Math topic中文

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use of:

  • (a+b)2=a2+2ab+b2(a + b)^{2} = a^{2} + 2ab + b^{2}
  • (a−b)2=a2−2ab+b2(a - b)^{2} = a^{2} - 2ab + b^{2}
  • a2−b2=(a+b)(a−b)a^{2} - b^{2} = (a + b)(a - b)

Practice questions

  1. 1Factorise completely 9b2−6ab+a2−x29b^{2} - 6ab + a^{2} - x^{2}. Simplify 52x2−7x−4−84−x\dfrac{5}{2x^{2} - 7x - 4} - \dfrac{8}{4 - x}.
  2. 2设(5a+3b)2=(5a−3b)2+A(5a+3b)^{2}=(5a-3b)^{2}+A，则A=()A=() A. 30ab30ab B. 60ab60ab C. 15ab15ab D. 12ab12ab
  3. 3n is a positive integer. Explain whether, for all n, (7n+2)2−(7n−1)2(7n+2)^{2}-(7n-1)^{2} is a multiple of 3. Answer ___
  4. 4下列算式中，其中正确的有()①(−12)−3=−18()①(- \dfrac {1}{2})^{-3}=- \dfrac {1}{8}，②a2+2a−1=(a−1)2②a^{2}+2a-1=(a-1)^{2}，③a8÷a8=1(a≠0)③a^{8}÷a^{8}=1(a\neq 0)，④(a−b)2=a2−b2④(a-b)^{2}=a^{2}-b^{2}…
  5. 5Adam has some 50-cent and 20-cent coins in his wallet. Given that he has a total of 73 coins adding up to a value of $2…
  6. 6(1)先化简，再求值：(a−b)2+b(a−b)(a-b)^{2}+b(a-b)，其中a=2a=2，b=−12b=- \dfrac {1}{2}． (2)先化简，再求值：(3x+2)(3x−2)−5x(x−1)−(2x−1)2(3x+2)(3x-2)-5x(x-1)-(2x-1)^{2}，其中x=−13x=- \dfrac {1}{3}…
  7. 7把多项式ax2−4ay2a{{x}^{2}}-4a{{y}^{2}}分解因式的结果是___.
  8. 8Factorise completely. abc−3c−6+2ababc - 3c - 6 + 2ab Answer ... 80x4−580x^{4} - 5 Answer ...
  9. 9三角形的三边长分别为aa、bb、cc，且满足等式：(a+b)2−c2=2ab(a+b)^{2}-c^{2}=2ab，则此三角形是()() A. 锐角三角形 B. 直角三角形 C. 钝角三角形 D. 等腰三角形
  10. 102x2−5xy−12y2x2−16y2\displaystyle \frac{2x^2-5xy-12y^2}{x^2-16y^2} Simplify. Answer ...
  11. 11已知定义：在数列{an}\{a_{n}\}中，若an2−an−12=p(n⩾2,n∈N∗,pa_{n}^{2}-a_{n-1}^{2}=p(n\geqslant 2,n∈N^{*},p为常数))，则称数列{an}\{a_{n}\}为等方差数列，下列判断： ①{(−1)n}①\{(-1)^{n}\}…
  12. 12已知实数a,b,ca,b,c满足(a−b)2+b2+c2−8b−10c+41=0.(1){{(a-b)}^{2}}+{{b}^{2}}+{{c}^{2}}-8b-10c+41=0.(1)分别求a,b,ca,b,c的值； (2)如果aa、bb、cc是正数，且满足a+b+c=9a+b+c=9…
  13. 13下列运箅正确的是()() A. 2a2−a=a2a^{2}-a=a B. (a+2)2=a2+4(a+2)^{2}=a^{2}+4 C. (a2)3=a6(a^{2})^{3}=a^{6} D. (−3)2=−3 \sqrt {(-3)^{2}}=-3
  14. 14Simplify 3a25b3+(12c25ab)2\frac{3a^2}{5b^3} + \left(\frac{12c}{25ab}\right)^2. nn is a positive integer. Show that, for all nn, (3n+2)2−(3n−2)2(3n+2)^2 - (3n-2)^2…
  15. 15等差数列{an}\{a_{n}\}中，an>0a_{n} > 0，a12+a72+2a1a7=4a_{1}^{2}+a_{7}^{2}+2a_{1}a_{7}=4，则它的前77项的和等于()() A. 52 \dfrac {5}{2} B. 55…
  16. 16解下列方程 (1)6x−2=1x+3(1) \dfrac {6}{x-2}= \dfrac {1}{x+3}； (2)x+1x−1−4x2−1=1(2) \dfrac {x+1}{x-1}- \dfrac {4}{x^{2}-1}=1．
  17. 17先化简，再求值：x2−2x+1x2+3x÷(1−4x+3)\frac { x ^ { 2 } -2x+1 } { x ^ { 2 } +3x }÷\left ( { 1-\frac { 4 } { x+3 } } \right )，然后从−3-3，0，1，3中选…
  18. 18已知x1,x2\mathbf{x}_{\mathbf{1}}\mathbf{,}\mathbf{x}_{\mathbf{2}}是一元二次方程4kx2−4kx+k+1=0\mathbf{4}\mathbf{k}\mathbf{x}^{\mathbf{2}}\mathbf{-4}\mathbf{kx+k+1=0}…
  19. 19a+2−42−a=a+2- \dfrac {4}{2-a}= ___ ．
  20. 20将多项式mn2+2mn+mmn^{2}+2mn+m因式分解的结果是 ___ ．
  21. 21关于xx的一元二次方程x2−kx+4=0x^{2}-kx+4=0的两个实数根分别是x1x_{1}、x2x_{2}，且满足x12+x22−2x1−2x2−7=0x_{1}^{2}+x_{2}^{2}-2x_{1}-2x_{2}-7=0，则kk的值为 ___.
  22. 22在△ABC\triangle ABC中，若b2=acb^{2}=ac，B=60∘B=60^{\circ}，则A=A=( ) A. 120∘120^{\circ} B. 60∘60^{\circ} C. 45∘45^{\circ} D.…
  23. 23化简(a+3a−4a−3)(1a−2−1)\left(a+\dfrac{3a-4}{a-3}\right)\left(\dfrac{1}{a-2}-1\right)的结果等于 ().(\quad). A. −a−2-a-2 B. a−2a−3\dfrac{a-2}{a-3}…
  24. 24已知a2−3a+1=0a^{2}-3a+1=0，求a23a4+4a2+3= \dfrac {a^{2}}{3a^{4}+4a^{2}+3}= ___ ．
  25. 25下列函数中，满足f(x2)=[f(x)]2f(x^{2})=[f(x)]^{2}的是()() A. f(x)=ln⁡xf(x)=\ln x B. f(x)=∣x+1∣f(x)=|x+1| C. f(x)=x3f(x)=x^{3} D. f(x)=exf(x)=e^{x}
  26. 26Solve the equation 3−2x7=13(2x−1).\displaystyle \frac{3-2x}{7} = \frac{1}{3}(2x-1). Answer x = ... Simplify 25a2−9b215a−10ab+6b2−9b\displaystyle \frac{25a^{2}-9b^{2}}{15a-10ab+6b^{2}-9b}…
  27. 27下列各式中，不能用平方差公式计算的是()(\quad) A. (x3−y3)(x3+y3)(x^{3}-y^{3})(x^{3}+y^{3}) B. (−a−b)(a−b)(-a-b)(a-b) C. (c2−d2)(d2+c2)(c^{2}-d^{2})(d^{2}+c^{2}) D.…
  28. 28Solve x/5+(2x−3)/4=−6x/5 + (2x-3)/4 = -6. Answer x = ... Simplify 2(p−3q)/(10r)+(3q−p)2/r2(p-3q)/(10r) + (3q-p)^2/r. Answer ... Solve the inequality x+1≤7x−24<5x+195x+1 \le \dfrac{7x-2}{4} < \dfrac{5x+19}{5}…
  29. 29下列运算正确的是()() A. x3−3x2=−2xx^{3}-3x^{2}=-2x B. (−13x3)2=19x6(- \dfrac {1}{3}x^{3})^{2}= \dfrac {1}{9}x^{6} C. 6x3÷2x−2=3x6x^{3}÷2x^{-2}=3x D.…
  30. 30(1)12+27+1448−1513(2)(23+5)2−(23−5)2(3){4a−9b=−12a+3b=2(4){2x−y=143x+2y=21(1) \sqrt {12}+ \sqrt {27}+ \dfrac {1}{4} \sqrt {48}-15 \sqrt { \dfrac {1}{3}}(2)(2 \sqrt {3}+ \sqrt {5})^{2}-(2 \sqrt {3}- \sqrt {5})^{2}(3) \begin{cases} \overset{2a+3b=2}{4a-9b=-1}\end{cases}(4) \begin{cases} \overset{3x+2y=21}{2x-y=14}\end{cases}…
  31. 31Solve the inequality 4x+13>3−2x5\frac{4x+1}{3} > \frac{3-2x}{5}. Simplify (16a12b8)−14\left(\dfrac{16a^{12}}{b^8}\right)^{-\tfrac{1}{4}}, l…
  32. 32Solve the inequality −3x+42<5−x7\frac{-3x+4}{2} < \frac{5-x}{7}. Answer ... Simplify (64a6b−15)−13\left(\dfrac{64a^{6}}{b^{-15}}\right)^{-\tfrac{1}{3}}…
  33. 33若点P(a+2,3−a)P(a+2,3-a)在第二象限，则(a−3)33−a2+4a+4=\sqrt[3]{(a-3)^{3}}-\sqrt{a^{2}+4a+4}=___.
  34. 34(x+y)2(x−2y)4(x+y)^{2}(x-2y)^{4}的展开式中x2y4x^{2}y^{4}的系数为()(\quad) A. 8888 B. 104104 C. −40-40 D. −24-24
  35. 35下列运算正确的是()() A. a10+a5=a2a^{10}+a^{5}=a^{2} B. (a3)4=a7(a^{3})^{4}=a^{7} C. (x−y)2=x2−y2(x-y)^{2}=x^{2}-y^{2} D. x3⋅(−x3)=−x6x^{3}⋅(-x^{3})=-x^{6}…
  36. 36用配方法解下列方程： (1)x2−6x+10=0(1){{x}^{2}}-6x+10=0； (2)2x2−7x+6=0.(2)2{{x}^{2}}-7x+6=0.
  37. 37Factorise 9x2+24xy+16y29x^2 + 24xy + 16y^2 completely. Answer ... Hence factorise 144a8−(9a8+24a4y+16y2)144a^8 - (9a^8 + 24a^4 y + 16y^2) completely. A…
  38. 38勾股定理最早出现在商高的《周髀算经》：“勾广三，股修四，经隅五”．观察下列勾股数：33，44，55；55，1212，1313；77，2424，2525；…，这类勾股数的特点是：勾为奇数，弦与股相…
  39. 39如果两数和的平方的结果是x2+(a−1)x+25x^{2}+(a-1)x+25，那么aa的值是()(\quad) A. −9-9 B. −9-9或1111 C. 99或−11-11 D. 1111
  40. 40不论aa，bb为何有理数，a2+b2−2a−4b+ca^{2}+b^{2}-2a-4b+c的值总是非负数，则cc的最小值是()(\quad) A. 44 B. 55 C. 66 D. 无法确定
  41. 41若 x  ⁣2+mx+9x{\,\!}^{2}+mx+9是完全平方式，则 mm 的值为 ．
  42. 42设m∈Rm∈R，在平面直角坐标系中，已知向量a→=(mx,y+1) \overrightarrow{a}=(mx,y+1)，向量b→=(x,y−1) \overrightarrow{b}=(x,y-1)，a→⊥b→ \overrightarrow{a}⊥ \overrightarrow{b}…
  43. 43根据下列各式，回答问题： ①11×29=202−92②12×28=202−82③13×27=①11\times 29={{20}^{2}}-{{9}^{2}}②12\times 28={{20}^{2}}-{{8}^{2}}③13\times 27= ___ ④14×26=202−62⑤15×25=202−52⑥16×24=202−42⑦17×23=④14\times 26={{20}^{2}}-{{6}^{2}}⑤15\times 25={{20}^{2}}-{{5}^{2}}⑥16\times 24={{20}^{2}}-{{4}^{2}}⑦17\times 23=…
  44. 44在直角坐标系xOyxOy中，圆CC的方程为(x−3)2+(y+1)2=9(x- \sqrt {3})^{2}+(y+1)^{2}=9，以OO为极点，xx轴的非负半轴为极轴建立极坐标系． (1)求圆CC的极坐标方程； (2)直线OPOP…
  45. 45计算12a2−9+23−a\dfrac{12}{{{a}^{2}}-9}+\dfrac{2}{3-a}的结果是___.
  46. 46把下列各式分解因式： (1)9x2−6x+1(1)9x^{2}-6x+1； (2)(x+y)2+4(x+y)+4.(2)(x+y)^{2}+4(x+y)+4.
  47. 47定义：若一个整数能表示成a2+b2(a,ba^{2}+b^{2}(a,b是正整数))的形式，则称这个数为“完美数”． 例如：因为13=32+2213=3^{2}+2^{2}，所以1313是“完美数”； 再如：因为a2+2ab+2b2=(a+b)2+b2a^{2}+2ab+2b^{2}=(a+b)^{2}+b^{2}…
  48. 48先化简，再求值： [(x+y)(x−y)+2y(x−y)−(x−y)2]÷(2y)[(x+y)(x-y)+2y(x-y)-(x-y)^{2}]÷(2y)，其中x=1x=1，y=2y=2．
  49. 49在∆ABC∆ABC 中，边，aa，bb，cc分别是角AA，BB，CC的对边，且满足等式bcos⁡C=(2a+c)xos(π−B)b\cos C=(2a+c)xos(π-B) ． (1)求角BB的大小;(2);(2)若b=13b= \sqrt{13} …
  50. 50Show that (3n+1)2+2(3n + 1)^2 + 2 is always divisible by 3 for all integer values of nn. Answer
  51. 51先化简，再求值：a(a−4)+(1−a)(1+a)a(a-4)+(1-a)(1+a)，其中a=34a= \dfrac {3}{4}．
  52. 52Express x2−6x−2x^2-6x-2 in the form (x−a)2+b(x-a)^2 + b where aa and bb are constants. Answer ___ Hence, state the minimum value…
  53. 53阅读下面问题： 12+1=1×(2−1)(2+1)(2−1)=2−1\dfrac{1}{ \sqrt{2}+1} = \dfrac{1×\left( \sqrt{2}-1\right)}{\left( \sqrt{2}+1\right)\left( \sqrt{2}-1\right)} = \sqrt{2}-1…
  54. 54Expand and simplify (3x−y)(y+x).(3x - y)(y + x). Factorise completely 3−12a23 - 12a^2.
  55. 55推理是数学的基本思维方式，若推理过程不严谨，则推理结果可能产生错误． 例如，有人声称可以证明“任意一个实数都等于00”，并证明如下： 设任意一个实数为xx，令x=mx=m， 等式两边都乘以xx，得x2=mx.x^{2}=mx.…
  56. 56若代数式x2−4x+bx^{2}-4x+b可化为(x−a)2−3(x-a)^{2}-3，则b=b= ___ ．
  57. 57不解方程，判断方程根的情况 (1)2x2+3x−4=0(1)2x^{2}+3x-4=0； (2)3x2−5x+4=0(2)3x^{2}-5x+4=0； (3)ax2+bx=0(a≠0)(3)ax^{2}+bx=0(a\neq 0)； (4)3x2+2=26x(4){3}{{x}^{{2}}}+{2}={2}\sqrt{{6}}x…
  58. 58Express 2x2−6x−122x^2 - 6x - 12 in the form k(x−a)2−bk(x - a)^2 - b, where kk is an integer. Answer: ... Hence, solve 2x2−6x−12=02x^2 - 6x - 12 = 0…
  59. 59Factorise a2b4−ab2.Answer...Expandandsimplifya^2b^4 - ab^2. Answer ... Expand and simplify (3x + 2)^2 + 4x(2 - x). Answer ...
  60. 60Factorise a4−b2a^4 - b^2. Answer ... Factorise completely 8xy−1−4x+2y8xy - 1 - 4x + 2y. Answer ...