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2021 CMC (Math Major) Set B

Source: 2021 CMC (Math Major) Set B

November 26, 2022
functionalgebradomaingeometry3D geometryspherepyramid

Problem Statement

1. Let sphere S:x2+y2+z2=1S:x^2+y^2+z^2=1. Find the equation of pyramid with vertex M0(0,0,a)M_0(0,0,a)(a∈Ra\in\mathbb{R}, ∣a∣>1|a|>1) and cut SS. 2. Same as 2021 CMC (Math Major) Set A Problem 2. 3. Same as 2021 CMC (Math Major) Set A Problem 3. 4. Let R=0,1,−1R={0,1,-1} and S={det(aij)3×3∣aij∈R}S=\{ det(a_{ij})_{3\times 3}|a_{ij}\in R\}. Prove that S=−4,−3,−2,−1,0,1,2,3,4S={-4,-3,-2,-1,0,1,2,3,4}. 5. Let function ff is defined in [−1,1][-1,1] and ff is continuously differentiable in some domain of x=0x=0. Also, lim⁡x→0f(x)x=a>0\lim_{x \to 0} \frac{f(x)}{x}=a>0. Prove that if series ∑n=1∞(−1)nf(1n)\sum_{n=1}^{\infty}(-1)^nf(\frac{1}{n}) converges, then ∑n=1∞f(1n)\sum_{n=1}^{\infty}f(\frac{1}{n}) diverges. 6. Let f(x)=ln∑n=1∞enxn2f(x)=ln\sum_{n=1}^{\infty}\frac{e^{nx}}{n^2}. Prove that function ff is strictly convex in (−∞,0)(- \infty ,0) and for any ξ∈(−∞,0)\xi\in(-\infty ,0), there exists x1x_1, x2∈(−∞,0)x_2\in(-\infty,0) such that f′(ξ)=f(x2)−f(x1)x2−x1f'(\xi)=\frac{f(x_2)-f(x_1)}{x_2-x_1}.
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