Math topic中文

Operations rationalising

four operations on surds, including rationalising the denominator

Practice questions

  1. 1下列式子中，是最简二次根式的是( )． A. 4 \sqrt[]{4} B. 6 \sqrt[]{6} C. 8 \sqrt[]{8} D. 12 \sqrt[]{12}
  2. 2The diagram shows a prism in which its cross-section is a regular hexagon of side (5+3)(5 + \sqrt{3}) cm. Find an expressio…
  3. 3The triangle ABC is such that its area is 16+7102 cm2\dfrac{16 + 7\sqrt{10}}{2}\ \text{cm}^2, the length of AB is (32+5) cm(3\sqrt{2} + \sqrt{5})\ \text{cm}…
  4. 4Given that sin⁡A=−35\sin A = -\frac{3}{5} and cos⁡B=−817\cos B = -\frac{8}{17}, where angle AA and angle BB are in the same quadrant,…
  5. 5In ΔABC\Delta ABC, AC=12 cmAC = 12\text{ cm}, ∠ABC\angle ABC is a right angle, ∠ACB=π6\angle ACB = \frac{\pi}{6} radians and MM is the…
  6. 6Do not use a calculator in this question. Express sin⁡22.5∘\sin 22.5^\circ in the form of a−aa\frac{\sqrt{a - \sqrt{a}}}{a},…
  7. 7Find the coordinates of the points of intersection of the curve x+3=2xyx + 3 = 2xy and the curve y=2xy = \frac{2}{\sqrt{x}}.
  8. 8The diagram shows a chocolate bar in the form of a triangular prism and the cross-section of the chocolate bar is an iso…
  9. 9Show that 1 is a solution of the equation x3−5x2=2−6xx^3 - 5x^2 = 2 - 6x. Hence solve the equation x3−5x2=2−6xx^3 - 5x^2 = 2 - 6x complete…
  10. 10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
  11. 11当xx ___ 时，二次根式2−x \sqrt {2-x}有意义．
  12. 12Without the use of calculator, show that sin⁡75∘=2+64\sin 75^\circ = \frac{\sqrt{2} + \sqrt{6}}{4}.
  13. 13It is given that f(x)=x3+ax2+bx+5f(x)=x^3+ax^2+bx+5, where aa and bb are constants, has a factor of x−1x-1 and leaves a remainder of…
  14. 14若x<2x < 2，化简(X−2)2+∣3−x∣ \sqrt{(X-2{)}^{2}} +|3-x|的正确结果是 ．
  15. 15下列四条线段中，不能成比例的是()(\quad) A. a=3a=3，b=6b=6，c=2c=2，d=4d=4 B. a=1a=1，b=2b=\sqrt{2}，c=6c=\sqrt{6}，d=3d=\sqrt{3} C.…
  16. 16化简820\dfrac{\sqrt{8}}{\sqrt{20}}的结果是___.
  17. 17It is given that aa and bb are the roots of the quadratic equation x2−2x−1=0x^2 - 2x - 1 = 0 and that a>ba > b. Show that ab=−3−22\frac{a}{b} = -3 - 2\sqrt{2}…
  18. 18下列运算正确的是()(\quad) A. 2+3=5\sqrt{2}+\sqrt{3}=\sqrt{5} B. 81=±9\sqrt{81}=±9 C. 32−2=33\sqrt{2}-\sqrt{2}=3 D. 2×3=6\sqrt{2}×\sqrt{3}=\sqrt{6}…
  19. 19(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}…
  20. 20A farmer uses 160 m of fencing to enclose a plot of his land in a shape that comprises an isosceles triangle and a recta…
  21. 212+1 \sqrt {2}+1与2−1 \sqrt {2}-1的等差中项是()() A. 11 B. −1-1 C. 2 \sqrt {2} D. ±1±1
  22. 22已知当1<a<21 < a < 2时，代数式(a−2)2−∣1−a∣ \sqrt {(a-2)^{2}}- |1-a|的值是___．
  23. 23下列运算正确的是()(\quad) A. 2+3=5\sqrt{2}+\sqrt{3}=\sqrt{5} B. 2×3=6\sqrt{2}×\sqrt{3}=\sqrt{6} C. 23−3=22\sqrt{3}-\sqrt{3}=2 D. 6÷2=3\sqrt{6}÷2=\sqrt{3}…
  24. 24下列计算正确的是()(\quad) A. 6x2=12\sqrt{6}x\sqrt{2}=\sqrt{12} B. 13÷3=13\sqrt{\dfrac{1}{3}}÷\sqrt{3}=\dfrac{1}{3} C. 12+6=32\sqrt{12}+\sqrt{6}=3\sqrt{2}…
  25. 25计算：2×8−3÷13=\sqrt{2}×\sqrt{8}-\sqrt{3}÷\sqrt{\dfrac{1}{3}}= ___ .
  26. 26与−5-\sqrt{5}是同类二次根式的是 ()(\quad) A. 10\sqrt{10} B. 15\sqrt{15} C. 20\sqrt{20} D. 25\sqrt{25}
  27. 27已知y=4−x+x−4+3y=\sqrt{4-x}+\sqrt{x-4}+3，则yx\dfrac{\text{y}}{x}的值为()(\quad) A. 43\dfrac{4}{3} B. -43\text{-}\dfrac{4}{3} C…
  28. 28Factorise 64−(x+1)364 - (x + 1)^3. Hence, solve 64−(x+1)3=15(3−x)64 - (x + 1)^3 = 15(3 - x), expressing non-integer roots in surd form.
  29. 29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…
  30. 30The diagram below shows part of the graph of y=x2+2x+5x+3y = \frac{x^2 + 2x + 5}{x + 3}. Find the coordinates of the minimum point…
  31. 31直线x+3y−1=0x+ \sqrt {3}y-1=0的斜率为()() A. 33 \dfrac { \sqrt {3}}{3} B. 3 \sqrt {3} C. −33- \dfrac { \sqrt {3}}{3} D. −3- \sqrt {3}…
  32. 32Without using a calculator, find the values of aa and bb for which the solution of the equation x24=x3+6x\sqrt{24} = x\sqrt{3} + \sqrt{6}…
  33. 33Solve the equation 8−ext11−2x=−x8 - ext{\sqrt{11 - 2x}} = -x, giving your answer in the form a+bca + b\sqrt{c}, where aa, bb and cc…
  34. 34Without using the calculator, simplify the following trigonometric functions. sin⁡(x+4π3)\sin \left(x + \frac{4\pi}{3}\right) cos⁡75∘\cos 75^\circ…
  35. 35计算：(−2022)0+6×(−12)+8÷2.(-2022)^{0}+6×(-\dfrac{1}{2})+\sqrt{8}÷\sqrt{2}.
  36. 36A miniature model in the shape of a triangular prism is shown above. A piece of wire, 36 cm in length, runs through all…
  37. 37已知a=5−1a= \sqrt {5}-1，则2a3+7a2−2a−122a^{3}+7a^{2}-2a-12的值等于 ___ ．
  38. 38计算：(1−2)0−tan⁡60∘+(−12)−1+23−1{\left(1- \sqrt{2}\right)}^{0}-\tan 60^{\circ}+{\left(- \dfrac{1}{2}\right)}^{-1}+ \dfrac{2}{ \sqrt{3}-1} ；
  39. 39化简 x3−925−x2\displaystyle\frac{\sqrt{x^3 - 9}}{\sqrt{25 - x^2}} 并求其定义域。
  40. 40下列运算正确的是()(\quad) A. 22⋅2=42\sqrt{2}\boldsymbol{⋅}\sqrt{2}=4 B. 2+2=222+\sqrt{2}=2\sqrt{2} C. 2+4=2+2\sqrt{2+4}=\sqrt{2}+2 D.…
  41. 41下列计算正确的是()(\quad) A. 2+3=5\sqrt{2}+\sqrt{3}=\sqrt{5} B. 32−2=33\sqrt{2}-\sqrt{2}=3 C. 18−82=9−4=3−1=1\dfrac{\sqrt{18}-\sqrt{8}}{2}=\sqrt{9}-\sqrt{4}=3-1=1…
  42. 42计算：16×4−1+18÷2−(5−5)0−∣−2∣16×4^{-1}+ \sqrt {18}÷ \sqrt {2}-( \sqrt {5}-5)^{0}-|-2|．
  43. 43计算： (1)(2+3)×6−412(1)\left( \sqrt{2}+\sqrt{3} \right)\times \sqrt{6}-4\sqrt{\dfrac{1}{2}}； (2)(548−627+415)÷3(2)\left( 5\sqrt{48}-6\sqrt{27}+4\sqrt{15} \right)\div \sqrt{3}…
  44. 44下列二次根式中，与3\sqrt{3}能合并的是()(\quad) A. 24\sqrt{24} B. 32\sqrt{32} C. 54\sqrt{54} D. 34\sqrt{\dfrac{3}{4}}
  45. 45Given that sin⁡A=−45\sin A = -\frac{4}{5}, tan⁡B=−512\tan B = -\frac{5}{12} and cos⁡A>0\cos A > 0, where AA and BB are in different quadra…
  46. 46Without using a calculator, show that cot⁡15∘=2+3\cot 15^\circ = 2 + \sqrt{3}. Hence show that csc⁡215∘=4cot⁡15∘\csc^2 15^\circ = 4 \cot 15^\circ…
  47. 47Find the values of xx and yy which satisfy the equations 2x÷4y=18,2^x \div 4^y = \frac{1}{8}, 9y(3)2x=33.9^y (\sqrt{3})^{2x} = 3\sqrt{3}.…
  48. 48In triangle ABCABC shown below, sides ABAB and ACAC are (p−62) cm(p - 6\sqrt{2})\text{ cm} and (4−2) cm(4 - \sqrt{2})\text{ cm} respec…
  49. 49下列各式中，运算正确的是( ) A. (−2)2=−2\sqrt{{{\left( -2 \right)}^{2}}}=-2 B. 2+8=10\sqrt{2}+\sqrt{8}=\sqrt{10} C. 2×8=4\sqrt{2}\times \sqrt{8}=4…
  50. 50计算： (1)(612−827)÷23(1)(6 \sqrt {12}-8 \sqrt {27})÷2 \sqrt {3}； (2)412−612+8+48(2)4 \sqrt { \dfrac {1}{2}}-6 \sqrt { \dfrac {1}{2}}+ \sqrt {8}+ \sqrt {48}…
  51. 51计算. (1)50×328−8(1)\dfrac{\sqrt{50}×\sqrt{32}}{\sqrt{8}}-\sqrt{8}；
  52. 52A calculator must not be used in this question. Show that tan⁡15∘=2−3\tan 15^\circ = 2 - \sqrt{3}. Use the result from part (a) t…
  53. 53化简37−2\dfrac{3}{\sqrt{7}-2}的结果是()(\quad) A. 7−2\sqrt{7}-2 B. 7+2\sqrt{7}+2 C. 3(7−2)3(\sqrt{7}-2) D. 3(7+2)3(\sqrt{7}+2)
  54. 54A right circular cylinder has a volume of (6+23)π cm3(6 + 2\sqrt{3})\pi\text{ cm}^3 and a base radius of (1+3) cm(1 + \sqrt{3})\text{ cm}…
  55. 55A and B are acute angles, such that cos⁡A=35\cos A = \frac{3}{5} and tan⁡B=125\tan B = \frac{12}{5}.\n\nFind the exact values of tan⁡2A\tan 2A…
  56. 56By writing 105∘105^\circ as a sum of two angles, without using a calculator, show that sin⁡105∘=24(3+1).\sin 105^\circ = \frac{\sqrt{2}}{4}(\sqrt{3} + 1).…
  57. 57已知y=x−1+5−x(x,yy= \sqrt {x-1}+ \sqrt {5-x}(x,y均为实数))，则yy的最大值与最小值的差为()() A. 22−22 \sqrt {2}-2 B. 4−224-2 \sqrt {2} C. 3−223-2 \sqrt {2}…
  58. 58下列计算正确的是()() A. 4×6=46 \sqrt {4}× \sqrt {6}=4 \sqrt {6} B. 4+6=10 \sqrt {4}+ \sqrt {6}= \sqrt {10} C. 40÷5=22 \sqrt {40}÷ \sqrt {5}=2 \sqrt {2}…
  59. 59先化简，再求值： ​xx2−2x+1÷(x+1x2−1+1)\dfrac{x}{{x}^{2}-2x+1}÷( \dfrac{x+1}{{x}^{2}-1}+1) ，其中 x=2+1.x= \sqrt{2}+1 .
  60. 60Show that (sin⁡θ−cos⁡θ)(1+sin⁡θcos⁡θ)cos⁡3θ=tan⁡3θ−1.\frac{(\sin\theta-\cos\theta)(1+\sin\theta\cos\theta)}{\cos^{3}\theta}=\tan^{3}\theta-1. Hence, for 0<θ<2π0<\theta<2\pi…