ScholarHigher Mathematics

Revision SessionThursday 18th May 7:30pm

You will need a pencil, paper and a calculator for some of the activities

SCHOLAR online tutor for Maths

and

Author of the SCHOLAR National 5 & Higher courses

Margaret Ferguson

Hints & Tips on the Exam Optimisation Logs & Exponentials Vectors Functions Trigonometry The Straight Line Integration The circle

Tonight’s Session will cover

You will need paper, pencil and a calculator for some of the activities.

The New Higher Maths ExamPaper I Paper I is the non-calculator paper and lasts for 1 hour 10 minutes There should be about 12 questions and the paper is worth 60 marksPaper II Paper II is the calculator paper and lasts for 1 hour 30 minutes There should be about 10 questions and the paper is worth 70 marks

The Old Higher Maths ExamPaper I Paper I is the non-calculator paper and lasts for 1 hour 30 minutes Allow 50 mins (+5 mins) for part A, 20 objective questions (40 marks) This leaves 35 minutes for the 3 questions in part B (30 marks)Paper II Paper II is the calculator paper and lasts for 1 hour 30 minutes There should be about 10 questions and the paper is worth 70 marks

PQRS is a rectangle formed according to the following conditions:• it is bounded by the lines x = 6 and y = 12• P lies on the curve with equation y = between (1,8) and (4,2)• R is the point (6,12)

(a) Express the lengths of PS and RS in terms of x, the x-coordinate of P.

PS = RS = 6 - x

Hence show than the area, A units2,

of PQRS is given by

A = PS x RS = (6 – x)

= 72

=

- 12x + 8

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PQRS is a rectangle formed according to the following conditions:• it is bounded by the lines x = 6 and y = 12• P lies on the curve with equation y = between (1,8) and (4,2)• R is the point (6,12)

(b) Find the greatest and least values of A and the corresponding values of x for which they occur.

What do you notice about the stationary points? x = -2 is outside our closed interval.

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How do we do this?

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+ 0 -

When x = 1, A = 20When x = 2, A = 32When x = 4, A = 20Hence the greatest area is 32 units2 when x = 2 and the least area is 20 units2 when x = 1 or 4.

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How do we determine the nature of the stationary point?

shape

x 2

PQRS is a rectangle formed according to the following conditions:• it is bounded by the lines x = 6 and y = 12• P lies on the curve with equation y = between (1,8) and (4,2)• R is the point (6,12)

Find the greatest and least values of A and the corresponding values of x for which they occur.

A 0

B 14

C 26

D 28

Vote for the correct answer now

Vectors p and q are such that |p| = 3, |q| = 4 and p.q = 10. What is the value of q.(p + q)?

q.(p + q) = q.p + q.q

= 10 + 42

= 26

✓= p.q + q2

It is claimed that a wheel is made from wood which is over 1000 years old. To test the claim, carbon dating is used.The formula A(t) = A0e-0.000124t is used to determine the age of wood where :• A0 is the amount of carbon in any living tree• A(t) is the amount of carbon in the wood being dated • t is the age of the wood in yearsFor the wheel it was found that A(t) was 88% of the amount of carbon in a living tree. Is the claim true?What is A(t) expressed in terms of A0? A(t) = 0.88A0

What is the formula expressed in terms of A0? 0.88A0 = A0e-0.000124t

Simplify 0.88 = e-0.000124t How do we determine t? loge 0.88 = -0.000124t

t = 1030.9 Hence the claim is true since 1030.9 > 1000.

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A

B

C

D

What is the solution of the equation

? Vote for the correct answer now

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AST C

In which quadrant is our solution?

x =

x = 120°

60°

Two variables, x and y, are connected by the law y = ax. The graph of log4 y against x is a straight line passing through the origin and the point A(6,3). Find the value of a.

What is the gradient of the straight line?m = ½

What is the equation of the straight line? Y = ½x log4 y = ½xExpress y = ax in terms of log4. log4 y = log4 ax

log4 y = x log4 a

What is log4 a equal to? log4 a = ½

4½ = aSo a = 2

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The diagram shows a wire framework in the shape of a cuboid with edges parallel to the axes.Relative to those axes A, B, C and H have coordinates (1,3,4), (2,3,4), (2,7,4) and (1,7,9) respectively.State the lengths of AB, AD and AE. AB = 1 AD = 4 AE = 5 Write down the components of HB and HC and hence or otherwise calculate the size of angle BHC.

✓HB = b - h

HC = c - h

|HB| =

|HC| =

HB . HC = 12 + (-4) x 0 + (-5)2 = 26

= 38.1°

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A

B

C

D

The diagram shows the line L.The angle between L and the positive direction of the x-axis is 135° as shown.What is the gradient of L?

Vote for the correct answer now

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What is the associated acute angle?

mL = tan 135°

= - tan 45°

= -1

Functions f, g and h are defined on suitable domains by f(x) = x2 – x + 10, g(x) = 5 – x and h(x) = log2 x.

Find expressions for h(f(x)) and h(g(x)).

h(f(x)) = h(x2 – x + 10)

= log2 (x2 – x + 10)

h(g(x)) = h(5 – x)

= log2 (5 – x)Hence solve h(f(x)) - h(g(x)) = 3.h(f(x)) – h(g(x)) = log2 (x2 – x + 10) – log2 (5 – x)

So log2 (x2 – x + 10) – log2 (5 – x) = 3

=>

40 – 8x = x2 – x + 10 0 = x2 + 7x - 30

(x – 3)(x + 10) = 0x = 3 or x = -10

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The graphs of y = f(x) and y = g(x) are shown in the diagram.f(x) = -4cos(2x) + 3 and g(x) if of the form g(x) = mcos(nx).What are the values of m and n?

m = n = 3 2 ✓Find correct to 1 decimal place, the coordinates of the points of intersection of the two graphs in the interval 0 ≤ x ≤ π.

-4cos(2x) + 3 = 3cos(2x)-7cos(2x) = -3

How do we proceed?

2x = 64.6°

AST C

2x = 295.4°x = 32.3° = 0.6 radians

x = 147.7° = 2.6 radians

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g(32.3) = 1.3 g(147.7) = 1.3

Hence the points of intersection are (0.6,1.3) and (2.6,1.3)

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The graphs of y = f(x) and y = g(x) are shown in the diagram.f(x) = -4cos(2x) + 3 and g(x) = 3cos(2x).Calculate the shaded area.

How do we proceed?

= 10.9 – (-1.5) = 12.4

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✓ This is why the angles must be in radians

Circle P has equation x2 + y2 – 8x – 10y + 9 = 0Circle Q has centre (-2,-1) and radius .Do circles P and Q touch, giving a reason for your answer?What are the radius and the coordinates of the centre of circle P? Centre of circle P = (4,5)

Radius of circle P =

What is the distance between the centres of circles P and Q?

What is the sum of the radii?Since the sum of the radii is equal to the distance between the centres, circles P and Q touch externally.

Question Time• Work through the exam questions sequentially• Use the paper provided like a book• If you can’t do a question write down the question

number and leave a good amount of space • This will make it easier to come back to it when you

have finished the paper• Always answer all the questions that you can first• If you have any questions about tonight’s session or the

exam please ask• Carol will provide a link for you to give us feedback