Maths Assignment: Quadratic Equations, Inequalities, and Index Numbers

This document presents a comprehensive solution to a maths assignment, encompassing a wide range of topics. The assignment begins with basic arithmetic problems involving fractions and percentage change. It then progresses to more complex algebraic concepts, including solving linear and quadratic equations using various methods, such as the quadratic formula and factorization. The solution also includes problems related to index numbers and chain base relatives. Furthermore, the assignment explores the graphical representation of quadratic equations, requiring the sketching of graphs and identification of key features. The assignment concludes with linear programming problems involving constraints, gradients, and optimization, including finding the feasible region defined by a set of inequalities.

Maths Sobs

Table of Contents
SOB211 (Threshold)........................................................................................................................1
Evaluate the following, giving your answer as a fraction in lowest terms.............................1
8/15+8/15................................................................................................................................1
SOB212 (Threshold)........................................................................................................................1
Evaluate the following, giving your answer as a fraction in lowest terms.............................1
Solution: cancel denominator with numerator that has same value 10..................................1
SOB221 (Typical)............................................................................................................................1
In 2015, 693 serious crimes were recorded in Mathsborough. In 2016 the number was 529.
What was the percentage change? Give your answer to two decimal places.........................1
The price of a washing machine in December was 642 pounds. During the January sales the
price fell by 29% before increasing by 45% in February. What was the overall percentage
change from December to February?.....................................................................................1
SOB 414 -Threshold........................................................................................................................3
Calculate the index number that represents change between the old value (Vo) and new value
(Vn).........................................................................................................................................3
Vo - 56 Vn - 54........................................................................................................................3
SOB 426 -Typical............................................................................................................................4
Calculate set of fixed base indices for income using the January figure as the base.............4
SOB 427 -Typical............................................................................................................................4
Calculate the chain base relative for the data given below....................................................4
SOB 514 -Threshold........................................................................................................................5
Find the equation of the straight line passing through the following two points:..................5
(4,-22) and (-5,-4)...................................................................................................................5
SOB 711 -Threshold........................................................................................................................5

Solve the quadratic equation x2 + 8x +2=0 using the quadratic formula. Give your answers to
two decimal place...................................................................................................................5
SOB231 (Excellent).........................................................................................................................6
Sketch the graph of above quadratic. Your sketch should indicate the coordinates of the
maximum or minimum points and any points with the curve crosses the axis......................6
SOB 421 -Typical............................................................................................................................6
How many roots does the quadratic equation 3x^2+ 13x+12=0 have?..................................6
SOB 422 -Typical............................................................................................................................6
Factorise the above equation and hence solve the equation...................................................6
SOB 712 -Threshold........................................................................................................................7
Solve the following system of simultaneous equations:.........................................................7
SOB732 (Excellent).........................................................................................................................7
Solve the region in the plane described by the following set of inequalities:........................7

SOB211 (Threshold)
Evaluate the following, giving your answer as a fraction in lowest terms
8/15+8/15
Solution: LCM (15, 15) = 15
Therefore, 8+8 = 16
15 15
SOB212 (Threshold)
Evaluate the following, giving your answer as a fraction in lowest terms
9 × 10
10 11
Solution: cancel denominator with numerator that has same value 10
Then, = 9
11
SOB221 (Typical)
In 2015, 693 serious crimes were recorded in Mathsborough. In 2016 the number
was 529. What was the percentage change? Give your answer to two decimal
places
Solution: percentage change= original value (Vo) – new value (Vn)× 100
Original value
= 693-529×100
693
= 23.66%
SOB731 (Excellent)
The price of a washing machine in December was 642 pounds. During the January sales the price
fell by 29% before increasing by 45% in February. What was the overall percentage
change from December to February?
Solution: percentage change= original value (Vo) – new value (Vn)× 100
Original value
1

For January sales the price fell by 29%
29 = (642 – X) × 100
642
X= 455.82
For February sales the price increase by 45%
45 = (455.82 – Y) × 100
455.82
Y= 660.93
The overall percentage change from December to February
= (642 – 660.93) × 100
642
= 3 % increase.
2

SOB 414 -Threshold
Calculate the index number that represents change between the old value (Vo) and new value
(Vn).
Vo - 56 Vn - 54
Solution: i = (Vn / Vo) × 100
= (56/54) × 100
= 96
3

SOB 426 -Typical
Calculate set of fixed base indices for income using the January figure as the base
Jan Feb Mar Apr May
5275 5358 5440 5512 5604
Solution: price relative to current month (Pon) = Price of current month (Pn)
Price of base month (Po)
Month Income Pon= Pn× 100
Po
Fixed base index
Jan 5275 (5275/5275)×100 100
Feb 5358 (5358/5275)×100 101.57
Mar 5440 (5440/5275)×100 103.12
Apr 5512 (5512/5275)×100 104.49
May 5604 (5604/5275)×100 106.23
SOB 427 -Typical
Calculate the chain base relative for the data given below
Jan Feb Mar Apr May June
2445 3323.5 4571.55 6245.015 8314.52 11029.88
Solution: Price relative to current month (P) = Price of current month (Pn) × 100
Price of previous month (Pn-1)
Month income P= Pn× 100
Pn-1
Chain base
relative
Index number
Jan 2445 100 100×100
100
100
4

Feb 3323.5 135.93 100×135.93
100
135.93
Mar 4271.55 137.55 135.93×137.55
100
186.97
Apr 6245.015 136.60 186.97×136.60
100
255.40
May 8314.52 133.13 255.40×133.13
100
340.01
June 11029.88 132.65 340.01×132.65
100
451.02
SOB 514 -Threshold
Find the equation of the straight line passing through the following two points:
(4,-22) and (-5,-4)
Solution: m= (y2-y1)
(x2-x1)
= -22 – 4
-4 – 5
= -26
Slope= -26
y = mx + b
y = -26x + b
taking point (4, -22)
x= 4, y= -22
-22 = -26* -22 + b
b= 78
equation of straight line is y = -26x+78
SOB 711 -Threshold
Solve the quadratic equation x2 + 8x +2=0 using the quadratic formula. Give your answers to two
decimal place
Solution: x2 + 8x +2=0
ax2+bx+c=0
Quadratic equation formula x = -b + √b2 – 4ac
2
x= -8 + √(64- 4*1*2)
5

2*1
= -8 + √(64- 8)
2
= -8 + 7.48
2
x1= (-8+ 7.48) /2 = -0.26
x2= (-8- 7.48)/2 = - 7.74
SOB231 (Excellent)
Sketch the graph of above quadratic. Your sketch should indicate the coordinates of the
maximum or minimum points and any points with the curve crosses the axis
SOB 421 -Typical
How many roots does the quadratic equation 3x^2+ 13x+12=0 have?
Solution:
Quadratic equation formula x = -b + √b2 – 4ac
2
x= -13 + ( √(13)2 – 4*3*12) / (2*3)
x= 13 + √(25))/ 6
= -1.33
x= 13 - ( √(25))/ 6
= -3
SOB 422 -Typical
Factorise the above equation and hence solve the equation
3x2+ 13x+12=0
Solution: 3x2- 4x - 9x+12=0
x (3x-4) - 3(x-4)
(x-3) (3x-4)
Hence, x = 3, 4/3
6

SOB 712 -Threshold
Solve the following system of simultaneous equations:
4x-4y=36
-2x-2y=10
Solution: 4x-4y=36 --- equation 1
-2x-2y=10--- equation 2
Multiple equation 2 with 2(-2x-2y =10) = -4x-4y = 20 --- equation 3
Subtracting equation 1 and equation 3
8y = 16
y = 2
Put value of y in equation 1
4x – 4*2 = 36
4x = 44
x =11
SOB732 (Excellent)
Solve the region in the plane described by the following set of inequalities:
y<=4-x,x>=0,y>=1.
7

All of the activities below refer to the following linear program:
Equation x coefficient y coefficient constraint gradient
Profit
Constraint 1
Constraint 2
Constraint 3
135
9
10
12
125
12
10
6
7180
6475
6040
8