Moderate Level Worksheet

Part A: Subjective Questions - Very Short Answer (1 Mark Each)

In this moderate level, we'll explore more complex expressions, higher degrees, and advanced operations.

Let's deepen our understanding of algebraic expressions!

1. Write the algebraic expression for: "5 times a number increased by 4".

Let the number be xn

5 times the number = 5xx⁵

Increased by 4 means addsubtract 4

Expression = 5x + 44 + 5x

Perfect! "Times" means multiply, "increased by" means add.

2. Identify the coefficients in 7a + 3b – 5.

Coefficient of a = 73

Coefficient of b = 37

–55 is a constant, not a coefficient

Excellent! Coefficients are numbers with variables.

3. Write the expression for: "x multiplied by y, added to 5".

x multiplied by y = xyx + y

Added to 5 = xy + 55xy

Correct! xy means x × y, then we add 5.

4. Simplify: 3x + 4x – 7.

Combine like terms: 3x + 4x = 7x7

Result = 7x – 77x + 7

Great! Add coefficients of x, keep the constant separate.

5. What is the degree of 2x³ + 5x² + 7?

The highest power of x is 32

Therefore, degree = 35

Perfect! Degree is the highest exponent of the variable.

6. Find the value of 5x – 3 when x = 2.

Substitute x = 25

5(2) – 3 = – 3

= 713

Excellent! Multiply first, then subtract.

7. What are like terms in 6m, 4n, 3m, –5?

Like terms with m: and

has different variable

is a constant

Correct! Like terms must have the same variable.

8. Write an expression for: "Twice a number decreased by 9".

Let number = xn

Twice the number = 2xx²

Decreased by 9 means subtractadd 9

Expression = 2x – 99 – 2x

Perfect! "Decreased by" means subtraction.

9. Identify the terms in 4p² + 3p + 2.

Term 1: ²

Term 2:

Term 3:

Total number of terms =

Great! This is a trinomial (three terms).

10. Simplify: 5x + 2y – x + y.

Combine x terms: 5x – x = 4x6x

Combine y terms: 2y + y = 3y2y

Result = 4x + 3y3y + 4x

Excellent! Group and simplify like terms separately.

Drag each description to its correct expression:

Part A: Section B – Short Answer Questions (2 Marks Each)

1. Simplify: (3x + 4y + 5) + (6x + 2y – 3).

Remove brackets and combine

3x + 4y + 5 + 6x + 2y – 3

Combine x terms: 3x + 6x =

Combine y terms: 4y + 2y =

Combine constants: 5 – 3 =

Final answer =

Perfect! Add all like terms separately.

2. Add: (5x – 3y + 2) and (2x + 4y – 5).

(5x – 3y + 2) + (2x + 4y – 5)

Combine x terms: 5x + 2x =

Combine y terms: –3y + 4y = y7y

Combine constants: 2 – 5 = –33

Result =

Excellent! Remember –3y + 4y = 1y = y.

3. Subtract: (3x – 5y – 2) from (7x + 2y + 8).

(7x + 2y + 8) – (3x – 5y – 2)

Change signs: 7x + 2y + 8 – 3x + 5y– 5y + 2– 2

Combine x: 7x – 3x =

Combine y: 2y + 5y =

Combine constants: 8 + 2 =

Result =

Great! Change ALL signs when subtracting.

4. Simplify: (2a + 3b + 4) – (a – 2b – 1).

Remove brackets: 2a + 3b + 4 – a + 2b– 2b + 1– 1

Combine a terms: 2a – a = a3a

Combine b terms: 3b + 2b =

Combine constants: 4 + 1 =

Result =

Perfect! Watch the signs carefully.

5. Find the value of 3x² + 2x + 1 when x = 2.

Substitute x =

3()² + 2() + 1

= 3() + 2(2) + 1

= + + 1

=

Excellent! Evaluate powers first, then multiply, then add.

6. Identify constants, coefficients, and variables in 7x³ + 5x² + 3x + 2.

Variable: appears with powers 3, 2, and 1

Coefficients: , ,

Constant:

Perfect! This is a polynomial of degree 3.

7. Write expressions for: (a) Area of rectangle = l × b (b) Perimeter of rectangle = 2(l + b)

(a) Area = l × bl + b or

Correct! Area is length times breadth.

(b) Perimeter = or

Excellent! Both forms are correct.

8. Add: (5x² + 2x – 3) and (4x² – 3x + 2).

Combine x² terms: 5x² + 4x² =

Combine x terms: 2x – 3x = –xx

Combine constants: –3 + 2 = –11

Result =

Great! Combine terms with same powers separately.

9. Simplify: 8x – 4y + 2x + 3y.

Combine x terms: 8x + 2x =

Combine y terms: –4y + 3y = –yy

Result =

Perfect! –4y + 3y = –1y = –y.

10. Subtract (7a + 2b – 5) from (10a – 3b + 4).

(10a – 3b + 4) – (7a + 2b – 5)

= 10a – 3b + 4 – 7a – 2b+ 2b + 5– 5

Combine a: 10a – 7a =

Combine b: –3b – 2b =

Combine constants: 4 + 5 =

Result =

Excellent! All short answer problems complete.

Part A: Section C – Long Answer Questions (4 Marks Each)

1. Simplify and find the value when x = 2, y = 3: (3x + 2y + 5) + (4x – y – 2).

Step 1: Simplify the expression

Remove brackets: 3x + 2y + 5 + 4x – y – 2

Combine x terms: 3x + 4x =

Combine y terms: 2y – y =

Combine constants: 5 – 2 =

Simplified expression =

Step 2: Substitute x = 2, y = 3

7() + + 3

= + 3 + 3

=

Perfect! Always simplify first, then substitute.

2. Simplify: (5x + 2y – 3) – (2x – 4y + 6).

Remove brackets and change signs:

5x + 2y – 3 – 2x + 4y– 4y – 6+ 6

Combine x terms: 5x – 2x =

Combine y terms: 2y + 4y =

Combine constants: –3 – 6 =

Result =

Excellent! Be careful with signs during subtraction.

3. Simplify: (3a + 4b – 5) + (2a – b + 8) – (a + 3b – 2).

Step 1: Add first two expressions

(3a + 4b – 5) + (2a – b + 8) = + +

Step 2: Subtract third expression

(5a + 3b + 3) – (a + 3b – 2)

= 5a + 3b + 3 – a – 3b+ 3b + 2– 2

Combine a: 5a – a =

Combine b: 3b – 3b =

Combine constants: 3 + 2 =

Final answer = or

Great! When terms cancel, we get 0 (not written).

4. Find the value of: (2x² + 3x – 4) + (x² – 2x + 6) for x = 2.

Step 1: Simplify

Combine x² terms: 2x² + x² =

Combine x terms: 3x – 2x =

Combine constants: –4 + 6 =

Simplified:

Step 2: Substitute x = 2

3()² + + 2

= 3() + 2 + 2

= + 2 + 2

=

Perfect! Simplify the expression before substituting.

5. Write an expression for the perimeter of a triangle with sides a, b, c. Then find the value if a = 3, b = 4, c = 5.

Perimeter expression:

Perimeter = a + b + cabc

Correct! Add all three sides.

Substitute values:

a = , b = , c =

Perimeter = 3 + 4 + 5 = units

Excellent! This is a right triangle (3-4-5).

Part B: Objective Questions - Test Your Knowledge!

Answer these multiple choice questions:

6. Coefficient of a² in 5a² + 3a + 2 is:

(a) 3 (b) 5 (c) 2 (d) a

Correct! The coefficient of a² is 5.

7. Value of 2x + 3 when x = 3 is:

(a) 5 (b) 7 (c) 9 (d) 15

Perfect! 2(3) + 3 = 6 + 3 = 9.

8. Simplify: (x + y) + (2x – 3y).

(a) 3x – 2y (b) 3x + 2y (c) x – 2y (d) 2x + 3y

Excellent! x + 2x = 3x and y – 3y = –2y.

9. The constant term in x² + 3x + 4 is:

(a) 1 (b) 3 (c) 4 (d) x

Correct! The constant term is 4 (no variable).

10. If x = 2, find 3x + 5.

(a) 6 (b) 8 (c) 9 (d) 11

Perfect! 3(2) + 5 = 6 + 5 = 11.

🌟 Excellent Progress! You've Mastered Intermediate Algebraic Expressions!

Here's what you've learned:

• Advanced Expression Writing:

  • "5 times a number increased by 4" → 5x + 4
  • "Thrice x decreased by 7" → 3x – 7
  • "Product of x and y, added to z" → xy + z
  • "Sum of squares of a and b" → a² + b²
  • Understanding complex word problems

• Polynomials and Degrees:

  Monomial: One term (5x, 3y², –7) Binomial: Two terms (3x + 5, 2a – 7b) Trinomial: Three terms (x² + 2x + 1) Polynomial: Multiple terms

  Degree: Highest power of variable

  • 2x³ + 5x² + 7 → Degree 3
  • 4x² + 3x → Degree 2
  • 5x + 2 → Degree 1
  • Constant (7) → Degree 0

• Complex Operations:

  Addition with multiple variables:

  • Group all like terms
  • (3x + 2y + 5) + (4x – y + 3) = 7x + y + 8

  Subtraction - KEY RULE:

  • Change ALL signs in second expression
  • (5x + 3y – 2) – (2x – y + 4)
  • = 5x + 3y – 2 – 2x + y – 4
  • = 3x + 4y – 6

  Multiple operations:

  • Work step by step
  • Combine like terms carefully
  • Check signs at each step

• Substitution with Powers:

  • For x² when x = 3: Calculate 3² = 9 first
  • Order: Powers → Multiplication → Addition/Subtraction
  • Example: 2x² + 3x + 1 when x = 2
    • = 2(2²) + 3(2) + 1
    • = 2(4) + 6 + 1
    • = 8 + 6 + 1 = 15

• Geometric Applications:

  • Perimeter of triangle: a + b + c
  • Perimeter of rectangle: 2(l + b) or 2l + 2b
  • Perimeter of square: 4a
  • Area of rectangle: l × b or lb
  • Area of square: a² or a × a

• Important Patterns to Remember:

  • x + x = 2x (not x²)
  • x × x = x² (not 2x)
  • 2y – 3y = –y (not –1)
  • –a + 2a = a (not 3a)
  • xy ≠ x + y (different operations)

• Step-by-Step Strategy:

  1. Remove all brackets
  2. Change signs if subtracting
  3. Group like terms together
  4. Add/subtract coefficients
  5. Write in standard form (decreasing powers)
  6. Verify your answer

• Common Mistakes to Avoid:

  • Forgetting to change signs when subtracting
  • Adding unlike terms (3x + 2y ≠ 5xy)
  • Wrong order of operations in substitution
  • Losing negative signs
  • Confusing x² and 2x
  • Not simplifying completely

Mastering these intermediate concepts prepares you for advanced algebra and equation solving!