8.2b Standard Normal Distribution Tables (Example 1)


8.2b Standard Normal Distribution Tables (Example 1)

Example 1:
Find the value of each of the following probabilities by reading the standard normal distribution tables.
(a) (Z > 0.600)
(b) (Z < –1.24)
(c) (Z > –1.1)
(d) (Z < 0.76)
 
Solution:
Standard Normal Distribution Table


*When reading the standard normal distribution tables, it involves subtraction of values.

(a)
From the standard normal distribution table,(Z > 0.600) = 0.2743


(b)
(Z < –1.24)
= (Z > 1.24)
= (1.24)
= 0.1075 ← (reading from the standard normal distribution table)


(*In the standard normal distribution table, all the values of are positive. As the curve is symmetrical about the vertical axis, the area of the shaded region in both of the graphs are the same.)


(c)
(Z > –1.1)
= 1 – Area P
= 1 – (–1.1)
= 1 – 0.1357 ← (reading from the standard normal distribution table)
= 0.8643

(d)
(Z < 0.76)
= 1 – Area P
= 1 – (0.76)
= 1 – 0.2236 ← (reading from the standard normal distribution table)
= 0.7764


4.3.1a Addition of Vectors (Examples)

Example 1:

Find
(a) The resultant vector of the addition of the two parallel vectors above
(b) The magnitude of the resultant vector.

Solution:
(a)
Resultant vector
= addition of the two vectors
=PQ+RS

(b)
Magnitude of the resultant vector
=|PQ|+|RS|=|62+82|+|62+82|=10+10=20units



Example 2:

The diagram above shows a parallelogram OABC. M is the midpoint of BC. Vector OA=a˜ and OC=c˜.  Find the following vectors in terms of a˜ and c˜.
(a)OB(b)MB(c)OM
 
Solution:
(a)
OB=OA+ABTriangle Law of addition=OA+OCAB=OC=a˜+c˜

(b)

MB=12CBM is the midpoint of CB =12OA =12a˜

(c)
OM=OC+CMTriangle Law of addition=OC+MBCM=MB=c˜+12a˜

3.6.1 Integration as the Summation of Volumes – Examples

Example 1:
Find the volume generated for the following diagram when the shaded region is revolved through 360° about the x-axis.


Solution:
Volume generated, Vx
Vx=πbay2dxVx=π42(3x8x)2dxVx=π42(3x8x)(3x8x)dxVx=π42(9x248+64x2)dxVx=π[9x3348x+64x11]42Vx=π[3x348x64x]42Vx=π[(3(4)348(4)644)(3(2)348(2)642)]Vx=π(16+104)Vx=88πunit3


Example 2:
Find the volume generated for the following diagram when the shaded region is revolved through 360° about the y-axis.


Solution:
Volume generated, Vy
Vy=πbax2dyVy=π21(2y)2dyVy=π21(4y2)dyVy=π214y2dyVy=π[4y11]21=π[4y]21Vy=π[(42)(41)]Vy=2πunit3


3.5.1 Integration as the Summation of Areas – Examples


Example 1
Find the area of the shaded region.


Solution:
Area of the shaded region=baydx=40(6xx2)dx=[6x22x33]40=[3(4)2(4)33]0=2623unit2


Example 2
Find the area of the shaded region.


Solution:
y = x -----(1)
x = 8yy2-----(2)
Substitute (1) into (2),
y = 8yy2
y2 – 7y = 0
y (y – 7) = 0
y = 0 or 7
From (1), x = 0 or 7
Therefore the intersection points of the curve and the straight line is (0, 0) and (7, 7).

Intersection point of the curve and y-axis is,
x = 8yy2
At y-axis, x = 0
0 = 8yy2
y (y – 8) = 0
y = 0, 8

Area of shaded region = (A1) Area of triangle + (A2) Area under the curve from y = 7 to y = 8.
=12×base×height +87xdy=12×(7)(7)+87(8yy2)dy=492+[8y22y33]87=2412+[4(8)2(8)33][4(7)2(7)33]=2412+85138123=2816unit2

3.4b Laws of Definite Integrals


3.4b Laws of Definite Integrals



Example:
Given that 73f(x)dx=5 , find the values for each of the following:

(a)736f(x)dx(b)73[3f(x)]dx(c)372f(x)dx(d)43f(x)dx+54f(x)dx+73f(x)dx(e)73f(x)+72dx


Solution:
(a)736f(x)dx=673f(x)dx=6(5)=30(b)73[3f(x)]dx=733dx73f(x)dx=[3x]735=[3(7)3(3)]5=7(c)372f(x)dx=732f(x)dx=273f(x)dx=2(5)=10(d)43f(x)dx+54f(x)dx+73f(x)dx=73f(x)dx=5(e)73f(x)+72dx=73[12f(x)+72]dx=7312f(x)dx+7372dx=1273f(x)dx+[7x2]73=12(5)+[7(7)27(3)2]=52+14=1612


Short Questions (Question 10)


Question 10 (4 marks):
Diagram 10 shows the position of three campsites A, B and C at a part of a riverbank drawn on a Cartesian plane, such that A and B lie on the same straight riverbank.

Diagram 10

Sam wants to cross the river from campsite C to the opposite riverbank where the campsites A and B are located.
Find the shortest distance, in m, that he can take to cross the river. Give your answer correct to four decimal places.

Solution:

Let the shortest distance from campsite Cto opposite riverbank is CD.Area of ABC=12|2     7     4     3     5   3  2  3|=12|(2)(5)+(7)(3)+(4)(3)(3)(7)(5)(4)(3)(2)|=12|66|=33 units2Distance of AB=(27)2+(35)2=85 mArea of ABC=3312(AB)(CD)=3312(85)(CD)=33CD=33(2)85CD=7.1587 (4 d.p.)Thus, the shortest distance is 7.1587 m.

Short Question 9 & 10


Question 9 (2 marks):
(a) Given C6n>1, list out all the possible values of n.(b) Given Cym=Cyn, express y interms of m and n.

Solution:
(a)
n = 1, 2, 3, 4, 5

(b)
y = m + n



Question 10 (4 marks):
Danya has a home decorations shop. One day, Danya received 14 sets of cups from a supplier. Each set contained 6 pieces of cups of different colours.

(a)
Danya chooses 3 sets of cups at random to be checked.
Find the number of different ways that Danya uses to choose those sets of cups.

(b)
Danya takes a set of cups to display by arranging it in a row.
Find the number of different ways the cups can be arranged such that the blue cup is not displaced next to the red cup.

Solution:
(a)
Number of different ways 3 sets of cups at random to be checked
= 14C3
 =364


(b)


Number of ways (Blue cup and red cup are next to each other)
= 5! × 2!
= 240

Number of different ways the cups can be arranged such that the blue cup is not displaced next to the red cup
= 6! – 240
= 720 – 240
= 480


Short Question 19


Question 19 (4 marks):
It is given that cos α = t where t is a constant and 0o ≤ α ≤ 90o.
Express in terms of t
(a) sin (180o + α),
(b) sec 2α.

Solution:
(a)




sin(180o+α)=sin180cosα+cos180sinα=0sinα=sinα=1t2

(b)
sec2α=1cos2α =12cos2α1 =12t21


Short Questions (Question 5 & 6)


Question 5 (4 marks):
Diagram shows a circle with centre O.

Diagram

PR
and QR are tangents to the circle at points P and Q respectively. It is given that the length of minor arc PQ is 4 cm and OR=5α cm.  
Express in terms of α,
(a) the radius, r, of the circle,
(b) the area, A, of the shaded region.

Solution:
(a)
Given sPQ=4   rα=4  r=4α cm

(b)

PR=(5α)2(4α)2PR=9α2PR=3αA= Area of shaded regionA= Area of quadrilateral OPRQArea of sector OPQ=2(Area of  OPR)12r2θ=2[12×3α×4α][12×(4α)2×α]=12α28α=128αα2 cm2


Question 6 (3 marks):
Diagram shows two sectors AOD and BOC of two concentric circles with centre O.

Diagram

The angle subtended at the centre O by the major arc AD is 7α radians and the perimeter of the whole diagram is 50 cm.
Given OB = r cm, OA = 2OB and ∠BOC = 2α, express r in terms of α.

Solution:

Length of major arc AOD=2r×7α=14rαLength of minor arc BOC=r×2α=2rαPerimeter of the whole diagram=50 cm14rα+2rα+r+r=5016rα+2r=508rα+r=25r(8α+1)=25r=258α+1

SPM Practice Question 15 & 16


Question 15 (2 marks):
It is given that the nth term of a geometric progression is Tn=3rn12, rk.  
State
(a) the value of k,
(b) the first term of progression.

Solution:
(a)
k = 0, k = 1 or k = -1 (Any one of these answer).

(b)
Tn=32rn1T1=32r11  =32r0  =32(1)  =32



Question 16 (4 marks):
It is given that p, 2 and q are the first three terms of a geometric progression.
Express in terms of q
(a) the first term and the common ratio of the progression.
(b) the sum to infinity of the progression.

Solution:
(a)
T1=p, T2=2, T3=qT2T1=T3T22p=q2p=4qFirst term, T1=p=4qCommon ratio=q2

(b)
a=4q, r=q2S=a1r=4q1q2=4q÷[1q2]=4q÷[2q2]=4q×22q=82qq2