Significant figure rules

* Non-zero digits are always significant (1,2,3,4,5,6,7,8,9)

* Any zeros between two significant digits are significant

* A final zero or trailing zeros in the decimal portion ONLY are significant e.g .500 or .63200 the zeros are significant

.006 or .000968 the zeros are NOT significant

Applying the above rules to 9453 x ^-6

Change it to fraction

Since the exponent is negative, you have to move to the left 6 times and place the decimal point there with leading zeros before the digits 9453

9453 x ^-6 = 0.009453

The significant figures are 9,4,5 and 3. Since it is 3 significant figures, it ends on 5

Thus 0.00945 is the answer

NB: The leading zeros before 9453 are not significant as explained by the above rule. Hence they were not counted as part of the significant digits.

Change the fraction and decimal into percentage so that they will all be in percentages.

Percentage is simply multiplying by 100.

0.45 ⇒ 0.45 x 100 = 45%

¾ ⇒ ¾ x 100 = 75%

25% is already in percentage

Now arrange in ascending order

25%, 45%, 75%

45% = 0.45

¾ = 75%

Hence the ascending order is 25%,0.45,¾

NB: Ascending is from lowest to highest

Applying log_a^bc = c x log_a^b = clog_a^b

log₁₀^2y = y x log₁₀²

But log₁₀^2y = 1.8060

⇒ y x log₁₀² = 1.8060

But log₁₀² = 0.3010

Hence y x 0.3010 = 1.8060

Dividing both sides by 0.3010

⇒ y = 1.8060/0.3010 = 6

3 can go into 48, hence express 48 as a multiplication of 3 and 16

√48 = √(16x3)

Applying √(axb) = √a x √b

⇒ √(16x3) = √16 x √3

√16 = 4

Hence √16 x √3 = 4 x √3 = 4√3

NB: 4√3 - √3 is similar to 4x - 3x, thus you can simplify like terms.

Express 72 too as a multiplication of 24 and 3 so we can cancel the √3 at the numerator and denominator

√72 = √(24 x 3) = √24 x √3

24 = 4 x 6

√24 = √(4 x 6) = √4 x √6

√4 = 2

√24 = 2 x √6 = 2√6

⇒ √72 = √(24 x 3) = √24 x √3 = 2√6 x √3

⇒ √72 /3√3 = 2√6 x √3 / 3√3

√3 cancels each other

Hence 2√6/3

Prime number: A number that is divisible only by itself and 1

NB: 1 is not a prime number hence the option is not A

Integer: A whole number (not a fraction) that can be positive, negative or zero. e.g ...,-5,-4,-3,-2,-1,0,1,2,3...

P is a set of positive (No negative) integers and less than 7, hence the correct answer is B

Rational Number: A number such as 5/8 that can be expressed as quotient or fraction p/q of two integers.

Real Number: Any positive or negative number. This includes all integers and all rational and irrational numbers.

y ∝ 1 / x

NB: Inversely = 1 / the relation

Varies directly = Relation 1 ∝ Relation 2

Find the constant of the relation

Let k = the constant

y = k x 1/x

When y = 6, x is 3

⇒ 6 = k/3

Multiplying both sides by 3

⇒ k = 6 x 3 = 18

Hence y = 18/x

When x = 9

⇒ y = 18 / 9 = 2

Geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, none-one number called the common ratio

Rules

For the sequence a₁, a₂, a₃, a₄ ...

Where a₁ = first term

Where a₂ = second term

Where a₃ = third term

Where a₄ = forth term

r = a₂ / a₁ = a₃ / a₂ = a₄ / a₃

Where r = common ratio

Thus the common ratio (r) is the same for all calculations

Hence 4/x = 16/4

4/x = 4

Multiplying both sides by x

x x 4/ x = 4 x x

4x = 4

Dividing both sides by 4

x = 1

16/4 = y/16

4 = y/16

Multiplying both sides by 16

16 x 4 = y/16 x 16

y = 64

⇒ x:y = 1:64

Move the z to the left side, + becomes - and - becomes +

E - z = kx²/2y

Multiply both sides by 2y to get rid of the 2y at the denominator

2y x (E-z) = kx²

2y(E-z) = kx²

Divide both sides by k

2y(E-z)/k = x²

To get rid of a square on an alphabet, take the square root of both sides

Thus

Change the mixed fractions to improper fractions

Rule

Rules

Thus, reciprocate/switch the numerator and denominator at the right and change the division to multiplication

Thus, the numerators multiply each other and the denominators multiply each other

⇒

Multiply both sides by 96

96 x 9/16 = x

16 goes into 96, 6 times

6 x 9 = x

x = 54

Change the mixed fraction to improper fraction

Rule

If α and β are the two roots of a quadratic equation, then the formula to construct the quadratic equation is:

x² - (α + β)x + αβ = 0

Thus, x² - (sum of roots)x + product of roots = 0

Since

and

are the roots

⇒ x² - (-1/2 + 3/2)x + (-1/2 x 3/2) = 0

-1/2 + 3/2 = (-1+3)/2 = 2/2 = 1

Rule

-1/2 x 3/2 = (-1x3)/(2x2) = -3/4

⇒ x² - (1)x + (-3/4) = 0

x² - x - 3/4 = 0

NB: + - = - ⇒ + - 3/4 = -3/4

Multiply both sides by 4

4x - 4x - 3 = 0

Trace any point on the graph where the line passes and use that x and y coordinates to substitute and find y

The line passes through the points (0,1), hence x = 0 and y = 1

y = d - x ⇒ 1 = d - 0 ⇒ 1 = d ⇒ d = 1

NB: The line also passes through the points (1,0), (4, -3 ). Any point on the line you use should arrive at the same answer

Since 3 can go into 6 and 9, split -6pqr into -3pqr and -3pqr

⇒ p²q² - 6pqr + 9r² = p²q² -3pqr -3pqr + 9r²

Now factorize

p²q² -3pqr -3pqr + 9r²

pq(pq - 3r) -3r(pq-3r)

(pq-3r)(qp-3r)

since pq-3r = pq-3r, the multiplication makes it square

(pq-3r)²

The equation above is similar to the general factorization rule of (a-b)²

(a-b)² = a² - 2ab + b²

Thus the first term square then the sign (either + or -) and 2 times the first and second terms + the second term square

Similarly (a+b)² = a² +2ab + b²

The first term for the above equation is pq and the second term is 3r and the sign is -

You can apply the (a-b)² = a²-2ab+b² to expand the (pq-3r)², you will realize you will get the expression you were asked to factorize

Let h = height of the water

Volume of a cuboid = Length x Breadth x Height

Since the length and breadth are in cm, convert the litres to cm³

1 litre = 1000 cm³

⇒ 180 litres = 180 x 1000 cm³

Length = 250 cm, breadth = 120 cm

Volume = 180 litres = 180 x 1000 cm³

⇒ 250 cm x 120 cm x h cm = 180 x 1000 cm³

Divide both sides by 250 cm x 120 cm

h = (180 x 1000) / (250 x 120) = 6

Diagram Illustration

Area = 49 cm²

a = 6cm, b = 7cm

⇒ 49 cm² = 1/2 x (6cm + 7 cm) x h

49cm² = 1/2 x (13cm) x h

Multiply both sides by 2

2 x 49cm² = 1 x (13cm) x h

Divide both sides by 13cm

98 cm² / 13 cm = 1 x h

⇒ h = 98/13 = 7.53846 = 7.5 (One decimal place)

Where θ in radians

Area of the sector = 231 cm²

NB: respectively means in the order they were mentioned

⇒ 231 cm² = θ/2 x r²

Multiply both sides by 2

2 x 231 cm² = θ x r²

462 cm² = θ x r²

Length of its arc = 66cm

⇒ 66cm = θ x r

θ x r² = θ x r x r

But θ x r = 66cm

∴ θ x r² = 66cm x r

But 462 cm² = θ x r²

∴ 462 cm² = 66cm x r

Divide both sides by 66cm

462 cm² / 66 cm = r

∴ r = 7

Represent them by diagrams

First Ladder

Second Ladder

The angle θ can be calculated using any of SOH, CAH , TOA

SOH = sin θ = Opposite (O) / Hypotenuse (H)

CAH = cos θ = Adjacent (A) / Hypotenuse (H)

TOA = tan θ = Opposite (O) / Adjacent (A)

You have to know which of the sides are known (Adjacent, Opposite or Hypotenuse)

The hypotenuse is the side facing the right angle triangle (90^o), the ladder

The adjacent is the side adjacent to the angle θ, ground

The opposite is the side opposite (facing) the angle θ, the building

From both diagrams, the Opposite and Hypotenuse sides are known

Hence SOH can be used to calculate the angle

SOH = sin θ = Opposite / Hypotenuse

For the first ladder

Opposite = 8 m and Hypotenuse = 10 m

Hence sin θ = 8 m / 10 m = 8 / 10

For the second ladder

Opposite = h, Hypotenuse = 12 m

Hence sin θ = h m/ 12 m

Since the angle (θ) is the same as indicated in the question, it means the value of sin θ is the same for both the first and second ladders

Hence 8/10 = h/12

Multiply both sides by 12

12 x 8/10 = h

∴ h = 9.6