CfE Higher Mathematics Formal Homework - Wikispaces Higher Mathematics... · II. The gradient of ST= 4 3. ... All others ⃝c Higher Still Notes CfE Higher Mathematics: Circles Formal

Dunoon Grammar School

Mathematics Department

CfE Higher Mathematics

Formal Homework

Introduction Homework should be completed to the same high standard as class work. You should always attempt a rough draft of your homework before giving a final copy to your teacher. The copy you submit to your teacher should represent your best work. Homework should be completed well in advance of any deadline so that assistance can be sought if required. All working must be shown. For multiple-choice questions especially, marks will only be awarded for correct answers where full working is shown. Please note that formal homework scores are recorded and used to track progress throughout the course. Contents

• Straight Lines Page 1 • Sequences Page 4 • Differentiation 1 Page 7 • Differentiation 2 Page 10 • Circles Page 11 • Integration 1 Page 13 • Integration 2 Page 14 • Functions & Graphs Page 16 • Trigonometry 1 Page 18 • Polynomials Page 22 • Quadratics Page 24 • Trigonometry 2 Page 26 • Further Calculus Page 28 • Vectors Page 30 • Exponentials & Logarithms Page 32 • Wave Functions Page 34

Page twoPage two

FORMULAE LIST

Circle:

The equation x2 + y2

+ 2gx + 2fy + c = 0 represents a circle centre (−g, −f ) and radius

The equation (x − a)2 + (y − b)

2 = r2 represents a circle centre (a, b) and radius r.

Scalar Product: a.b = |a||b| cos θ, where θ is the angle between a and b

or a.b = a1b

1 + a

2b

2 + a

3b

3 where a =

1 1

2 2

3 3

=and

a ba ba b

b⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠

.

Trigonometric formulae: sin (A ± B) = sin A cos B ± cos A sin B

cos (A ± B) = cos A cos B

±

sin A sin B

sin 2A = 2 sin A cos A

cos 2A = cos2 A − sin

2 A

= 2 cos2 A − 1

= 1 − 2 sin2 A

Table of standard derivatives:

Table of standard integrals:

f (x) f ′(x)

sin axcos ax

a cos ax–a sin ax

f (x) ∫ f (x)dx

sin ax

cos ax

cos ax + c

sin ax + c

1a1a

2 2

Higher Mathematics

The Straight Line

1. Here are two statements about the points S(2, 3) and T(5,−1) :

I. The length of ST = 5 units;

II. The gradient of ST = 43 .

Which of the following is true?

A. neither statement is correct

B. only statement I is correct

C. only statement II is correct

D. both statements are correct 2

2. A line l has equation 3y+ 2x = 6.

What is the gradient of any line parallel to l ?

A. −2

B. − 23

C. 32

D. 2 2

3. The line with equation y = ax+ 4 is perpendicular to the line with equation3x+ y+ 1 = 0.

What is the value of a?

A. −3

B. − 13

C. 13

D. 3 2

4.[SQA]

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CfE Higher Mathematics: Straight LinesFormal Homework

CfE Higher Mathematics DGSMaths.comPage 1

Higher Mathematics

5.[SQA] The points A and B have coordinates (a, a2) and (2b, 4b2) respectively. Determinethe gradient of AB in its simplest form. 2

6.[SQA]

7.[SQA]

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Higher Mathematics

8.[SQA]

[END OF QUESTIONS]

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Higher Mathematics

Sequences

1. A sequence is defined by the recurrence relation un+1 =14un + 8 with u0 = 32.

Evaluate u2 .

A. 10

B. 12

C. 16

D. 32 2

2. A sequence is generated by the recurrence relation un+1 =14un + 7, with

u0 = −2.

What is the limit of this sequence as n→ ∞?

A. 128

B. 285

C. 283

D. 28 2

3.[SQA] A sequence is defined by the recurrence relation un+1 = 0·3un + 5 with first termu1 .

(a) Explain why this sequence has a limit as n tends to infinity. 1

(b) Find the exact value of this limit. 2

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CfE Higher Mathematics: SequencesFormal Homework


Higher Mathematics

4.[SQA]

5.[SQA]

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Higher Mathematics

6.[SQA] Two sequences are generated by the recurrence relations un+1 = aun + 10 andvn+1 = a2vn + 16.

The two sequences approach the same limit as n→ ∞ .

Determine the value of a and evaluate the limit. 5

[END OF QUESTIONS]



Higher Mathematics

Differentiation 1

1.[SQA]

2. If s(t) = t2 − 5t+ 8, what is the rate of change of s with respect to t when t = 3?

A. −5

B. 1

C. 2

D. 9 2

3. Differentiate 2 3√x with respect to x .

A. 6√x

B. 323√x4

C. −4

3 3√x2

D.2

3 3√x2

2

4. If y = 3x−2+ 2x32 , x > 0, determine dydx .

A. −6x−3 + 45x52

B. −3x−1 + 3x12

C. −6x−3 + 3x12

D. −3x−1 + 45x52 2

5.[SQA] Differentiate 2√x(x+ 2) with respect to x . 4

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CfE Higher Mathematics: Differentiation 1Formal Homework


Higher Mathematics

6.[SQA] Finddy

dxwhere y =

4

x2+ x

√x . 4

7.[SQA] Given f (x) = 3x2(2x− 1) , find f ′(−1) . 3

8.[SQA] Find f ′(4) where f (x) =x− 1√x. 5

9.[SQA] If f (x) = kx3 + 5x− 1 and f ′(1) = 14, find the value of k . 3

10. A curve has equation y = 5x3− 12x .

What is the gradient of the tangent at the point (1,−7)?

A. −7

B. −5

C. 3

D. 5 2

11.[SQA] The point P(−1, 7) lies on the curve with equation y = 5x2+ 2. Find the equationof the tangent to the curve at P. 4

12.[SQA] For what values of x is the function f (x) = 13x3 − 2x2− 5x− 4 increasing? 5

13.[SQA] A ball is thrown vertically upwards. The height h metres of the ball t seconds afterit is thrown, is given by the formula h = 20t− 5t2 .

(a) Find the speed of the ball when it is thrown (i.e. the rate of change of heightwith respect to time of the ball when it is thrown). 3

(b) Find the speed of the ball after 2 seconds.

Explain your answer in terms of the movement of the ball. 2

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Higher Mathematics

14.[SQA]

[END OF QUESTIONS]



Higher Mathematics

Differentiation 2

1.[SQA] A function f is defined by the formula f (x) = (x− 1)2(x+ 2) where x ∈ R .

(a) Find the coordinates of the points where the curve with equation y = f (x)crosses the x - and y-axes. 3

(b) Find the stationary points of this curve y = f (x) and determine their nature. 7

(c) Sketch the curve y = f (x) . 2

2.[SQA]

3.[SQA]

[END OF QUESTIONS]


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CfE Higher Mathematics: Differentiation 2Formal Homework


Higher Mathematics

The Circle

1. A circle has equation x2 + y2 + 8x+ 6y− 75 = 0.

What is the radius of the circle?

A. 5

B. 10

C.√75

D.√175 2

2.[SQA]

3.[SQA] For what range of values of c does the equation x2+ y2− 6x+ 4y+ c = 0 representa circle? 3

4.[SQA]

5.[SQA]

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CfE Higher Mathematics: CirclesFormal Homework


Higher Mathematics

6.[SQA]

7.[SQA]

8.[SQA] The line y = −1 is a tangent to a circle which passes through (0, 0) and (6, 0) .

Find the equation of this circle. 6

[END OF QUESTIONS]



x + y - 10x - 12y + 45 = 0 and the 2 2

Higher Mathematics

Integration 1

1.[SQA] Find!

"

2x2 + 3#

dx . 3

2.[SQA] Find!

"

3x3 + 4x#

dx . 3

3.[SQA] Find!

x2 − 5x√xdx . 4

4.[SQA] Find!

(x2 − 2)(x2 + 2)x2

dx , x ̸= 0. 4

5.[SQA] The graph of y = g(x) passes through the point (1, 2) .

Ifdy

dx= x3 +

1

x2−1

4, express y in terms of x . 4

6.[SQA] Find the value of! 4

1

√x dx . 4

7.[SQA] Evaluate! 2

1

$

x2 +1

x

%2

dx . 5

8.[SQA] Find the value of! 2

1

u2 + 22u2

du . 5

[END OF QUESTIONS]



CfE Higher Mathematics: Integration 1Formal Homework


Higher Mathematics

Integration 2

1.[SQA]

(a) Find the value of! 2

1

"

4− x2#

dx . 3

(b) Sketch a graph and shade the area represented by the integral in (a). 2

2.[SQA]

3.[SQA]

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CfE Higher Mathematics: Integration 2Formal Homework


Higher Mathematics

4.[SQA]

5.[SQA] Calculate the shaded areaenclosed between theparabolas with equationsy = 1 + 10x − 2x2 andy = 1+ 5x− x2 . 6

O x

y

y = 1+ 10x− 2x2

y = 1+ 5x− x2

6.[SQA]

[END OF QUESTIONS]



Higher Mathematics

Functions Graphs

1.[SQA] On a suitable set of real numbers, functions f and g are defined by f (x) =1

x+ 2

and g(x) =1

x− 2.

Find f!

g(x)"

in its simplest form. 3

2.[SQA] f (x) = 2x− 1, g(x) = 3− 2x and h(x) = 14(5− x) .

(a) Find a formula for k(x) where k(x) = f!

g(x)"

. 2

(b) Find a formula for h!

k(x)"

. 2

(c) What is the connection between the functions h and k? 1

3.[SQA]

4.[SQA]

5.[SQA] Given f (x) = x2 + 2x− 8, express f (x) in the form (x+ a)2 − b . 2

6.[SQA] Express x2+ 6x+ 11 in the form (x+ a)2 + b and hence state the maximum value

of1

x2 + 6x+ 11. 4

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CfE Higher Mathematics: Functions & GraphsFormal Homework


Higher Mathematics

7.[SQA]

(a) Show that f (x) = 2x2 − 4x + 5 can be written in the formf (x) = a(x+ b)2 + c . 3

(b) Hence write down the coordinates of the stationary point of y = f (x) andstate its nature. 2

8.[SQA] Functions f and g are given by f (x) = 3x+ 1 and g(x) = x2 − 2.

(a) (i) Find p(x) where p(x) = f (g(x)) .

(ii) Find q(x) where q(x) = g( f (x)) . 3

(b) Solve p′(x) = q′(x) . 3

9.[SQA] A function f is defined on the set of real numbers by f (x) =x

1− x, x ̸= 1.

Find, in its simplest form, an expression for f!

f (x)"

. 3

[END OF QUESTIONS]



[END OF QUESTIONS]

10. If = , find f (x) 5x – 2 f (x).-1 3

Higher Mathematics

Trigonometry 1

1. A line makes an angle of 30◦ with the positive direction of the x -axis as shown.

30°

O x

y

What is the gradient of the line?

A. 1√3

B. 1√2

C. 12

D.√32 2

2.[SQA] Find the size of the angle a◦ that the linejoining the points A(0,−1) and B(3

√3, 2)

makes with the positive direction of thex -axis. 3

O x

y

A(0,−1)

a◦

B(3√3, 2)

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CfE Higher Mathematics: Trigonometry 1Formal Homework


Higher Mathematics

3. The diagram shows the graph with equation of the form y = a cos bx for0 ≤ x ≤ 2π .

2

π 2π

–2

O x

y

What is the equation of this graph?

A. y = 2 cos 3x

B. y = 2 cos 2x

C. y = 3 cos 2x

D. y = 4 cos 3x 2

4. The diagram shows the curve with equation of the form y = cos(x+ a) + bfor0 ≤ x ≤ 2π .

–2

2π

π

6

7

6

π

Ox

y

What is the equation of this curve?

A. y = cos!

x− π

6

"

− 1

B. y = cos!

x− π

6

"

+ 1

C. y = cos!

x+ π

6

"

− 1

D. y = cos!

x+ π

6

"

+ 1 2

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Higher Mathematics

5. How many solutions does the equation

(4 sin x−√5)(sin x+ 1) = 0

have in the interval 0 ≤ x < 2π?

A. 4

B. 3

C. 2

D. 1 2

6. Functions f and g are defined on suitable domains by f (x) = cos x andg(x) = x+ π

6 .

What is the value of f!

g!

π

6

""

?

A. 12 + π

6

B.√32 + π

6

C.√32

D. 12 2

7. Find the maximum value of

2− 3 sin#

x−π

3

$

and the value of x where this occurs in the interval 0 ≤ x ≤ 2π .

max value x

A. −1 11π6

B. 5 11π6

C. −1 5π6

D. 5 5π6 2

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Higher Mathematics

8.[SQA]

9.[SQA] The diagram shows the graph of acosine function from 0 to π .

(a) State the equation of the graph. 1

(b) The line with equationy = −

√3 intersects this

graph at point A and B.

Find the coordinates of B. 3

O x

y

A B y = −√3

π

2π

2

−2

10.[SQA] Find the values of t , where 0 < t < 2π , for which 4 cos!

2t− π

4

"

has its maximumvalue. 4

11.[SQA] Solve the equation 2 cos2 x = 12 , for 0 ≤ x ≤ π . 3

12.[SQA] Solve 2 sin 3x◦ − 1 = 0 for 0 ≤ x ≤ 180. 4

13.[SQA] Solve the equation 2 sin#

2x− π

6

$

= 1, 0 ≤ x < 2π . 4

[END OF QUESTIONS]



Higher Mathematics

Polynomials

1. A function f is defined on the set of real numbers by f (x) = x3 − x2 + x+ 3.

What is the remainder when f (x) is divided by (x− 1)?

A. 0

B. 2

C. 3

D. 4 2

2. Given p(x) = x2 + x− 6, which of the following are true?

I. (x+ 3) is a factor of p(x) .

II. x = 2 is a root of p(x) = 0.

A. Neither I nor II is true

B. Only I is true

C. Only II is true

D. Both I and II are true 2

3. (a)[SQA] Given that x+ 2 is a factor of 2x3 + x2 + kx+ 2, find the value of k . 3

(b) Hence solve the equation 2x3 + x2 + kx+ 2 = 0 when k takes this value. 2

4.[SQA]

(a) The function f is defined by f (x) = x3 − 2x2− 5x+ 6.

The function g is defined by g(x) = x− 1.

Show that f!

g(x)"

= x3 − 5x2+ 2x+ 8. 4

(b) Factorise fully f!

g(x)"

. 3

(c) The function k is such that k(x) =1

f!

g(x)" .

For what values of x is the function k not defined? 3

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CfE Higher Mathematics: PolynomialsFormal Homework


Higher Mathematics

5.[SQA] When f (x) = 2x4− x3+ px2+ qx+ 12 is divided by (x− 2) , the remainder is 114.

One factor of f (x) is (x+ 1) .

Find the values of p and q . 5

6.[SQA] Factorise fully 2x3+ 5x2 − 4x− 3. 4

7.[SQA]

[END OF QUESTIONS]



Higher Mathematics

Quadratics

1. If f (x) = (x− 3)(x+ 5) , for what values of x is the graph of y = f (x) above thex -axis?

A. −5 < x < 3

B. −3 < x < 5

C. x < −5, x > 3

D. x < −3, x > 5 2

2. What is the solution of x2+ 4x > 0, where x is a real number?

A. −4 < x < 0

B. x < −4, x > 0

C. 0 < x < 4

D. x < 0, x > 4 2

3. Solve 6− x− x2 < 0.

A. −3 < x < 2

B. x < −3, x > 2

C. −2 < x < 3

D. x < −2, x > 3 2

4.[SQA] Find the values of x for which the function f (x) = 2x3− 3x2 − 36x is increasing. 4

5.[SQA] The roots of the equation (x− 1)(x+ k) = −4 are equal.

Find the values of k . 5

6.[SQA] Show that the roots of the equation (k− 2)x2 − (3k− 2)x+ 2k = 0 are real. 4

7.[SQA] Show that the equation (1 − 2k)x2 − 5kx − 2k = 0 has real roots for all integervalues of k . 5

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CfE Higher Mathematics: QuadraticsFormal Homework


Higher Mathematics

8.[SQA]

[END OF QUESTIONS]



Higher Mathematics

Trigonometry 2

1. The diagram shows two right-angled triangles with sides and angles as given.

2

3

2

pq

1

5

What is the value of sin(p+ q)?

A. 2√5+ 23

B. 2√5+

√53

C. 23 + 2

3√5

D. 43√5+ 13 2

2.[SQA]

3.[SQA] Solve the equation sin 2x◦ + sin x◦ = 0, 0 ≤ x < 360. 5

4.[SQA] Solve the equation 3 cos 2x◦ + cos x◦ = −1 in the interval 0 ≤ x ≤ 360. 5

5. Solve 2 cos 2x− 5 cos x− 4 = 0 for 0 ≤ x < 2π . 5

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CfE Higher Mathematics: Trigonometry 2Formal Homework


Higher Mathematics

6.[SQA]

7.[SQA]

(a) Show that 2 cos 2x◦ − cos2 x◦ = 1− 3 sin2 x◦ . 2

(b) Hence solve the equation 2 cos 2x◦ − cos2 x◦ = 2 sin x◦ in the interval0 ≤ x < 360. 4

8.[SQA]

[END OF QUESTIONS]



Higher Mathematics

Further Calculus

1.[SQA] Given that f (x) = (5x− 4) 12 , evaluate f ′(4) . 3

2.[SQA] Given that f (x) = 5(7− 2x)3 , find the value of f ′(4) . 4

3.[SQA] Find!

1

(7− 3x)2dx . 2

4.[SQA] Evaluate! 0

−3

"

2x+ 3#2dx . 4

5.[SQA] The graph of y = f (x) passes through the point$

π

9 , 1%

.

If f ′(x) = sin(3x) express y in terms of x . 4

6.[SQA] Find the equation of the tangent to the curve y = 2 sin(x− π

6 ) at the point wherex = π

3 . 4

7.[SQA] Differentiate sin 2x+2√xwith respect to x . 4

8.[SQA] Find!

"

6x2 − x+ cos x#

dx . 4

9.[SQA] Given f (x) = cos2 x− sin2 x , find f ′(x) . 3

10.[SQA] Differentiate 4√x+ 3 cos 2x with respect to x . 4

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CfE Higher Mathematics: Further CalculusFormal Homework


Higher Mathematics

11.[SQA]

[END OF QUESTIONS]



Higher Mathematics

Vectors

1.[SQA]

2.[SQA]

3.[SQA]

4.[SQA]

5.[SQA]

6.[SQA] The point Q divides the line joining P(−1,−1, 0) to R(5, 2,−3) in the ratio 2 : 1.

Find the coordinates of Q. 3

7.[SQA] Show that the vectors a = 2i + 3 j − k and b = 3i − j + 3k are perpendicular. 3

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CfE Higher Mathematics: VectorsFormal Homework


Higher Mathematics

8.[SQA] The vector ai + b j + k is perpendicular to both the vectors i − j + k and−2i + j + k .

Find the values of a and b . 3

9.[SQA]

10.[SQA]

[END OF QUESTIONS]



Higher Mathematics

Exponentials Logarithms

1. The diagram shows the graph of y = f (x) where f is a logarithmic function.

(6, 1)

(4, 0)

= f ( )

O

x

x

y

y

What is f (x)?

A. f (x) = log6(x− 3)

B. f (x) = log3(x+ 3)

C. f (x) = log3(x− 3)

D. f (x) = log6(x+ 3) 2

2.[SQA]

3.[SQA] Evaluate log5 2+ log5 50− log5 4. 3

4.[SQA] Given x = log5 3+ log5 4, find algebraically the value of x . 4

5.[SQA] Find x if 4 logx 6− 2 logx 4 = 1. 3

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CfE Higher Mathematics: Exponentials & LogarithmsFormal Homework


Higher Mathematics

6.[SQA] Find the x -coordinate of the point where the graph of the curve with equationy = log3(x− 2) + 1 intersects the x -axis. 3

7. Solve logb x− logb 7 = logb 3 for x > 0.

A. x = 21

B. x = 10

C. x = 73

D. x = 37 2

8.[SQA]

9.[SQA]

10.[SQA]

[END OF QUESTIONS]



x.


Higher Mathematics

Wave Functions

1. k and a are given by

k sin a◦ = 1

k cos a◦ =√3

where k > 0 and 0 ≤ a < 90.

What are the values of k and a?

k a

A. 2 60

B. 2 30

C.√10 60

D.√10 30

2

2.[SQA] Express 8 cos x◦ − 6 sin x◦ in the form k cos(x◦ + a◦) where k > 0 and 0 < a < 360. 4

3.[SQA]

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CfE Higher Mathematics: Wave FunctionsFormal Homework


Higher Mathematics

4.[SQA]eplacements

5.[SQA] Solve the equation 2 sin x◦ − 3 cos x◦ = 2·5 in the interval 0 ≤ x < 360. 8

[END OF QUESTIONS]


All others c⃝ Higher Still NotesPage 35

Documents

CfE Higher Mathematics Formal Homework - Wikispaces Higher Mathematics... · II. The gradient of ST= 4 3. ... All others ⃝c Higher Still Notes CfE Higher Mathematics: Circles Formal