Math topic中文

Interpreting notations

interpreting notations:

  • abab as a×ba \times b
  • ab\dfrac{a}{b} as a÷ba \div b or a×1ba \times \dfrac{1}{b}
  • a2a^{2} as a×aa \times a, a3a^{3} as a×a×aa \times a \times a, a2ba^{2}b as a×a×ba \times a \times b, …\ldots
  • 3y3y as y+y+yy + y + y or 3×y3 \times y
  • 3(x+y)3(x + y) as 3×(x+y)3 \times (x + y)
  • 3+y5\dfrac{3 + y}{5} as (3+y)÷5(3 + y) \div 5 or 15×(3+y)\dfrac{1}{5} \times (3 + y)

Practice questions

  1. 1The equation of a curve is y=x2e2xy = x^2 e^{2x}. Obtain an expression for dydx\dfrac{dy}{dx}. Find the range of values of xx…
  2. 2n 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 ___
  3. 3下列计算正确的是(    )(\:\:\:\:) A. a2+a3=a5a ^{2} +a ^{3} =a ^{5} B. a3⋅a3=a9a ^{3} \boldsymbol{⋅}a ^{3} =a ^{9} C. (a3)2=a6(a ^{3} ) ^{2} =a ^{6}…
  4. 4Simplify 3x2×4y53x^2 \times 4y^5. Answer ... 3(2x−1)−53(2x-1)-5. Answer ...
  5. 5Simplify 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…
  6. 6下列各式中正确的是()(\quad) A. a4⋅a2=a8a^{4}\cdot a^{2}=a^{8} B. (−2a2)3=−6a6(-2a^{2})^{3}=-6a^{6} C. 6a2+13a=2a+1\dfrac{6a^{2}+1}{3a}=2a+1 D. a(a−3)=a2−3aa(a-3)=a^{2}-3a…
  7. 7The equation of a curve is given by y=ln⁡3x32x−1y = \ln \sqrt{\dfrac{3x^3}{2x-1}}, where x>12x > \tfrac{1}{2}. Find the coordinates…
  8. 8Expand and simplify (2x+5)(3+4x)(2x+5)(3+4x).
  9. 9The force, F units, between two objects is inversely proportional to the square of the distance, d units, between them.…
  10. 10Solve 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}…
  11. 11Solve 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}…
  12. 12用代数式表示“2m与5的差”为（ ） A. 2m－5 B. 5－2m C. 2(m－5) D. 2(5－m)
  13. 13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.…
  14. 14Solve 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…
  15. 15Simplify (x327y6)43\left(\frac{x^3}{27y^6}\right)^{\frac{4}{3}}.
  16. 16Simplify Simplify 5−2(6x−1)5-2(6x-1). Simplify (27b9a6)−1/3\left(\dfrac{27b^9}{a^6}\right)^{-1/3}.
  17. 17Factorise completely. 24a2b+12ab2−ab24a^2b + 12ab^2 - ab Answer ... mn−18−9m+2nmn - 18 - 9m + 2n Answer ... Expand and simplify (−2x+3q)(x−2q)(-2x+3q)(x-2q)…
  18. 18Study the pattern below. Write down the values of aa, bb and cc in Figure 11. Answer aa = ... bb = ... cc = ... Wr…
  19. 19Complete the table of values for y=2x+20x−15y = 2x + \dfrac{20}{x} - 15. | x | 1 | 2.5 | 5 | 10 | 12.5 | 15 | 20 | |---|---|---|-…
  20. 20The first four terms in a sequence of numbers are given below. Find T5T_5. Answer ... Find an expression, in terms of n,…
  21. 21Write down, and simplify the terms in the expansion of (2+x)5(2 + x)^5 in ascending powers of xx. Hence find the coefficien…
  22. 22The variables x and y are connected by the equation y=x35−2x+2y=\frac{x^3}{5}-2x+2. Some corresponding values of x and y are giv…
  23. 23T=2πLgT = 2\pi\sqrt{\dfrac{L}{g}} Rearrange the formula to make g the subject. Answer g = ...
  24. 24Express 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…
  25. 25Expand and simplify (3x−y)(y+x).(3x - y)(y + x). Factorise completely 3−12a23 - 12a^2.
  26. 26若2x3y22x^{3}y^{2}和−ynx3m-y^{n}x^{3m}是同类项，则(m−n)2018=(m-n)^{2018}=___.
  27. 27Alan drove a distance of xx km and his car used 7.5 litres of fuel. Write down an expression, in terms of xx, for the…
  28. 28Factorise a2b4−ab2.Answer...Expandandsimplifya^2b^4 - ab^2. Answer ... Expand and simplify (3x + 2)^2 + 4x(2 - x). Answer ...
  29. 29Factorise a4−b2a^4 - b^2. Answer ... Factorise completely 8xy−1−4x+2y8xy - 1 - 4x + 2y. Answer ...
  30. 30Solve 5(2−x)2+1x−2=4\dfrac{5}{(2-x)^2}+\dfrac{1}{x-2}=4. Answer x = ...
  31. 31下列计算正确的是()(\quad) A. x5−x3=x2x^{5}-x^{3}=x^{2} B. (x+2)2=x2+4(x+2)^{2}=x^{2}+4 C. (−2x2)3=−8x6(-2x^{2})^{3}=-8x^{6} D. x2⋅x3=x6x^{2}\boldsymbol{⋅}x^{3}=x^{6}…
  32. 32n is a positive integer. Show that, for all n, (5n+3)2−(5n−3)2(5n+3)^2-(5n-3)^2 is a multiple of 4.
  33. 33单项式−2πx3y23- \dfrac {2πx^{3}y^{2}}{3}的系数是 ___
  34. 34Let p=12x2−3yx2p = \frac{1}{2} \sqrt{\dfrac{x^2 - 3y}{x^2}} (a) Evaluate pp when x=−12x = -12 and y=4y = 4, giving your answer corre…
  35. 35Given 3x−yx+2y=13\dfrac{3x - y}{x + 2y} = \dfrac{1}{3}, find the ratio x : y. Answer x : y = ... Solve the inequality 2−3x3<2x−16\dfrac{2 - 3x}{3} < \dfrac{2x - 1}{6}…
  36. 36多项式3x2y−6xy−13x^{2}y-6xy-1的次数和常数项分别是()(\quad) A. 22和11 B. 22和−1-1 C. 33和11 D. 33和−1-1
  37. 37Simplify. a33a2×27a6\dfrac{a^3}{3a^2}\times 27a^6 Answer ... 416x3\sqrt{\dfrac{4}{16x^3}} Answer ... 492a7b=3434c\dfrac{49^{2a}}{7^b}=343^{4c}…
  38. 38n is a positive integer. Show that, for all n, (5n+3)2−(5n−3)2(5n+3)^2-(5n-3)^2 is a multiple of 4. Answer ...
  39. 39下列代数式中，次数为33的单项式是()(\quad) A. −a3b-a^{3}b B. 3a2b23a^{2}b^{2} C. 4a3−34a^{3}-3 D. x2y4\dfrac{x^{2}y}{4}
  40. 40Simplify 5a2−(3a−2b)25a^2 - (3a - 2b)^2. Answer ...
  41. 41Factorise completely m2−2mn+n2−p2m^2 - 2mn + n^2 - p^2. Answer ... Express as a single fraction in its simplest form 72x−3+x+16−4x\dfrac{7}{2x-3} + \dfrac{x+1}{6-4x}…
  42. 42By using factorisation, solve (x - 3)(2x + 8) = -12. Answer: x = ... Solve the equation $$\frac{4}{2x-3} - \frac{3}{x+2} = 1.…
  43. 43Sketch the graph of y=2ln⁡xy = 2\ln x. In order to solve the equation xe2x=13xx e^{2x} = \dfrac{1}{3x}, a suitable straight line ha…
  44. 44Expand and simplify a(5a−2b)(5a+2b)a(5a-2b)(5a+2b).
  45. 45Simplify the following expressions, leaving your answers in positive index. (−a2)3÷4b0(-a^2)^3 \div 4b^0 Answer ... (a−1b)2×(b)3(a^{-1}b)^2 \times (\sqrt{b})^3…
  46. 46若A是一个五次多项式，B是一个四次多项式，则A+B一定是（ ） A. 次数不超过五次的多项式 B. 五次多项式或单项式 C. 九次多项式 D. 次数不低于五次的多项式
  47. 47Solve the equation ex−2−14(1e)x−2=5e^{x-2} - 14\left(\frac{1}{e}\right)^{x-2} = 5. Solve the equation log2(3x+1)−2log⁡22=log⁡2(2−x2)log_{2}(3x+1) - \dfrac{2}{\log_{\sqrt{2}}2} = \log_{2}(2 - x^{2})…
  48. 48在式子x+yzx+yz，4a4a，mn3+ma+bmn^{3}+ma+b，−x-x，11，3xy23xy^{2}，15m\dfrac{1}{5m}，m+n4\dfrac{m+n}{4}，m+nab\dfrac{m+n}{ab}中()(\quad)…
  49. 49解二元一次方程组{2x−y=5①y=x+3②\begin{cases}{2x-y=5①}\\ {y=x+3②}\end{cases}，把②代入①，结果正确的是()(\quad) A. 2x−x+3=52x-x+3=5 B. 2x+x+3=52x+x+3=5 C. 2x−(x+3)=52x-(x+3)=5…
  50. 50Simplify 25m2−44−4m−15m2\dfrac{25m^2-4}{4-4m-15m^2}. Given that r=3−p24−qr = 3 - \sqrt{\dfrac{p^2}{4} - q}, evaluate rr when p=−8p = -8 and q=7q = 7…
  51. 51Write as a single fraction in its simplest form 3(1−2h)2+82h−1\dfrac{3}{(1-2h)^2}+\dfrac{8}{2h-1}. Answer ...
  52. 52下列语句不正确的是()() A. 00是代数式 B. aa是整式 C. xx的33倍与yy的14 \dfrac {1}{4}的差表示为3x−14y3x- \dfrac {1}{4}y D. s=πr2s=πr^{2}…
  53. 53下列各组中，是同类项的是()(\quad) A. −2x2y-2x^{2}y和xy2xy^{2} B. x2yx^{2}y和x2zx^{2}z C. 2mn2mn和4nm4nm D. −ab-ab和abcabc
  54. 54a=3−bb+4+2a = \frac{3 - b}{b + 4} + 2 Find a when b = -6. Answer a = ... Rearrange the formula to make b the subject. Answer b =…
  55. 55化简(−2a)⋅a−(−2a)2(-2a)·a-(-2a)^{2}的结果是()() A. 00 B. 2a22a^{2} C. −6a2-6a^{2} D. −4a2-4a^{2}
  56. 56Write as a single fraction in the simplest form 3x2x2−50−1x−5\frac{3x}{2x^2-50}-\frac{1}{x-5}. x=25y7z+23x = \sqrt[3]{\frac{25y}{7z+2}} Fin…
  57. 57Simplify 16a6b25−8a325ab3\displaystyle\frac{16a^6 b^2}{5} - \displaystyle\frac{8a^3}{25ab^3}. Answer ___ (i) Express −4x+x2−6-4x + x^2 - 6 i…
  58. 58Rearrange the formula 1a−12b=13c\dfrac{1}{a}-\dfrac{1}{2b}=\dfrac{1}{3c} to make bb the subject. Answer b = ...
  59. 59单项式−a2h-a^{2}h的次数为 ___.
  60. 60Simplify x2−9y23x2+7xy−6y2\dfrac{x^2-9y^2}{3x^2+7xy-6y^2}.