Math topic中文

System of linear equations

Solving a system of linear equations using a graphing calculator or a graphing software.

Practice questions

1It is given that $f(x)=ax^3+bx^2+cx+d$ , where $a$ , $b$ , $c$ and $d$ are constants. The curve $C$ with equation $y=f(x)$ …
2Find the exact solution of the inequality $\frac{2x+1}{x^2-1} \le 1.$ A curve $G$ has equation $ay^2 + bx^3 + cx = 2$ , w…
3A sequence of numbers $u_1, u_2, u_3, \dots$ has a sum $S_n$ where $S_n = \sum_{r=1}^{n} u_r$ . It is given that $S_n = An^2(n + 1) - Bn$ …
4A curve $C$ has equation $y = \frac{a}{x} + bx^2 + c$ , where $a$ , $b$ and $c$ are constants. It is given that $C$ passes…
5A random variable $X$ has the probability distribution given in the following table. | $x$ | 1 | 4 | 6 | 8 | |---|---|--…
6There are 3 bike-sharing companies in the current market. For each ride, $\alpha$ -bike charges a certain amount per 5 mi…
7A home interior designer is looking to design a pull-down wardrobe lift. To test his concept, the designer built a small…
8The diagram below shows a tetrahedron $ABCD$ . The equation of the plane $ABD$ is $4x + y + 2z = 16$ . Given that $A$ is o…
9The curve C is given by the equation $\dfrac{y^2}{x^2}=\dfrac{3a-x}{a+x},\quad x \ne 0,$ where $a$ is a positive const…
10A temporary isolation centre is built to manage the increasing number of COVID-19 cases. The roof takes the shape of a t…
11Do not use a calculator in answering this question. Given that $x = 1 + 3\mathrm{i}$ is a root of the equation $x^3 + ax^2 + 18x + b = 0,$ …
12In the triangle shown below, one vertex is origin O, and the two other vertices are A and B where $\overrightarrow{OA}=\mathbf{a}$ …
13Let f(x) be a cubic polynomial where f(2) = 6 and f(-1) = 15. Also, f(x) takes on stationary values at x = -2 and x = 1.…
14A hyperbola has the equation $a(x-2)^2 - b(y+1)^2 = c$ where $a$ , $b$ , and $c$ are positive integers. It passes through…
15A ball is rolling in a straight line such that its distance away from the starting point, $s$ cm, can be modelled using…
16The sum of the first $n$ terms of a sequence, $S_n$ , is given by $S_n = an^3 + bn^2 + cn + d$ , where $a$ , $b$ , $c$ and $d$ …
17The first four terms of a sequence are given by $v_1 = -9$ , $v_2 = 7$ , $v_3 = 47$ and $v_4 = 141$ . It is given that $v_n$ …
18The probability distribution of a discrete random variable $X$ is given as follows: | $x$ | $-2$ | $-1$ | $0$ | $1$ | $3$ …
19A curve $C$ has equation $y = px + \sqrt{qx} - \ln(rx)$ , where $p$ , $q$ and $r$ are constants. Given that $C$ crosses th…
20Payprice Supermarket held a sale at the "Fresh Food Corner". Ms Ng, Mdm Lim, Mr Yeo and Mrs Heng were at Payprice for th…
21By the end of 2022, the percentages of a population who has received V dose(s) of Covid-19 vaccine are shown in the tabl…
22Planes $p$ and $q$ are perpendicular. Plane $p$ has equation $2x - z = 10$ . Plane $q$ contains the line $l$ with equatio…
23In the annual Best Infantry Unit Competition, 3 top units - Units A, B and C, are shortlisted for the eventual champion.…
24The two planes $p_{1}$ and $p_{2}$ , given by the equations $2x+y=1$ and $8x+ay+z=4$ respectively, meet at a line $L$ whi…
25Solve the equation $iz^5=-32$ , giving your roots in the form $re^{i\theta}$ , where $r>0$ and $-\pi<\theta\le\pi$ . Sketch…
26A function $\text{f}$ is defined by $\text{f}(x) = x^3 + ax^2 + bx + c$ , where $a$ , $b$ and $c$ are constants. The graph…
27It is given that $\mathrm{f}(x) = ax^3 + bx^2 + cx + d$ , where $a$ , $b$ , $c$ and $d$ are constants. The curve $C$ with e…
28A café sells three types of cake: banana, chocolate and walnut. The café gives 12% discount on total cost to any custome…
29Adam bought a total of 50 fruits consisting of apples, oranges and pineapples. The apples, oranges and pineapples cost \…
30In the finals of a General Knowledge Quiz, a team is required to answer 25 questions. Each question that is correctly an…
31A confectionary bakes 20 banana cakes, 50 chocolate cakes and 30 durian cakes every day. The total price of 1 banana cak…
32A household's monthly utility bill is calculated by summing up the charges it incurs for its usage of electricity, gas a…
33A curve has equation $y=f(x)$ , where $f(x)=x^{3}+ax^{2}+bx+c$ . The curve $y=f(x)$ has a maximum point at $(1,-32)$ , whil…
34The point A has coordinates (1, -2, 4) and the plane $\pi_1$ has equation $2x - y + 2z = 5.$ Find the exact shortest dis…
35A nursery supplies plants to Evergreen Pte. Ltd. The heights, in centimetres, of plants supplied may be taken to follow…
36It is given that $\mathrm{f}(x) = \ln(1 + 2x^2)$ . A polynomial $p(x) = ax^2 + bx + c$ is used to approximate $\mathrm{f}(x)$ …
37The plane $\pi_1$ contains the point $(2,-2,1)$ and the line $\mathbf{r} = \begin{pmatrix}0 \\ 1 \\ 8\end{pmatrix} + \lambda \begin{pmatrix}0 \\ 2 \\ 3\end{pmatrix},$ …
38To build a Virtual Reality (VR) experience, a virtual world is created to contain geometric models. With reference to th…
39A sequence $T_n$ is given by $T_n = ext{ln}(n^a) + bn^2 + c$ for $n ext{ } oldsymbol{ m ackslash in} ext{ } oldsymbol{ m ackslash Z}^+$ …
40The plane $\pi_1$ has equation $\mathbf{r}\cdot\begin{pmatrix}2 \\ 5 \\ 1\end{pmatrix}=11.$ The plane $\pi_2$ passes t…
41A function $\text{f}$ is defined by $\text{f}(x) = ax^3 + bx^2 + cx + d$ . The graph of $y = \text{f}(x)$ passes through…
42An airfield on a horizontal flat piece of ground must manage the take-off flight paths of aircraft from its runways. The…
43Relative to the origin O, points A and B have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively, where $\mathbf{a}$ …
44In a city of 5 million people, the number of customers of a new company increases at a rate proportional to the number o…
45The diagram shows a pyramid DOABC. Taking unit vectors i, j, k as shown, the position vectors of A, B, C, D are given by…
46A renovation company wishes to obtain sand, stone and brick for construction work. The company sends its requirements fo…
47A small company makes wine glasses. Each day, n randomly chosen wine glasses are checked and the number of wine glasses…
48A curve $D$ has equation $y = ax + b \ln x + \frac{c}{x}, \quad x > 0,$ where $a$ , $b$ and $c$ are constants. It is gi…
49Four friends buy three different kinds of fruit in the market. When they get home they cannot remember the individual pr…
50The plane $\Pi_1$ contains the points $A(1,4,2),\ B(1,0,5)$ and $C(0,8,-1)$ . Find a cartesian equation of $\Pi_1$ . Find…
51A curve $G$ has equation $y = ax + b + \dfrac{c}{(x+1)^2}$ , where $a$ , $b$ and $c$ are positive integers. Let $R$ be the…
52**In this question, you should state clearly all the distributions that you use, together with the values of the appropr…
53The curve $C$ has equation $2x - y^2 = (x + y)^2$ . Find the equations of the tangents to $C$ which are parallel to the $x$ …
54The plane $p_1$ has equation $x - y + z = 0$ and the plane $p_2$ has equation $x + y + z = 2$ . Find the acute angle betw…
55In a city of 5 million people, the number of customers of a new company increases at a rate proportional to the number o…
56The equations of two planes $P_1$ and $P_2$ are $x - 2y + 3z = 4$ and $3x + 2y - z = 4$ respectively. The equation of a…
57The sum, $S_{n}$ , of the first $n$ terms of a sequence $u_{1}, u_{2}, u_{3}, \dots$ is given by $S_{n}=a(3^{n})+bn+c$ , w…
58A biased die in the form of a tetrahedron has its four faces labelled 1 to 4, with one number printed on each face. The…
59The lines l_1 and l_2 have equations $\mathbf{r}=\begin{pmatrix}2.5k\\k\\2k\end{pmatrix}+\lambda\begin{pmatrix}1\\3\\1\end{pmatrix}\quad\text{and}\quad \mathbf{r}=\begin{pmatrix}3\\0.5k\\0.5k\end{pmatrix}+\mu\begin{pmatrix}4\\7\\5\end{pmatrix},\;\lambda,\mu\in\mathbb{R}$ …
60The finishing times in a 10km race with a large number of runners follow a normal distribution. After 40 minutes, 10% of…