Math topic中文

Equations

solving equations involving surds

Practice questions

1Answer all questions. Given that $a = \sqrt{\frac{b^2+4ac}{2a}}$ , find the values of $b$ when $a = 4$ and $c = -1$ . Answ…
2The triangle ABC is such that its area is $\dfrac{16 + 7\sqrt{10}}{2}\ \text{cm}^2$ , the length of AB is $(3\sqrt{2} + \sqrt{5})\ \text{cm}$ …
3Find the coordinates of the points of intersection of the curve $x + 3 = 2xy$ and the curve $y = \frac{2}{\sqrt{x}}$ .
4It is given that $\frac{2^x \times 32(2^x)}{8^{x+1}} = \frac{9(5^{2x}) - 5^{2x+1}}{5^x - 5^{x-1}}$ . Evaluate $10^x$ with…
5The diagram shows part of the curve $y = \sqrt{3x + 4}$ . The line $AB$ is tangent to the curve at $A$ . The points $B$ an…
6The polynomial $f(x)$ is given by $f(x)=9x^3-30x^2-23x-4$ . Factorise $f(x)$ completely. Hence, prove that the equation $9x\sqrt{x}-23\sqrt{x}=4+30x$ …
7 $ABCD$ is a rectangle which fits inside a semicircle of radius 8 cm and centre $O$ . It is given that $AB = x$ cm and $BC = y$ …
8Without using a calculator, find the values of $a$ and $b$ for which the solution of the equation $x\sqrt{24} = x\sqrt{3} + \sqrt{6}$ …
9Solve the equation $8 - ext{\sqrt{11 - 2x}} = -x$ , giving your answer in the form $a + b\sqrt{c}$ , where $a$ , $b$ and $c$ …
10The diagram shows a prism in which the cross-section is a trapezium $ABCD$ . $AB = 4x\text{ cm}$ , $BC = DC = AD = 2x\text{ cm}$ …
11In triangle $ABC$ shown below, sides $AB$ and $AC$ are $(p - 6\sqrt{2})\text{ cm}$ and $(4 - \sqrt{2})\text{ cm}$ respec…
12已知 $\sqrt {2x-6}+|y+2|=0$ ，则 $A(x,y)$ 所在象限是 $()$ A. 第一象限 B. 第二象限 C. 第三象限 D. 第三象限
13Express $\frac{30+18\sqrt{3}}{2+\sqrt{3}}$ in the form $r + s\sqrt{3}$ , where $r$ and $s$ are integers. The diagram show…
14The diagram shows part of the graph of $y = 2\sqrt{2 + x}$ . It cuts the x-axis at $(-2,0)$ . The line $y = 2x$ meets the…
15The diagram shows a lake (shaded) bordered by a straight level road AXB. Point C is on the surface of the lake and B is…
16Solve $2x^3 - x^2 - 15x - 10 = 0$ , showing all necessary working. Solve the equation $x - 3x + 1 = 0$ .
17Solve $3x = \sqrt{1-x} - 1$ . A triangle has a base of length $(1+\sqrt{3})\text{ cm}$ and an area of $(14+8\sqrt{3})\text{ cm}^2$ …
18Solve $\sqrt{4x+12} - \sqrt{x+3} = 2$ . The volume of right circular cone is $4\pi\text{ cm}^3$ . The radius of its base i…
19It is given that $y = \frac{2x-1}{\sqrt{x+2}}$ for $x > q$ . State the value of $q$ . Find $\dfrac{dy}{dx}$ . State whether…
20Without using the calculator, find the values of the integer $a$ and $b$ such that $\frac{4 - \sqrt{3}}{a + b\sqrt{3}} = \frac{2 + \sqrt{3}}{4 - \sqrt{3}}.$ …
21 $ABCDEF$ is the cross-sectional plan of a temporary spectator stand, where $AB$ is parallel to $FC$ and $ED$ and $AE$ is…
22Show that $\sin 105^\circ = \dfrac{\sqrt{6} + \sqrt{2}}{4}$ . The diagram below shows triangle $ABC$ , such that $AB = 4\ \text{cm}$ …
23已知实数 $a$ 满足 $|2020-a|+\sqrt{a-2021}=a$ ，那么 $a-2020^{2}+1$ 的值是 ___.
24Solve the equation $\sqrt{2x+1} - \sqrt{x} = 1$ . Express $\frac{\left(2\sqrt{3}-5\right)^2}{\sqrt{3}-2}$ in the form $p\sqrt{3} + q$ …
25设 $x _{1}$ 、 $x _{2}$ 是方程 $x ^{2} -6x+a=0$ 的两个根，以 $x _{1}$ 、 $x _{2}$ 为两边长的等腰三角形只可以画出一个，则实数 $a$ 的取值范围是___．
26下列方程中没有实数解的是 $(\quad)$ A. $\sqrt{1-x}+\sqrt{x-1}=0$ B. $\sqrt{x+1}+2=0$ C. $2y+\dfrac{6}{y}=7$ D. $x^{2}-3x-6=0$ …
27Without using a calculator, find the values of the integers $a$ , $b$ and $c$ for which the solution to the equation $5x - \sqrt{3} = 4 + \sqrt{3}\,x$ …
28Answer all the questions. (a) Express $\dfrac{6\sqrt{2}-9}{\sqrt{2}-1}$ in the form of $a+b\sqrt{2}$ , where $a$ and $b$ …
29The equation of a circle, centre $C$ , is $x^2 + y^2 + px + \left(\frac{p}{2} + 4\right)y + k = 0$ , where $p$ and $k$ are…
30The diagram shows a solid prism with a triangular base, $ABC$ , with angle $BAC = 60^\circ$ and $AB = (2 + \sqrt{3})\text{ cm}$ …
31The area of a quadrilateral is $(13 - \sqrt{48})\text{ cm}^2$ . In the case where the quadrilateral is a rectangle with a…
32A square has an area $(17-8\sqrt{2})\ \text{cm}^2$ . The length of each side of the square can be expressed in the form $(a+b\sqrt{2})\ \text{cm}$ …
33Without using a calculator, find the values of the integers $a$ and $b$ for which the solution of the equation $xackslash√{18} - ackslash√{72} = xackslash√{3}$ …