Math topic中文

Multiplication division

multiplication and division of polynomials

Practice questions

1Express $\frac{2x^4 + x^3 + 9x^2 + 7x + 3}{(x+1)(x^2+5)}$ in partial fractions.
2It 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…
3Express $\dfrac{2x^{2}+3x}{x^{2}+3x+2}$ in partial fractions. Hence evaluate $\displaystyle \int_{1}^{3} \dfrac{2x^{2}+3x}{x^{2}+3x+2}\,dx$ …
4Find, in ascending powers of $x$ , the first three terms in the expansion of $(1+4x)^7$ , $(2-x)^7$ . Hence, find the coeff…
5The diagram shows part of the curve $y = (1 - x)(x - 4)$ and two lines $l_1$ and $l_2$ . The lines $l_1$ and $l_2$ are ta…
6The first two non-zero terms in the expansion of $(1 + bx)(1 + ax)^6$ are $1$ and $-\frac{21}{4}x^2$ . Given that $a < b$ …
7Solve the equation $x + 1 + \sqrt{5 - x} = 0$ . Solve the equation $\log_x(2x-1) = \log_{x^2}(4-5x)$ .
8In the expansion of $\left(x^2 - \frac{m}{2x}\right)^{12}$ , where $m$ is a positive constant, the independent term of $x$ …
9如果 $3a^{3}b^{2}÷\dfrac{1}{3}ab=A$ ，那么 $A$ 是 $(\quad)$ A. $a^{3}b^{4}$ B. $a^{2}b$ C. $3a^{4}b^{3}$ D. $9a^{2}b$
10Solutions to this question by accurate drawing will not be accepted. The diagram, which is not drawn to scale, shows…
11Solutions to this question by accurate drawing will not be accepted. The diagram shows a trapezium $ABCD$ in which $AB$ …
12Show that the equation $3e^{2x}+7e^{x}=6$ has only one solution and find its value correct to 2 significant figures. Fac…
13Write down the first four terms in the expansion of $\left(1 - 4x\right)^{6}$ . Hence find the coefficient of $x^{3}$ in…
14Factorise $64 - (x + 1)^3$ . Hence, solve $64 - (x + 1)^3 = 15(3 - x)$ , expressing non-integer roots in surd form.
15The cubic polynomial $\text{f}(x)$ is such that the coefficient of $x^3$ is $-1$ and the roots of $\text{f}(x) = 0$ are…
16Solve the equation $8 - ext{\sqrt{11 - 2x}} = -x$ , giving your answer in the form $a + b\sqrt{c}$ , where $a$ , $b$ and $c$ …
17已知 $a= \sqrt {5}-1$ ，则 $2a^{3}+7a^{2}-2a-12$ 的值等于 ___ ．
18The line $y = x - 4$ and the curve $y = \frac{2}{x - 3}$ intersect at the points $A$ and $B$ . Find the length of $A B$ .
19State the third term in the expansion of $(2+x)^n$ in terms of $a$ and $n$ . Write down the first three terms in the expa…
20The first two non-zero terms in the expansion of $(1 + bx)(1 + ax)^6$ in ascending powers of $x$ are $1$ and $-\frac{21}{4}x^2$ …
21By considering the general term in the binomial expansion of $\left(x^2 - \frac{1}{3x}\right)^9$ , explain why the powers…
22The polynomial $\mathrm{f}(x) = Ax^3 - (2A + B)x^2 + 2x + B$ , where $A$ and $B$ are constants, is exactly divisible by $(x - 1)$ …
23Write down, and simplify, the first 3 terms in the expansion of $\left(2 - x\right)^7$ in ascending powers of $x$ . Find…
24计算： -28x4y2÷7x3y＝
25The diagram shows part of the curve $y = \frac{15}{(0.5x+1)}$ and a straight line $y = 1.25x - 10$ . The straight line in…
26先化简，再求值． $(a-2b)(a+2b)+a{{b}^{3}}\div(-ab)$ ，其中 $a=\sqrt{2}$ ， $b=-1.$
27Given that $\frac{(\log_y x)^2}{\log_x y} + 64 = 0$ , express $y$ in terms of $x$ . Solve the equation $\log_2 x + 3 = 2\log_2(x - 1)$ …
28Given that $\log_x p = 9$ and $\log_x q = 6$ , find the value of $\log_{\frac{p}{q}} x^2$ . Solve the equation $\lg(3x + 1) + \lg(2x - 1) = 2 - \lg 2$ …
29Show that the roots of the equation $ax^2 + (3a + b)x + 3b = 0$ are real for all real values of $a$ and $b$ . Find the ra…
30The diagram shows three points $P(-6, 0)$ , $Q(4k, k-2)$ and $R(8, 5)$ where $PQ$ is perpendicular to $QR$ and $k < 0. Show that$ …
31Express $\frac{2x^3 - 5}{x^2(x - 1)}$ in partial fractions.
32Given that $y = \sec x$ , show that $\frac{dy}{dx} = \sec x \tan x$ . Hence show that $y^4 - y^2 = \left(\frac{dy}{dx}\right)^2$ …
33已知一个多项式除以多项式 $2a^{2}+3a-4$ 所得的商式为 $3a+2$ ，余式为 $5a+9$ ，求这个多项式.
34The diagram shows a lake (shaded) bordered by a straight level road AXB. Point C is on the surface of the lake and B is…
35By using long division, show that $2x^2+1$ is a factor of $2x^3-4x^2+x-2$ . Express $\dfrac{-5x^2+11x-11}{2x^3-4x^2+x-2}$ …
36It is given that $\log_2\left(4 - x^2\right) - \log_{\sqrt{2}}(x - 1) = 1$ . Explain clearly why $1 < x < 2$ . Hence, solv…
37Factorise $\left(2x+1\right)^3 - \left(2x-1\right)^3$ completely. Hence show that the equation $\left(2x+1\right)\bigl[\left(2x+1\right)^3 - \left(2x-1\right)^3\bigr]=0$ …
38Find the remainder when $3x^3 - x^2 + 4x + 2$ is divided by $x^2 + 1$ . The remainder when $x^3 + ax$ , where $a$ is a con…
39The equation of a curve is $y = x^2(6 - x)$ . Find the coordinates of the stationary points of the curve. Use the second…
40Solve $3x = \sqrt{1-x} - 1$ . A triangle has a base of length $(1+\sqrt{3})\text{ cm}$ and an area of $(14+8\sqrt{3})\text{ cm}^2$ …
41The diagram shows part of the curve $y=(9-x)(x-3)$ and the line $y=k-3$ , where $k>3$ . The line through the maximum point…
42Factorise $x^4 - 1$ completely. Given that $\frac{x^4}{x^4 - 1} = a + \frac{b}{x^4 - 1}$ , find the values of $a$ and of…
43Express $\frac{9x^2 + 17x - 16}{3x^2 + 5x - 2}$ in partial fractions. Hence, find $\int \frac{9x^2 + 17x - 16}{3x^2 + 5x - 2} \, dx$ …
44Express $\frac{8x^3 + 3x + 1}{x(2x + 1)^2}$ as a sum of a polynomial and a proper fraction. Hence, express $\frac{8x^3 + 3x + 1}{x(2x + 1)^2}$ …
45Integrate $\left(x - \frac{5}{2x}\right)^2$ with respect to $x$ .
46Express $\frac{2x^3 + x^2 + x}{(x-1)(x^2 + 1)}$ in partial fractions. Differentiate $\ln(x^2 + 1)$ with respect to $x$ .…
47State the third term in the expansion of $(2+x)^n$ in terms of $a$ and $n$ . Write down the first three terms in the expa…
48The equation of a curve is $y = e^{x^2 - 5x}$ . Find the expressions for $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$ . Find th…
49Given that $\text{f}(x) = x^3 + 3x^2 - 4$ , show that $x+2$ is a factor of $\text{f}(x)$ , and hence factorise $\text{f}(x)$ …
50Find the term independent of $x$ and the $\frac{1}{x^3}$ term in the binomial expansion of $\left(x^2 - \frac{2}{x}\right)^9.$ …
51Express $\frac{2x^3 - 4x^2 - 5x + 3}{(x-3)(x^2-9)}$ in partial fractions.
52Write down and simplify the first three terms in the expansion, in ascending powers of $x$ , of $\left(2 + \frac{x}{2}\right)^n$ …
53Show that the solution of the equation $3^{2x+4}\times5^{x}=3^{3x}\times25^{x}$ is $x=\dfrac{\lg81}{\lg15}$ . Express the…
54Given that $y = \frac{\ln(x+3)^3}{3x+9}$ for $x > -3$ , find the set of values of $x$ for which $y$ is an increasing func…
55Express $\frac{10x}{3-5x}$ in the form $a + \frac{b}{3-5x}$ , where $a$ and $b$ are integers. Hence, integrate $\frac{10x}{3-5x}$ …
56By using long division, divide $4x^3 + 5x^2 + x - 1$ by $x^2 (x+1)$ . Hence, express $\dfrac{4x^3 + 5x^2 + x - 1}{x^2 (x+1)}$ …
57一个长方体的体积为 $x^{2}-2xy+x$ ，高是 $x$ ，则这个长方体的底面积是 $()$ A. $x-2y$ B. $x+2y$ C. $x-2y-1$ D. $x-2y+l$
58Express $\frac{x^2 + 1}{x^2 - 1}$ in partial fractions.
59The diagram shows part of the curve of $y = x(x-2)^2$ , which passes through the point $P(1, 1)$ . Show that the equation…
60Express $\frac{8x^2 - x}{4x^2 - 1}$ in partial fractions. Use your results in part (i) to find $\int \frac{8x^2 - x}{4x^2 - 1} \, dx$ …