mathcapeunit 1&2 2011-2006q

CAPE Unit OneExamination 2011

2

Question 1

(a) Without using calculators, find the exact value of

(i)(√

75 +√

12)2 −

(√75 −

√12

)2[3]

(ii) 271/4 × 93/8 × 811/8. [3]

(b) The diagram below, not drawn to scale, represents a segment of the

graph of the function

f (x) = x3 + mx2 + nx + p

where m, n and p are constants.

Q 0 1 2

(0, 4)

f(x)

x

Find

(i) the value of p [2]

(ii) the values of m and n [4]

(iii) the x-coordinate of the point Q. [2]

(c) (i) By substituting y = log2 x, or otherwise, solve for x, the equation

√

log2 x = log2

√x. [6]

(ii) Solve, for real values of x, the inequality

x2 − |x| − 12 < 0. [5]

3

Question 2

(a) The quadratic equation x2 − px + 24 = 0, p ∈ R, has roots α and β.

(i) Express in terms of p,

a) α + β [1]

b) α2 + β2. [4]

(ii) Given that α2 + β2 = 33, find the possible values of p. [3]

(b) The function f (x) has the property that

f (2x + 3) = 2f (x) + 3, x ∈ R.

If f (0) = 6, find the value of

(i) f (3) [4]

(ii) f (9) [2]

(iii) f (−3) . [3]

(c) Prove that the product of two consecutive integers k and k + 1 is an

even integer. [2]

(d) Prove, by mathematical induction, that n (n2 + 5) is divisible by 6 for

all positive integer n. [6]

4

Question 3

(a) (i) Let a = a1i + a2j and b = b1i + b2j with |a| = 13 and |b| = 10.

Find the value of (a + b) · (a − b) [5]

(ii) If 2b − a = 11i , determine the possible values of a and b. [5]

(b) The line L has equation x − y + 1 = 0 and the circle C has equation

x2 + y2 − 2y − 15 = 0.

(i) Show that L passes through the centre of C. [2]

(ii) If L intersects C at P and Q, determine the coordinates of P and

Q. [3]

(iii) Find the constants a, b and c such that x = b + a cos θ and y =

c + a sin θ are parametric equations (in parameter θ) of C. [3]

(iv) Another circle C2, with the same radius as C, touches L at the

centre of C. Find the possible equations of C2. [7]

5

Question 4

(a) By using x = cos2 θ, or otherwise, find all values of the angle θ such

that

8 cos4 θ − 10 cos2 θ + 3 = 0, 0 ≤ θ ≤ π. [6]

(b) The diagram below, not drawn to scale, shows a rectangle PQRS

with sides 6 cm and 8 cm inscribed in another rectangle ABCD.

θ

P

8 cm

6 cm

Q

R

S

A B

CD

(i) The angle that SR makes with DC is θ. Find in terms of θ, the

length of the side BC. [2]

(ii) Find the value of θ if |BC| = 7 cm. [5]

(iii) Is 15 cm a possible value for |BC|? Give a reason. [2]

(c) (i) Show that1 − cos 2θ

sin 2θ= tan θ. [3]

(ii) Hence, show that

a)1 − cos 4θ

sin 4θ= tan 2θ [3]

b)1 − cos 6θ

sin 6θ= tan 3θ. [2]

(iii) Using the results in (c) (i) and (ii) above, evaluaten

∑

r=1

(tan rθ sin 2rθ + cos 2rθ)

where n is a positive integer. [2]

6

Question 5

(a) Find limx→−2

x2 + 5x + 6

x2 − x − 6. [4]

(b) The function f on R is defined by

f (x) =

{

x2 + 1 if x ≥ 2bx + 1 if x < 2.

Determine

(i) f (2) [2]

(ii) limx→2+

f (x) [2]

(iii) limx→2−

f (x) in terms of the constant b [2]

(iv) the value of b such that f is continuous at x = 2. [4]

(c) The curve y = px3 + qx2 + 3x + 2 passes through the point T (1, 2) and

its gradient at T is 7. The line x = 1 cuts the x-axis at M, and the

normal to the curve at T cuts the x-axis at N.

Find

(i) the values of the constants p and q. [6]

(ii) the equation of the normal to the curve at T [3]

(iii) the length of MN. [2]

7

Question 6

(a) The diagram below, not drawn to scale, is a sketch of the section of

the function f (x) = x (x2 − 12) which passes through the origin O. A

and B are stationary points on the curve.

O

f(x) = x(x2 − 12)

A

B

x

y

b

b

Find

(i) the coordinates of each of the stationary points A and B [8]

(ii) the equation of the normal to the curve f (x) = x (x2 − 12) at the

origin, O. [2]

(iii) the area between the curve and the positive x-axis. [6]

(b) (i) Use the result

∫ a

0

f (x) dx =

∫ a

0

f (a − x) dx, a > 0,

to show that

∫ π

0

x sin xdx =

∫ π

0

(π − x) sin xdx. [2]


a)

∫ π

0

x sin xdx = π

∫ π

0

sin xdx +

∫ π

0

x sin xdx [2]

b)

∫ π

0

x sin xdx = π. [5]

10

Question 1

(a) Find the values of the constant p such that (x − p) is a factor of

f(x) = 4x3 − (3p + 2)x2 −(

p2 − 1)

x + 3.

[5]

(b) Solve for x and y, the simultaneous equations

log(x − 1) + 2 log y = 2 log 3

log x + log y = log 6.

[8]

(c) Solve, for x ∈ R, the inequality

2x − 3

x + 1− 5 > 0.

[5]

(d) By putting y = 2x, or otherwise, solve

4x − 3(

2x+1)

+ 8 = 0.

[7]

11

Question 2

(a) (i) Use the fact that Sn =n

∑

r=1

r = 12n (n + 1) to express S2n =

2n∑

r=1

r

in terms of n. [2]

(ii) Find constants p, q such that

S2n − Sn = pn2 + qn.

[5]

(iii) Hence, or otherwise, find n such that

S2n − Sn = 260.

[5]

(b) The diagram below (not drawn to scale) shows the graph of y =

x2(3−x). The coordinates of P and Q are (2, 4) and (3, 0) respectively.

x

y

1

P

32

Q

O

(i) Write down the solution set of the inequality x2(3 − x) ≤ 0. [4]

(ii) Given that the equation x2(3 − x) = k has three real roots for x,

write down the possible values of k. [3]

(iii) The functions f and g are defined as follows:

f : x → x2(3 − x), for 0 < x < 2

g : x → x2(3 − x), for 0 < x < 3

By using (b) (ii) above, or otherwise, show that

a) f has an inverse,

b) g does NOT have an inverse. [6]

12

Question 3

(a) The vectors p and q are given by

p = 6i + 4j

q = −8i − 9j.

(i) Calculate, in degrees, the angle between p and q. [5]

(ii) a) Find a non-zero v such that p · v = 0.

b) State the relationship between p and v. [5]

(b) The circle C1 has (−3, 4) and (1, 2) as endpoints of a diameter.

(i) Show that the equation of C1 is x2 + y2 + 2x − 6y + 5 = 0. [6]

(ii) The circle C2 has equation x2 + y2 + x − 5y = 0. Calculate the

coordinates of the points of intersection of C1 and C2. [9]

13

Question 4

(a) (i) Solve the equation cos 3A = 0.5 for 0 ≤ A ≤ π. [4]

(ii) Show that cos 3A = 4 cos3 A − 3 cosA. [6]

(iii) The THREE roots of the equation 4p3−3p−0.5 = 0 all lie between

−1 and 1. Use the results in (a) (i) and (ii) to find these roots.[4]

(b) The following diagram, not drawn to scale, represents a painting of

height h metres, that is fastened to a vertical wall at a height d metres

above and x metres away from the level of an observer, O.

Painting

α

O

h m

β

x m

d m

The viewing angle of the painting is (α−β), where α and β are respec-

tively the angles of inclination, in radians, from the level of the observer

to the top and base of the painting.

(i) Show that

tan(α − β) =hx

x2 + d (d + h).

[6]

(ii) The viewing angle of the painting, (α−β), is at a maximum when

x =√

d (d + h). Calculate the maximum viewing angle, in radians,

when d = 3h. [5]

14

Question 5

(a) Find

(i) limx→3

x2 − 9

x3 − 27[4]

(ii) limx→0

tan x − 5x

sin 2x − 4x. [5]

(b) The function f on R is defined by

f (x) =

{

3x − 7 if x > 41 + 2x if x ≤ 4.

(i) Find

a) limx→4+

f (x) [2]

b) limx→4−

f (x) . [2]

(ii) Deduce that f(x) is discontinuous at x = 4. [2]

(c) (i) Evaluate

∫ 1

−1

(

x − 1

x

)2

dx. [6]

(ii) Use the substitution u = x2 +4, or otherwise, find

∫

x√

x2 + 4dx.

[4]

15

Question 6

(a) Differentiate with respect to x

(i) y = sin(3x + 2) + tan 5x [3]

(ii) y =x2 + 1

x3 − 1. [4]

(b) The function f(x) satisfies

∫ 4

1

f (x) dx = 7.

(i) Find

∫ 4

1

[3f (x) + 4] dx. [4]

(ii) Using the substitution u = x + 1, evaluate

∫ 3

0

2f (x + 1) dx. [4]

(c) In the diagram below (not drawn to scale), the line x + y = 2 inter-

sects the curve y = x2 at the points P and Q.

Q

O

x + y = 2

x

y

P

y = x2

(i) Find the coordinates of the points P and Q. [5]

(ii) Calculate the area of the shaded portion of the diagram bounded

by the curve and the straight line. [5]

18

Question 1

(a) Without using tables or a calculator, simplify√

28 +√

343 in the

form k√

7 where k is an integer. [5]

(b) Let x and y be positive real numbers such that x 6= y.

(i) Simplifyx4 − y4

x − y. [6]

(ii) Hence, or otherwise, show that

(y + 1)4 − y4 = (y + 1)3 + (y + 1)2y + (y + 1)y2 + y3.

[4]

(iii) Deduce that (y + 1)4 − y4 < 4(y + 1)3. [2]

(c) Solve the equation log4 x = 1 + log2 2x, x > 0. [8]

19

Question 2

(a) The roots of the quadratic equation 2x2 + 4x + 5 = 0 are α and β.

Without solving the equation, find the quadratic equation whose

roots are2

αand

2

β. [6]

(b) The coach of an athletic club trains six athletes u, v, w, x, y and z, in

this training camp. He makes an assignment, f , of athletes u, v, w, x, y

and z to physical trainers 1, 2, 3 and 4 according to the diagram below

in which A = {u, v, w, x, y, z} and B = {1, 2, 3, 4} .

v

y

x

u

w

z

f

2

1

3

4

BA

(i) Express f as a set of ordered pairs. [4]

(ii) a) State TWO reasons why f is NOT a function. [2]

b) Hence, with MINIMUM changes to f , construct a function

g : A → B as a set of ordered pairs. [4]

c) Determine how many different functions are possible for g in

(ii) b) above. [2]

(c) The function f on R, is defined by

f (x) =

{

x − 3 if x ≤ 3x

4if x > 3

Find the value of

(i) f [f(20)] [3]

(ii) f [f(8)] [2]

(iii) f [f(3)]. [2]

20

Question 3

(a) The circle C has equation

(x − 3)2 + (y − 4)2 = 25.

(i) State the radius and the coordinates of the centre of C. [3]

(ii) Find the equation of the tangent at the point (6, 8) on C. [4]

(iii) Find the coordinates of the points of intersection of C with the

straight line y = 2x + 3. [7]

(b) The points P and Q have position vectors, relative to the origin O,

given respectively by

p = −i + 6j and q = 3i + 8j.

(i) a) Calculate, in degrees, the size of the angle, θ, between p and

q. [5]

b) Hence, calculate the area of triangle POQ. [2]

(ii) Find, in terms of i and j, the position vector of

a) M , where M is the mid-point of PQ. [2]

b) R, where R is such that PQRO, labelled clockwise, forms a

parallelogram. [3]

21

Question 4

(a) The diagram below, which is not drawn to scale, shows a quadri-

lateral ABCD in which BC = 9 cm, AD = x cm, AB = 4 cm and

∠BAD = ∠BCD = θ and ∠CDA is a right-angle.

CD

A

B

θ

θ

9 cm

4 cm

x cm

(i) Show that x = 4 cos θ + 9 sin θ. [4]

(ii) By expressing x in the form r cos(θ − α), where r is positive and

0 ≤ α < 12π, find the maximum possible value of x. [6]

(b) Given that A and B are acute angles such that sin A = 35

and cos B =513

, find, without using tables or calculators, the EXACT value of

(i) sin(A + B) [3]

(ii) cos(A − B) [3]

(iii) cos 2A. [2]

(c) Prove that

tan(x

2+

π

4

)

= sec x + tanx.

[7]

22

Question 5

(a) Find

limx→2

x3 − 8

x2 − 6x + 8

[5]

(b) The function f on R, is defined by

f (x) =

{

3 − x if x ≥ 11 + x if x < 1.

(i) Sketch the graph of f(x) for the domain −1 ≤ x ≤ 2. [2]

(ii) Find

a) limx→1+

f (x) [2]

b) limx→1−

f (x) . [2]

(iii) Deduce that f(x) is continuous at x = 1. [3]

(c) Differentiate from first principles, with respect to x, the function y =1

x2. [6]

(d) The function f(x) is such that f ′(x) = 3x2+6x+k where k is a constant.

Given that f(0) = −6 and f(1) = −3, find the value of f(x). [5]

23

Question 6

(a) Given that y = sin 2x + cos 2x, show that

d2y

dx2+ 4y = 0.

[6]

(b) Given that

∫ a

0

(x + 1) dx = 3

∫ a

0

(x − 1) dx, a > 0,

find the value of the constant a. [6]

(c) The diagram below (not drawn to scale) represents a piece of thin

cardboard 16 cm by 10 cm. Shaded squares, each of side x cm, are

removed from each corner. The remainder is folded to form a tray.

16 cm

x

x

10 cm

(i) Show that the volume, V cm3, of the tray is given by

V = 4(

x3 − 13x2 + 40x)

.

(ii) Hence, find the possible value of x such that V is a maximum.[8]

26

Question 1

(a) The roots of the cubic equation x3 + 3px2 + qx + r = 0 are 1,−1 and

3. Find the values of the real constants p, q and r. [7]

(b) Without using tables or calculators, show that

(i)

√6 +

√2√

6 −√

2= 2 +

√3 [5]

(ii)

√6 +

√2√

6 −√

2+

√6 −

√2√

6 +√

2= 4. [5]

(c) (i) Show that

n∑

r=1

r (r + 1) =1

3n(n + 1)(n + 2), n ∈ N.

[5]

(ii) Hence, or otherwise, evaluate

50∑

r=31

r (r + 1) .

[3]

27

Question 2

(a) The roots of the quadratic equation

2x2 + 4x + 5 = 0

are α and β.

Without solving the equation,

(i) write down the values of α + β and αβ [2]

(ii) calculate

a) α2 + β2 [2]

b) α3 + β3 [4]

(iii) find the equation whose roots are α3 and β3. [4]

(b) (i) Solve for x the equation x1/3 − 4x−1/3 = 3. [5]

(ii) Find x such that log5(x + 3) + log5(x − 1) = 1. [5]

(iii) Without the use of calculators or tables, evaluate

log10

(

1

2

)

+log10

(

2

3

)

+log10

(

3

4

)

+· · ·+log10

(

8

9

)

+log10

(

9

10

)

.

[3]

28

Question 3

(a) The lines y = 3x + 4 and 4y = 3x + 5 are inclined at angles α and β

respectively to the x-axis.

(i) State the value of tan α and tanβ. [2]

(ii) Without using tables or calculators, find the tangent of the

angle between the two lines. [4]

(b) (i) Prove that sin 2θ − tan θ cos 2θ = tan θ. [3]

(ii) Express tan θ in terms of sin 2θ and cos 2θ. [2]

(iii) Hence show, without using tables or calculators, that tan 22.5◦ =√2 − 1. [4]

(c) (i) Given that A, B and C are the angles of a triangle, prove that

a) sinA + B

2= cos

C

2[3]

b) sin B + sin C = 2 cosA

2cos

B − C

2. [2]


sin A + sin B + sin C = 4 cosA

2cos

B

2cos

C

2.

[5]

29

Question 4

(a) In the cartesian plane with origin O, the coordinates of the points P

and Q are (−2, 0) and (8, 8) respectively. The midpoint of PQ is M .

(i) Find the equation of the line which passes through M and is per-

pendicular to PQ. [8]

(ii) Hence, or otherwise, find the centre of the circle through P, O and

Q. [9]

(b) (i) Prove that the line y = x + 1 is a tangent to circle

x2 + y2 + 10x − 12y + 11 = 0.

[6]

(ii) Find the coordinates of the point of contact of this tangent to the

circle. [2]

30

Question 5

(a) Find

limx→3

x3 − 27

x2 + x − 12.

[4]

(b) A chemical process is controlled by the function

P =u

t+ vt2

where u and v are constants.

Given that P = −1 when t = 1 and the rate of change of P with respect

to t is −5 when t = 12, find the values of u and v. [6]

(c) The curve C passes through the point (−1, 0) and its tangent at any

point (x, y) is given bydy

dx= 3x2 − 6x.

(i) Find the equation of C. [3]

(ii) Find the coordinates of the stationary points of C and determine

the nature of EACH point. [7]

(iii) Sketch the graph of C and label the x-intercepts. [5]

31

Question 6


(i) x√

2x − 1 [3]

(ii) sin2(x3 + 4). [4]

(b) (i) Given that

∫ 6

1

f (x) dx = 7, evaluate

∫ 6

1

[2 − f (x) dx] . [3]

(ii) The area under the curve y = x2 + kx − 5, above the x-axis and

bounded by the lines x = 1 and x = 3, is 1423

units2.

Find the value of the constant k. [4]

(c) The diagram below (not drawn to scale) represents a can in the

shape of a closed cylinder with a hemisphere at one end. The can has

a volume of 45 units3.

r

h

(i) Taking r units as the radius of the cylinder and h as its height,

show that

a) h =45

r2− 2r

3[3]

b) A =5πr2

3+

90π

r, where A is the external surface area of the

can. [3]

(ii) Hence, find the value of r for which A is a minimum and the

corresponding minimum value of A. [5]

[Volume of sphere = 43πr3, surface area of sphere = 4πr2]

[Volume of cylinder πr2h, curved surface area of cylinder = 2πrh.]

34

Question 1

(a) Let g(x) = x4 − 9, x ∈ R. Find

(i) all the real factors of g(x) [3]

(ii) all the real roots of g(x) = 0. [1]

(b) The function f is defined by

f(x) = x4 − 9x3 + 28x2 − 36x + 16, x ∈ R,

and u = x +4

x, x 6= 0.

(i) Express u2 in terms of x. [3]

(ii) By writing

f(x) = x2

[

x2 − 9x + 28 − 36

x+

16

x2

]

and using the result from (b) (i) above, show that if f(x) = 0,

then u2 − 9u + 20 = 0. [6]

(iii) Hence, determine the values of x ∈ R, for which f(x) = 0. [7]

35

Question 2

(a) Let Sn =

n∑

r=1

r for n ∈ N. Find the value of n for which 3S2n = 11Sn.

Note:

n∑

r=1

r = 12n (n + 1) . [4]

(b) The quadratic equation x2 − px + 24 = 0, p ∈ R, has roots α and β,

and the quadratic equation x2 − 8x + q = 0, q ∈ R, has roots 2α + β

and 2α − β.

(i) Express p and q in terms of α and β. [2]

(ii) Find the values of α and β. [4]

(iii) Hence, determine the values of p and q. [2]

(c) Prove by Mathematical Induction that n2 > 2n for all integers n ≥ 3.

[8]

36

Question 3

The circle shown in the diagram below (not drawn to scale) has centre C

at (5,−4) and touches the y-axis at the point D. The circle cuts the x-axis

at the points A and B. The tangent at B cuts the y-axis at the point P .

y

xOA

(5,−4)

B

CD

P

b

(a) Determine

(i) the length of radius of the circle [2]

(ii) the equation of the circle [1]

(iii) the coordinates of the points A and B, at which the circle cuts the

x-axis. [6]

(iv) the equation of the tangent at B [4]

(v) the coordinates of P [2]

(b) Show by calculation that PD = PB. [5]

37

Question 4

(a) (i) Prove that cos 2θ =1 − tan2 θ

1 + tan2 θ. [4]

(ii) Hence, show, without using calculators, that

tan 671

2

◦

= 1 +√

2.

[7]

(b) In the triangle shown below, (not drawn to scale), sin q = 35

and

cos p = 513

.

q

r tp

60◦

Determine the exact value of

(i) cos q [1]

(ii) sin p [1]

(iii) sin r [3]

(iv) cos(p + t). [4]

38

Question 5

(a) Given that y =√

5x2 + 3,

(i) obtaindy

dx[4]

(ii) show that ydy

dx= 5x [2]

(iii) hence, or otherwise, show that yd2y

dx2+

(

dy

dx

)2

= 5. [4]

(b) At a certain port, high tides and low tides occur daily. Suppose t

minutes after high tide, the height, h metres, of the tide above a fixed

point is given by

h = 2

(

1 + cosπt

450

)

, 0 ≤ t.

Note: High tide occurs when h has its maximum value and

low tide occurs when h has it minimum value.

Determine

(i) the height of the tide when high tide occurs for the first time [2]

(ii) the length of time which elapses between the first high tide and

the first low tide [3]

(iii) the rate, in metres per minute, at which the tide is falling 75

minutes after high tide. [5]

39

Question 6

(a) (i) Use the result

∫ a

0

f (x) dx =

∫ a

0

f (a − x) dx, a > 0, to show that

if

I =

∫ π/2

0

sin2 xdx, then I =

∫ π/2

0

cos2 xdx.

[2]

(ii) Hence, or otherwise, show that I =π

4. [6]

(b) (i) Sketch the curve y = x2 + 4. [4]

(ii) Calculate the volume created by rotating the plane figure bounded

by x = 0, y = 4, y = 5 and the curve y = x2+4 through 360◦ about

the y-axis. [8]

42

Question 1

(a) Solve the simultaneous equations

x2 + xy = 6

x − 3y + 1 = 0.

[8]

(b) The roots of the equation x2 +4x+1 = 0 are α and β. Without solving

the equation,

(i) state the value of α + β and αβ [2]

(ii) find the value of α2 + β2 [3]

(iii) find the equation whose roots are 1 +1

αand 1 +

1

β. [7]

Question 2

(a) Prove by Mathematical Induction, that

n∑

r=1

r = 12n (n + 1) . [10]

(b) Express, in terms of n and in the simplest form,

(i)

2n∑

r=1

r [2]

(ii)

2n∑

r=n+1

r [4]

(c) Find n if

2n∑

r=n+1

r = 100. [4]

43

Question 3

(a) (i) Find the coordinates of the centre and the radius of the circle

x2 + 2x + y2 − 4y = 4.

[4]

(ii) By writing x + 1 = 3 sin θ, show that the parametric equations of

this circle are x = −1 + 3 sin θ and y = 2 + 3 cos θ. [5]

(iii) Show that the x-coordinates of the points of intersection of this

circle with the line x + y = 1 are x = −1 ± 32

√2. [4]

(b) Find the general solution of the equation cos θ = 2 sin2 θ − 1. [7]

Question 4

(a) Given that 4 sinx − cos x = R sin(x − α); R > 0 and 0◦ < α < 90◦,

(i) find the values of R and α correct to one decimal place [7]

(ii) hence, find ONE value of x between 0◦ and 360◦ for which the

curve y = 4 sin x − cos x has a stationary point. [2]

(b) Let z1 = 2 − 3i and z2 = 3 + 4i.

(i) Find in the form a + bi, a, b ∈ R,

a) z1 + z2 [1]

b) z1z2 [3]

c)z1

z2. [5]

(ii) Find the quadratic equation whose roots are z1 and z2. [2]

44

Question 5

(a) (i) State the value of limδx→0

sin δx

δx. [1]

(ii) Given that

sin 2(x + δx) − sin 2x ≡ 2 cosA sin B,

find A and B in terms of x and/or δx. [2]

(iii) Hence, or otherwise, differentiate with respect to x, from first prin-

ciples, the function y = sin 2x. [7]

(b) The curve y = hx2 +k

xpasses through the point P (1, 1) and has a

gradient of 5 at P . Find

(i) the values of the constants h and k. [5]

(ii) the equation of the tangent to the curve at the point where x = 12.

[5]

45

Question 6

(a) In the diagram given below (not drawn to scale), the area S under

the line y = x, for 0 ≤ x ≤ 1, is divided into a set of (n − 1) rectangular

strips, each of width1

nunits.

y

x

y = x

1n0

(1, 0)

2n

nn

3n

n−1n

4n

(i) Show that the area S is approximately

1

n2+

2

n2+

3

n2+ · · ·+ n − 1

n2.

[6]

(ii) Given thatn−1∑

r=1

r = 12n (n − 1) , show that S ≈ 1

2

(

1 − 1

n

)

. [2]

(b) (i) Show that for f(x) =2x

x2 + 4,

f ′(x) =8 − 2x2

(x2 + 4)2 .

[4]

(ii) Hence evaluate∫ 1

0

24 − 6x2

(x2 + 4)2dx.

[3]

(c) Find the value of u > 0, if∫ 2u

u

1

x4dx =

7

192.

[3]

Part I

CAPE UNIT TWO

CAPE Unit TwoExamination 2011

50

Question 1

(a) Finddy

dxif

(i) x2 + y2 − 2x − 2y − 14 = 0 [3]

(ii) y = ecos x [3]

(iii) y = cos2 6x + sin2 8x. [3]

(b) Let y = x sin1

x, x 6= 0.

Show that

(i) xdy

dx= y − cos

1

x[3]

(ii) x4 d2y

dx2+ y = 0. [3]

(c) A curve is given by the parametric equations x =√

t, y − t =1√t.

(i) Find the gradient of the tangent to the curve at the point where

t = 4. [7]

(ii) Find the equation of the tangent to the curve at the point where

t = 4. [3]

51

Question 2

(a) Let Fn (x) =1

n!

∫ x

0

tne−tdt.

(i) Find F0 (x) and Fn (0) , given that 0! = 1. [3]

(ii) Show that Fn (x) = Fn−1 (x) − 1

n!xne−x. [7]

(iii) Hence, show that if M is an integer greater than 1, then

exFM (x) = −(

x +x2

2!+

x3

3!+ · · · + xM

M !

)

+ (ex − 1) . [4]

(b) (i) Express2x2 + 3

(x2 + 1)2in partial fractions. [5]

(ii) Hence, find

∫

2x2 + 3

(x2 + 1)2dx. [6]

52

Question 3

(a) The sequence of positive terms {xn} is defined by xn+1 = x2n +

1

4,

x1 <1

2, n ≥ 1,

(i) Show, by mathematical induction, that xn <1

2for all positive

integers n. [5]

(ii) By considering xn+1 − xn, show that xn < xn+1. [3]

(b) (i) Find constants A and B such that

2 − 3x

(1 − x) (1 − 2x)≡ A

1 − x+

B

1 − 2x. [3]

(ii) Obtain the first FOUR non-zero terms in the expansion of each of

(1 − x)−1 and (1 − 2x)−1 as a power series of x in ascending order.

[4]

(iii) Find

a) the range of values of x for which the series expansion of

2 − 3x

(1 − x) (1 − 2x)

is valid. [2]

b) the coefficient of xn in (iii) a) above. [2]

(iv) The sum, Sn, of the first n terms of a series is given by

Sn = n (3n − 4) .

Show that the series is an Arithmetic Progression (A.P.) with com-

mon difference 6. [6]

53

Question 4

(a) A Geometric Progression (G.P.) with first term a and common ratio r,

0 < r < 1, is such that the sum of the first three terms is26

3and their

product is 8.

(i) Show that r + 1 +1

r=

13

3. [5]

(ii) Hence find

a) the value of r [4]

b) the value of a [1]

c) the sum to infinity of the G.P. [2]

(b) Expand2

ex + e−x, |x| < 1

in ascending powers of x as far as the term in x4. [5]

(c) Let f (r) =1

r (r + 1), r ∈ N.

(i) Express f (r) − f (r + 1) in terms of r. [3]

(ii) Hence, or otherwise, find

Sn =

n∑

r=1

1

r (r + 1) (r + 2). [4]

(iii) Deduce the sum to infinity of the series in (c) (ii) above. [2]

54

Question 5

(a)

(

n

r

)

is defined as the number of ways of selecting r distinct objects

from a set of n distinct objects. From the definition, show that

(i)

(

n

r

)

=

(

n

n − r

)

[2]

(ii)

(

n + 1r

)

=

(

n

r

)

+

(

n

r − 1

)

[4]

(iii) Hence, prove that

[(

86

)

+

(

85

)]

×[(

83

)

+

(

82

)]

is a perfect square. [3]

(b) (i) Find the number of 5-digit numbers greater than 30, 000 which can

be formed with the digits 1, 3, 5, 6 and 8, if no digit is repeated.[3]

(ii) What is the probability that one of the numbers chosen in (b) (i)

being even? [5]

(c) (i) a) Show that (1 − i) is one of the square roots of −2i. [2]

b) Find the other square root. [1]

(ii) Hence, find the roots of the quadratic equation

z2 − (3 + 5i) z + (8i − 4) = 0. [5]

55

Question 6

(a) The matrix A =

1 1 12 −3 2

−1 −3 −2

.

(i) Show that |A| = 5. [3]

(ii) The matrix A is changed to form new matrices B, C and D. Write

down the determinant of EACH of the new matrices, giving a

reason for your answer in EACH case.

a) Matrix B is formed by interchanging row 1 and row 2 of A

and then interchanging column 1 and column 2 of the resulting

matrix. [2]

b) Row 1 of C is formed by adding row 2 to row 1 of matrix A.

The other rows remain unchanged. [2]

c) Matrix D is formed by multiplying each element of A by 5.

[2]

(b) Given the matrix M =

12 −1 52 −1 0

−9 2 −5

,

find

(i) AM [3]

(ii) the inverse A−1, of A . [2]

(c) (i) Write the system of equations

x + y + z = 52x − 3y + 2z = −10−x − 3y − 2z = −11

in the form Ax = b. [1]

(ii) Show that x = A−1b. [2]

(iii) Hence, solve the system of equations. [2]

56

(iv) a) Show that (x, y, z) = (1, 1, 1) is a solution of the following

system of equations:

x + y + z = 32x + 2y + 2z = 63x + 3y + 3z = 9

[1]

b) Hence, find the general solution of the system. [5]

58

Question 1

(a) The temperature of water x◦C, in an insulated tank at time, t hours,

may be modelled by the equation x = 65 + 8e−0.02t.

Determine the

(i) initial temperature of the water in the tank [2]

(ii) temperature at which the water in the tank will eventually stabilize

[2]

(iii) time when the temperature of the water in the tank is 70◦C. [4]

(b) It is given that y = etan−1(2x), where −12π < tan−1(2x) < 1

2π.

(i) Show that(

1 + 4x2) dy

dx= 2y.

[4]


(

1 + 4x2)2 d2y

dx2= 4y (1 − 4x) .

[4]

(c) Determine

∫

4

ex + 1dx

(i) by using the substitution u = ex [6]

(ii) by first multiplying both the numerator and the denominator of

the integrand by e−x before integrating. [3]

59

Question 2

(a) (i) Given that n is a positive integer, findd

dx[x (ln x)n] . [4]

(ii) Hence, or otherwise, derive the reduction formula

In = x(ln x)n − nIn−1,

where In =

∫

(ln x)ndx. [4]

(iii) Use the reduction formula in (a) (ii) above to determine

∫

(ln x)3dx.

[6]

(b) The amount of salt, y kg, that dissolves in a tank of water at time t

minutes satisfies the differential equation

dy

dt+

2y

t + 10= 3.

(i) Using a suitable integrating factor, show that the general solution

of this differential equation is

y = t + 10 +c

(t + 10)2,

where c is an arbitrary constant. [7]

(ii) Given that the tank initially contains 5 kg of salt in the liquid,

calculate the amount of salt that dissolves in the tank of water at

t = 15. [4]

60

Question 3

(a) The first four terms of a sequence are

2 × 3 5 × 5 8 × 7 11 × 9.

(i) Express, in terms of r, the rth term of the sequence. [2]

(ii) If Sn denotes the series formed by summing the first n terms of

the sequence, find Sn in terms of n. [5]

(b) The 9th term of an A.P. is three times the 3rd term and the sum of the

first 10 terms is 110.

Find the first term a and the common difference d. [6]

(c) (i) Use the binomial theorem to expand (1+2x)1/2 as far as the term

in x3, stating the values of x for which the expansion is valid. [5]

(ii) Prove that

x

1 + x +√

1 + 2x=

1

x

(

1 + x −√

1 + 2x)

for x > 0. [4]

(iii) Hence or otherwise, show that, if x is small so that the term in x3

and higher powers of x can be neglected, the expansion in (c) (ii)

above is approximately equal to 12x (1 − x) . [3]

61

Question 4

(a) (i) By expressing nCr and nCr−1 in terms of factorials, prove that

nCr + nCr−1 = n+1Cr.

[6]

(ii) a) Given that r is a positive integer and f(r) =1

r!, show that

f(r) − f(r + 1) =r

(r + 1)!.

[3]

b) Hence, or otherwise, find the sum

Sn =

n∑

r=1

r

(r + 1)!.

[5]

c) Deduce the sum to infinity of Sn in (ii) (b) above. [2]

(b) (i) Show that the function f(x) = x3 − 6x + 4 has a root x in the

closed interval [0, 1] [5]

(ii) By taking x1 = 0.6 as a first approximation, use the Newton-

Raphson method to obtain a second approximation x2 in the in-

terval [0, 1]. [4]

62

Question 5

(a) Calculate

(i) the number of different permutations of the 8 letters of the word

SYLLABUS. [3]

(ii) the number of different selections of 5 letters which can be made

from the letters of the word SYLLABUS. [5]

(b) The events A and B are such that P (A) = 0.4, P (B) = 0.45 and

P (A ∪ B) = 0.68.

(i) Find P (A ∩ B). [3]

(ii) Stating a reason in each case, determine whether or not the events

A and B are

a) mutually exclusive, [3]

b) independent. [3]

(c) (i) Express the complex number

(2 + 3i) +i − 1

i + 1

in the form a + ib where a and b are both real numbers. [4]

(ii) Given that 1− i is a root of the equation z3 + z2 −4z +6 = 0, find

the remaining roots. [4]

63

Question 6

(a) A system of equations is given by

x + y + z = 02x + y − z = −1x + 2y + 4z = k

where k is a real number.

(i) Write the augmented matrix of the system. [2]

(ii) Reduce the augmented matrix to echelon form. [3]

(iii) Deduce the value of k for which the system is consistent. [2]

(iv) Find ALL solutions corresponding to the value of k obtained in

(iii) above. [4]

(b) Given that

A =

0 −1 1−1 0 1

1 1 1

,

(i) find

a) A2 [4]

b) B = 3I + A − A2. [4]

(ii) Calculate AB. [4]

(iii) Deduce the inverse, A−1 of the matrix A. [2]

66

Question 1

(a) Finddy

dxif

(i) y = sin2 5x + sin2 3x + cos2 3x [3]

(ii) y =√

cos x2 [4]

(iii) y = xx. [4]

(b) (i) Given that y = cos−1 x, where 0 ≤ cos−1 x, prove that

dy

dx=

−1√1 − x2

.

Note: cos−1 x ≡ arccos x. [7]

(ii) The parametric equations of a curve are defined in terms of the

parameter t by

y =√

1 − t, x = cos−1 t, where 0 ≤ t < 1.

a) Show thatdy

dx=

√1 + t

2. [4]

b) Hence, findd2y

dx2in terms of t, giving your answer in simplified

form. [3]

67

Question 2

(a) Sketch the region whose area is defined by the integral

∫ 1

0

√1 − x2dx.

[3]

(b) Using FIVE vertical strips, apply the trapezium rule to show that

∫ 1

0

√1 − x2dx ≈ 0.759.

[6]

(c) (i) Use integration by parts to show that if I =

∫ √1 − x2dx, then

I = x√

1 − x2 − I +

∫

1√1 − x2

dx.

[9]

(ii) Deduce that

I =x√

1 − x2 + sin−1 x

2+ C,

where C is an arbitrary constant of integration.

Note: sin−1 x ≡ arcsin x. [2]

(iii) Hence, find

∫ 1

0

√1 − x2dx. [3]

(iv) Use the results in parts (b) and (c) (iii) above to find an approxi-

mation to π. [2]

68

Question 3

(a) A sequence {tn} is defined by the recurrence relation

tn+1 = tn + 5, t1 = 11 for all n ∈ N.

(i) Determine t2, t3 and t4. [3]

(ii) Express tn in terms of n. [5]

(b) Find the range of values of x for which the common ratio r of a con-

vergent geometric series is2x − 3

x + 4. [8]

(c) Let f(r) =1

r + 1for r ∈ N.

(i) Express f(r) − f(r + 1) in terms of r. [3]

(ii) Hence, or otherwise, find

Sn =

n∑

r=1

4

(r + 1) (r + 2).

[4]

(iii) Deduce the sum to infinity of the series in (c) (ii) above. [2]

69

Question 4

(a) (i) Find n ∈ N, such that 5(nC2) = 2(n+2C2). [5]

(ii) The coefficient of x2 in the expansion of

(1 + 2x)5(1 + px)4

is −26. Find the possible values of the real number p. [7]

(b) (i) Write down the first FOUR non-zero terms of the power series

expansion of ln(1 + 2x), stating the range of values of x for which

the series is valid. [2]

(ii) Use Maclaurin’s theorem to obtain the first THREE non-zero

terms in the power series expansion in x of sin 2x. [7]

(iii) Hence, or otherwise, obtain the first THREE non-zero terms in

the power series expansion in x of ln(1 + sin 2x). [4]

70

Question 5

(a) A committee of 4 persons is to be chosen from 8 persons, including Mr.

Smith and his wife. Mr. Smith will not join the committee without his

wife, but his wife will join the committee without him.

Calculate the number of ways in which the committee of 4 persons can

be formed. [5]

(b) Two balls are drawn without replacement from a bag containing twelve

balls numbered 1 to 12.

Find the probability that

(i) the numbers on BOTH balls are even [4]

(ii) the number on one ball is odd and the number on the other ball

is even. [4]

(c) (i) Find complex numbers u = x + iy such that x and y are real

numbers and

u2 = −15 + 8i

[7]

(ii) Hence, or otherwise, solve for z the equation

z2 − (3 + 2i)z + (5 + i) = 0.

[5]

71

Question 6

(a) Solve for x the equation

∣

∣

∣

∣

∣

∣

x − 3 1 −11 x − 5 1−1 1 x − 3

∣

∣

∣

∣

∣

∣

= 0.

[10]

(b) (i) Given the matrices

A =

1 −1 11 −2 41 3 9

, B =

30 −12 25 −8 3

−5 4 1

,

a) find AB [4]

b) hence deduce the inverse A−1, of the matrix A. [3]

(ii) A system of equations is given by

x − y + z = 1x − 2y + 4z = 5x + 3y + 9z = 25.

a) Express the system in the form

Ax = b

where A is a square matrix and x and b are column vectors.

[3]

b) Hence, or otherwise, solve the system of equations. [5]

74

Question 1


(i) e4x cos πx [4]

(ii) ln

(

x2 + 1√x

)

. [4]

(b) Given that y = 3−x, show, by using logarithm, that

dy

dx= −3−x ln 3.

[5]

(c) (i) Express in partial fractions

2x2 − 3x + 4

(x − 1)(x2 + 1).

[7]

(ii) Hence, find∫

2x2 − 3x + 4

(x − 1)(x2 + 1)dx.

[5]

75

Question 2

(a) Solve the differential equation

dy

dx+ y = e2x.

[5]

(b) The gradient at the point (x, y) on a curve is given by

dy

dx= e4x.

Given that the curve passes through the point (0, 1), find its equation.

[5]

(c) Evaluate∫ e

1

x2 ln xdx,

writing your answer in terms of e. [7]

(d) (i) Use the substitution v = 1 − u to find

∫

du√1 − u

.

[3]

(ii) Hence, or otherwise, use the substitution u = sin x to evaluate

∫ π/2

0

√1 + sin xdx.

[5]

76

Question 3

(a) A sequence {un} is defined by the recurrence relation

un+1 = un + n, u1 = 3, for n ∈ N.

(i) State the first FOUR terms of the sequence. [3]

(ii) Prove by mathematical induction, or otherwise, that

un =n2 − n + 6

2.

[8]

(b) A GP with first term a and common ratio r has sum to infinity 81 and

the sum of the first four terms is 65. Find the values of a and r. [6]

(c) (i) Write down the first FIVE terms in the power series expansion of

ln(1 + x), stating the range of values of x for which the series is

valid. [3]

(ii) a) Using the result of (c) (i) above, obtain a similar expansion

for ln(1 − x). [2]

b) Hence, prove that

ln

(

1 + x

1 − x

)

= 2

(

x +1

3x3 +

1

5x5 + · · ·

)

.

[3]

77

Question 4

(a) (i) Show that the function f(x) = x3 − 3x + 1 has a root α in the

closed interval [1, 2]. [3]

(ii) Use the Newton-Raphson method to show that if x1 is a first ap-

proximation to α in the interval [1, 2], then a second approximation

to α in the interval [1, 2] is given by

x2 =2x3

1 − 1

3x21 − 3

.

[5]

(b) (i) Use the binomial theorem or Maclaurin’s theorem to expand (1 +

x)−1/2 in ascending powers of x as far as the term in x3, stating

the values of x for which the expansion is valid. [4]

(ii) Obtain a similar expansion for (1 − x)1/2. [4]

(iii) Prove that if x is so small that x3 and higher powers of x can be

neglected, then√

1 − x

1 + x≈ 1 − x +

1

2x2.

[5]

(iv) Hence, by taking x = 117

, show, without using calculators or

tables, that√

2 is approximately equal to 16351156

. [4]

78

Question 5

(a) A cricket selection committee of 4 members is to be chosen from 5

former batsmen and 3 former bowlers.

In how many ways can this committee be selected so that the committee

includes AT LEAST

(i) ONE former batsman? [8]

(ii) ONE batsman and ONE bowler? [3]

(b) Given the matrices

A =

3 1 01 0 10 −1 0

, B =

1 −1 10 0 −1

−1 3 −1

and M =

1 0 10 0 −3

−1 3 −1

,

(i) Determine each of the following matrices:

a) A − B [2]

b) AM . [3]

(ii) Deduce from (i) b) above, the inverse A−1 of the matrix A. [3]

(iii) Find the matrix X such that AX + B = A. [6]

79

Question 6

(a) (i) Express the complex number

2 − 3i

5 − i

in the form λ(1 − i). [4]

(ii) State the value of λ. [1]

(iii) Verify that

(

2 − 3i

5 − i

)4

is a real number and state its value. [5]

(b) The complex number z is represented by the point T in an Argand

diagram.

Given that z =1

3 + it, where t is a variable and z denotes the complex

conjugate of z, show that

(i) z + z = 6zz [7]

(ii) as t varies T lies on a circle, and state the coordinates of the centre

of this circle. [8]

82

Question 1

(a) Solve, for x > 0, the equation

3 log8 x = 2 logx 8 − 5. [8]

(b) (i) Copy and complete the table below for values 2x and e−x, using a

calculator, where necessary. Approximate all values to 2 decimal

places.

x −1.0 0 0.5 1.0 1.5 2.0 2.5 3.02x 1.00 1.41 2.00 4.0 8.00e−x 2.72 0.37 0.22 0.05

[3]

(ii) On the same pair of axes and using a scale of 4 cm for 1 unit on

the x-axis, draw the graphs of the two curves y = 2x and y = e−x

for −1 ≤ x ≤ 3, x ∈ R. [5]

(iii) Use the graphs to find

a) the value of x satisfying 2x − e−x = 0 [3]

b) the range of values of x for which 2x − e−x < 0. [6]

Question 2

(a) Show that for n ≥ 2,

tann x = tann−2 x sec2 x − tann−2 x. [3]

(b) Finddy

dxwhen y = tann x. [3]

(c) Let

In =

∫ π/4

0

tann xdx, n ≥ 2.

(i) By using the result in (a) above, show that

In + In−2 =1

n − 1. [7]

(ii) Hence evaluate I4. [7]

83

Question 3

(a) The sequence {un} is given by u1 = 1 and un+1 = (n + 1)un, n ≥ 1.

Prove by Mathematical Induction that un = n! ∀n ∈ N. [9]

(b) Given that the sum of the first n terms of a series, S, is 9 − 32−n,

(i) find the nth term of S [5]

(ii) show that S is a geometric progression [2]

(iii) find the first term and common ratio of S [2]

(iv) deduce the sum to infinity of S. [2]

Question 4

(a) The function f is given by f : x → x4 − 4x + 1. Show that

(i) f(x) = 0 has a root α in the interval (0, 1) [4]

(ii) if x1 is a first approximation to f(x) = 0 in (0, 1), the Newton-

Raphson method gives a second approximation x2 in (0, 1) satis-

fying

x2 =3x4

1 − 1

4 (x31 − 1)

.

[5]

(b) John’s father gave him a loan of $10 800 to buy a car. The loan was to

be repaid by 12 unequal monthly installments, starting with an initial

payment of $P in the first month. There is no interest charges on the

loan, but the installments increase by $60 per month.

(i) Show that P = 570. [5]

(ii) Find, in terms of n, 1 ≤ n ≤ 12, an expression for the remaining

debt on the loan after John has paid the nth installment. [6]

84

Question 5

(a) A bag contains 5 white marbles and 5 black marbles. Six marbles are

chosen at random.

(i) Determine the number of ways of selecting the six marbles if there

are no restrictions. [2]

(ii) Find the probability that the marbles chosen contain more black

marbles than white marbles. [4]

(b) The table below summarises the programme preference of 100 television

viewers.

Preference Number of Males Number of Females Total

Matlock 20 10 30News 14 18 32Friends 18 20 38Total 52 48 100

Determine the probability that a person selected at random

(i) is a female [2]

(ii) is a male or likes watching the News [4]

(iii) is a female that likes watching Friends [2]

(iv) does not like watching Matlock. [2]

(c) The table below lists the probability distribution of the number of ac-

cidents per week on a particular highway.

Number of accidents 0 1 2 3 4 5Probability 0.25 0 0.10 p 0.30 0.15

(i) Calculate the value of p. [2]

(ii) Determine the probability that there are more than 3 accidents in

a week. [2]

85

Question 6

(a) A system of equations is given by

x + y + z = 103x − 2y + 3z = 352x + y + 2z = α

where α is a real number.

(i) Write the system in matrix form. [1]

(ii) Write down the augmented matrix. [1]

(iii) Reduce the augmented matrix to echelon form. [3]

(iv) Deduce the value of α for which the system is consistent. [1]

(v) Find ALL solutions corresponding to this value of α. [4]

(b) Given A =

0 −1 1−1 0 1

1 1 1

,

find

(i) kI −A, where I is the identity matrix and k is a real number. [3]

(ii) the values of k for which |kI − A| = 0. [7]

88

Question 1

(a) If f(x) = x3 ln2 x, show that

(i) f ′(x) = x2 ln x(3 ln x + 2), [5]

(ii) f ′′(x) = 6x ln2 x + 10x lnx + 2x. [5]

(b) The enrolment pattern of membership of a country club follows an

exponential logistic function N,

N =800

1 + ke−rt, k ∈ R, r ∈ R,

where N is the number of members enrolled t years after the formation

of the club. The initial membership was 50 persons and after one year

there are 200 persons enrolled in the club.

(i) What is the LARGEST number reached by the membership of the

club? [2]

(ii) Calculate the EXACT value of k and r. [6]

(iii) How many members will there be in the club 3 years after its

formation? [2]

Question 2

(a) (i) Express1 + x

(x − 1)(x2 + 1)in partial fractions. [6]

(ii) Hence, find∫

1 + x

(x − 1)(x2 + 1)dx. [3]

(b) Let In =

∫ 1

0

xnexdx where n ∈ N.

(i) Evaluate I1. [4]

(ii) Show that In = e − nIn−1. [4]

(iii) Hence, or otherwise, evaluate I3, writing your answer in terms of

e. [3]

89

Question 3

(a) (i) Show that the terms ofm

∑

r=1

ln 3r

are in arithmetic progression. [3]

(ii) Find the sum of the first 20 terms of this series. [4]

(iii) Hence, show that

2m∑

r=1

ln 3r =(

2m2 + m)

ln 3.

[3]

(b) The sequence of positive terms, {xn}, is defined by

xn+1 = x2n +

1

4, x1 <

1

2.

(i) Show, by mathematical induction, or otherwise, that xn < 12

for

all positive integers n. [7]

(ii) By considering xn+1 − xn, or otherwise, show that xn < xn+1. [3]

Question 4

(a) Sketch the functions y = sin x and y = x2 on the SAME axes. [5]

(b) Deduce that the function f(x) = sin x − x2 has EXACTLY two real

roots. [3]

(c) Find an interval in which the non-zero root of f(x) lies. [4]

(d) Starting with a first approximation of at x1 = 0.7, use one iteration

of the Newton-Raphson method to obtain a better approximation to 3

decimal places. [8]

90

Question 5

(a) (i) How many numbers lying between 3 000 and 6 000 can be formed

from the digits 1, 2, 3, 4, 5, 6, if no digit is used more than once in

forming a number? [5]

(ii) Determine the probability that a number in 5 (a) (i) above is even.

[5]

(b) In an experiment, p is the probability of success and q is the probability

of failure in a single trial. For n trials, the probability of x successes

and (n − x) failures is represented by

nCxpxqn−x, n > 0.

Apply this formula to the following problem.

The probability that John will hit the target at a firing practice is 56.

He fires 9 shots. Calculate the probability that he will hit the target

(i) AT LEAST 8 times [7]

(ii) NO MORE than seven times. [3]

91

Question 6

(a) If

A =

1 2 −1−1 2 1

1 −2 3

and B =

2 −1 11 1 00 1 1

,

(i) find AB [3]

(ii) deduce A−1. [3]

(b) A nursery sells three brands of grass-seed mix, P, Q and R. Each brand

is made from three types of grass, C, Z and B. The number of kilograms

of each type of grass in a bag of each brand is summarized in the table

below.Grass seed Type of grass (kg)

Mix C-grass Z-grass B-grassBrand P 2 2 6Brand Q 4 2 4Brand R 0 6 4Blend c z b

A blend is produced by mixing p bags of Brand P , q bags of Brand Q

and r bags of Brand R.

(i) Write down an expression in terms of p, q and r, for the number

of kilograms of Z-grass in the blend. [1]

(ii) Let c, z and b be represent the number of kilograms of C-grass, Z-

grass and B-grass in the blend. Write down a set of three equations

in p, z, r to represent the number of kilograms of EACH type of

grass in the blend. [3]

(iii) Rewrite the set of three equations in (b) (ii) above in the matrix

form MX = D where M is a 3 by 3 matrix, and X and D are

column matrices. [3]

(iv) Given that M−1 exists, write X in terms of M−1 and D. [3]

92

(v) Given that

M−1 =

−0.2 −0.2 0.30.35 0.1 −0.15

−0.05 0.2 −0.05

,

calculate how many bags of EACH brand P, Q and R, are re-

quired to produce a blend containing 30 kilograms of C-grass, 30

kilograms of Z-grass and 50 kilograms of B-grass. [4]

mathcapeunit 1&2 2011-2006q

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