Math topic中文

Quadratic inequalities

solving quadratic inequalities, and representing the solution on the number line

Practice questions

  1. 1The curve CC has equation y=x2+x+3x−2y = \frac{x^2 + x + 3}{x - 2}. Without using a calculator, find the set of values that yy…
  2. 2The equation of a curve is y=ax2−3x+4−ay = ax^2 - 3x + 4 - a, where aa is a constant. In the case where a=−2a = -2, find the set of…
  3. 3Given that y=6x22−5xy = \frac{6x^2}{2-5x}, find the range of values of xx for which yy is a decreasing function of xx.
  4. 4The equation of a curve is y=16−ax−x2y=16-ax-x^{2}, where aa is positive constant. Given that yy can be expressed in the form…
  5. 5已知条件p：a≥3，条件q：a（a-3）≥0，则p是q 条件．
  6. 6已知[x][x]表示不超过xx的最大整数，称为高斯取整函数，例如[3.4]=3[3.4]=3，[−4.2]=−5[-4.2]=-5，方程[2x2−x]=0[2x^{2}-x]=0的解集为AA，集合B={x∣6x2−5ax+a2>0}B=\{x|6x^{2}-5ax+a^{2}>0\}…
  7. 7The equation of a curve is given by y=5xe2x+1y = 5xe^{2x+1}. Find dydx\frac{dy}{dx}. Hence, find ∫xe2x+1 dx\int xe^{2x+1} \text{ d}x. Det…
  8. 8Find the range of values of xx for which 2x(x−3)≥3−5x2x(x-3) \ge 3-5x. Find the greatest value of kk such that the line 2y+k=x2y+k=x…
  9. 9函数y=x2−2x+3+log⁡3(x+2)y= \sqrt{{x}^{2}-2x+3}+{\log }_{3}(x+2) 的定义域为( ) A. (−,∞−1)∪(3,+∞)(-,∞-1)∪(3,+∞) B. (−∞,−1)∪[3,+∞)(-∞,-1)∪[3,+∞) C. (−2,−1](-2,-1] D.…
  10. 10已知不等式ax2−3x+2>0a{x}^{2}-3x+2>0的解集为{x∣x<1或x>b}.(1)\{x|x< 1或x>b\}.(1)求aa、bb的值； (2)解关于xx的不等式x2−b(a+c)x+4c>0.{x}^{2}-b(a+c)x+4c>0.
  11. 11已知函数f(x)f(x)是定义在RR上的偶函数，且当x⩾0x\geqslant 0时，f(x)=x2+x.f(x)=x^{2}+x.若f(a)f(a)十f(−a)<4f(-a) < 4，测实数aa的取值范围为___．
  12. 12已知p(x)p(x)：x2−5x+6<0x^{2}-5x+6 < 0，则使p(x)p(x)为真命题的xx取值范围为 ___ ．
  13. 13设两个向量a→\overrightarrow{a} ，b→\overrightarrow{b} ，满足∣a→∣=2|\overrightarrow{a} |=2，∣b→∣=1|\overrightarrow{b} |=1 ． (1)若(a→+2b→)⋅(a→−b→)=1(\overrightarrow{a} +2\overrightarrow{b} )·(\overrightarrow{a} -\overrightarrow{b} )=1…
  14. 14集合P={x∣xx−3<0}P=\{x|\dfrac{x}{x-3}< 0\}，M={x∣x2<9}M=\{x|x^{2}< 9\}，则P∩M=P∩M=___.
  15. 15已知不等式a+3b⩽∣x2−1∣a+ \sqrt{3}b\leqslant \left|{x}^{2}-1\right| 对于满足条件a2+b2=1{a}^{2}+{b}^{2}=1 的任意实数aa，bb恒成立，实数xx的取值范围是 ．
  16. 16A curve has the equation y=x3+kx2+kx+8y = x^3 + kx^2 + kx + 8. Find the set of values of kk such that the curve is an increasing f…
  17. 17已知椭圆x2a2+y2b2=1(a>b>0) \dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1(a > b > 0)的左、右焦点分别为F1(−c,0)F_{1}(-c,0)，F2(c,0)F_{2}(c,0)，若椭圆上存在点PP使asin⁡∠PF1F2=csin⁡∠PF2F1 \dfrac {a}{\sin \angle PF_{1}F_{2}}= \dfrac {c}{\sin \angle PF_{2}F_{1}}…
  18. 18A curve has equation y=x2+x+2k−4y = x^2 + x + 2k - 4 and a line has equation y−2x=ky - 2x = k, where kk is a constant. Find the set…
  19. 19The equation of a curve is y=3x2+5x+1y = 3x^2 + 5x + 1. Find the set of values of xx for which the curve lies completely above…
  20. 20已知集合M={x∣2x2−5x−3<0}M=\{x|2x^{2}-5x-3< 0\}，N={−2,1,2,4}N=\{-2,1,2,4\}，则M∩N=()M∩N=(\quad) A. {−2,1,2}\{-2,1,2\} B. {1,2}\{1,2\} C. {1,4}\{1,4\} D. {1,2,4}\{1,2,4\}…
  21. 21若不等式x2−(a+1)x+a⩽0x^{2}-(a+1)x+a\leqslant 0的解集是[−4,3][-4,3]的子集，则aa的取值范围是()() A. [−4,1][-4,1] B. [−4,3][-4,3] C. [1,3][1,3] D. [−1,3][-1,3]…
  22. 22设集合M={x∣0⩽x<2}M=\{x|0\leqslant x< 2\}，N={x∣(x−3)(x+1)<0}N=\{x|(x-3)(x+1)< 0\}，则集合M∩N=M\cap N= A. {x∣0⩽x<2}\left\{ x|0\leqslant x< 2\right\} B. {x∣0⩽x<3}\left\{ x|0\leqslant x< 3\right\}…
  23. 23已知函数f(x)=ax2−3x+bf\left({x}\right)=ax^{2}-3x+b，其中aa， b∈Rb\in R . (1)若关于xx的不等式f(x)>0f\left({x}\right)>0的解集为(−4,1)\left({-4,1}\right)…
  24. 24已知集合A={x∣2x2−5x−12⩽0}A=\{x|2x^{2}-5x-12\leqslant 0\}，B={−3,−1,1,3}B=\{-3,-1,1,3\}，则A∩B=()A∩B=(\quad) A. {−3,−1,1}\{-3,-1,1\} B. {−1,1}\{-1,1\} C. {−1,1,3}\{-1,1,3\}…
  25. 25Find the range of values of xx such that the graph y=ln⁡(x−2x−3)2y = \ln \left( \frac{x-2}{x-3} \right)^2 is decreasing.
  26. 26若关于x的不等式（a-x）（b-x）＞0的解集为{x|x＜a或x＞b}，则实数a，b的大小关系是 ．
  27. 27The equation of a curve is x2+4y2=20x^2 + 4y^2 = 20 and the equation of a line is y=k−xy = k - x, where kk is a constant. Find th…
  28. 28A ball is thrown vertically upwards. Its height hh m, above the ground at time tt seconds after being thrown is given…
  29. 29The equation of a curve is y=2x2+5y = 2x^2 + 5. Find the set of values of xx for which y>3x2y > 3x^2 and represent this set on…
  30. 30已知复数z=(2m2−m−1)+(m2+2m−3)i,m∈R(1)z=(2{m}^{2}-m-1)+({m}^{2}+2m-3)i,m\in R(1)当 mm 取什么值时，复数 zz 是纯虚数； (2)当复数 zz 在复平面内对应的点位于第四象限时，求 mm 的取值…
  31. 31化简 x3−925−x2\displaystyle\frac{\sqrt{x^3 - 9}}{\sqrt{25 - x^2}} 并求其定义域。
  32. 32It is given that y=x3+px2+qx+10y = x^3 + px^2 + qx + 10 where pp and qq are integers. The only values of xx for which yy is a de…
  33. 33设函数f(x)={2x,x⩽0,log2x,x>0,f(x)=\left\{\begin{array}{l}2^{x},x\leqslant 0,\\log_{2}x,x>0,\end{array}\right.对任意给定的y∈(2,+∞)y\in(2,+∞)，都存在唯一的x∈Rx\in R…
  34. 34已知zz是复数，若z+2iz+2i为实数(i(i为虚数单位))，且z−4z-4为纯虚数． (1)求复数zz； (2)若复数(z+mi)2(z+mi)^{2}在复平面上对应的点在第四象限，求实数mm的取值范围．
  35. 35The equation of a curve is y=−2x2−kx−6+ky = -2x^2 - kx - 6 + k, where kk is a constant. Find the range of values of kk for which…
  36. 36已知a < 0,a\text{ < }0,函数f(x)=acos⁡x+1+sin⁡x+1−sin⁡xf\left(x\right)=a\cos x+\sqrt{1+\sin x}+\sqrt{1-\sin x}，其中x∈[−π2,π2].(1)x{\in}\left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right].(1)…
  37. 37已知函数f(x)=x2+ax+b−a(a,b∈R)f(x)=x^{2}+ax+b-a(a,b∈R)． (1)若关于xx的不等式f(x)>0f(x) > 0的解集为(−∞,−1)∪(3,+∞)(-∞,-1)∪(3,+∞)，求实数aa，bb的值； (2)设a=2a=2，若不等式f(x)>b2−3bf(x) > b^{2}-3b…
  38. 38Given that y=16x1+x2y = \frac{16x}{1+x^2}, find the range of values of xx for which yy is increasing. Another curve has equat…
  39. 39Given that f(x)=x2−ax+3f(x)=x^2-ax+3, where aa is a constant, find the range of values of aa for which f(x)>x−1f(x)>x-1, for all real…
  40. 40已知复数z1=2+aiz_{1}=2+ai，z2=2−iz_{2}=2-i，若∣z1∣<∣z2∣|z_{1}| < |z_{2}|，则实数aa的取值范围是 ___ ．
  41. 41已知直线x−2y−2k=0x-2y-2k=0与两坐标轴围成的三角形面积不大于11，则实数kk的取值范围是___.
  42. 42若关于xx的不等式m(x−1)>x2−xm(x-1) > {{x}^{2}}-x的解集为{x∣1<x<2}\left\{ x|1 < x < 2 \right\}，则实数mm的值为 ．
  43. 43已知函数f(x)={x2+4x⩾04x−x2<0f(x)=\begin{cases}{x}^{2}+4x\geqslant 0 \\ 4x-{x}^{2} < 0\end{cases} ，若f(2−a2)>f(a)f\left(2-{a}^{2}\right) > f\left(a\right) …
  44. 44设集合S={x||x|＜5}，T={x|（x+7）（x-3）＜0}．则S∩T=（ ） A. {x|-7＜x＜5 } B. {x|3＜x＜5 } C. {x|-5＜x＜3 } D. {x|-7＜x＜-5 }
  45. 45The points A(5,7)A(5, 7) and B(−11,15)B(-11, 15) lie on a circle CC. The centre of the circle OO lies on the line y=−2x−3y = -2x - 3.…
  46. 46Find the range of xx such that y=3x2e5−2xy = 3x^2 e^{5-2x} is a decreasing function.
  47. 47Find the range of values of kk such that 3x2−5x+k3x^2 - 5x + k is always positive. Hence, solve the inequality x2−x−23x2−5x+4<0\frac{x^2 - x - 2}{3x^2 - 5x + 4} < 0…
  48. 48Given that log⁡a8+3log⁡aN=3\log_a 8 + 3\log_a N = 3 and N>14N > \frac{1}{4}, find the range of values of aa. Solve the equation log⁡3x−log⁡x3x−2=0\log_3 x - \log_x 3x - 2 = 0…
  49. 49已知实数a>0a > 0，设函数f(x)=∣x+1a∣+∣x−a∣f(x)=\left|x+ \dfrac{1}{a}\right|+\left|x-a\right| ． ((Ⅰ))证明：f(x)⩾2f(x)\geqslant 2； ((Ⅱ))…
  50. 50The function ff is defined, for all values of xx, by f(x)=(x+3)(1−2x)2f(x)=(x+3)(1-2x)^2. Find the range of values of xx for which ff…
  51. 51Given that 4x2−xh−3=04x^2 - \frac{x}{h} - 3 = 0, where h≠0h \ne 0, show that there are no real values of hh for which the equati…
  52. 52不等式−1⩽−14x2+x−1<0-1\leqslant-\dfrac{1}{4}x^{2}+x-1< 0的解集是 ___.
  53. 53Find the set of values of xx for which 5x2+12x>3x+25x^2 + 12x > 3x + 2.
  54. 54已知函数f(x)f(x)的定义域为[2,8][2,8]，则函数h(x)=f(2x)+9−x2h(x)=f(2x)+\sqrt{9-x^{2}}的定义域为()(\quad) A. [4,16][4,16] B. (−∞,1]∪[3,+∞)(-∞,1]∪[3,+∞) C. [3,4][3,4]…
  55. 55设函数f(x)=ax+bxf(x)=ax+ \dfrac {b}{x}，曲线y=f(x)y=f(x)在点M(3,f(3))M( \sqrt {3},f( \sqrt {3}))处的切线方程为2x−3y+23=02x-3y+2 \sqrt {3}=0． ((Ⅰ))求f(x)f(x)…
  56. 56Answer ALL the questions The function ff is defined, for all values of xx, by f(x)=(2x−x2)exf(x) = (2x - x^2)e^x. Find the ran…
  57. 57设全集为RR，A={x∣x2−5x−6>0}A=\{x|x^{2}-5x-6>0\}，B={x∣∣x−5∣<a}(aB=\{x||x-5|< a\}(a为常数))，11∈B11\in B，则()(\quad) A. (∁RA)∪B=R(∁_{R}A)∪B=R B. A∪(∁RB)=RA∪(∁_{R}B)=R…
  58. 58若不等式ax2+bx+2>0ax^{2}+bx+2 > 0的解集为{x∣−12<x<13}\{x|- \dfrac {1}{2} < x < \dfrac {1}{3}\}，则a−ba-b的值为 ___ ．
  59. 59Show that the roots of the equation ax2+(3a+b)x+3b=0ax^2 + (3a + b)x + 3b = 0 are real for all real values of aa and bb. Find the ra…
  60. 60关于xx的不等式组{x−1⩾a2x−4<2a \begin{cases} x-1\geqslant a^{2} \\ x-4 < 2a\end{cases}有解，则实数aa的取值范围是()() A. [−3,1][-3,1] B. (−3,1)(-3,1)…