Linear systems

A piece of alloy made from gold, silver and bronze weighs 136 g. A second piece of alloy weighs 323 g and contains the same amount of gold, twice as much silver, and three times as much bronze as the first piece. A third piece of alloy weighs 238 g and contains 4 times as much gold, one third as much silver, and same amount of bronze as the second piece.…

Mary runs a fruit stall and keeps a record of the daily sales revenue. The weights of fruits and total amounts received from the sales of three particular types of imported fruits in the past three days are shown in the following table.

A Sumo wrestler would like to have fish fillet, salad and fries for breakfast. As he is on a special diet, he must make sure that his intake (in grams) of protein, carbohydrates and fats per meal is in the ratio of 4:8:3. The table below shows the nutritional breakdown for one serving of each item.

A curve $C$ has equation $y=a(2x+1)^2+bx+ce^{2x}$, where $a,b,c$ are constants. It is given that $C$ passes through the point $(1,17-e^2)$ and the gradient of $C$ at the point $(0,1)$ is 5. Find the equation of $C$.

The first four terms of a sequence are given by $T_1=1,T_2=2,T_3=4,T_4=8$. Given that $T_r$ is a cubic polynomial in $r$, find $T_r$ in terms of $r$.

An online shopper from Singapore (currency Singapore Dollar, symbol …

The cubic curve $C$ has a maximum point $A$ and a minimum point $B$. The line segment $AB$ is the diameter of a circle with equation $(x-5)^2+y^2=25$. Given that the $x$-coordinate of $B$ is 1, find the equation of $C$.

The diagram below shows a portion of a quadratic curve $y=f(x)$. The area of the shaded region bounded by the axes, the curve $y=f(x)$ and the line $x=1$ is $\dfrac{19}{6}$ units$^2$. Given also that the curve passes through the points $(1,\dfrac{13}{2})$ and $(-2,\dfrac12)$, determine the equation of the curve.

$N$ is a three-digit positive number that is 26 times the sum of its digits. When the digits of $N$ are reversed, the new number is 198 more than $N$. Given that the digits of $N$ are 3 consecutive terms of an arithmetic progression, find $N$.

Given the polynomial $x^4+ax^2+bx+c$ has a factor $(x-2)$ and gives remainders 12 and 26 when divided by $(x-3)$ and $(x-4)$ respectively, find the values of $a,b,c$.

The usual selling price of a T-box game console, a Kinect sensor and a game DVD is 499 dollars in total. During the Great Singapore Sale, two companies, P and Q, offered the following discounts to their customers:

A candy shop is having a Halloween sale. The items that are on promotion are chocolate bars, gummy bears and lollipops. There is a 20% discount for every chocolate bar purchased, a 2 dollar discount for every 3 bags of gummy bears purchased and every 6 lollipops can be purchased at the price of 5 lollipops.

A small garment factory manufactures dresses, blouses and T-shirts. The production process includes cutting, sewing and packaging. The table below shows the amounts of time in minutes required to produce each piece of garment.

In a petroleum refinery, 2 types of petrol: regular and premium, are obtained by blending naphtha and reformed gasoline in a certain ratio.

The diagram below shows the cross-section of a single track tunnel entrance for steam-engine in a model train system. The elliptical entrance of the tunnel can be modelled by the equation $x^2+Ay^2+By+C=0$.