初三數(shù)學視角題目及答案

一、單項選擇題(共10題,每題2分)1.下列方程是一元二次方程的是(）A.\(x^2+1=0\)B.\(x+1=0\)C.\(\frac{1}{x}+x=0\)D.\(x^2+y=1\)答案:A2.拋物線\(y=(x-2)^2+3\)的頂點坐標是()A.\((2,3)\)B.\((-2,3)\)C.\((2,-3)\)D.\((-2,-3)\)答案:A3.一元二次方程\(x^2-4x+3=0\)的根是()A.\(x_1=1\),\(x_2=3\)B.\(x_1=-1\),\(x_2=-3\)C.\(x_1=1\),\(x_2=-3\)D.\(x_1=-1\),\(x_2=3\)答案:A4.在\(\text{Rt}\triangleABC\)中,\(\angleC=90^\circ\),\(AC=3\),\(BC=4\),則\(\sinA\)的值為()A.\(\frac{3}{5}\)B.\(\frac{4}{5}\)C.\(\frac{3}{4}\)D.\(\frac{4}{3}\)答案:B5.已知反比例函數(shù)\(y=\frac{k}{x}\)的圖象經(jīng)過點\((2,-3)\),則\(k\)的值為()A.\(-6\)B.\(6\)C.\(-5\)D.\(5\)答案:A6.二次函數(shù)\(y=ax^2+bx+c\)的圖象開口向下,對稱軸在\(y\)軸右側,且與\(y\)軸正半軸相交,則\(a\)、\(b\)、\(c\)的符號分別為()A.\(a<0\),\(b>0\),\(c>0\)B.\(a<0\),\(b<0\),\(c>0\)C.\(a<0\),\(b>0\),\(c<0\)D.\(a<0\),\(b<0\),\(c<0\)答案:A7.若點\(A(-2,y_1)\),\(B(1,y_2)\),\(C(2,y_3)\)在反比例函數(shù)\(y=-\frac{1}{x}\)的圖象上,則\(y_1\)、\(y_2\)、\(y_3\)的大小關系是()A.\(y_1>y_2>y_3\)B.\(y_3>y_2>y_1\)C.\(y_2>y_3>y_1\)D.\(y_1>y_3>y_2\)答案:D8.一元二次方程\(x^2-2x-3=0\)配方后可化為()A.\((x-1)^2=4\)B.\((x+1)^2=4\)C.\((x-1)^2=2\)D.\((x+1)^2=2\)答案:A9.在\(\triangleABC\)中,\(\angleA=60^\circ\),\(AB=3\),\(AC=4\),則\(BC\)的長為()A.\(\sqrt{13}\)B.\(\sqrt{15}\)C.\(\sqrt{17}\)D.\(5\)答案:A10.函數(shù)\(y=\sqrt{x-1}\)的自變量\(x\)的取值范圍是()A.\(x>1\)B.\(x\geq1\)C.\(x<1\)D.\(x\leq1\)答案:B二、多項選擇題(共10題,每題2分)1.下列函數(shù)中是二次函數(shù)的是(）A.\(y=3x-1\)B.\(y=ax^2+bx+c\)C.\(y=x^2+2\)D.\(y=(x-1)^2-x^2\)答案:C2.關于\(x\)的方程\(ax^2+2x+1=0\)有實數(shù)根,則\(a\)的取值可能是()A.\(0\)B.\(1\)C.\(-1\)D.\(2\)答案:ABC3.下列圖形中,既是軸對稱圖形又是中心對稱圖形的是()A.矩形B.菱形C.等邊三角形D.圓答案:ABD4.在\(\triangleABC\)中,\(\angleC=90^\circ\),\(\sinA=\frac{3}{5}\),則下列正確的是()A.\(\cosA=\frac{4}{5}\)B.\(\tanA=\frac{3}{4}\)C.\(\cosB=\frac{3}{5}\)D.\(\tanB=\frac{4}{3}\)答案:ABC5.反比例函數(shù)\(y=\frac{k}{x}\)的圖象在二、四象限,則\(k\)可能的值是()A.\(-1\)B.\(0\)C.\(1\)D.\(2\)答案:A6.二次函數(shù)\(y=x^2-2x-3\)的圖象與\(x\)軸的交點坐標為()A.\((3,0)\)B.\((-1,0)\)C.\((1,0)\)D.\((-3,0)\)答案:AB7.下列說法正確的是()A.一元二次方程的一般形式是\(ax^2+bx+c=0\)(\(a\neq0\)）B.拋物線\(y=x^2-2x+1\)的頂點坐標是\((1,0)\)C.反比例函數(shù)\(y=\frac{2}{x}\)的圖象在一、三象限D.正弦函數(shù)\(\sin60^\circ=\frac{\sqrt{3}}{2}\)答案:ABCD8.在\(\text{Rt}\triangleABC\)中,\(\angleC=90^\circ\),\(AC=5\),\(BC=12\),則下列正確的是()A.\(AB=13\)B.\(\sinA=\frac{12}{13}\)C.\(\cosB=\frac{5}{13}\)D.\(\tanA=\frac{12}{5}\)答案:ABD9.若點\(A(1,y_1)\),\(B(2,y_2)\)在反比例函數(shù)\(y=-\frac{2}{x}\)的圖象上,則()A.\(y_1<y_2\)B.\(y_1>y_2\)C.\(y_1=-2\)D.\(y_2=-1\)答案:BD10.一元二次方程\(x^2-5x+6=0\)的兩根為\(x_1\)、\(x_2\),則()A.\(x_1+x_2=5\)B.\(x_1x_2=6\)C.\((x_1-1)(x_2-1)=2\)D.\((x_1-2)(x_2-2)=0\)答案:ABD三、判斷題(共10題,每題2分)1.一元二次方程一定有兩個實數(shù)根。(×)2.拋物線\(y=2(x+1)^2-3\)的開口向下。(×)3.反比例函數(shù)\(y=\frac{3}{x}\)的圖象經(jīng)過點\((1,3)\)。(√)4.在\(\triangleABC\)中,\(\angleA=90^\circ\),則\(\sinA=1\)。（√)5.二次函數(shù)\(y=x^2+2x+3\)的最小值是\(2\)。(√)6.方程\(x^2=4x\)的解是\(x=4\)。(×)7.點\((2,-3)\)在反比例函數(shù)\(y=-\frac{6}{x}\)的圖象上。(√)8.銳角三角形中,\(\sinA=\cos(90^\circ-A)\)。（√)9.一元二次方程\(x^2-3x+1=0\)的判別式\(\Delta=5\)。(√)10.函數(shù)\(y=\sqrt{x+2}\)的自變量\(x\)的取值范圍是\(x\geq-2\)。(√）四、簡答題(共4題,每題5分)1.解一元二次方程\(x^2-4x+3=0\)。答:因式分解得\((x-1)(x-3)=0\),所以\(x_1=1\),\(x_2=3\)。2.求拋物線\(y=x^2-2x-3\)的頂點坐標。答:配方得\(y=(x-1)^2-4\),所以頂點坐標為\((1,-4)\)。3.在\(\text{Rt}\triangleABC\)中,\(\angleC=90^\circ\),\(AC=6\),\(BC=8\),求\(\sinA\)的值。答：\(AB=\sqrt{6^2+8^2}=10\),\(\sinA=\frac{BC}{AB}=\frac{8}{10}=\frac{4}{5}\)。4.已知反比例函數(shù)\(y=\frac{k}{x}\)經(jīng)過點\((2,3)\),求\(k\)的值。答：把點代入得\(3=\frac{k}{2}\),所以\(k=6\)。五、討論題(共4題,每題5分)1.討論二次函數(shù)\(y=ax^2+bx+c\)的圖象與系數(shù)\(a\)、\(b\)、\(c\)的關系。答:\(a\)決定開口方向(\(a>0\)開口向上,\(a<0\)開口向下);對稱軸為\(x=-\frac{2a}\);\(c\)是與\(y\)軸交點的縱坐標。2.討論一元二次方程\(ax^2+bx+c=0\)(\(a\neq0\))根的情況與判別式\(\Delta\)的關系。答：\(\Delta>0\)時,方程有兩個不等實根；\(\Delta=0\)時,有兩個相等實根；\(\Delta<0\)時,無實根。3.討論反比例