优优班--学霸训练营 > 知识点挑题
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            • 1.
              已知向量\( \overrightarrow{a}\),\( \overrightarrow{b}\)满足\(| \overrightarrow{a}|=1\),且对任意实数\(x\),\(y\),\(| \overrightarrow{a}-x \overrightarrow{b}|\)的最小值为\( \dfrac { \sqrt {3}}{2}\),\(| \overrightarrow{b}-y \overrightarrow{a}|\)的最小值为\( \sqrt {3}\),则\(| \overrightarrow{a}+ \overrightarrow{b}|=(\)  \()\)
              A.\( \sqrt {7}\)
              B.\( \sqrt {5+2 \sqrt {3}}\)
              C.\( \sqrt {7}\)或\( \sqrt {3}\)
              D.\( \sqrt {5+2 \sqrt {3}}\)或\( \sqrt {5-2 \sqrt {3}}\)
            • 2.
              已知向量\( \overrightarrow{a}\),\( \overrightarrow{b}\)满足\(| \overrightarrow{a}|=1\),\(| \overrightarrow{b}|=2\),\( \overrightarrow{a}- \overrightarrow{b}=( \sqrt {3}, \sqrt {2})\),则\(|2 \overrightarrow{a}+ \overrightarrow{b}|=(\)  \()\)
              A.\(2 \sqrt {2}\)
              B.\( \sqrt {17}\)
              C.\( \sqrt {15}\)
              D.\(2 \sqrt {5}\)
            • 3.
              在直角坐标平面\(xOy\)上的一列点\(A_{1}(1,a_{1})\),\(A_{2}(2,a_{2})\),\(…\),\(A_{n}(2,a_{n})\),\(…\),简记为\(\{A_{n}\}\)若由\(b_{n}= \overrightarrow{A_{n}A_{n+1}}\cdot \overrightarrow{j}\)构成的数列\(\{b_{n}\}\)满足\(b_{n+1} > b_{n}\),\(n=1\),\(2\),\(…\),其中\( \overrightarrow{j}\)为方向与\(y\)轴正方向相同的单位向量,则称\(\{A_{n}\}\)为\(T\)点列\(.\)有下列说法
              \(①A_{1}(1,1),A_{2}(2, \dfrac {1}{2}),A_{3}(3, \dfrac {1}{3}),…,A_{n}(n. \dfrac {1}{n}),…\),为\(T\)点列;
              \(②\)若\(\{A_{n}\}\)为\(T\)点列,且点\(A_{2}\)在点\(A_{1}\)的右上方\(.\)任取其中连续三点\(A_{k}\)、\(A_{k+1}\)、\(A_{k+2}\),则\(\triangle A_{k}A_{k+1}A_{k+2}\)可以为锐角三角形;
              \(③\)若\(\{A_{n}\}\)为\(T\)点列,正整数若\(1\leqslant m < n < p < q\),满足\(m+q=n+p\),则\(a_{q}-a_{p}\geqslant (q-p)b_{p}\);
              \(④\)若\(\{A_{n}\}\)为\(T\)点列,正整数若\(1\leqslant m < n < p < q\),满足\(m+q=n+p\),则\( \overrightarrow{A_{n}A_{q}}\cdot \overrightarrow{j} > \overrightarrow{A_{m}A_{p}}\cdot \overrightarrow{j}\).
              其中,正确说法的个数为\((\)  \()\)
              A.\(1\)
              B.\(2\)
              C.\(3\)
              D.\(4\)
            • 4.
              设\(x\),\(y∈R\),向量\( \overrightarrow{a}=(x,1)\),\( \overrightarrow{b}=(1,y)\),\( \overrightarrow{c}=(2,-4)\)且\( \overrightarrow{a}⊥ \overrightarrow{c}\),\( \overrightarrow{b}/\!/ \overrightarrow{c}\),则\(| \overrightarrow{a}+ \overrightarrow{b}|=(\)  \()\)
              A.\( \sqrt {5}\)
              B.\( \sqrt {10}\)
              C.\(2 \sqrt {5}\)
              D.\(10\)
            • 5.
              已知\(M\)是\(\triangle ABC\)内的一点,且\( \overrightarrow{AB}\cdot \overrightarrow{AC}=2 \sqrt {3}\),\(∠BAC=30^{\circ}\),若\(\triangle MBC\),\(\triangle MCA\)和\(\triangle MAB\)的面积分别为\( \dfrac {1}{2}\),\(x\),\(y\),则\( \dfrac {1}{x}+ \dfrac {4}{y}\)的最小值是\((\)  \()\)
              A.\(20\)
              B.\(18\)
              C.\(16\)
              D.\(9\)
            • 6.
              如图,在平面内放置两个相同的三角板,其中\(∠A=30^{\circ}\),且\(B\),\(C\),\(D\)三点共线,则下列结论不成立的是\((\)  \()\)
              A.\( \overrightarrow{CD}= \sqrt {3} \overrightarrow{BC}\)
              B.\( \overrightarrow{CA}\cdot \overrightarrow{CE}=0\)
              C.\( \overrightarrow{AB}\)与\( \overrightarrow{DE}\)共线
              D.\( \overrightarrow{CA}\cdot \overrightarrow{CB}= \overrightarrow{CE}\cdot \overrightarrow{CD}\)
            • 7.
              如图,以\(AB\)为直径在正方形内部作半圆\(O\),\(P\)为半圆上与\(A\),\(B\)不重合的一动点,下面关于\(| \overrightarrow{PA}+ \overrightarrow{PB}+ \overrightarrow{PC}+ \overrightarrow{PD}|\)的说法正确的是\((\)  \()\)
              A.无最大值,但有最小值
              B.既有最大值,又有最小值
              C.有最大值,但无最小值
              D.既无最大值,又无最小值
            • 8.
              已知点\(C(1,-1)\)、\(D(2,x)\),若向量\( \overrightarrow{a}=(x,2)\)与\( \overrightarrow{CD}\)的方向相反,则\(| \overrightarrow{a}|=(\)  \()\)
              A.\(1\)
              B.\(-2\)
              C.\(2 \sqrt {2}\)
              D.\( \sqrt {2}\)
            • 9.
              已知\(\triangle ABC\)是边长为\(4\)的等边三角形,\(P\)为平面\(ABC\)内一点,则\( \overrightarrow{PA}\cdot ( \overrightarrow{PB}+ \overrightarrow{PC})\)的最小值为\((\)  \()\)
              A.\(-3\)
              B.\(-6\)
              C.\(-2\)
              D.\(- \dfrac {8}{3}\)
            • 10.
              对于向量\(a\),\(b\),定义\(a×b\)为向量\(a\),\(b\)的向量积,其运算结果为一个向量,且规定\(a×b\)的模\(|a×b|=|a||b|\sin θ(\)其中\(θ\)为向量\(a\)与\(b\)的夹角\()\),\(a×b\)的方向与向量\(a\),\(b\)的方向都垂直,且使得\(a\),\(b\),\(a×b\)依次构成右手系\(.\)如图,在平行六面体\(ABCD-EFGH\)中,\(∠EAB=∠EAD=∠BAD=60^{\circ}\),\(AB=AD=AE=2\),则\(( \overrightarrow{AB}× \overrightarrow{AD})\cdot \overrightarrow{AE}=(\)  \()\)
              A.\(4\)
              B.\(8\)
              C.\(2 \sqrt {2}\)
              D.\(4 \sqrt {2}\)
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