QRT

Question 1

Multiplying Fractions

a/b x x/y

Answer 1

Multiply numerators together and denominator together

= ax/by

Question 2

Dividing Fractions

a/b ÷ x/y

Answer 2

Flip the fraction you’re dividing by then multiply

= a/b x y/x = ay/bx

Question 3

Solving a Proportion

x/5 = 3/4

Answer 3

Cross-multiply

4x = 15

4 = 15/4

Question 4

(x^m)(xⁿ)

Answer 4

=x^m+n

Question 5

x^m / xⁿ

Answer 5

=x^m-n

Question 6

(x^m)ⁿ

Answer 6

x^mn

Question 7

(xⁿ)(yⁿ)

Answer 7

xyⁿ

Question 8

xⁿ / yⁿ

Answer 8

(x/y)ⁿ

Question 9

Multiplying Polynomials

(x² + 3x + 4)(x + 5)

Answer 9

Multiply each term in the first polynomial with each term in the second

=x²(x + 5) + 3x(x + 5) + 4(x + 5)

=x³ + 5x² + 3x² + 15x + 4x + 20

x³ + 8x² + 19x + 20

Question 10

Dividing Polynomials

(2x³ + 13x² + 11x -16) ÷ (x + 5)

Answer 10

Use long division:

= 2x2 + 3x - 4

Question 11

Factoring

ax + ay

Answer 11

a(x + y)

Question 12

Factoring

a² - b²

Answer 12

Difference of squares:

(a + b)(a - b)

eg 4x² - 9 = (2x + 3)(2x - 3)

Question 13

Factoring

a² + 2ab + b²

Answer 13

Squares of binomials:

(a + b)²

Question 14

Factoring

a² -2ab + b²

Answer 14

Squares of binomials

(a - b)²

Question 15

For all x ≠ ±3,

(3x² - 11x + 6) / (9 - x²)

Answer 15

1. Factorising: denominator - difference of squares -

(9 - x²) = (3 - x)(3 + x)

2. Factorising: numerator -

(3x² - 11x + 6) = (x -3)(3x - 2)

3. Factorising numerator for -1:

(x -3)(3x - 2) = (-1)(3 - x)(3x - 2)

4. Cancelling (3 - x):

(-3x + 2)/(3 + x)

Question 16

Unknown in a Denominator

1 + 1/x = 2 - 1/x

Answer 16

Cross multiply:

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

x² + x = 2x² - x

2x² - x² = x + x

x² = 2x

Dividing both sides by x:

x = 2

Question 17

Unknown In an Exponent

If 8^x = 16^x-1, x = ?

Answer 17

Since both bases are factors of two, can be reexpressed as powers of two:

(2³)^x = (2⁴)^x-1

3x = 4x - 4

x = 4

Question 18

Quadratic Formula

Answer 18

Question 19

‘In terms of’

If a = (b + x)(c + x), what is the value of x in terms of a, b and c

Answer 19

b + x = ac + ax

ax - x = b - ac

x(a - 1) = b - ac

x = (b - ac)/(a - 1)

Question 20

Simulataneous Equations

2x - 9y = 11 and x + 12y = -8

What is the value of x + y?

Answer 20

Add the equations:

3x + 3y = 3

x + y = 1

Question 21

Absolute Value

|x - 12| = 3

Answer 21

x - 12 = 3

x - 12 = -3

x = 15 or 9

Question 22

Inequalities

|whatever| < p

Answer 22

-p < |whatever| < +p

Question 23

Inequalities

|2x - 3| < 7

Answer 23

= -7 < 2x - 3 < 7

2x - 3 > -7, 2x > -4, x > -2

2x - 3 < 7, 2x < 10, x < 5

Question 24

Sum of Exterior Angles of a Triangle

Answer 24

360º

Question 25

Triangle Inequality Theorum

Answer 25

The length of any one side of a triangle must be greater than the positive difference between the two other sides and less that their sum For example, if one side is 3 and another is 7, then the length of the third must be between 4 \< x \< 10

Question 26

Similar Triangles Area od A in tems of B if length of sides of A are twice that of B

Answer 26

A:B = 2:1 Areas = 2²:1² = 4:1

Question 27

Area of an Equilateral Triangle

Answer 27

= (s²√3) / 4 where s is the side length

Question 28

Special Right Triangles

Answer 28

1. 3-4-5 2. 5-12-13 3. 45º-45º-90º - sides in ratio 1:1:√2 (if sides are 3, 3, x, x = 3√2) 4. 30º-60º-90º - sides in ratio 1:√3:2 (if sides 3, 6, x, x = 3√3)

Question 29

Area of a Trapezoid

Answer 29

= ((base1 + base2) / 2) x height

Question 30

Area of a Parallelogram

Answer 30

= base x height

Question 31

Area of a Hexagon

Answer 31

=(3s²√3)/2

Question 32

Length of an Arc

Answer 32

= (n/360) x 2πr

Question 33

Area of a Sector

Answer 33

= (n/360) x π r²

Question 34

Area of a Cone

Answer 34

= π r² + π rl r = base radius l = slant length

Question 35

Volume of a Cone

Answer 35

= 1/3 π r² h

Question 36

Surface Area of a Sphere

Answer 36

= 4π r²

Question 37

Volume of a Sphere

Answer 37

= 4/3 π r³

Question 38

Volume of a Pyramid

Answer 38

= 1/3 Bh B = area of base h = height

Question 39

Coordinate Geometry Distance Between Two Points

Answer 39

√(x₁-x₂)² + (y₁-y₂)²

Question 40

Slope Intercept Which of the following has no intercept with the line y = 4x + 5 (A) y = 0.25x - 5 (B) y = -0.25x -5 (C) y = 4x + 0.2 (D) y = -4x + 0.2 (E) y = -4x -0.5

Answer 40

Must be parallel - gradient 4x | (C)

Question 41

Cotangent

Answer 41

= 1/tangent =Adjacent/Opposite

Question 42

Secant

Answer 42

=1/cos =Hypotenuse/Adjacent

Question 43

Cosecant

Answer 43

=1/sine =Hypotenuse/Opposite

Question 44

Trigonometry Important Identities

Answer 44

1. sin2 ^x + cos² x = 1 2. tan² x + 1 = sec² x 3. 1 + cot² x = csc² x

Question 45

Radian in Degrees

Answer 45

= 180/

Question 46

Degrees in Radians

Answer 46

One degree = π/180 radians