S3 Algebra

Question 1

a² - b² =

Answer 1

(a+b)(a-b)

Question 2

a² + 2ab + b² =

Answer 2

(a+b)²

Question 3

a² - 2ab + b² =

Answer 3

(a-b)²

Question 4

Under what conditions does x² + px + q = (x+A)(x+B)?

Answer 4

1. AB = q
2. A + B = p (Ax + Bx on RHS = px on LHS)

Question 5

a³ + b³ =

Answer 5

(a+b)(a²-ab+b²)

Question 6

a³ - b³ =

Answer 6

(a-b)(a²+ab+b²)

Question 7

If a ≠ 0, a^(-n) =

Answer 7

1/aⁿ

Question 8

If a ≠ 0, a⁰ =

Answer 8

1

Question 9

If a^m = aⁿ, then

Answer 9

m=n

Question 10

For numbers in scientific notation (a x 10^b), what is the range of a?

Answer 10

1<=a<10
a can be a decimal but b must be an integer.

Question 11

For positive a and positive integral n, arrange the following in ascending order:
1. a x 10ⁿ
2. (-a) x 10^(-n)
3. a x 10^(-n)
4. (-a) x 10ⁿ
5. 5a x 10 ^(n-1)

Answer 11

4<2<3<5<1

2: -a/10ⁿ
3: a/10ⁿ
5: 5a x 10^(n-1) < a x 10ⁿ

Question 12

For positive a, b>1 and positive integral m>n, arrange the following in ascending order:
1. (a x 10^m) x (b x 10ⁿ)
2. (a x 10ⁿ) / (b x 10^m)
3. -(a x 10ⁿ) / (b x 10^m)
4. -(a x 10^m) / (b x 10ⁿ)

Answer 12

4<3<2<1

1: a x b x 10^(m+n)
2: a/(b x 10^(m-n))
3: -a/(b x 10^(m-n))
4: -(a x 10^(m-n)) /b

Question 13

What is the place value and value of digit of “3” in the number 1234?

Answer 13

place value: 10¹
value of digit: 30

Question 14

Express 4+8+32 in binary system.

Answer 14

101100₂

Question 15

Express 100.01₂ in expanded form.

Answer 15

1 x 2² + 0 x 2¹ + 0 x 2⁰ + 0 x 2^(-1) + 1 x 2^(-2)

Question 16

How many digits are there in the binary representation of a number between 3000₁₀ and 4000₁₀?

Answer 16

12

2¹¹ = 2048 and 2¹² = 4096. Since any denary between 3000 and 4000 is between these two numbers, the largest term in its expanded form should be 1 x 2¹¹. Since the number of digits equals to the highest power of 2 plus 1, there are 12 digits in total.

Question 17

Convert 1101 0111 0001₂ to hexadecimal (base 16).

Answer 17

D71₁₆
Since 2⁴ = 16, each digit of the hexadecimal number corresponds to 4 digits in the binary number.
Last digit: a x 16⁰, depends on last 4 digits of binary number = 1₁₀ = 1₁₆
Second digit: depends on middle 4 digits of binary number = 7₁₀ = 7₁₆
First digit: depends on first 4 digits of binary number = 14₁₀, which is D₁₆.

Question 18

Convert 23 30 11 02₄ to hexadecimal (base 16).

Answer 18

BC52₁₆
Since 4² = 16, each digit in the hexadecimal number corresponds to 2 digits in the numbner in base 4.
Last digit: depends on last 2 digits of base 4 = 2₁₀ = 2₁₆
Second-last digit = 1 + 1 x 4 = 5₁₀ = 5₁₆
Second digit = 3 x 4 + 0 = 12₁₀ = C₁₆
First digit = 2 x 4 + 3 = 11₁₀ = B₁₆

Question 19

What is the meaning of principal, interest, and amount?

Answer 19

Principal: the money we deposit in a bank
Interest: the extra money paid by the bank
Amount: the sum of principal and interest
A = P + I

Question 20

What is the interest equation for simple interest?

Answer 20

I = P x R% x T
Interest = principal x annual interest rate (R%) x time (in years)

Question 21

What is the amount equation for simple interest?

Answer 21

A = P + I = P(1 + R% x T)

Question 22

What is the amount equation for compound interest compounded yearly?

Answer 22

A = P(1 + R%)ⁿ, where n equals the number of years.

Question 23

What is the amount equation for compound interest compounded every k weeks, with annual interest rate of R%? (assume that there are 52 weeks in a year)

Answer 23

A = P(1+(k x R%/52))^52/k

Question 24

What are the definitions of growth factor and decay factor?

Answer 24

If a value increases at a constant rate of R% per period, (1+R%) is defined as the growth factor.
If a value decreases at a constant rate of R% per period, (1-R%) is defined as the decay factor.

Question 25

If a value P decreases at a constant rate of R% per period to an amount A, what are the depreciation and depreciation rate respectively?

Answer 25

Depreciation: original value - new value = P-A
Depreciation rate: depreciation/original value x 100% = R%

Question 26

What is the definition of Net Chargeable Income in taxation?

Answer 26

NCI = annual income - allowance

Question 27

Additive property of inequalities

Answer 27

if x>y, then x + c>y + c
(c can be a negative number)

Question 28

Multiplicative property of inequalities

Answer 28

if x>y and c>0, then xc>yc and x/c>y/c
if x>y and c<0, then xc<yc and x/c<y/c

Question 29

Transitive property of inequalities

Answer 29

if x>y and y>z, then x>z

Question 30

Reciprocal property of inequalities

Answer 30

if x>y>0, then 1/x<1/y (sign change)
if x<y<0, then 1/x>1/y (sign change)
if x<0<y, then 1/x<1/y (no sign change)

Question 31

Non-negative property of inequalities

Answer 31

If x>y>0, then xⁿ>yⁿ
If x<y<0, then x⁽²ⁿ⁾ > y⁽²ⁿ⁾, but x^(2n-1) < y^(2n-1)
If x<0<y, it depends on the absolute value of x and y.