TheMathBooklets

Tuesday, 21 April 2020

S-L1 Algebra Practice

ALGEBRA

Recap

Simplify

1) a + a 1a) 2a + 3a

2) 2a + a 2a) 2a - 5a

3) 2a - a 3a) -a + 4a

4) a - a 4a) 2a - 2a

5) a - 0 5a) 4a - 4a

6) 0 - a 6a) 0 - 7a

7) 2a + 3a - a 7a) 6a +a - a

8) 5a - 2a + a 8a) -9a + 6a

9) 2a + a - a - 2a 9a) 2a + a + 3a + 2a

10) 2a + 2a + 3b 10a) a + b + 2a + 2b

11) 2a - a - b + 2b 11a) -a + b -2a - 4b

12) 3a + 2a - b 12a) a - a + b - b

13) 1 x a 13a) 2 x a

14) 0 x a 14a) 2a x b x 0

15) 1 x -a 15a) -2a x 0

16) a x a 16a) a x -a

17) 2a x a 17a) -2a x -a

18) a x 3a 18a) -a x 3a

Expand the followings

1) a(a + 1)

2) 2a(1 - a)

3) a² (b + 2)

4) (a + 1) (a + 2)

5) (a + 1) (a - 1)

Simplify

1) ab + 2a

2) 2a + 4b

3) 5ab - 5

4) 4ab + 2a

5) 8ab - b

(1) Notations, Addition/Subtraction of Linear Expression

(A) Write the algebraic expression for the following:

1. Add 2x and y and 5
2. Subtract 2w from 34
3. Product of 21 and g
4. Divide 6 by n
5. Sum of 7, 2p and q

6. Sally had x books. Her friend gives her 2 books. How many books does she have?

7. A box has x cookies. How many cookies are there in 12 such boxes?

(B) Simplify:

1. 4y + 2x -y + x

2. 5x - 3y - y - 2x

3. 7x + y - 3y - 4y + x

4. 2x + y - x - y - x

1. 3ab + a
2. 4ab + 2b
3. ab + 2ab

4. 3a²b + 6b

5. 2a + 3b

5. 5
6. 3ab - ab

8 4
7. a + a

2 2

8. a + a

(D). Simplification of Linear Expression

1. Express each of the following as two separate terms.

a. 2x – 7 b. 15x – 8

4 10

2. Simplify the following

a. x+ x b. -3x + 2x c. 3(y -1 ) + 5(2y – 3)

2 5 4 6 8 12

3. Lily is 5x years old. Annie is 2(x – 1)/5 years old. Find the sum and difference of Lily and Annie.

(2) Expansion of 2 linear Expressions/Extract Common Factors

1. Expand the followings:

a. 4 ( x + y)

b. (2x + y)

c. 3(2x - 3y)

2. Simplify the following:

a. -8 + 4y – 4y² -2 + y²

b. (-3u² + 5u – 8) – (4u²– 6u + 5)

c. -6(-3y² + 3y + 1) – 4(2y2 – 2y – 9)

3. Subtract the sum of -3(n² – 2n +5) and 2(-6n2 + n + 2) from 4(-3n² -3n – 1)

4. Car X travels at a speed of (2x² + 5x – 7) km/h for 4 hours. Car Y travels at a speed of (x² – 2x + 9) km/h for 3 hours. Which car has covered more distance? By how many km more?

5. Expand a. (5m + n)(7m + 3)

b. (2m – 5n)(2m + 5n)

c. –c(-6c – d)(-4y + 3z)

d. (2p + 5q – 9)(4p – 5q)

6. Expand and simplify

a. (2p – q)(2p + 3q) – (p + q)(p – 5q)

b. (2p + q)(3x – 2y) + (p – 2q)(3x + 4y)

7. Expand each of the following.

a. (12p + 1/3q)2

b.(5a -3b)2

c. (11p – 8q)(11p + 8q)

d. (3u – 4v)² – (2u + 9v)(2u +9v)

8. Expand (x + 1/x)². If x² + 1/x² = 7, and x > 0, find the value of

a. x + 1/x

b. (x + 1/x)²

(3) Factorisation

1. Factorise each of the following expressions

a. a² + 15a + 44

b. 21 – 4b – b2

c. 9q2 + 30q + 25

d. 4 – 2d – 12d2

e. (3x – 4)(2x – 5) + 8(2x – 5)

f. (3x – 4)2 – (2x + a)2

g. y + 24yz – 81yz2

2. Factorise

a. p² + 8pq + 16q2

b. s²t² – 16st + 64

c. 45x² – 320y²

3. Find the value of

a. 143²- 142²

b. √200.5 ² – 199.5 ²

c. 13.5 ² – 6.5 ²

4. Factorise

a. 100 – y² + 6yz – 9z²

b. 3x² – 48x

5. Factorise

a. (a + 3b)x + (a + 3b)y

b. m(3p – q) – 2n(3p – q)

c. 10px + 15qz + 8py + 12qy

d. 36ax – 63ay – 4bx + 7by

e. 54p2 – 6p – s + 9ps

f. 21 mx – 7kx – 2ky + 6my

Expansion and Factorisation of algebraic expression

1. Simplify the following:

a. -8 + 4y – 4y² -2 + y²

b. (-3u² + 5u – 8) – (4u²– 6u + 5)

c. -6(-3y² + 3y + 1) – 4(2y2 – 2y – 9)

2. Subtract the sum of -3(n² – 2n +5) and 2(-6n2 + n + 2) from 4(-3n² -3n – 1)

3. Car X travels at a speed of (2x² + 5x – 7) km/h for 4 hours. Car Y travels at a speed of (x² – 2x + 9) km/h for 3 hours. Which car has covered more distance? By how many km more?

4. Expand a. (5m + n)(7m + 3)

b. (2m – 5n)(2m + 5n)

c. –c(-6c – d)(-4y + 3z)

d. (2p + 5q – 9)(4p – 5q)

5. Expand and simplify

a. (2p – q)(2p + 3q) – (p + q)(p – 5q)

b. (2p + q)(3x – 2y) + (p – 2q)(3x + 4y)

Exercise 1

1. Factorise bx - 2by + 3ax - 6ay

2. Simplify -4(x - 2) + 8x

3. Write 2 + 4 as a fraction in its simplest form.

(x+1)² x + 1

4. Simplify 3a² x (2a²)⁴

5. Simplify 5a³÷ 25a

2 8

Exercise 2

1. Simplify 3(2x - 5) + 8x

2. Find the value of -5y - 2 = 3

3. Simplify the following, expressing your answers with positive indices

(a) y² x (y²)⁴

(b) (xy)⁰

4. Evaluate the following without using calculator

(a) 81^5/4

(b) 1000 ^-2/3

5. Solve 4^5x = 512

Answers

Recap

1. a + a = 2a 1a) 2a + 3a = 5a

2. 2a + a = 3a 2a) 2a - 5a = -3a

3. 2a - a = a 3a) -a + 4a = 3a

4. a - a = 0 4a) 2a - 2a = 0

5. a - 0 = a 5a) 4a - 4a = 0

6. 0 - a = -a 6a) 0 - 7a = -7a

7. 2a + 3a - a = 5a - a = 4a 7a) 6a + a - a = 6a

8. 2a + 3a - a = 3a + a = 4a 8a) -9a + 6a = -3a

9. 2a - a -a - 2a = 3a - a - 2a = 0 9a) 8a

10. 2a + 2a + 3b = 4a + 3b 10a) 3a + 3b

11. 2a - a -b + 2b = a + b 11a) -3a - 3b

12. 3a + 2a - b = 5a - b 12a) 0

13. a x a = a 13a) 2a

14. 0 x a = 0 14a) 0

15. 1 x -a = -a 15a) 0

16. a x a = a² 16a) -aa²

17. 2a x a = 2 a² 17a) 2a²

18. a x 3a = 3a² 18a) -3a²

Expand the followings

1) a(a + 1)

= a x a + a

= a² + a

2) 2a(1 - a)

= 2a x 1 - 2a x a

= 2a - 2a²

3) a² (b + 2)

=a² x b + a² x 2

= a²b + 2a²

4) (a + 1) (a + 2) = a x a + a + 2a + 2

= a² + 3a + 2

5) (a + 1) (a - 1) = a x a + a - a - 1x1

= a² - 1

Simplify

1) ab + 2a = a(b + 2)

2) 2a + 4b = 2(a + 2b)

3) 5ab - 5 = 5(ab - 1)

4) 4ab + 2a = 2a(2b + 1)

5) 8ab - b = b(8a - 1)

C4. 3a²b + 6b Step 1 : Take out common factors 3b

3b(a² + 2) Step 2 : bracket

5. 2a + 3b Step 1 : Take out common factors 1/5

5 5

1 (2a + 3b) Step 2 : bracket

S3TN Completing the Square, Graphical Method - Quadratic Equations

Solve by Completing the square

Completing the square "change" the equation from

ax² + bx + c = (x + d)² + e =0 where d and e are numerical.

ax² + bx + c = 0

Divide the equation by a

=> x² + bx/a + c/a = 0

Applying (x + ky)² = (x +k/2)² - (K/2)²

(x + b/2a)² - (b/2)² + c/a = 0

(x+ b/2a)² = (b/2)² - c/a

Solving the equation for x:

x + b/2a = + √ (b/2)² - c/a

x = -b/2a + √ (b/2)² - c/a

x = -b/2a + √ (b/2)² - c/a

Or x = -b/2a - √ (b/2)² - c/a

Formula : ax²+ bx + c = a(x+ b/a)²– (b/a)²+ c/a = 0

(x+ b/2a)² = (b/2a)² - c/a

=> x = -b/2a + √ (b/2)2 - c/a Or

x = -b/2a - √ (b/2)2 - c/a

Example

Solve 2x²+ 4x + 1 = 0

Using completing the square ax²+ bx + c = a(x+ b/a)²– (b/a)²+ c

2x² + 6x + 1 = 2(x² + 4x + 1/2) = 0

x²+ 4x + 1/2 = 0

(x + 4/2)²– (4/2)²+ 1/2 = 0

(x + 2)²= (2)²- 1/2

(x + 2)²= 4 – 1/2

(x + 2)²= 7/2

x + 2 = +√ 7/2

x = -2+√ 7/2

x = -2 + √ 7/2 or x = -2 - √ 7/2

Graphical Method

Using the above equation x² + 4x + 1 = 0 to plot a graph,

(x + 2)²= 7/2

=> The minimum point is (-2, - 7/2)

To solve 2x²+ 4x + 1 = 0

=> y = 0

When y = 0

x = -2 + √ 7/2 or x = -2 - √ 7/2

Secondary Maths : Practices

TOPICAL

Number And Algebra

Numbers and Operations

Ratio & Proportion, Percentage , Rate & Speed

Algebra Expressions and Formulae

Functions and Graph

Equations and Inequalities

Geometry and Measurement

1. Angles, Triangles and Polygons

2. Congruence and Similarity

3. Properties of Circles

4. Pythagoras Theorem and Trigonometry

5. Mensurations

6. Coordinate Geometry

Statistics and Probability

1. Data Analysis

2. Probability

SHORT/DAILY

Daily Practice (Sec 2 - 4)

S-L1 Numbers and Operations Practice

NUMBERS

LEVEL 1

1. Fill in the blanks, stating what the value of each digit is

a. 91296 = _____ + 8000 + 200 + ___ + ___

b. 1 604 301 = 1 600 000 + _____ + 4000 + 300 + ___

2. State the place value of the digit 5 in

a. 1 572 423

b. 0.521

3. Write the value of the digit 2 in the following

a. 285

b. 95200

c. 1520

d. 5402

4. Arrange the following numbers in descending order

-1.2 , 150% , -1 1/9 , 1.5

5. Arrange the following numbers in descending order.

a. 5.3, 5.25, 5.205

b. 2.91, 2.903, 2.95

6. Write down the following in numerals

a. 5.24 million

b. 6.022 billion

7. Round off each of the following numbers to 1 decimal place, 2 decimal places and 3 decimal places.

(a) 36.7247 (b) 2.8734 (c) 100.283

(d) 1.0049 (e) 0.9999 (f) 9.959

(f) 0.2139 (g) 5.3027

8. Round off each of the following numbers to 2 significant figures, 3 significant figures and 4 significant figures.

(a) 483.86 (b) 248999 (c) 15.795

(d) 0.59 974 (e) 18932 (f) 37.649

9. Round (1/13/1/3/T)

a. 25.6389 correct to 2 decimal places

b. 506423 correct to 3 significant figures

c. 0.00678472 correct to 4 significant figures

10. Correct 975.38 to

a. 1 decimal place

b. 1 significant figure

c. to the nearest ten

11. Write down the next three items of the following sequences

a. 13, 16, 19, . . .

b. 6, 17, 28, 39

12. Express 1 2/5 as

a. a decimal

b. a percentage

13. Express 1/25 as a decimal. (1/13/p1/1b)

14. Convert 100 kilometres per hour into metres per second. (2/13/1/2/T)

15. Fill in the boxes with < , > or =

16. Use <, > or = to complete each of these statements. (1/14/2/4/T)

a. 1/3 ___ 0.3

b. 12 ½% ______ 1/8

c. 7/12 _______ 5/9

d. -2 [ ] -3

17. Complete the number line.

<——|——|——|——|——|——|——|——|——|

-6 ___ ___ 3. 6. ___ 12 15 ____

18. List all the prime numbers between 20 and 35

19. Find the HCF and LCM 36 and 80.

20. Write all the factors of number 118.

21. Find the prime factorization of 214.

22. Write the prime factorization of 18 x 21.

23. Find the LCM of the following numbers 42 and 124

LEVEL 2

1a. Express 180 as the product of its prime factors (1/10/p1/3/A)

1b. Write down the smallest positive integers, k, such that 180k is a perfect square.

2. Write all the factors, prime factorization and index notation of 64.

3. Find the fraction exactly halfway between ¾ and 3/5. Give your answer in its simplest form. (2/11/p1/8/A)

4a. Write 156 as a product of its prime factors. [1]

4b. Find the highest common factor of 156 and 390 [2]

FOUR OPERATIONS

Level 1

1. Find the value of the following.

a. 3 + 6 ÷ (2.5 + 2.5)

b. 12 - 8 ÷ (3 + 1)

2. By rounding each number to the nearest whole number, estimate the value of

a. 38.4 - 2.0 x (8.5 + 2.2). Show your working.

b. 30.1 - 6.0 x (0.5 + 2.4) - 4.8

3. By rounding each number to 1 significant figure, estimate the value of

a. 12 x 27.25

19 + 11

b. 21.2 x 2.3

33 - 11

4. A pen costs $2.75. Round the cost of the pen to the nearest dollars and estimate the cost of 8 pens.

5. Amy has $4.80 in her pencil case and $22.50 in her wallet.She bought 2 pens that cost $1.80 and 1 pencil that cost $0.85

a. How much money has she left?

b. Round the amount she left to the nearest dollar, and find the amount of money she needed to have $50.

Wednesday, 1 April 2020

S3T3 Solving Quadratic Equations by Factorisation/Formula

Recap

To solve for x, we ‘put’ x on the LHS (left-hand-side) and the number value to the RHS (Right-hand-side),

and we find the value of x.

Example

Solve for 7x + 3 = 17

7x = 17 – 3

= 14

x = 14/7 = 2

Solving Quadratic Equation

(1) What is the meaning of Solve x² + 3x + 2 = 0

=> to find the values (two or less) of x where it will make the equation = 0.

(2) Any number multiply by 0 is equal to 0.

=> For any (x + a)(x – b) = 0, as long as any one of (x + a) or (x – b) = 0, equation is solved because

When x = -a

(-a + a)( x- b)

= 0 x (x – b) = 0

When x = b

(x+ a)( b - b) = 0

= (x + a) x 0 = 0

= 0

(3) Solving the quadratic equation

(x + P)(x – Q) = 0

When (x + P) = 0

x + P = 0

x = -P

When (x – Q) = 0

x – Q = 0

x = Q

Values of x are

x = -P or x = Q

Quadratic equations are usually solved by putting the equation into the form

(Jx + P)(Kx + Q) = 0 where J, K, P and Q are numerical values.

A quadratic equation can have 0, 1 or 2 values.

Recap : Factorisation (x ² + bx + c)

Factorise x ² + 5x + 6

=> To find P and Q where x ² + bx + c = x ² + 5x + 6 = (x + P)(x + Q)

=> b = 5 = P + Q, c = 6 = PQ

Step 1 : Draw the cross X

<< Since is it x² , fill in x² >>

x [What is P?]

\ /

/ \

x [What is Q?]

Step 2 : Find factors of c (P X Q) for b (P + Q)

When there is more than 1 set of factors,

Find the factor of c = 6 (P x Q)

6 = 1 x 6,

= 2 x 3

Using the X to cross-multiply,

x 2 x 1

\ / \ /

/ \ / \

x 3 x 6

x x 3 = 3 x, 2 x x = 2 x x x 1 = 3 x, x x 6= 2 x

2 x + 3 x = 5 x x + 6 x = 7 x

P = 2 , Q = 3

Step 3 : Factorise

=> ( x +2 )

\ /

/ \

( x +3)

Thus,

x ² + 5x + 6 = (x + 2) (x + 3)

Solving Quadratic Equation by Factorisation

Example:

Solve y² + 3y + 2 = 0

(y + 2)(y + 1) = 0 By Factorisation y \ / 2

(y + 2) = 0 or (y+ 1) = 0 y / \ 1

y = -2 or y = -1

Solving by Formula

For Equation: ax2 + bx + c = 0

formula x = -b + √b2 – 4ac

Example:

Solve 2x²+ 6x + 1 = 0 (ax2 + bx + c = 0)

a = 2 , b = 6, c = 1

x = -6 + √6²– 4 x 2 x 1

2x2

= -6 + √28

= -6 + 2/4√7

= -6 + 7/2