Higher Mathematics – Homework A

Non–calcuator section:

1. Find the equation of the perpendicular bisector of the line joining the points

A(-3,1) and B(5,-3)

2. Find the equation of the tangent to the circle

x2 + y

2 – 8x + 4y – 33 = 0 at the point P(1,-4).

3. A recurrence relation is defined as un+1 = 0.8un + 10, u1 = 26

(a) Find the value of uo and u2.

(b) State why this relation has a limit and calculate this limit.

4. The diagram opposite shows the graph

y = asin bx + c.

(a) Write down the values of a, b and c.

(b) Find the points of intersection between

this curve and the line y = 2 for 0 ≤ x ≤ 360

5. f(x) = x

3x - x3

find f /(4)

6. (a) Express 4 – 6x – x2 in the form (x + a)

2 + b.

(b) Hence sketch the graph of f(x) = 4 – 6x – x2 showing clearly its turning point and

where it cuts the y-axis.

(c) On the same diagram sketch the graph of y = 6 – f(x).

7. (a) Given that (x – 2) and (x + 2) are both factors of f(x) = x3 + x

2 + px + q, find the

values of p and q.

(c) Solve f(x) = 0 for these values.

P

Calculator section:

8. Show that the roots of (t – 1)x2 + 2tx + 4 = 0 are real for all values of t.

9. Given tan x = 125 , find the exact value of cos 2x.

10. The diagram shows the graph of y = x3 – x

2 – 6x – 5

and a tangent to this curve at the point A(1,-11).

(a) Find the equation of this tangent.

(b) Find the coordinates of B, the point where

the tangent meets the curve again.

11. f /(x) = x

2 – 4x = 6 and f(3) = 4. Find a formula for f(x).

12. Solve the equation 2cos 2x – 3cos x + 1 = 0 for 0 ≤ x ≤ 360

Higher Mathematics – Homework B

Non-calculator section:

1. (a) Given y = x2 – 12x + 1, express y in the form (x + a)

2 + b.

(b) Hence, or otherwise, write down the turning point of y.

2. Given (x – 2) is a factor of f(x) = x3 – x

2 + kx + 12, find the value of k.

Hence factorise f(x) completely.

3. The roots of mx2 + 4mx + 16 = 0 are equal. Find the value of m given m ≠ 0.

4. The recurrence relations

un + 1 = 0.8un + 12 and vn + 1 = avn + 18

have the same limit. Find the value of a.

5. A triangle ABC has vertices A(2,1), B(12,1) and C(6,-7).

(a) Find the equation of the median BD.

(b) Write down the equation of the perpendicular

bisector of AB.

(c) Find the coordinates of the point of intersection

of these two points.

6. Using the information opposite show that

the exact value of cos(x + y) is 2 5 - 2

3 5

7. (a) Show that the function f(x) = x3 + 3x

2 + 3x – 15 is never decreasing.

(b) Find the coordinates of the stationary point of f(x).

A(2,1) B(12,1)

C(6,-7)

D

2

1

2

x

y

Calculator section:

8. (a) Find the equation of the tangent to the circle

x2 + y

2 + 10x – 2y – 19 = 0 at the point A(1,4).

(b) Show that this tangent is also a tangent to the parabola

y = 2x2 – 10x + 14 and find the point of contact.

9. A curve has equation x

8 - x4 y = . Find the equation of the tangent to this curve

at the point where x = 4.

10. Solve the equation 3sin 2x = 3cos x for 0 ≤ x ≤ 360

11. x

x- 5x

dx

dy 2

= . Given y = ⅓ when x = 1, find a formula for y.

A(1,4)

Higher Mathematics - Homework C

Non-calculator section:

1. (a) Show that (x + 4) is a factor of f(x) = x3 + 2x

2 – 7x + 4.

(b) Hence solve f(x) = 0.

2. Find the equation of the line through the point (-1,5) which is perpendicular to the

line with equation 2x + 3y = 1.

3. (a) Express f(x) = 7 – 4x – x2 in the form f(x) = a + (x + b)

2.

(b) Sketch the graph of f(x) showing clearly its turning point and where it crosses

the y-axis.

4. (a) Find the equation of the tangent to the circle

x2 + y

2 – 18y + 64 = 0 at the point A(4,8).

(b) Show that this tangent is also a tangent to

the parabola y = x2 – 6x + 17 and find the

point of contact.

5. Given tan x = 4

3, find the exact value of

(a) cos 2x

(b) cos 4x

6. Show that the roots of the equation mx2 + (m – 2)x – (m + 1) = 0

are real for all values of m.

7. Find the equation of the tangent to the curve y = 21 x

4 – 15x + 20 which makes

an angle of 450 with the positive direction of the x-axis.

Calculator section:

8. The diagram opposite shows the graphs of

y = acos bx and y = 3sin x.

(a) Write down the values of a and b.

(b) Find the coordinates of P and Q.

A(4,8)

3

4 x

9. Given x6 4x dx

dy+= and y = 50 when x = 4, find a formula for y.

10. A recurrence relation is defined as un = 0.85un - 1 + 30, uo = 40.

(a) Find the smallest value of n such that un > 110.

(b) Find the limit of this recurrence relation, stating why a limit exists.

11. The diagram opposite shows the graph of f(x).

(a) Find a formula for f(x).

(b) Find the coordinates of the turning point P.

(c) Calculate the shaded area.

12. Part of the graph of y = 2sin x + 5cos x

is shown in the diagram.

(a) Express 2sin x + 5cos x in the form

ksin(x + a)° where k > 0 and 0 ≤ a ≤ 360.

(b) Find the coordinates of the minimum

turning point P.

x

y

0

P

3

(4,8)

x360

P

y = 2sin x + 5cos x

Higher Mathematics - Homework D

Non-calculator section:

1. Find the equation of the tangent to the curve x2 - 3x y 2= at the point where x = 1.

2. f(x) = x3 – px + q. Given (x – 2) and (x + 4) are both factors of f(x), find p and q.

3. Solve the equation 4cos2 x – 1 = 1 for 0 ≤ x ≤ 360

4. A triangle has vertices A(1,3), B(1,9) and C(5,5).

(a) Write down the equation of the perpendicular

bisector of AB.

(b) Find the equation of the median from A to BC.

(c) Find the point of intersection of these two lines.

5. The diagram opposite shows the graph of y = f(x).

Sketch the graph of f /(x).

6. Given cos x = 23

(a) Show that (i) sin 2x = 9

54 (ii) cos 2x = -91

(b) Hence, or otherwise, find the value of tan 2x.

7. f /(x) = 4x(x

2 – 1) and f(-1) = 0. Find a formula for f(x).

A(1,3)

B(1,9)

C(5,5)

(4,2)

(0,-8)

(-1,-12)

y=f(x)

8. The diagram opposite shows the curves

y = x3 – x

2 + x and y =

32 x

3 + x

2 – 3x + 1.

The point A lies on the curve y = x3 – x

2 + x

and B lies on the curve y = 32 x

3 + x

2 – 3x + 1.

At the points A and B, the gradients of the curves

are equal.

Find the coordinates of A and B.

9. The functions f(x) = sin x and g(x) = 2x + 2

π are defined on suitable domains.

(a) h(x) = f(g(x)). Show that h(x) = cos 2x.

(b) Solve the equation h(x) = 2

1 for 0 ≤ x ≤ 360

10. The diagram opposite shows the design for

the blades of a windmill.

All 4 blades are equal in size and are made

from aluminium.

y = 6x

A single blade can be described as the area y = 2x2

between the line y = 6x and the parabola

y = 2x2, as shown.

On the diagram each square unit

represents 3m2

Calculate the total area of aluminium

needed to make the blades.

11. (a) A circle has centre (a,0) and radius 3. Write down the equation of this circle.

(b) The line y = x is a tangent to this circle. Show that the exact value of a is ± 23

x

y

Higher Mathematics – Homework E

Non-calculator section:

1. A is the point (3,-1) and B is (7,1). Find the equation of the perpendicular

bisector of AB.

2. (a) f(x) = 10 + 6x – x2. Express f(x) in the form a – (x + b)

2.

(b) Sketch the graph of 4 – f(x), showing clearly its turning point and where

it cuts the y-axis.

3. (a) P has coordinates (2,-1,4) and R has coordinates (7,4,-1). Q divides PR in the

ratio 3:2. Find the coordinates of Q.

(b) T is (-1,0,-3) and U is (-10,-3,-9). Show that Q,T and U are collinear, stating

the ratio of QT:TU.

4. y = x

2x - x 2

. Find the equation of the tangent to y at the point where x is 4.

5. (a) A circle has centre (6,5) and radius 17 . Show that the equation of this

circle can be written in the form

x2 + y

2 – 12x – 10y + 44 = 0

(b) Show that the line y = 4x – 2 is a tangent to this circle and find the point of

contact.

Calculator section:

6. (a) Express 5 cos x + 2sin x in the form kcos (x – a) where k > 0 and 0 ≤ a ≤ 360

(b) Hence write down the maximum value of 2 + 5 cos x + 2sin x and the

corresponding value of x in the range 0 ≤ x ≤ 360.

7. The graph of y = g(x) passes through the point (1,2).

If 4

1 -

x

1 x

dx

dy2

3+= , express y in terms of x.

8. (a) The diagram opposite shows the graph of

y = asin bx + c. Write down the values of

a, b and c.

(b) The line y = 2 is also drawn on the diagram.

Find the x – coordinates of the points P and Q.

9. A curve has equation y = x4 – 4x

3 + 3.

Find algebraically the coordinates of the stationary points of y and determine

their nature.

10. (a) Show that the equation (x – 1)(x + k) = -4 can be written in the form

x2 + x(k – 1) + 4 – k = 0

(b) The roots of the equation (x – 1)(x + k) = -4 are equal. Find the values

of k.

11. Solve the equation 3cos 2x + 4cos x + 1 = 0 for 0 ≤ x ≤ 360

12. The diagram opposite shows the line y = 3 – 3x

and the parabola f(x).

(a) Find a formula for f(x). y = 3 – 3x

(b) Calculate the shaded area.

1 4

12

Higher Mathematics - Homework F

Non-calculator section:

1. (a) Express f(x) = 2x2 – 12x – 3 in the form f(x) = a(x + b)

2 + c

(b) Sketch the graph of y = 3 – f(x), showing clearly its turning point and

where it cuts the y-axis.

2. A curve has equation y = (3x + 2)4. Find the equation of the tangent to this

curve at the point where x = -1.

3. Using the information opposite, show that

sin (a + b) = 2

1

4. Solve for x > 0, log2 (x – 4) + log2 x = 5

5. A function has equation f(x) = 21 x

4 + ax

2 + 24x – 1.

(a) f(x) has a stationary point when x = -2. Find the value of a.

(b) Show that f(x) has no other stationary points.

6. (a) P is the point (1,2,-4) and Q is the point (-4,2,6). A divides PQ in the ratio

2:3. Find the coordinates of A.

(b) B is the point (u,-1,-1) and C is (11,-7,-3). Given A, B and C are collinear,

Find u.

x2 + y

2 – 2x – 2y – 56 = 0 y = x – 4

7. (a) The line y = x – 4 intersects the circle with equation

x2 + y

2 – 2x – 2y – 56 = 0 at two points A and B.

Find the coordinates of A and B.

(b) Find the equation of the circle which has AB as

diameter.

1

3 2

a b

A

B

8. (a) Express 3 cos x – sin x in the form kcos(x + a) where k > 0 and 0 ≤ x ≤ 360

(b) Hence solve the equation 3 cos x – sin x = -1 for 0 ≤ x ≤ 360

Calculator section:

9. P has coordinates (1,-1,-1), Q is (3,0,1) and R is (7,4,3).

Calculate the size of angle PQR.

10. The diagram opposite shows the graph of

y = alog3 (x + b).

Find the values of a and b.

11. Two vectors u and v are such that 6 and 2 == v� .

Given that 2u.(u + v) = - 4, show that angle θ = 1200.

u

v

θ

P

Q R

Higher Mathematics – Homework G

Non – calculator section:

1. u =

−3

1

2

and v =

1

2

4

.

(a) Find the vectors 2u + v and u – v

(b) Show that the vectors 2u + v and u – v are perpendicular.

2. Solve the equation log2 (7x + 4) – log2 x = 3

3. f(x) = 1 – x and g(x) = 6 – x2

(a) h(x) = g(f(x)). Show that h(x) = 5 + 2x – x2

(b) Express h(x) in the form a – (x + b)2

(c) Hence, or otherwise, write down the turning point of h(x), stating whether

it is a maximum or minimum.

4. (a) Express cos x + 3 sin x in the form kcos(x – a) where k > 0 and 0 ≤ a ≤ 360.

(b) Hence solve the equation cos x + 3 sin x = 3 for 0 ≤≤≤≤ x < 90.

5. The diagram shows the graph of

y = logb (x – a).

Find a and b.

6. Show that the tangent to the curve y = 4sin(3x - 3π ) at the point

where x = 6π has equation y – 2 = 6 3 (x -

6π )

7. A is the point (-1,2,4), B is (0,4,2) and C 1s (-4,0,2).

Calculate the size of angle ABC.

A

B

C

Calculator section:

8. A recurrence relation is defined as un + 1 = aun + b.

(a) Given u1 = 32, u2 = 20 and u3 = 17, find the values of a and b.

(b) The limit of the recurrence relation in part (a) is the same as the

limit of vn + 1 = pvn + 10. Find the value of p.

9. Find the values of x for which the function f(x) = 61 (2x – 3)

3 – x is increasing.

10. Solve the equation 2cos 2x – 1 = cos x 0 ≤ x ≤ 360

11. A circle, centre Q, has equation x2 + y

2 – 2y – 1 = 0.

(a) Find the equation of the tangent to this circle

at the point P(1,2).

(b) There are two tangents to the circle which are

parallel to the radius PQ.

Find the equations of these tangents.

12. (a) The parabola opposite cuts the x-axis at

-1 and p and the y-axis at -2p.

Show that the parabola has equation

y = 2x2 +2x(1 – p) – 2p.

(b) The shaded area has a value

equal to 3

p10−.

Calculate the value of p.

x

y

P(1,2)

Q

x

y

- 1 p

- 2 p