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OCR Plenary Tasks

Pure Core 3

by

Pam Charlton

Commissioned by The PiXL Club Ltd.

Date October 2013

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This resource is strictly for the use of member schools for as long as they remain members of The PiXL Club. It may not be copied, sold nor transferred to a third party or used by the school after membership ceases. Until such time it may be freely used within the member school.

All opinions and contributions are those of the authors. The contents of this resource are not connected with nor endorsed by any other company, organisation or institution.

© Copyright The PiXL Club Ltd, 2013

Functions

f(x) = 4x – 3

g(x) = x + 1

h(x) = x2

Evaluate

f(-1), g(5) and h(2)

Find

hf(x)

The domain of f(x) is xe"3 Find the range of f(x)

Find

fh(x)

Find the inverse function of

[pic]

State the domain and range for the The domain of f(x) is x≥3 Find the range of f(x)

Find

fh(x)

Find the inverse function of

[pic]

State the domain and range for the inverse function

Draw f(x) and f-1(x) on the same axis.

What is the transformation that moves f(x) onto f-1(x)?

Find f-1(x)

Sketch the curve of

[pic]

State the natural domain and range for the function

Evaluate

fh(-2)

Transformations

y = f(x)

A (-1,2), B(0,1) and C(4,-1)

Describe the sequence of transformations that maps the graph of y=f(x) onto the graph of y= 2 – f(x).

Find the image of A, B and C under this sequence of transformations.

Describe the sequence of transformations that maps the graph of y=f(x) onto the graph of y=f(3x) – 2

Find the image of A, B and C under this sequence of transformations.

Describe the sequence of transformations that map the graph of y=f(x) onto the graph of y=4f(x+1).

Find the image of A, B and C under this sequence of transformations.

Sketch the graphs of

y=f(x) and y=4f(x+1)

on the same axes

Sketch the graphs of

y=f(x) and y=f(3x) – 2

on the same axes

Sketch the graphs of

y=f(x) and y= 2 – f(x)

on the same axes

Sketch the graphs of

y=f(x) and y= [pic]

on the same axes

A

B

C

Modulus functions

f(x) = cosx

g(x) = 4x – 3

h(x) = x2

Find gf(x)

Solve the equation

[pic]

Solve the inequality

[pic]

Sketch the graph of

y = h(x) – g(x).

Hence sketch the graph of

[pic] on the same axes

Find the function

y = h(x) – g(x)

Sketch the graph of y = g(x).

Hence sketch the graph of

[pic] on the same axes

Solve the inequality

[pic]

Sketch the graph of y = gf(x).

Hence sketch the graph of

[pic] on the same axes

Solve the inequality

[pic]

0 < x < 3600

Natural Logarithms and Exponentials

Draw the graph of

y = ex and y = ln x

on the same axes

Solve

ln(x + 1) + ln(x – 1) = 0

Solve

e2x+1 = 5

Write your answer in log form

Solve

(5 = ex

Giving the exact value in the form plnq

Draw the graph of

y = ex and y = 5 + e2x

on the same axes

Draw the graph of

y = ex and y = e2x – 3

on the same axes

Draw the graph of

y = ex and y = 5 – e – x

on the same axes

The temperature T0C of a cup of tea m minutes after it has been made is modelled by this equation.

[pic]

What is the initial temperature?

What is the temperature after 8 minutes?

Find the time taken for the temperature to drop to 300C

Trigonometrical Functions and Graphs

Draw the graph of y = sinx for – π ≤ x ≤ 2π.

State the range of this function.

Draw the graphs of y = cosx and y = cos-1x on the same axes for 0 ≤ x ≤ π

What is the transformation that maps

y = cosx onto y = cos-1x

Draw the graph of y = sin-1x

For -1 ( x ( 1

Why is it necessary to restrict the domain?

Draw the graph of y = tan-1x

State the natural domain and range for this function

Draw the graph of y = tanx

for – π < x < π

State the range of this function

Draw the graph of y = cosecx

For 0 ( x ( 2(

Draw the graph of y = secx

For -180 ( x ( 180

Draw the graph of y = cotx

For -180 ( x ( 180

Trigonometrical Identities

Solve the equation

[pic]

In the interval 0 ( ( ( 2(

Solve the equation

[pic]

In the interval -180 ( x ( 180

Solve the equation

[pic]

In the interval -( ( ( ( (

Solve the equation

[pic]

In the interval 0 ( y ( 360

Solve the equation

[pic]

In the interval 0 ( ( ( 2(

Solve the equation

[pic]

In the interval 0 ( y ( 360

Solve the equation

[pic]

In the interval 0 ( ( ( (

Prove the identity

[pic]

Solve the equation

[pic]

In the interval 0 ≤ x ≤1800

Use the identity [pic]

To find the exact value of tan150 in the form a + b√3,

where a and b are integers.

Prove the identity

[pic]

Use the expansion of cos(A+B) to show that

cos3x ≡ 4cos3x – 3cosx

Trigonometrical Identities

Solve the equation

4cos3x – 3cosx = 0.7

In the interval 0 ≤ x ≤ (

Prove the identity

sin3xcosx + cos3xsinx ( sin4x

Solve the equation

2sin3xcosx + 2cos3xsinx = (3

Write 7cos θ + 6sinθ

in the Rcos(θ – α) form

0 < α < 900

Solve the equation

7cosθ + 6sinθ = 2

In the interval 0 ≤ θ ≤ 3600

What is the maximum value of 7cosθ + 6sinθ

State the value of θ for which the expression is a maximum

Solve the equation

2cos θ + 5sinθ = 3

In the interval – π ≤ θ ≤ π

What is the minimum value of 2cos θ + 5sinθ

State the value of θ for which this expression is a minimum

Write 2cos θ + 5sinθ

in the Rsin(θ + α) form

0 < α ≤ [pic]

Solving Equations Using the Compound Angle Formulae

Chain Rule

f(x) = ln ( 3x + 2)

g(x) = e2x – 5

Find h’(x)

Find the equation of the normal to the curve

y = f(x)

when x = [pic]

Find the gradient of the curve

y = g(x)

when x = 3

Show that there are no stationary points on the curve

y = g(x)

Find g’’(x)

Find g’(x)

Find f’(x)

Exponentials and Calculus

y = 2e2x – 8x – 1

Find [pic]

Find the equation of the tangent to the curve when x = 1

Find the values of x for which y is an increasing function.

Find the gradient of the curve when x = 1

Find the coordinates of the stationary point on the curve and determine its nature.

Find [pic]

Find the area of the region bounded by the curve the axes and the line x=1

Find [pic]

Find [pic]

Natural logs

f(x) =lnx

[pic]

If y = f(x), find [pic]

Find the equation of the normal to the curve

y = f(x) when x = 2

Draw the graph of

y = f(x)

State the domain and range of this function

Find the gradient of the curve y = f(x) when x = 2

If y = f(x), find [pic]

Find [pic]

Find [pic]

Find the area bounded by the curve y = g(x), the x axis and the lines x = 1 and x = e

Draw the graph of

y = f-1(x)

State the domain and range of this function

Product Rule

f(x) = x2 + 5

g(x) = e2x

h(x) = ln x

Find the equation of the normal to the curve

y = f(x) h(x)

when x = 1

Show that the function

y= f(x) g(x)

has no stationary points.

Find the equation of the tangent to the curve

y = g(x) h(x)

when x = 1

Find [pic] when

y = f(x) g(x)

Find [pic] when

y = h(x) g(x)

Find [pic] when

y = f(x) h(x)

Find [pic] when

y = f(x) g(x)

Quotient Rule

f(x) = x2

g(x) = e2x

h(x) = 2x + 5

Find the equation of the normal to the curve

[pic]

when x = -1

Find the coordinates of the stationary points on the curve

[pic]

and determine their nature.

Find the equation of the tangent to the curve

[pic]

when x = 0

Find [pic] when

[pic]

Find [pic] when

[pic]

Find [pic] when

[pic]

Find [pic] when

[pic]

Differentiation

x = ey, find [pic]

x = ey find [pic] in terms of x

[pic]

Rearrange this equation to find x in terms of y

[pic] find [pic]

[pic] find [pic] in terms of x

y = lnx by writing x in terms of y

Find [pic] in terms of x

The rate at which the area of a circle changes is 2cm2 per minute.

Find the rate at which the radius of the circle changes.

The rate at which the surface area of a cube changes is 10cm2 per second.

Find the rate at which the volume of the cube changes when the edge is of length 12cm.

Integration by Substitution

Find [pic] using a suitable substitution or by inspection

Using the substitution u = x2 + 4x find [pic]

Using the substitution u = 1 – x3 find [pic]

Using the substitution u = x + 4 find [pic]

Find [pic]

using a suitable substitution or by inspection

Find [pic]

using a suitable substitution or by inspection

Find [pic]

using a suitable substitution or by inspection

Find [pic]

Volume of Revolution

Find the area enclosed by the curve [pic], the axes and the line x = 1

Using the substitution u = x2 find [pic]

Find the volume of the solid produced by rotating the curve y = x √( x + 1 ) through 3600 about the x axis for 0 ≤ x ≤ 2

Find the volume of the solid produced by rotating the curve [pic]through 3600 about the x axis for 0 ≤ x ≤ 2

If [pic] use the substitution u = x2

to find [pic]

Find the volume of the solid produced by rotating the curve y = lnx through 3600 about the y axis for 1 ≤ x ≤ 2

Find

[pic]

Numerical Solutions to Equations

By drawing a suitable graph show that the equation x = 3-x has a root between 0.5 and 1

Use the iterative formula

[pic]with x1 = 0.2

to find x2, x3 and x4 to six decimal places.

Use the iterative formula

[pic] with x1 = 0.2

to find the solution to the equation correct to 4 d.p.

The graph of y = x3 – 5x intersects the graph of y = x2 – 3 where x = α show that α satisfies the equation

[pic]

Draw a diagram with x1 = 2 to show how x converges to α

The graph of y = x3 – 5x intersects the graph of y = x2 – 3 where x = α show that α satisfies the equation

[pic]

Use a suitable iterative formula and x1 = 0.1 to find α correct to 5 d.p.

Write a suitable iterative formula to solve the equation [pic]

Using x1= 1.8, find the solution to the equation correct to 4 d.p.

By looking for a sign change show that the equation x = 3-x has a root between 0.5 and 1

Numerical Integration

Simpson’s Rule

By considering rectangular strips of width 0.5 use Simpson’s rule to obtain an approximation for

[pic]

Giving your answer to 3 d.p.

By considering 4 rectangular strips of equal width use Simpson’s rule to obtain an approximation for

[pic]

Giving your answer to 4 d.p.

Use Simpson’s rule with five ordinates to find an approximation to

[pic]

Giving your answer to 5 d.p.

Use Simpson’s rule with seven ordinates to find an approximation to

[pic]

Giving your answer to 4 s.f.

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