Straight line graphs

A company is trying to determine the most profitable selling price for a new toy. Given that

• if the selling price of each toy is £30, the company expects to sell 1500 toys in one year

• if the selling price of each toy is £50, the company expects to sell 300 toys in one year

Using a linear model, with y being the expected number of toys sold in one year and x pounds being the selling price of the toy,

find an equation for y in terms of x. Given that

• the cost of making each toy is £10

• the company has additional costs of £8 000 per year

show that, according to the model, the yearly profit, P, in thousands of pounds, is given by

Use the model given in part (b) to answer parts (c), (d) and (e). Given that the company wishes to make a profit on sales of the toy,

find the range of possible selling prices of the toy.

Hence, or otherwise, deduce the selling price of the toy that maximises the profit. In one particular year, the company sold the toy for £35 and made £21 750 profit.

Use this information to evaluate the suitability of the model.

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

The curve C has equation y = f(x) where x

Given that

• the curve has a stationary point with x coordinate α

• α is small

use the small angle approximation for cosx to estimate the value of α to 3 decimal places.

The point P(0, 3) lies on C

Find the equation of the tangent to the curve at P, giving your answer in the form y = mx + c, where m and c are constants to be found.

A small factory makes bars of soap. On any day, the total cost to the factory, £ y, of making x bars of soap is modelled to be the sum of two separate elements:

a cost that is proportional to the number of bars of soap that are made that day

Write down a general equation linking y with x, for this model.

The bars of soap are sold for £2 each. On a day when 800 bars of soap are made and sold, the factory makes a profit of £500 On a day when 300 bars of soap are made and sold, the factory makes a loss of £80 Using the above information,

show that y = 0.84 x + 428

With reference to the model, interpret the significance of the value 0.84 in the equation.

Assuming that each bar of soap is sold on the day it is made,

find the least number of bars of soap that must be made on any given day for the factory to make a profit that day.

The circle C has centre A with coordinates (7, 5).

The line l, with equation y 2x 1, is the tangent to C at the point P, as shown in Figure 3.

Show that an equation of the line PA is 2y x 17

Find an equation for C.

The line with equation y 2x + k, k 1 is also a tangent to C.

Find the value of the constant k.

The curve C has equation

Find in its simplest form.

Find an equation of the tangent to C at the point where x = –1 Give your answer in the form ax + by + c = 0, where a, b and c are integers.

Figure 2 shows a right angled triangle LMN. The points L and M have coordinates (–1, 2) and (7, –4) respectively.

Find an equation for the straight line passing through the points L and M. Give your answer in the form ax + by + c = 0, where a, b and c are integers.

Given that the coordinates of point N are (16, p), where p is a constant, and angle LMN = 90°,

find the value of p.

Given that there is a point K such that the points L, M, N, and K form a rectangle,

find the y coordinate of K.

Find the gradient of L,. (2) The line L, is perpendicular to L, and passes through the point (2, 5).

The straight line L₁ passes through the points (–1, 3) and (11, 12).

Find an equation for L₁ in the form ax + by + c = 0, where a, b and c are integers.

The line L₂ has equation 3y + 4x – 30 = 0.

Find the coordinates of the point of intersection of L₁ and L₂.

The line l₁ has equation y = 2x + 3 The line l₂ is perpendicular to l₁ and passes through the point (5, 6).

Find an equation for l₂ in the form ax + by + c = 0, where a, b and c are integers.

The line l₂ crosses the x-axis at the point A and the y-axis at the point B.

Find the x-coordinate of A and the y-coordinate of B.

Given that O is the origin,

find the area of the triangle OAB.

The point P (4, –1) lies on the curve C with equation y = f(x), x > 0, and

Find the equation of the tangent to C at the point P, giving your answer in the form y = mx + c, where m and c are integers.

Find f(x).

The line L₁ has equation 4y + 3 = 2x The point A (p, 4) lies on L₁

Find the value of the constant p.

The line L₂ passes through the point C (2, 4) and is perpendicular to L₁

Find an equation for L₂ giving your answer in the form ax + by + c = 0, where a, b and c are integers.

The line L₁ and the line L₂ intersect at the point D.

Find the coordinates of the point D.

Show that the length of CD is 5 A point B lies on L₁ and the length of AB = The point E lies on L₂ such that the length of the line CDE = 3 times the length of CD.

Find the area of the quadrilateral ACBE.

Find an equation of the line joining A (7, 4) and B (2, 0), giving your answer in the form ax + by + c =0, where a, b and c are integers.

Find the length of AB, leaving your answer in surd form.

The point C has coordinates (2, t), where t > 0, and AC = AB.

Find the value of t.

Find the area of triangle ABC.

The curve C has equation y = f(x), x > 0, where

Given that the point P (4, 5) lies on C, find

f(x),

an equation of the tangent to C at the point P, giving your answer in the form ax + by + c = 0, where a, b and c are integers.

The line l₁ has equation 3x 5y 2 0

Find the gradient of l₁.

The line l₂ is perpendicular to l₁ and passes through the point (3, 1).

Find the equation of l₂ in the form y = mx + c, where m and c are constants.

The curve C has equation

Find an equation of the tangent to C at the point where x = 2

Factorise completely x³ 4x

Sketch the curve C with equation

showing the coordinates of the points at which the curve meets the x-axis.

The point A with x-coordinate 1 and the point B with x-coordinate 3 lie on the curve C.

Find an equation of the line which passes through A and B, giving your answer in the form y = mx + c, where m and c are constants.

Show that the length of AB is k 10, where k is a constant to be found.

The points A and B have coordinates (6, 7) and (8, 2) respectively. The line l passes through the point A and is perpendicular to the line AB, as shown in Figure 1.

Find an equation for l in the form ax + by + c = 0, where a, b and c are integers.

Given that l intersects the y-axis at the point C, find

the coordinates of C,

the area of OCB, where O is the origin.

The line l₁ passes through the point A (2, 5) and has gradient ––

Find an equation of l₁, giving your answer in the form y = mx + c.

The point B has coordinates (–2, 7).

Show that B lies on l₁.

Find the length of AB, giving your answer in the form k√5, where k is an integer.

The point C lies on l₁ and has x-coordinate equal to p. The length of AC is 5 units.

Show that p satisfies

The points Q (1, 3) and R (7, 0) lie on the line l₁, as shown in Figure 2. The length of QR is a5.

Find the value of a.

The line l₂ is perpendicular to l₁, passes through Q and crosses the y-axis at the point P, as shown in Figure 2. Find

the coordinates of P,

The point A (–6, 4) and the point B (8, –3) lie on the line L.

Find an equation for L in the form ax + by + c = 0, where a, b and c are integers.

Find the distance AB, giving your answer in the form k√5, where k is an integer.