清华大学2020年强基计划招生考试数学试题

清华大学2020年强基计划招生考试数学试题

本卷共35道不定项选择题,每题5分,错选或漏选均不得分. (学生只回忆了其中的20道题)

1.若$x^2+y^2leqslant 1$,则$x^2+xy-y^2$的取值范围是
egin{tasks}(4)
ask $left[ -frac{sqrt{3}}{2},frac{sqrt{3}}{2} ight]$
ask $left[ -1,1 ight]$
ask $left[ -frac{sqrt{5}}{2},frac{sqrt{5}}{2} ight]$
ask $left[ -2,2 ight]$
end{tasks}

2.在非等边三角形$ABC$中, $CA=CB$,若$O,P$分别为$ riangle ABC$的外心和内心,点$D$在线段$BC$上,且满足$ODot BP$,则下列说法正确的是
egin{tasks}(2)
ask $O,C,P$三点共线
ask $ODparallel AC$
ask $B,D,O,P$四点共圆
ask $PDparallel AC$
end{tasks}

3.已知集合$A,B,Csubseteq {1,2,3,cdots,2020}$,且$Asubseteq C,Bsubseteq C$,则有序集合组$(A,B,C)$的个数是
egin{tasks}(4)
ask $2^{2020}$
ask $3^{2020}$
ask $4^{2020}$
ask $5^{2020}$
end{tasks}

4.已知数列${a_n}$满足$a_0=1,|a_{i+1}|=|a_i +1\, (iin mathbb{N})$,则$A=left|sum_{k=1}^{20}a_k ight|$的值可能是
egin{tasks}(4)
ask $0$
ask $2$
ask $10$
ask $12$
end{tasks}

5.已知$P$在椭圆$frac{x^2}{4}+frac{y^2}{3}=1$上, $A(1,0),B(1,1)$,则$|PA|+|PB|$的最大值是
egin{tasks}(4)
ask $4$
ask $4+sqrt{3}$
ask $4+sqrt{5}$
ask $6$
end{tasks}

6.已知$ riangle ABC$的三条边长均为整数,且面积为有理数,则$|AB|$的可能值有
egin{tasks}(4)
ask $1$
ask $2$
ask $3$
ask $4$
end{tasks}

7.己知$P$为双曲线$frac{x^2}{4}-y^2=1$上一点, $A(-2,0),B(2,0)$,令$angle PAB=alpha,angle PBA=eta$, $ riangle PAB$的面积为$S$,则下列表达式为定值的是
egin{tasks}(4)
ask $ analpha an eta$
ask $ anfrac{alpha}{2} anfrac{eta}{2}$
ask $S an(alpha+eta)$
ask $Scot(alpha+eta)$
end{tasks}

8.甲、乙、丙三人一起做同一道题,甲说:“我做错了.”乙说:“甲做对了.”丙说:“我做错了.”而事实上仅有一人做对题目且仅有一人说谎了,那么谁可能做对了题目.
egin{tasks}(4)
ask 甲
ask 乙
ask 丙
ask 没有人
end{tasks}

9.在直角$ riangle ABC$中, $angle ABC=90^circ,AB=sqrt{3},BC=1$,且$frac{overrightarrow{PA}}{left| overrightarrow{PA} ight|}+frac{overrightarrow{PB}}{left| overrightarrow{PB} ight|}+frac{overrightarrow{PC}}{left| overrightarrow{PC} ight|}=overrightarrow{0}$,则下列说法正确的是
egin{tasks}(4)
ask $angle APB=120^circ$
ask $angle BPC=120^circ$
ask $PC=2PB$
ask $PA= 2PC$
end{tasks}

10.求值: $lim_{n oinfty}left(sum_{k=1}^{n}
arctanfrac{2}{k^2} ight)=$
egin{tasks}(4)
ask $frac{pi}{2}$
ask $frac{3pi}{4}$
ask $frac{5pi}{4}$
ask $frac{3pi}{2}$
end{tasks}

11.从$0-9$这十个数中任取五个数组成一个 五位数$overline{ABCDE}$ ($A$可以为$0$),则$396mid overline{ABCDE}$的概率是
egin{tasks}(4)
ask $frac{1}{396}$
ask $frac{1}{324}$
ask $frac{1}{315}$
ask $frac{1}{210}$
end{tasks}

12.随机变量$X(=1,2,3,cdots),Y(=0,1,2)$,满足$P(X =k)=frac{1}{2^k}$且$Yequiv X(mod\,3)$,则$E(Y)=$
egin{tasks}(4)
ask $frac{4}{7}$
ask $frac{8}{7}$
ask $frac{12}{7}$
ask $frac{16}{7}$
end{tasks}

13.已知向量$overrightarrow{a},overrightarrow{b},overrightarrow{c}$满足$left| overrightarrow{a} ight|leqslant 1,left| overrightarrow{b} ight|leqslant 1,left| overrightarrow{a}+2overrightarrow{b}+overrightarrow{c} ight|leqslant left| overrightarrow{a}-2overrightarrow{b} ight|$,则下列说法正确的是
egin{tasks}(2)
ask $left| overrightarrow{c} ight|$的最大值是$4sqrt{2}$
ask $left| overrightarrow{c} ight|$的最大值是$2sqrt{5}$
ask $left| overrightarrow{c} ight|$的最小值是$0$
ask $left| overrightarrow{c} ight|$的最小值是$2$
end{tasks}

14.若存在$x,yin mathbb{N}^ast$,使得$x^2+ky,y^2+ kx$均为完全平方数,则$k$的可能值是
egin{tasks}(4)
ask $2$
ask $4$
ask $5$
ask $6$
end{tasks}

15.求值: $sin left( mathrm{arc} an 1+mathrm{arc}cos frac{3}{sqrt{10}}+mathrm{arc}sin frac{1}{sqrt{5}} ight)=$
egin{tasks}(4)
ask $0$
ask $frac{1}{2}$
ask $frac{sqrt{2}}{2}$
ask $1$
end{tasks}

16.已知正四棱锥中,相邻两侧面构成的二面角为$alpha$,侧棱和底面夹角为$eta$,则
egin{tasks}(2)
ask $cosalpha + an^2 eta=1$
ask $secalpha + an^2 eta=-1$
ask $cosalpha + 2 an^2 eta=1$
ask $secalpha + 2 an^2 eta=-1$
end{tasks}

17.已知函数$f(x)=frac{2e^x}{e^x+e^{-x}}+sin x\, (-2leqslant xleqslant 2)$,则$f(x)$的最大值与最小值的和是
egin{tasks}(4)
ask $2$
ask $e$
ask $3$
ask $4$
end{tasks}

18.已知函数$f(x)$的图像如右图所示,记$y=f(x),x=a,x=t\,(a<t<c)$
及$x$轴围成的曲边梯形面积为$S(t)$,则下列说法正确的是
egin{tasks}(4)
ask $S(t)<cf(b)$
ask $S'(t)leqslant f(a)$
ask $S'(t)leqslant f(b)$
ask $S'(t)leqslant f(c)$
end{tasks}

19.我们称数列${a_n}$为好数列,若对于任意$nin mathbb{N}^ast$,存在$min mathbb{N}^ast$,使得$a_m=sum_{i=1}^{m}a_i$,则下列说法正确的是
egin{tasks}(1)
ask 若$a_n=left{ egin{array}{c}
1,n=1\
2^{n-2},ngeqslant 2\
end{array} ight.$,则数列${a_n}$为好数列
ask 若$a_n=kn$ ($k$为常数),则数列${a_n}$为好数列
ask 存在任意两项均不相同的好数列${a_n}$,且对于任意$nin mathbb{N}^ast$, $|a_n|<n$
ask 对于任意等差数列${a_n}$,存在好数列${b_n},{c_n}$,使得对于任意$nin mathbb{N} ^ast$,有$a_n=b_n+c_n$
end{tasks}

20.求值: $int_{0}^{2pi}frac{sin^2x}{sin^4x+cos^4x}dx=$
egin{tasks}(4)
ask $pi$
ask $sqrt{2}pi$
ask $2pi$
ask $sqrt{5}pi$
end{tasks}

解.
egin{align*}
int_0^{2pi}{frac{sin ^2x}{sin ^4x+cos ^4x}dx} &=4int_0^{frac{pi}{2}}{frac{sin ^2x}{sin ^4x+cos ^4x}dx}=4int_0^{frac{pi}{2}}{frac{ an ^2x}{ an ^4x+1}frac{1}{cos ^2x}dx}
\
&=4int_0^{infty}{frac{u^2}{u^4+1}du}=4left[ int_0^1{frac{u^2}{u^4+1}du}+int_1^{infty}{frac{u^2}{u^4+1}du} ight]
\
&=4left[ int_0^1{frac{u^2}{u^4+1}du}+int_0^1{frac{1}{u^4+1}du} ight] =4int_0^1{frac{u^2+1}{u^4+1}du}
\
&=4int_0^1{frac{1+frac{1}{u^2}}{u^2+frac{1}{u^2}}du}=4int_0^1{frac{1}{left( u-frac{1}{u} ight) ^2+2}dleft( u-frac{1}{u} ight)}
\
&=2sqrt{2}int_0^1{frac{1}{left( frac{u-frac{1}{u}}{sqrt{2}} ight) ^2+1}dfrac{u-frac{1}{u}}{sqrt{2}}}=2sqrt{2}left[ mathrm{arc} an frac{u-frac{1}{u}}{sqrt{2}} ight] _{0}^{1}=sqrt{2}pi.
end{align*}