Problem Analysis #13

Grok-4.1-fast · Run 1

29.23s

6,529 toks

Problem Statement

Let $L_1$ be the line of intersection of the planes given by the equations

2x + 3y + z = 4 \quad \text{and} \quad x + 2y + z = 5.

Let $L_2$ be the line passing through the point $P(2, -1, 3)$ and parallel to $L_1$ . Let $\Pi$ denote the plane given by the equation

2x + y - 2z = 6.

Suppose that the line $L_2$ meets the plane $\Pi$ at the point $Q$ . Let $R$ be the foot of the perpendicular drawn from $P$ to the plane $\Pi$ .If the length of the line segment $PQ$ is $\ell$ and the area of $\triangle PQR$ is $A$ , find the value of $4\ell^2 A^2$ .\

Ground Truth

511758

Execution Trace

Reasoning Process

[THOUGHT] First, find direction of L1 by cross product of normals <2,3,1> and <1,2,1>.

Then parametric for L2 through P(2,-1,3) with dir <1,-1,1>.

Intersect with plane 2x+y-2z=6 to find Q.

Then foot R from P to plane along normal <2,1,-2>.

Then compute PQ vector, PR vector, cross product magnitude /2 for A.

PQ mag for ell.

Then compute 4 ell^2 A^2.

Use sympy or numpy for vectors, but since simple, use numpy for cross and norms. [/THOUGHT] [PYTHON] import numpy as np

Direction of L1

n1 = np.array([2,3,1]) n2 = np.array([1,2,1]) d = np.cross(n1, n2) print('Direction:', d)

L2 parametric: point P

P = np.array([2, -1, 3])

Find Q: solve 2(2+t)-1-t -2(3+t)=6, but use param

t from plane eq

Or solve linear

Plug parametric into plane

coeffs: normal dot dir = 21 +1(-1) + (-2)*1 =2-1-2= -1

plane at P: 22 +(-1) -23 =4-1-6= -3, plane=6 so -3 + t*(-1) =6

t*(-1)=9, t=-9 yes

t = -9 Q = P + t * d print('Q:', Q)

R: foot, s=1 as calculated

R = np.array([4,0,1]) print('R:', R)

PQ vec

PQ = Q - P print('PQ:', PQ) ell = np.linalg.norm(PQ) print('ell:', ell)

PR

PR = R - P print('PR:', PR)

cross

cross = np.cross(PQ, PR) print('cross:', cross) mag_cross = np.linalg.norm(cross) A = 0.5 * mag_cross print('A:', A)

ell2 = ell2 A2 = A2 result = 4 * ell2 * A2 print('4 l^2 A^2:', result) [/PYTHON]

Tool Output

[PYTHON OUTPUT] Direction: [ 1 -1 1] Q: [-7 8 -6] R: [4 0 1] PQ: [-9 9 -9] ell: 15.588457268119896 PR: [ 2 1 -2] cross: [ -9 -36 -27] A: 22.94558781116753 4 l^2 A^2: 511758.0 [/PYTHON OUTPUT]

Reasoning Process

[THOUGHT] The Python output gives 4 l^2 A^2: 511758.0, which is exactly 511758. That's the answer. [/THOUGHT] $\boxed{511758}$

Explanation

The reasoning follows a systematic vector

Final Model Output

511758

Comparative Analysis (P13) (Average across runs)

Avg Time Taken

K-EXAONE-236B-A23B

479.1

Deepseek-V3.2

97.6

Kanana-2-30B-Thinking

71.3

Gemini-3-Pro-Preview

51.8

Llama-VARCO-8B-Instruct

51.0

Solar-Open-100B

48.1

Kanana-2-30B-Thinking-2601

45.8

Solar-Pro-2 (31B)(high)

38.9

A.X-4.0 (72B)

35.8

HCX-007(high)

35.3

K-EXAONE-236B-A23B

31.5

Grok-4.1-fast

29.2

GPT-5.1 (high)

22.2

Claude-Opus-4.5

21.5

GPT-5.2 (high)

11.5

EXAONE-4.0.1-32B (high)

7.0

Avg Token Usage

K-EXAONE-236B-A23B

51,665

Kanana-2-30B-Thinking

11,762

Kanana-2-30B-Thinking-2601

10,530

Solar-Open-100B

10,064

Solar-Pro-2 (31B)(high)

7,713

Gemini-3-Pro-Preview

7,454

GPT-5.1 (high)

7,218

Grok-4.1-fast

6,529

HCX-007(high)

5,083

Claude-Opus-4.5

4,957

Deepseek-V3.2

4,661

EXAONE-4.0.1-32B (high)

4,487

K-EXAONE-236B-A23B

4,111

GPT-5.2 (high)

3,541

A.X-4.0 (72B)

2,921

Llama-VARCO-8B-Instruct

1,020