Math topic中文

Discriminant roots

conditions for a quadratic equation to have:

(i) two real roots
(ii) two equal roots
(iii) no real roots
and related conditions for a given line to:
(i) intersect a given curve
(ii) be a tangent to a given curve
(iii) not intersect a given curve

Practice questions

1The curve $C$ has equation $y = \frac{x^2 + x + 3}{x - 2}$ . Without using a calculator, find the set of values that $y$ …
2The equation of a curve is $y = ax^2 - 3x + 4 - a$ , where $a$ is a constant. In the case where $a = -2$ , find the set of…
3A curve has the equation $y = \frac{2x - 3}{3x + 4}$ , where $x \ne -\frac{4}{3}$ . The normal to the point $P$ on the cur…
4The radius of a circle increases at a rate of 3 cm s^{-1}. Find, in terms of $\pi$ , the rate of increase of the area whe…
5A particle starts from rest from a point $O$ and moves in a straight line such that its velocity $v$ m/s, is given by $v = 24t - 6t^2$ …
6关于 $x$ 的方程 $x^{2}-bx+4=0$ 有两个相等的正实数根，则 $b$ 的值为 $(\quad)$ A. $4$ B. $-4$ C. $-4$ 或 $4$ D. $0$
7The solution of the equation $3^{2x-1}=4(3^x)$ can be expressed as $\dfrac{\ln p}{\ln q}$ . Find the value of $p$ and o…
8Show that 1 is a solution of the equation $x^3 - 5x^2 = 2 - 6x$ . Hence solve the equation $x^3 - 5x^2 = 2 - 6x$ complete…
9关于 $x$ 的一元二次方程 $a{{x}^{2}}+bx+1=0.(1)$ 当 $b=a+2$ 时，利用根的判别式判断方程根的情况； (2)若方程有两个相等的实数根，写出一组满足条件的 $a$ ， $b$ 的值，并求此时方程的根.
10Find the range of values of $x$ for which $2x(x-3) \ge 3-5x$ . Find the greatest value of $k$ such that the line $2y+k=x$ …
11已知△ABC是等腰三角形，BC=8，AB，AC的长是关于x的一元二次方程x2﹣10x+k=0的两根，则（） A. k=16 B. k=25 C. k=﹣16或k=﹣25 D. k=16或k=25
12若关于 $x$ 的一元二次方程 $(m-1)x^{2}+2mx+m+3=0$ 有两个不等的实根，则 $m$ 的取值范围 $(\quad)$ A. $m< \dfrac {3}{2}$ B. $m< \dfrac {3}{2}$ 且 $m≠1\:\:\:\:\:$ …
13已知 $\mathbf{x}_{\mathbf{1}}\mathbf{,}\mathbf{x}_{\mathbf{2}}$ 是一元二次方程 $\mathbf{4}\mathbf{k}\mathbf{x}^{\mathbf{2}}\mathbf{-4}\mathbf{kx+k+1=0}$ …
14若抛物线 $y=x^{2}-2x+k$ 和 $x$ 轴有交点，则 $k$ 的取值范围是 ___.
15关于 $x$ 的一元二次方程 $x^{2}-kx+4=0$ 的两个实数根分别是 $x_{1}$ 、 $x_{2}$ ，且满足 $x_{1}^{2}+x_{2}^{2}-2x_{1}-2x_{2}-7=0$ ，则 $k$ 的值为 ___.
16在平面直角坐标系中，如果点 $P$ 的横坐标与纵坐标相等，则称点 $P$ 为和谐点，例如：点 $P(1,1)$ 、 $(-2,-2)$ 、 $(0.5,0.5)$ …，都是和谐点，若二次函数 $y=ax^{2}+7x+c(a≠0)$ 的图象…
17A particle $P$ moves in a straight line so that, $t$ seconds after passing through a fixed point $O$ , its velocity $v$ m…
18In the diagram, A, B and D are fixed points such that $AB = 25\text{ cm}$ , $AD = 35\text{ cm}$ and angle $BAD = 90^\circ$ …
19A particle moves in a straight line, so that, $t$ seconds after leaving a fixed point $O$ its velocity, $v$ is given by…
20The diagram shows the line $y = 24$ and part of the graph of $y = |x^2 - 16x + 39|$ . The graph crosses the line $y = 24$ …
21It is given that $a$ and $b$ are the roots of the quadratic equation $x^2 - 2x - 1 = 0$ and that $a > b$ . Show that $\frac{a}{b} = -3 - 2\sqrt{2}$ …
22当 $m$ 为何值时，关于 $x$ 的一元二次方程 $(2m+1)x^{2}+4mx+2m-3=0$ ． (1)有两个不相等的实数根； (2)有两个相等的实数根； (3)没有实数根．
23Solve the equation $x + 1 + \sqrt{5 - x} = 0$ . Solve the equation $\log_x(2x-1) = \log_{x^2}(4-5x)$ .
24如果关于 $x$ 的一元二次方程 $kx^{2}-3x-1=0$ 有两个不相等的实根，那么 $k$ 的取值范围是 ___ ．
25A particle moves along the curve $y = \frac{3(x+6)}{x+4}$ , where $x \ne -4$ , in such a way that the $y$ -coordinate of th…
26By considering the general term in the binomial expansion of $\left(x + \frac{1}{ax^2}\right)^7$ , where $a$ is a constan…
27Solve the equation $2^{4p+1} + 20(4^{p-1}) = 3$ . Explain why $2^{4p+1} + 20(4^{p-1}) = k$ has no solution if $k < -3\frac{1}{8}$ …
28Solutions to this question by accurate drawing will not be accepted. The diagram shows a trapezium $ABCD$ in which $AB$ …
29Solutions to this question by accurate drawing will not be accepted. The diagram shows an isosceles triangle PQR in whic…
30By using an appropriate substitution, solve the equation $e^{\frac{1}{2}x} = 2 + 24 e^{-\frac{1}{2}x}$ .
31A curve has the equation $y = x^3 + kx^2 + kx + 8$ . Find the set of values of $k$ such that the curve is an increasing f…
32The lines $y=9$ , $y=-1$ and $3y=4x+9$ are tangents to a circle. The x-coordinate of the centre of the circle is positive…
33It is given that $f(x) = ax^3 + 3x^2 + bx + 1$ where $a$ and $b$ are constants. Given that $f(x)$ is divisible by $x + 1$ …
34A curve is defined by the equation $y = \ln\left(\frac{\text{e}^x}{x^2 + 10}\right)$ . Show that the curve has no turning…
35A curve has equation $y = x^2 + x + 2k - 4$ and a line has equation $y - 2x = k$ , where $k$ is a constant. Find the set…
36The equation of a curve is $y = 3x^2 + 5x + 1$ . Find the set of values of $x$ for which the curve lies completely above…
37已知关于 $x$ 的一元二次方程 $(x-2)(x-5)-m^{2}=0$ ． (1)求证：对于任意实数 $m$ ，方程总有两个不相等的实数根； (2)若方程的一个根是 $1$ ，求 $m$ 的值及方程的另一个根．
38Show that the equation $x^2 + px + 2p = 2x + 8$ has real roots for all real values of $p$ .
39Factorise $64 - (x + 1)^3$ . Hence, solve $64 - (x + 1)^3 = 15(3 - x)$ , expressing non-integer roots in surd form.
40The cubic polynomial $\text{f}(x)$ is such that the coefficient of $x^3$ is $-1$ and the roots of $\text{f}(x) = 0$ are…
41若函数 $y=(a-1)x^{2}-4x+2a$ 的图象与 $x$ 轴有且只有一个交点，则 $a$ 的值为 ___ ．
42The diagram below shows part of the graph of $y = \frac{x^2 + 2x + 5}{x + 3}$ . Find the coordinates of the minimum point…
43Solve the equation $8 - ext{\sqrt{11 - 2x}} = -x$ , giving your answer in the form $a + b\sqrt{c}$ , where $a$ , $b$ and $c$ …
44不解方程，判断方程根的情况 $(1)2x^{2}+3x-4=0$ ； $(2)3x^{2}-5x+4=0$ ； $(3)ax^{2}+bx=0(a\neq 0)$ ； $(4){3}{{x}^{{2}}}+{2}={2}\sqrt{{6}}x$ …
45圆 $x^{2}+y^{2}-mx+y+m=0$ 在 $x$ 轴上截得的弦长是它在 $y$ 轴上截得的弦长的 $2$ 倍，则实数 $m$ 的值是 $(\quad)$ A. $-6-2\sqrt{10}$ B. $-6+2\sqrt{10}$ …
46已知函数 $f\left({x}\right)=ax^{2}-3x+b$ ，其中 $a$ ， $b\in R$ . (1)若关于 $x$ 的不等式 $f\left({x}\right)>0$ 的解集为 $\left({-4,1}\right)$ …
47Factorise $x^3 - 125$ . The expression $-2x^3 + 7x^2 + ax + b$ , where $a$ and $b$ are integers, has a factor of $x - 5$ a…
48已知 $a > 0$ ，函数 $f(x)=x{\,\!}^{2}+a\ln x- ax$ 在 $(0,+∞)$ 上是增函数，则 $a$ 的范围为__
49The equation of a curve is $y = ax^2 + 4x + 3a - 7$ , where $a$ is a constant. The line $y = 2ax - 5$ is a tangent to the…
50Show that the solution of the equation $2^{3x+4}\times 5^{2x-1}=16^{x}\times 5^{3x}$ is $\lg\dfrac{16}{5}$ . Express $2\log_{2}x-\log_{2}(x-4)=3$ …
51A particle P moves along a horizontal straight line so that t seconds after motion has begun, its displacement, s m, fro…
52The equation of a curve is $x^2 + 4y^2 = 20$ and the equation of a line is $y = k - x$ , where $k$ is a constant. Find th…
53A ball is thrown vertically upwards. Its height $h$ m, above the ground at time $t$ seconds after being thrown is given…
54The equation of a curve is $y = 2x^2 + 5$ . Find the set of values of $x$ for which $y > 3x^2$ and represent this set on…
55A particle moves in a straight line so that its velocity, $v$ m/s is given by $v = t^2 - 4t + 3$ , where $t$ is the time…
56A particle travels in a straight line so that, $t$ seconds after leaving a fixed point $O$ , its velocity, $v\text{ m s}^{-1}$ …
57The equation of a curve is $y = \frac{5x}{6 - 2x}$ . Find the $x$ -coordinates of the points at which the gradient of the…
58The line $y = x - 4$ and the curve $y = \frac{2}{x - 3}$ intersect at the points $A$ and $B$ . Find the length of $A B$ .
59下列方程中有两个相等实数根的是 $(\quad)$ A. $7x^{2}-x-1=0$ B. $9x^{2}=4(3x-1)$ C. $x^{2}+7x+15=0$ D. $2x^{2}-\sqrt{3}x-2=0$
60The equation of a curve is $y = -2x^2 - kx - 6 + k$ , where $k$ is a constant. Find the range of values of $k$ for which…