Quadratic equations

Given that

• f(x) = 2x³ + 3x² – 16x + 16

• f(–4) = 0

write f(x) in the form (x + a) Q (x) where a is a constant and Q(x) is a quadratic expression.

Hence prove that –4 is the only real root of the equation f(x) = 0 (Solutions relying on calculator technology are not acceptable.)

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.

The functions f and g are defined by

Find fg(2)

Find g^–1

Find gf(x), giving your answer as a simplified fraction.

Deduce the range of gf(x).

The function h is defined by

where k is a constant.

Find the range of values of k for which the equation

f(x) = h(x)

has no real solutions.

Jamie takes out an interest‑free loan of £8100

Jamie makes a payment every month to pay back the loan.

Jamie repays £400 in month 1, £390 in month 2, £380 in month 3, and so on, so that the amounts repaid each month form an arithmetic sequence.

Show that Jamie repays £290 in month 12

After Jamie’s Nth payment, the loan is completely paid back.

Show thatN² – 81N + 1620 = 0

Hence find the value of N.

A sequence u₁, u₂, u₃,... is defined by

where k is a positive constant.

Given that u₃ = –1

show that

Hence

find the value of k,

The graph in Figure 3 shows the path of a small ball.

The ball travels in a vertical plane above horizontal ground.

The ball is thrown from the point represented by A and caught at the point represented by B.

The height, H metres, of the ball above the ground has been plotted against the horizontal distance, x metres, measured from the point where the ball was thrown.

With respect to a fixed origin O, the point A has coordinates (0, 2) and the point B has coordinates (20, 0.8), as shown in Figure 3.

The ball reaches its maximum height when x = 9

A quadratic function, linking H with x, is used to model the path of the ball.

Find H in terms of x.

Give one limitation of the model.

Chandra is standing directly under the path of the ball at a point 16 m horizontally from O.

Chandra can catch the ball if the ball is less than 2.5 m above the ground.

Use the model to determine if Chandra can catch the ball.

The curve C has equation

Find an expression for in terms of x and y.

The point P(1, 0) lies on C.

Show that the normal to C at P has equation

Prove that the normal to C at P does not meet C again.

You should use algebra for your proof and make your reasoning clear.

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

Solutions relying entirely on calculator technology are not acceptable.

where a is a positive constant.

Given (x – a) is a factor of f(x),

show that

Hence

find the value of a

use algebra to find the exact solutions of the equation

f(x) = 3

A circle C has equation

where k is a constant.

Find in terms of k,

the coordinates of the centre of C

the radius of C

The line with equation y = 2x – 1 intersects C at 2 distinct points.

Find the range of possible values of k.

The function f is defined by

where k is a positive constant.

Show that

where g(x) is a function to be found. Given that the curve with equation y = f(x) has at least one stationary point,

find the range of possible values of k.

Given that

express f (x) in the form (x + a)² + b where a and b are integers to be found. The curve with equation y = f (x)

• meets the y-axis at the point P

• has a minimum turning point at the point Q

Write down

the coordinates of P

the coordinates of Q

Given that

express f (x) in the form (x + a)² + b where a and b are integers to be found.

The curve with equation y = f (x)

• meets the y-axis at the point P

• has a minimum turning point at the point Q

Write down

the coordinates of P

the coordinates of Q

The sequence u₁, u₂, u₃, … is defined by

where k is an integer.

Given that u₁ + 2u₂ + u₃ = 0

show that

Find the value of k, giving a reason for your answer.

Find the value of u₃

The circle C has equation

Find

the coordinates of the centre of C,

the exact radius of C, giving your answer as a simplified surd.

The line l has equation y = 3x + k where k is a constant.

Given that l is a tangent to C,

find the possible values of k, giving your answers as simplified surds.

Figure 3 is a graph of the trajectory of a golf ball after the ball has been hit until it first hits the ground.

The vertical height, H metres, of the ball above the ground has been plotted against the horizontal distance travelled, x metres, measured from where the ball was hit.

The ball is modelled as a particle travelling in a vertical plane above horizontal ground.

Given that the ball

• is hit from a point on the top of a platform of vertical height 3 m above the ground

• reaches its maximum vertical height after travelling a horizontal distance of 90 m

• is at a vertical height of 27 m above the ground after travelling a horizontal distance of 120 m

Given also that H is modelled as a quadratic function in x

find H in terms of x

Hence find, according to the model,

the maximum vertical height of the ball above the ground,

the horizontal distance travelled by the ball, from when it was hit to when it first hits the ground, giving your answer to the nearest metre.

The possible effects of wind or air resistance are two limitations of the model. Give one other limitation of this model.

f(x) = 2x² + 4x + 9 x

Write f(x) in the form a(x + b)² + c, where a, b and c are integers to be found.

Sketch the curve with equation y = f(x) showing any points of intersection with the coordinate axes and the coordinates of any turning point.

Describe fully the transformation that maps the curve with equation y = f(x) onto the curve with equation y = g(x) where

g(x) = 2(x – 2)² + 4x – 3 x

Find the range of the function

A curve C has parametric equations

Show that all points on C satisfy y 6(x 3)²

Sketch the curve C.

Explain briefly why C does not include all points of y 6(x 3)², x

The line with equation x y k, where k is a constant, intersects C at two distinct points.

State the range of values of k, writing your answer in set notation.

Calculate f (2)

Write f (x) as a product of two algebraic factors.

Using the answer to (a)(ii),

prove that there are exactly two real solutions to the equation

deduce the number of real solutions, for 7π θ <10π, to the equation

Figure 1 is a graph showing the trajectory of a rugby ball.

The height of the ball above the ground, H metres, has been plotted against the horizontal distance, x metres, measured from the point where the ball was kicked.

The ball travels in a vertical plane.

The ball reaches a maximum height of 12 metres and hits the ground at a point 40 metres from where it was kicked.

Find a quadratic equation linking H with x that models this situation.

The ball passes over the horizontal bar of a set of rugby posts that is perpendicular to the path of the ball. The bar is 3 metres above the ground.

Use your equation to find the greatest horizontal distance of the bar from O.

Give one limitation of the model.

Given that y = 2^x,

express 4^x in terms of y.

Hence, or otherwise, solve

Find the set of values of x for which

2(3 x + 4) > 1 – x

4x² + 8x + 3 ax + b² + c

Find the values of the constants a, b and c.

On the axes on page 27, sketch the curve with equation y = 4x² + 8x + 3, showing clearly the coordinates of any points where the curve crosses the coordinate axes.

4x – 5 – x² = q – (x + p)² where p and q are integers.

Find the value of p and the value of q.

Calculate the discriminant of 4x – 5 – x²

On the axes on page 17, sketch the curve with equation y = 4x – 5 – x² showing clearly the coordinates of any points where the curve crosses the coordinate axes.

Show that x² +6x + 11 can be written as

where p and q are integers to be found.

In the space at the top of page 7, sketch the curve with equation y x² 6x 11, showing clearly any intersections with the coordinate axes.

Find the value of the discriminant of x² +6x + 11

f(x) x² 4kx (311k), where k is a constant.

Express f(x) in the form (x + p)² + q, where p and q are constants to be found in terms of k.

Given that the equation f(x) = 0 has no real roots,

find the set of possible values of k.

Given that k = 1,

sketch the graph of y = f(x), showing the coordinates of any point at which the graph crosses a coordinate axis.