Math topic中文

Simultaneous one linear

solving simultaneous equations in two variables by substitution, with one of the equations being a linear equation

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 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…
3The expression $f(x) = ax^3 + (a - 3b)x^2 + 3bx + c$ is exactly divisible by $x^2 + 3x$ . When $f(x)$ is divided by $x + 2$ …
4It is given that $\left(x + \dfrac{k}{x^2}\right)^n$ is a binomial expansion where $k$ and $n$ are positive constants. W…
5A circle with centre O passes through the points P(-1,7) and Q(0,8). State the relationship between the perpendicular bi…
6Find the coordinates of the points of intersection of the curve $x + 3 = 2xy$ and the curve $y = \frac{2}{\sqrt{x}}$ .
7A drug is administered to a patient through an injection. The concentration of drug, $x$ mg/ $l$ in a patient's bloodstre…
8It is given that $f(x)=x^3+ax^2+bx+5$ , where $a$ and $b$ are constants, has a factor of $x-1$ and leaves a remainder of…
9Find the range of values of $x$ for which $2x(x-3) \ge 3-5x$ . Find the greatest value of $k$ such that the line $2y+k=x$ …
10A cone of height, $h$ and radius, $r$ fits exactly inside a sphere of radius $5\text{ cm}$ as shown below. $\left(\text{Volume of cone} = \frac{1}{3}\pi r^2 h\right)$ …
11Given that $f(x) = 2x^3 + px^2 + 7x + q$ . Find the value of $p$ and of $q$ for which $(x-3)$ and $(x-2)$ are factors of…
12Solve the simultaneous equations $\begin{aligned} y &= x^2 - 6, \\ y &= 2x - 3. \end{aligned}$ Prove that $\frac{3 \cos \theta + \cos 2\theta - 1}{\cos^2 \theta + 2 \cos \theta} = 2 - \sec \theta$ …
13The 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…
14The first two non-zero terms in the expansion of $(1 + bx)(1 + ax)^6$ are $1$ and $-\frac{21}{4}x^2$ . Given that $a < b$ …
15Solutions to this question by accurate drawing will not be accepted. In the trapezium $OABC$ , the point $A$ has coor…
16The point A lies on the curve $y = x \ln x^2$ , $x > 0$ . The tangent to the curve at A is parallel to the line $y - 3x = 1$ …
17Solutions to this question by accurate drawing will not be accepted. The diagram, which is not drawn to scale, shows…
18Solutions to this question by accurate drawing will not be accepted. The diagram shows an isosceles triangle PQR in whic…
19In the diagram, $OPQR$ is a trapezium with $OP$ parallel to $RQ$ and $OP$ perpendicular to $PQ$ . The point $P$ has coord…
20The points $P$ and $Q$ both lie on a circle and have coordinates $(-10, 15)$ and $(10, 0)$ respectively. The centre of t…
21The line $\frac{x}{a} + \frac{y}{b} = 1$ , where $a$ and $b$ are positive constants, intersects the x-axis at A and the y…
22A particle passes a fixed point $O$ and moves in a straight line such that, $t$ s after leaving $O$ , its velocity, $v\text{ m/s}$ …
23The 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…
24It is given that $f(x) = ax^3 + 3x^2 + bx + 1$ where $a$ and $b$ are constants. Given that $f(x)$ is divisible by $x + 1$ …
25Leroy has written down four numbers. The mean of these numbers is $2x + y$ , the median is $x + y$ and the mode is $x - y$ …
26A 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…
27The line $y + 2x = 12$ intersects the curve $y = 2x^2 - 6x - 4$ at two distinct points $A$ and $B$ . Find the coordinates…
28Factorise $64 - (x + 1)^3$ . Hence, solve $64 - (x + 1)^3 = 15(3 - x)$ , expressing non-integer roots in surd form.
29The curve $(x-1)^2 + (y-3)^2 = 5$ intersects the line $y - 3x = 5$ at two points. Find the coordinates of these two poin…
30Factorise $x^3 - 125$ . The expression $-2x^3 + 7x^2 + ax + b$ , where $a$ and $b$ are integers, has a factor of $x - 5$ a…
31The 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…
32The line $x + 2y = 3$ intersects the curve $x^2 + 12x + 5y^2 = 18$ at two points. Find the coordinates of the two points…
33The 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…
34The 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…
35The line $y = x - 4$ and the curve $y = \frac{2}{x - 3}$ intersect at the points $A$ and $B$ . Find the length of $A B$ .
36The first two non-zero terms in the expansion of $(1 + bx)(1 + ax)^6$ in ascending powers of $x$ are $1$ and $-\frac{21}{4}x^2$ …
37The diagram (not drawn to scale) shows a right-angled triangle PQR in which point P is (5, 4) and point R is (9, 2). Poi…
38The curve $y = \frac{5}{x} + 3$ and the line $x - 2y - 3 = 0$ intersect at the points $P$ and $Q$ . Find the coordinates…
39Given that $f(x)=x^2-ax+3$ , where $a$ is a constant, find the range of values of $a$ for which $f(x)>x-1$ , for all real…
40Find the values of $x$ and $y$ which satisfy the equations $2^x \div 4^y = \frac{1}{8},$ $9^y (\sqrt{3})^{2x} = 3\sqrt{3}.$ …
41When $f(x)$ is divided by $x - 1$ , the remainder is $61$ . The remainder is $-123$ when $f(x)$ is divided by $x + 3$ . Fin…
42Solutions to this question by accurate drawing will not be accepted. $ABCD$ is a trapezium where $AD$ is parallel to…
43In the diagram below, the line $OP$ makes an angle of $\alpha^\circ$ with the positive $x$ -axis such that $\tan \alpha = 0.2$ …
44The points $A(5, 7)$ and $B(-11, 15)$ lie on a circle $C$ . The centre of the circle $O$ lies on the line $y = -2x - 3$ .…
45The diagram shows the graphs of $y = 2x - 1$ and $y = kx^2$ , where $k$ is a positive constant. The graphs intersects at…
46Write down the general term in the binomial expansion of $\left(px - \dfrac{q}{x}\right)^n$ , where $p$ and $q$ are posit…
47已知双曲线 $x^{2}- \dfrac {y^{2}}{3}=1$ 上存在两点 $M$ ， $N$ 关于直线 $y=x+m$ 对称，且 $MN$ 的中点在抛物线 $y^{2}=18x$ 上，则实数 $m$ 的值为 ___ ．
48A and B are the points of intersection of the line $4y = 2x + 1$ and the curve $3y - x = 4xy$ . Find the coordinates of A…
49The polynomial $f(x)$ is such that $f(x)=6x^3+ax^2-50x+b$ , where $a$ and $b$ are integers. It is given that $f(x)$ is di…
50The diagram shows part of the curve $y = \frac{15}{(0.5x+1)}$ and a straight line $y = 1.25x - 10$ . The straight line in…
51Answer ALL the questions The function $f$ is defined, for all values of $x$ , by $f(x) = (2x - x^2)e^x$ . Find the ran…
52The diagram (not drawn to scale) shows a quadrilateral $ABCD$ such that $AB = BC$ and angle $BCD = 90^\circ$ . The point…
53Solve $9^{x+1} - 3^{x+2} = 2\left(3^{x+1} - 2\right)$ . Given that $\log_3(2x-1) = \log_9(2x-1) + 1$ , find the value of $x$ …
54In the expansion of $(2x-1)^2\left(1+\dfrac{p}{x}\right)^8$ , where $p$ is a positive constant, there is no term in $\dfrac{1}{x^3}$ …
55The straight line $2x + y = 1$ intersects the curve $x^2 - xy - y^2 + 5 = 0$ at the points $A$ and $B$ . Show that the le…
56The diagram shows part of the curve $y = \frac{5}{2x - 1}$ . The line $9y = -10x + 55$ intersects the curve at points $A$ …
57Show 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…
58Nurul took 2.5 hours to travel at a constant speed from Town $A$ to $B$ . On her journey back, she increased her constant…
59A virus is spreading in a country. The number of infected cases, $I$ , after $t$ days, can be modelled by $I = \frac{a}{b + 2^{20-t}}$ …
60A tangent to a circle at the point $(-3,3)$ passes through the origin. Find the equation of the normal to the circle at…