[
{
"id": "U_001",
"short_name": "间断点分类计数",
"question": "函数 f(x) = x^x / [(1-x)(2-x)] 的第一类间断点的个数为（ ）（A）3 （B）2 （C）1 （D）0",
"answer": "C",
"core_method": "逐一分析x=0,1,2处的极限类型：x=0处右极限存在（可去，第一类），x=1,2处为无穷间断点（第二类）",
"easy_mistake": "忽略x^x在x<0时实数域无定义，误将x=0判为第二类；或遗漏x=0处间断点",
"mastery": 1,
"error_count": 0,
"last_practice": "",
"suggested_time": "5min",
"solutions": ["逐一求各可疑点的左右极限并分类"],
"base_parts": [],
"tags": {
"板块": "高数",
"一级考点": "函数极限连续",
"二级考点": "间断点分类",
"难度": 2,
"考频权重": 4
},
"risk_warning": "暂无"
},
{
"id": "U_002",
"short_name": "参数方程与复合函数极限",
"question": "已知 x = t² + 1, y = e^(3t)，求 lim(x→2) [f(f(x)+2) - f(2)] / (x-2) 的值（A）e² （B）(4/3)e³ （C）(2/3)e³ （D）e³",
"answer": "A",
"core_method": "由参数方程确定x=2对应t=1，求dy/dx在t=1处的值，结合复合函数求导法则计算极限",
"easy_mistake": "忽略参数方程中t的符号选择（t=±1），或复合函数求导链式法则遗漏",
"mastery": 1,
"error_count": 0,
"last_practice": "",
"suggested_time": "8min",
"solutions": ["参数方程求导+链式法则"],
"base_parts": ["U_001"],
"tags": {
"板块": "高数",
"一级考点": "函数极限连续",
"二级考点": "参数方程确定的函数极限",
"难度": 3,
"考频权重": 3
},
"risk_warning": "暂无"
},
{
"id": "U_003",
"short_name": "变限积分奇偶性判定",
"question": "已知 f(x) = ∫(0到sinx) sin t dt，g(x) = ∫(0到x) f(t) dt，则（A）f奇g奇 （B）f奇g偶 （C）f偶g偶 （D）f偶g奇",
"answer": "D",
"core_method": "f(x)=1-cos(sinx)为偶函数；偶函数f的变限积分g(x)=∫(0到x)f(t)dt，通过换元u=-t可证g(-x)=-g(x)，故g为奇函数",
"easy_mistake": "混淆偶函数变限积分的奇偶性——偶函数f且f(0)=0时，∫(0到x)f(t)dt为奇函数；偶函数f但f(0)≠0时，∫(0到x)f(t)dt非奇非偶",
"mastery": 1,
"error_count": 0,
"last_practice": "",
"suggested_time": "6min",
"solutions": ["直接计算f(x)=1-cos(sinx)再判定奇偶性", "利用定理：偶函数f的∫(0到x)f(t)dt为奇函数（需f(0)条件验证）"],
"base_parts": [],
"tags": {
"板块": "高数",
"一级考点": "定积分",
"二级考点": "变限积分函数性质",
"难度": 3,
"考频权重": 4
},
"risk_warning": "暂无"
},
{
"id": "U_004",
"short_name": "数列发散与指数函数性质",
"question": "已知数列{aₙ}(aₙ≠0)发散，则（A）{aₙ+aₙ₊₁}发散 （B）{aₙ-aₙ₊₁}发散 （C）{e^(aₙ)+e^(aₙ₊₁)}发散 （D）{e^(aₙ)-e^(aₙ₊₁)}发散",
"answer": "D",
"core_method": "反证法：若{e^(aₙ)-e^(aₙ₊₁)}收敛，则{e^(aₙ)}为Cauchy列从而收敛，由ln连续性得{aₙ}收敛，矛盾。ABC均可构造反例排除",
"easy_mistake": "试图对每个选项找反例时忽略D的必要性证明；混淆数列收敛与子列收敛的关系",
"mastery": 1,
"error_count": 0,
"last_practice": "",
"suggested_time": "8min",
"solutions": ["反证法+Cauchy收敛准则", "逐项构造反例排除ABC"],
"base_parts": [],
"tags": {
"板块": "高数",
"一级考点": "函数极限连续",
"二级考点": "数列收敛性判定",
"难度": 4,
"考频权重": 3
},
"risk_warning": "暂无"
},
{
"id": "U_005",
"short_name": "多元函数可微性与偏导连续性",
"question": "已知函数 f(x,y) = (x²+y²)sin(1/(xy)), xy≠0; f(x,y)=0, xy=0，则在点(0,0)处（A）f_x连续且可微 （B）f_x连续且不可微 （C）f_x不连续且可微 （D）f_x不连续且不可微",
"answer": "C",
"core_method": "f_x(0,0)=0但f_x在(0,0)附近含sin(1/(xy))导数项振荡不连续；可微性由(Δx²+Δy²)sin(1/(ΔxΔy))=o(ρ)判定可微",
"easy_mistake": "混淆偏导存在/偏导连续/可微三者的关系；误认为偏导不连续则不可微",
"mastery": 1,
"error_count": 0,
"last_practice": "",
"suggested_time": "10min",
"solutions": ["分步验证：先求f_x(0,0)，再判f_x连续性，最后验证可微性定义"],
"base_parts": [],
"tags": {
"板块": "高数",
"一级考点": "多元函数微分学",
"二级考点": "可微性与偏导连续性关系",
"难度": 4,
"考频权重": 5
},
"risk_warning": "暂无"
},
{
"id": "U_006",
"short_name": "交换二重积分次序",
"question": "设f(x,y)是连续函数，则 ∫(0到π/6) dx ∫(1到2sinx) f(x,y) dy = （选项为交换积分次序后的形式）",
"answer": "D",
"core_method": "画出积分区域，确定边界y=2sinx的逆函数x=arcsin(y/2)，按y从小到大重新确定积分限",
"easy_mistake": "忽略y的范围需覆盖整个区域；arcsin定义域限制遗漏；交换后内层积分上下限方向错误",
"mastery": 1,
"error_count": 0,
"last_practice": "",
"suggested_time": "8min",
"solutions": ["画图法：先画区域再重新定限", "解析法：直接求反函数确定新积分限"],
"base_parts": [],
"tags": {
"板块": "高数",
"一级考点": "二重积分",
"二级考点": "交换积分次序",
"难度": 3,
"考频权重": 4
},
"risk_warning": "暂无"
},
{
"id": "U_007",
"short_name": "反常积分收敛性命题判定",
"question": "设非负函数f(x)在[0,+∞)上连续，给定三个命题：（1）若∫₀^∞f²(x)dx收敛，则∫₀^∞f(x)dx收敛；（2）若存在p>1使lim(x→+∞)x^p·f(x)存在，则∫₀^∞f(x)dx收敛；（3）若∫₀^∞f(x)dx收敛，则存在p>1使lim(x→+∞)x^p·f(x)存在。其中正确的个数是（A）0（B）1（C）2（D）3",
"answer": "B",
"core_method": "命题(2)由p-积分比较判别法正确；命题(1)反例f=1/x（∫f²收敛但∫f发散）；命题(3)反例f=1/(x·(lnx)²)收敛但x^p·f(x)→∞",
"easy_mistake": "误认为(1)成立（f²≤f需要f≤1，但f可以>1）；误认为(3)成立（收敛不一定有衰减率p>1）",
"mastery": 1,
"error_count": 0,
"last_practice": "",
"suggested_time": "10min",
"solutions": ["逐个命题构造反例或用比较判别法验证"],
"base_parts": [],
"tags": {
"板块": "高数",
"一级考点": "反常积分",
"二级考点": "反常积分收敛判别",
"难度": 4,
"考频权重": 4
},
"risk_warning": "暂无"
},
{
"id": "U_008",
"short_name": "矩阵合同变换逆推",
"question": "设A为三阶矩阵，P=1,0,0],[0,1,0],[1,0,1，若PᵀAP=diag(2,2,a+b+c²)（含参数），则A为（A）~（D）不同对称矩阵选项",
"answer": "A",
"core_method": "利用P的逆矩阵P⁻¹=1,0,0],[0,1,0],[-1,0,1，由A=(P⁻¹)ᵀ·D·P⁻¹逆推A的具体形式",
"easy_mistake": "混淆左乘右乘的顺序；Pᵀ与P⁻¹的转置关系搞错；初等矩阵的逆矩阵方向搞反",
"mastery": 1,
"error_count": 0,
"last_practice": "",
"suggested_time": "8min",
"solutions": ["直接求P⁻¹再计算A=(P⁻¹)ᵀDP⁻¹", "利用初等变换的逆变换逐步还原A"],
"base_parts": [],
"tags": {
"板块": "线代",
"一级考点": "矩阵",
"二级考点": "矩阵合同与初等变换",
"难度": 3,
"考频权重": 4
},
"risk_warning": "暂无"
},
{
"id": "U_009",
"short_name": "伴随矩阵与矩阵秩",
"question": "设A为四阶矩阵，A为A的伴随矩阵，若A(A-A)=O且A≠O，则r(A)的可能取值为（A）0或1（B）1或3（C）2或3（D）1或2",
"answer": "D",
"core_method": "A(A-A)=O展开为A²=AA=|A|E。若|A|=0则A²=O，幂零矩阵r(A)≤n/2=2，结合A≠O分析；若|A|≠0则r(A)=4。综合约束条件r(A)可能取1或2",
"easy_mistake": "混淆A(A-A)=O与AA=O的条件；A²=|A|E的展开方向错误；伴随矩阵秩公式r(A)=n-r(A)（仅当r(A)=n-1）或0（当r(A)≤n-2）适用条件记错",
"mastery": 1,
"error_count": 0,
"last_practice": "",
"suggested_time": "8min",
"solutions": ["展开A(A-A)=O得A²=|A|E，分|A|=0和|A|≠0讨论", "利用幂零矩阵秩的上界+伴随矩阵秩公式"],
"base_parts": [],
"tags": {
"板块": "线代",
"一级考点": "矩阵",
"二级考点": "伴随矩阵与秩的关系",
"难度": 4,
"考频权重": 4
},
"risk_warning": "暂无"
},
{
"id": "U_010",
"short_name": "交换矩阵可对角化条件",
"question": "设A,B均为2阶矩阵且AB=BA，则'A有两个不相等的特征值'是'B可对角化'的（A）充要条件（B）充分非必要条件（C）必要非充分条件（D）既非充分又非必要条件",
"answer": "B",
"core_method": "A有两个不等特征值→A可对角化→由AB=BA知B保持A的特征子空间不变→2阶时B可对角化（充分性）；反例A=E时B可对角化但A无两个不等特征值（非必要）",
"easy_mistake": "误认为是充要条件；忽略AB=BA这个前提条件在充分性证明中的关键作用",
"mastery": 1,
"error_count": 0,
"last_practice": "",
"suggested_time": "6min",
"solutions": ["充分性：A可对角化+AB=BA→B可对角化", "非必要性：构造反例A=E"],
"base_parts": [],
"tags": {
"板块": "线代",
"一级考点": "特征值与特征向量",
"二级考点": "矩阵可对角化条件",
"难度": 3,
"考频权重": 4
},
"risk_warning": "暂无"
},
{
"id": "U_011",
"short_name": "曲率圆方程求解",
"question": "曲线 y = x² 在点 (0,0) 处的曲率圆方程为_",
"answer": "x² + (y - 1/2)² = 1/4",
"core_method": "求y'=2x,y''=2，在(0,0)处K=2,R=1/2，曲率中心(0,1/2)，写出圆方程",
"easy_mistake": "曲率中心方向搞反（应在凹侧）；曲率公式分母忘记(1+y'²)^(3/2)而用(1+y'²)^(1/2)；曲率半径与曲率互为倒数混淆",
"mastery": 1,
"error_count": 0,
"last_practice": "",
"suggested_time": "6min",
"solutions": ["公式法：K=|y''|/(1+y'²)^(3/2), R=1/K, 曲率中心在凹侧法线方向距R处"],
"base_parts": [],
"tags": {
"板块": "高数",
"一级考点": "导数与微分",
"二级考点": "曲率与曲率圆",
"难度": 2,
"考频权重": 3
},
"risk_warning": "暂无"
},
{
"id": "U_012",
"short_name": "多元函数极值点求解",
"question": "函数 f(x,y) = x⁴ - 2x²y - (9/2)y² + 6xy + 12y - 24 的极值点是___",
"answer": "(-3, -2)",
"core_method": "联立f_x=0, f_y=0求驻点，再用Hessian判别式D=f_xx·f_yy-f_xy²判定极值性质",
"easy_mistake": "解驻点方程计算错误；Hessian判别时D>0且f_xx>0为极小、D>0且f_xx<0为极大记反；遗漏多个驻点",
"mastery": 1,
"error_count": 0,
"last_practice": "",
"suggested_time": "12min",
"solutions": ["直接求偏导联立求解", "消元法：从f_y解出y代入f_x"],
"base_parts": [],
"tags": {
"板块": "高数",
"一级考点": "多元函数微分学",
"二级考点": "多元函数极值与Hessian判别","难度": 3,"考频权重": 4},"risk_warning": "暂无"},{"id": "U_013","shortname": "一阶微分方程初值问题","question": "微分方程 y' = y²·(x+1)/x 满足初始条件 y(1) = 0 的解为","answer": "y = 0","core_method": "分离变量前先检查y=0是否为解。y≡0时y'=0且右端=0，满足方程和初值，即为解","easy_mistake": "直接分离变量dy/y²=(x+1)/x dx，忽略y=0时除以y²无意义；遗漏零解导致无法代入初值条件","mastery": 1,"error_count": 0,"last_practice": "","suggested_time": "5min","solutions": ["先检查平凡解y=0", "分离变量法（y≠0时）：-1/y=x+ln|x|+C，但初值y(1)=0无法代入，说明只有零解"],"base_parts": [],"tags": {"板块": "高数","一级考点": "常微分方程","二级考点": "一阶可分离变量方程初值问题","难度": 2,"考频权重": 4},"risk_warning": "暂无"},{"id": "U_014","short_name": "函数值差计算","question": "已知函数 f(x) = x²(x-1)eˣ，则 f(5) - f(1) = _","answer": "100e⁵","core_method": "直接代入计算：f(5)=25×4×e⁵=100e⁵，f(1)=1×0×e=0，差值为100e⁵","easy_mistake": "计算f(5)时x²=25和(x-1)=4的乘积算错；误认为题目涉及积分或其他复杂运算","mastery": 1,"error_count": 0,"last_practice": "","suggested_time": "3min","solutions": ["直接代入法"],"base_parts": [],"tags": {"板块": "高数","一级考点": "函数极限连续","二级考点": "函数值计算","难度": 1,"考频权重": 2},"risk_warning": "暂无"},{"id": "U_015","shortname": "平均速度与定积分求参数","question": "某物体以速度 v(t) = t + k·sin(πt) 做直线运动，若它从 t=0 到 t=3 的时间段内平均速度是 5/2，则 k = ","answer": "k = 3π/2","core_method": "平均速度=∫₀³v(t)dt/3=5/2。∫₀³(t+ksin(πt))dt=9/2+2k/π，除以3=5/2，解得2k/π=3，k=3π/2","easy_mistake": "平均速度公式用错（误用速度平均而非位移平均）；∫sin(πt)dt=-cos(πt)/π的符号计算错误；忽略cos3π=cosπ=-1与cos0=1的差值","mastery": 1,"error_count": 0,"last_practice": "","suggested_time": "6min","solutions": ["位移积分法：s=∫₀³(t+ksin(πt))dt=9/2+2k/π，平均速度=s/3=5/2"],"base_parts": [],"tags": {"板块": "高数","一级考点": "定积分应用","二级考点": "定积分求位移与平均速度","难度": 2,"考频权重": 3},"risk_warning": "暂无"},{"id": "U_016","short_name": "向量线性相关与参数求解","question": "设向量α₁=(a,1,-1,1)ᵀ, α₂=(1,1,b,a)ᵀ, α₃=(1,a,-1,1)ᵀ，若α₁,α₂,α₃线性相关，且其中任意两个向量均线性无关，则ab=___","answer": "ab = -4","core_method": "三个4维向量线性相关→构造齐次方程组c₁α₁+c₂α₂+c₃α₃=0有非零解，结合任意两个线性无关排除平凡情况，解出参数关系得ab=-4","easy_mistake": "忽略向量为4维而非3维（不能直接用3×3行列式）；参数a,b在向量各分量中的位置搞混；忽略'任意两个线性无关'的约束导致多解","mastery": 1,"error_count": 0,"last_practice": "","suggested_time": "10min","solutions": ["秩分析法：r(α₁,α₂,α₃)=2，任两向量秩=2，联立求解", "构造齐次方程组c₁α₁+c₂α₂+c₃α₃=0有非零解且任两系数不全为零"],"base_parts": [],"tags": {"板块": "线代","一级考点": "向量组","二级考点": "向量组线性相关性与参数求解","难度": 3,"考频权重": 4},"risk_warning": "暂无"},{"id": "U_017","short_name": "曲线围成区域二重积分","question": "设平面有界区域D位于第一象限，由曲线xy=1/3, xy=3与直线y=x/3, y=3x围成，计算∬_D(1+x-y)dxdy","answer": "(4/3)ln3","core_method": "令u=xy,v=y/x做变量替换，Jacobian=1/(2v)。利用D关于y=x的对称性，x-y部分积分为0，只需计算∬_D 1 dxdy","easy_mistake": "忽略区域对称性导致计算量暴增；变量替换的Jacobian计算错误；积分限确定错误","mastery": 1,"error_count": 0,"last_practice": "","suggested_time": "15min","solutions": ["变量替换u=xy,v=y/x+对称性化简", "直接在原坐标系下分块积分（计算量大但不需换元）"],"base_parts": ["U_006"],"tags": {"板块": "高数","一级考点": "二重积分","二级考点": "变量替换法与对称性求二重积分","难度": 4,"考频权重": 5},"risk_warning": "暂无"},{"id": "U_018","short_name": "Euler方程变量代换与定积分","question": "设y=y(x)满足方程x²y''+xy'-9y=0，且y(1)=2, y'(1)=6。（1）利用变换x=eˣ化简方程，并求y(x)的表达式；（2）求∫₁²y(x)√(4-x²)dx","answer": "（1）y=2x³；（2）22√3/5","core_method": "（1）令x=eˣ，Euler方程变换：xy'=dy/dt, x²y''=d²y/dt²-dy/dt，代入得y''_tt-9y=0，通解y=C₁e^(3t)+C₂e^(-3t)=C₁x³+C₂x⁻³，由初值C₁+C₂=2, 3C₁-3C₂=6解得C₁=2,C₂=0；（2）∫₁²2x³√(4-x²)dx，令x=2sinθ换元计算","easy_mistake": "Euler方程变换公式记错（x²y''=d²y/dt²-dy/dt而非d²y/dt²）；初值代入时y'=3C₁x²-3C₂x⁻⁴在x=1处为3C₁-3C₂=6；积分换元时√(4-x²)用x=2sinθ后dx=2cosθ dθ的积分限转换","mastery": 1,"error_count": 0,"last_practice": "","suggested_time": "20min","solutions": ["Euler方程标准代换x=eˣ→常系数ODE→初值定常数→积分换元", "猜测y=xⁿ形式直接代入方程验证n=±3"],"base_parts": ["U_013"],"tags": {"板块": "高数","一级考点": "常微分方程","二级考点": "Euler方程变量代换与定积分","难度": 4,"考频权重": 5},"risk_warning": "暂无"},{"id": "U_019","short_name": "旋转体体积最大值问题","question": "设t>0，求曲线y=xe^(-x)与直线x=t, x=2t及x轴所围平面图形绕x轴旋转所得旋转体体积V(t)的最大值","answer": "t = (3/2)ln2 时取最大值，V_max = (π/4)·[(2t²+2t+1)/8 - (8t²+4t+1)/64]，其中t=3ln2/2","core_method": "V(t)=π∫(t到2t)x²e^(-2x)dx，分部积分求出V(t)表达式，对t求导令V'(t)=0，解得t=3ln2/2","easy_mistake": "分部积分计算∫x²e^(-2x)dx时符号错误；求导时遗漏链式法则；忽略t>0的约束","mastery": 1,"error_count": 0,"last_practice": "","suggested_time": "18min","solutions": ["分部积分法求V(t)+求导令零求极值"],"base_parts": ["U_011"],"tags": {"板块": "高数","一级考点": "定积分应用","二级考点": "旋转体体积与最值问题","难度": 4,"考频权重": 5},"risk_warning": "暂无"},{"id": "U_020","short_name": "多元复合函数偏导与PDE求解","question": "设f(u,v)具有二阶连续偏导，g(x,y)=f(2x+y,3x-y)，且g_xx+g_xy-6g_yy=1。（1）求f_uv；（2）若f_u(u,0)=ue^(-u)且f(0,v)=v²/50-1，求f(u,v)表达式","answer": "（1）f_uv=1/25；（2）f(u,v)=uv/25-(u+1)e^(-u)+v²/50","core_method": "（1）链式法则展开g_xx=4f_uu+12f_uv+9f_vv, g_xy=2f_uu+f_uv-3f_vv, g_yy=f_uu-2f_uv+f_vv，代入g_xx+g_xy-6g_yy=(4+2-6)f_uu+(12+1+12)f_uv+(9-3-6)f_vv=25f_uv=1，得f_uv=1/25；（2）对f_uv=1/25关于v积分得f_u=v/25+φ(u)，由f_u(u,0)=ue^(-u)定φ(u)=ue^(-u)，再对u积分得f=uv/25-(u+1)e^(-u)+ψ(v)，由f(0,v)=v²/50-1定ψ(v)=v²/50","easy_mistake": "链式法则展开二阶偏导时交叉项系数错误；g_xx+g_xy-6g_yy的系数-6漏乘导致化简结果不对；积分时任意函数φ(u)和ψ(v)的确定顺序搞反；-(u+1)e^(-u)的积分需用分部积分法","mastery": 1,"error_count": 0,"last_practice": "","suggested_time": "20min","solutions": ["链式法则展开+系数化简得25f_uv=1", "双重积分：先对v积f_uv得f_u，定φ(u)，再对u积f_u得f，定ψ(v)"],"base_parts": ["U_005", "U_012"],"tags": {"板块": "高数","一级考点": "多元函数微分学","二级考点": "复合函数高阶偏导与PDE求解","难度": 5,"考频权重": 5},"risk_warning": "暂无"},{"id": "U_021","short_name": "二阶导数不等式证明","question": "设函数f(x)具有二阶导数，且f''(0)=f'(1)，|f(x)|≤1。证明：（1）当x∈(0,1)时，f(x)≤f(0)(1-x)+f(1)x+x(1-x)·f''(ξ)/2相关不等式；（2）∫₀¹f(x)dx ≤ 1/2+[f(1)-f(0)]/2","answer": "证明题，利用Taylor展开和构造法","core_method": "（1）Taylor展开到二阶+利用f''(0)=f'(1)条件+放缩；（2）对（1）的不等式积分+利用|f|≤1的约束","easy_mistake": "Taylor展开的余项形式选择错误（Peano余项vs Lagrange余项）；不等式放缩方向错误；忽略|f|≤1条件的运用","mastery": 1,"error_count": 0,"last_practice": "","suggested_time": "20min","solutions": ["Taylor展开+构造辅助函数法", "分部积分+利用凸性性质"],"base_parts": [],"tags": {"板块": "高数","一级考点": "微分中值定理","二级考点": "Taylor展开与不等式证明","难度": 5,"考频权重": 4},"risk_warning": "暂无"},{"id": "U_022","short_name": "线性方程组与二次型综合","question": "设矩阵A（2×3），B（3×2），二次型f(x₁,x₂,x₃)=xᵀBAx。已知Ax=0的解是Bᵀx=0的解但两方程组不同解。（1）求a,b的值；（2）求正交矩阵Q使x=Qy将f化为标准形","answer": "（1）a=1/2, b=2；（2）f=(3+3√17/4)y₁²+(3-3√17/4)y₂²，其中一个特征值为0（对应y₃方向自由）","core_method": "（1）由N(A)⊆N(Bᵀ)且N(A)≠N(Bᵀ)分析秩的关系：A的行空间⊆Bᵀ的行空间但不相等，解出a=1/2,b=2；（2）计算对称矩阵S=(BA+AᵀBᵀ)/2，求特征值0, 3+3√17/4, 3-3√17/4，对应单位特征向量组成正交矩阵Q","easy_mistake": "混淆Ax=0和Bᵀx=0的解空间包含关系方向（N(A)⊆N(Bᵀ)意味着A的约束更强）；B为3×2而非2×3的维度判断错误；二次型矩阵BA不一定对称，需用S=(BA+AᵀBᵀ)/2对称化；特征值计算复杂容易出错","mastery": 1,"error_count": 0,"last_practice": "","suggested_time": "25min","solutions": ["解空间分析+秩的关系求a,b", "对称化BA→求特征值→正交化特征向量→组成Q"],"base_parts": ["U_008", "U_009", "U_010", "U_016"],"tags": {"板块": "线代","一级考点": "二次型","二级考点": "线性方程组解空间与二次型标准形综合","难度": 5,"考频权重": 5},"risk_warning": "暂无"}]
发表于 2026-8-23 23:25
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发表于 2026-8-24 08:56
发表于 2026-8-24 10:51
