Hard Level Worksheet

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

This hard level explores advanced algebraic identities including cubic formulas and complex expansions.

Master these identities to excel in higher mathematics and competitive examinations.

1. Define algebraic identity.

An algebraic identity is an equation that is true for all valuesa universal equation of the variables. (Note: Any similar meaning)

Perfect! Examples: (a+b)² = a2 + 2ab + a2 is true for any a and b.

2. Write any two algebraic identities.

Identity 1: (a+b)² = a2 + 2ab + a2(a-b)² = a2 - 2ab + a2 (Note: Any identity)

Identity 2: (a+b)(a-b) = a2 - a2a+b3 = a3 + 3a2b + 3aa2 + b3 (Note: Any other identity)

Excellent! These are fundamental algebraic identities.

3. What is the degree of 4x3y2z⁴?

Answer:

Correct! Degree = 3 + 2 + 4 = 9.

4. Find the coefficient of x2y in 6x2y3.

Answer:

Correct! There's no x2y term (it's x2y3), so the coefficient is 0.

5. Expand (2x - 3y)^2.

= 2x2 - 2(2x)(3y) + 3y2

= x2 - xy + y2

Perfect! 2x−3y2 = 4x2 - 12xy + 9y2.

Drag each identity to its correct expansion:

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

1. Multiply (x + 2)(x + 3)(x + 4).

First, multiply (x + 2)(x + 3):

= x2 + 3x + 2x + 6 = x2 + 5x + 6

Now multiply by (x + 4):

= (x2 + 5x + 6)(x + 4)

= x3 + 4x2 + 5x2 + 20x + 6x + 24

= x3 + x2 + x +

Excellent! The product is x3 + 9x2 + 26x + 24.

2. Simplify: a+b2 + a−b2.

= (a2 + 2ab + a2) + (a2 - 2ab + a2)

= a2 + 2ab + a2 + a2 - 2ab + a2

= a2 + a2

= (a2 + a2)

Perfect! The simplified form is 2(a2 + a2) or 2a2 + 2a2.

3. Multiply (2x - 3y)(2x + 3y).

Using identity (a - b)(a + b) = a2 - a2

= 2x2 - 3y2

= x2 - y2

Excellent! The product is 4x2 - 9y2.

4. Expand and simplify (3x + 2y)^2 - x−y2.

(3x + 2y)^2 = 9x2 + 12xy + 4y2

(x - y)^2 = x2 - 2xy + y2

Difference = 9x2 + 12xy + 4y2 - x2 + 2xy - y2

= x2 + xy + y2

Perfect! The simplified form is 8x2 + 14xy + 3y2.

5. Multiply (p + 2q)(p - 3q).

= p(p - 3q) + 2q(p - 3q)

= p2 - 3pq + 2pq - 6q2

= p2 - pq - q2

Great! The product is p2 - pq - 6q2.

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

1. Verify the identity x+y+z2 = x2 + y2 + z2 + 2xy + 2yz + 2zx.

LHS: x+y+z2

= (x + y + z)(x + y + z)

= x(x + y + z) + y(x + y + z) + z(x + y + z)

= x2 + xy + xz + xy + y2 + yz + xz + yz + z^2

= x2 + y2 + z2 + xy + yz + xz

= RHS. Identity verified!

2. Prove that a+b3 - a−b3 = 2b(3a2 + a2).

LHS: a+b3 - (a - b)^3

a+b3 = a^3 + 3a2b + 3aa2 + b^3

a−b3 = a3 - 3a2b + 3aa2 - b3

Difference = (a3 + 3a2b + 3aa2 + b3) - (a3 - 3a2b + 3aa2 - b3)

= a3 + 3a2b + 3aa2 + b^3 - a3 + 3a2b - 3aa2 + b3

= a2b + b3

= b(a2 + a2)

= RHS. Identity proved!

3. Expand and simplify: 2x+3y−4z2.

Using a+b+c2 = a2 + a2 + c2 + 2ab + 2bc + 2ca

Here a = 2x, b = 3y, c = -4z

= 2x2 + 3y2 + −4z2 + 2(2x)(3y) + 2(3y)(-4z) + 2(-4z)(2x)

= x2 + y2 + z^2 + xy - yz - zx

Perfect! The expansion is 4x2 + 9y2 + 16z² + 12xy - 24yz - 16zx.

4. Simplify: x+y3 + x−y3 using algebraic identities.

x+y3 = x3 + 3x2y + 3xy2 + y3

x−y3 = x3 - 3x2y + 3xy2 - y3

Sum = (x3 + 3x2y + 3xy2 + y3) + (x3 - 3x2y + 3xy2 - y3)

= x3 + 3x2y + 3xy2 + y3 + x3 - 3x2y + 3xy2 - y3

= x³ + xy2

= x(x2 + y2)

Excellent! The simplified form is 2x(x2 + 3y2) or 2x³ + 6xy2.

Part B: Objective Questions - Test Your Knowledge!

Answer these multiple choice questions:

6. (p + q)³ - (p - q)³ =

(a) 6p²q + 2q³ (b) 2q(3p² + q²) (c) 3p³ + 3q³ (d) None

Correct! Expand both cubes and simplify: = 6p²q + 2q³ = 2q(3p² + q²).

7. (2x + 3y)² =

(a) 4x2 + 9y2 + 12xy (b) 4x2 + 9y2 + 6xy (c) 2x2 + 3y2 + 12xy (d) None

Perfect! (2x)² + 2(2x)(3y) + (3y)² = 4x2 + 12xy + 9y2.

8. (x + y)³ + (x - y)³ =

(a) 2x³ + 6xy2 (b) 2x(x2 + 3y2) (c) x³ + y³ (d) 2x³

Excellent! Both 2x³ + 6xy2 and 2x(x2 + 3y2) are correct - they're equivalent!

9. Which identity is used to expand (x + a)(x + b)?

(a) (a+b)² = a2 + 2ab + a2

(b) (x+a)(x+b) = x2 + (a+b)x + ab

(c) (a−b)² = a2 − 2ab + a2

(d) None

Perfect! This identity is very useful for factoring quadratics.

10. The product (2x + 3)(2x - 3) =

(a) 4x2 + 9 (b) 4x2 - 9 (c) 4x2 + 6x (d) None

Correct! Using (a+b)(a-b) = a2 - a2, we get (2x)² - 3² = 4x2 - 9.

🎉 Exceptional Achievement! You've Mastered Advanced Algebraic Identities!

Here's what you learned:

• Fundamental Identities:

  • (a + b)² = a2 + 2ab + a2
  • (a - b)² = a2 - 2ab + a2
  • (a + b)(a - b) = a2 - a2
  • (x + a)(x + b) = x2 + (a+b)x + ab

• Cubic Identities:

  • (a + b)³ = a³ + 3a2b + 3aa2 + b³
  • (a - b)³ = a³ - 3a2b + 3aa2 - b³
  • (a + b)³ + (a - b)³ = 2a(a2 + 3a2)
  • (a + b)³ - (a - b)³ = 2b(3a2 + a2)

• Trinomial Identity:

  • (a + b + c)² = a2 + a2 + c² + 2ab + 2bc + 2ca

• Advanced Techniques:

  • Combining multiple identities
  • Simplifying complex expressions
  • Verifying identities algebraically
  • Factoring using identities

• Problem-Solving Strategy:

  1. Identify which identity applies
  2. Substitute carefully (watch signs!)
  3. Expand systematically
  4. Combine like terms
  5. Factor when possible

• Applications:

  • Simplifying complex algebraic expressions
  • Solving higher-degree equations
  • Preparation for calculus and advanced mathematics
  • Competitive examination problems

These advanced identities are fundamental tools in higher mathematics, engineering, and scientific calculations!