学科网大联考五月数学新疆答案
SDL平面ABCD,BD是BE在平面ABCD上的射影,ACLBE()菜S19 D度BY1911图24分()解法1:如图1,由SD上平面ABCD知,\angle DBE= \varphi ,.SDL平面ABCD,CDc平面ABCD,SDLCD.又底面ABCD是正方形,CDLAD,而SDAAD=D,CD上平面SAD.连接AE、CE,过点D在平面SAD内作DELAE于F,连接CF,则CF \perp AE,故是二面角的平面角,即在Ht \triangle BDE中\because BD=2a,DE= \lambda a \therefore \tan \varphi = \frac {DE}{BD}= \frac { \lambda }{2}在R\triangle ADE中,AD= \sqrt {2a},DE= \lambda a, \therefore AE=a \sqrt { \lambda ^{2}+2}从1DF= \frac {AD \cdot DE}{AE}= \frac { \sqrt {2} \lambda a}{ \sqrt { \lambda ^{2}+2}}\tan \theta = \frac {CD}{DF}= \frac { \sqrt { \lambda ^{2}+2}}{ \lambda }.且\tan \theta \cdot \tan \phi =1,\frac { \sqrt { \lambda ^{2}+2}}{ \lambda } \cdot \frac { \lambda }{2}=1 \iff \sqrt { \lambda ^{2}+2}=2 \iff \lambda ^{2}=2.由\lambda \in(0,2],解得\lambda = \sqrt {2},即为所求.12分证法2:以D为原点,DA,DC,DD的方向分别作为x,y,z轴的正方向建立如\angle CDFC-AE-D\angle CDF= \theta.一.CABCAACDAB二11.\frac {2}{3}12.-1013.\frac {1}{3}14.\frac {k}{12},015.\frac {16}{3} \sqrt {3}6.16三、17.解:(1)由22 \cos ^{2} \frac {A-B}{2} \cos B- \s