comparative Analysis (Pcommon_17) (Average across runs)

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

Avg Time Taken

Union Alpha
43.7
Motif 3
15.8
Solar Pro 4
12.9
MiMo V2.6 Pro
11.2
Grok 4.7
10.3
Muse Spark 1.3
7.5
DeepSeek V4.1 Flash
3.2
Solar Mini 4 Preview
2.1

Avg Token Usage

Motif 3
5,511
Grok 4.7
4,722
Muse Spark 1.3
3,111
Solar Pro 4
3,069
Solar Mini 4 Preview
2,513
Union Alpha
2,389
DeepSeek V4.1 Flash
2,285
MiMo V2.6 Pro
2,004
    CSAT 2026 · Problem common_17 · Motif 3 | EntropyMath