Which identity is used to expand (2x - 3y)²?

(x + a)(x + b) = x² + (a + b)x + ab

What is the expanded form of (x + y + z)²?

x² + y² + z² + 2xy + 2yz + 2zx

The given diagram represents the factorisation of x² + 7x + 12 using algebra tiles. Which of the following statements is true about the arrangement?

The total area of the tiles forms a rectangle with sides (x+3) and (x+4).

The 7x term is split into 2x and 5x.

The total area of the tiles forms a rectangle with sides (x+3) and (x+4).

The diagram represents the expansion of (x+7)(x+12).

The diagram shows a perfect square trinomial.

Assertion (A): The expression (x - y)² is always greater than or equal to 0 for any real numbers x and y. Reason (R): The square of any real number is always non-negative.

A and R both correct, R explains A

Assertion (A): The expression x² + 5x + 6 can be factorised into (x + 2)(x + 3). Reason (R): For a quadratic trinomial x² + (a+b)x + ab, its factors are (x+a) and (x+b).

A and R both correct, R explains A

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This quiz contains 40 questions covering various aspects of Exploring Algebraic Identities, spread across easy, medium, and hard difficulty levels.

Is this quiz aligned with the NCERT syllabus?

Yes, all content and questions are strictly aligned with the CBSE NCERT Class 9 Maths syllabus for Chapter 4: Exploring Algebraic Identities.

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Currently, Shrutam quizzes require an internet connection to play. We are working on an offline mode for future updates.

What topics does this quiz cover?

This quiz covers visualising identities, factorisation using identities like (a+b)² and (a-b)², identities for three terms like (a+b+c)², and cubic identities like (a+b)³.

How can I improve my score in this chapter?

Regular practice of problems from the NCERT textbook, understanding the geometric interpretations of identities, and reviewing the explanations provided in this quiz will greatly help improve your understanding and score.

Home › CBSE › Class 9 › Maths › Exploring Algebraic Identities

CBSE · Class 9 · 🧮 Maths · Chapter 4

Exploring Algebraic Identities

Algebraic IdentitiesVisualising IdentitiesFactorisation using IdentitiesIdentities for three termsCubic Identities

Chapter 4, 'Exploring Algebraic Identities', delves into fundamental algebraic equations that hold true for all variable values. Students learn to visualise identities like (a+b)² and (a-b)² using geometrical models and algebra tiles. The chapter also covers factorisation of algebraic expressions using these identities, including trinomials of the form x² + (a+b)x + ab. Furthermore, it introduces advanced identities such as (a+b+c)² and (a+b)³, and their applications in simplifying rational expressions. Understanding these identities is crucial for building a strong foundation in algebra and solving complex mathematical problems efficiently.

Algebraic Identities ka Introduction

Algebraic Identity ek equation hoti hai jo variables ki har value ke liye true hoti hai. Equation aur Identity mein yahi main difference hai. Equation kuch specific values ke liye true hoti hai, jabki identity sabhi values ke liye.

Example:
\(x+1=5\) ek equation hai, jo sirf \(x=4\) ke liye true hai.
\((x+1)^2 = x^2+2x+1\) ek identity hai, jo \(x\) ki har value ke liye true hai.

Importance:
Calculations ko simplify karti hain.
Algebraic expressions ko factorise aur expand karne mein help karti hain.
Higher mathematics mein base provide karti hain.

📖Definition

Algebraic Identity: Ek equation jo usmein use hone wale variables ki har value ke liye true hoti hai.

⭐Remember

Identity \( = \) Always True Equation. Equation \( = \) Sometimes True Equation.

Basic Identities aur Geometrical Visualisation

Humne pehle bhi kuch basic identities padhi hain. Inko geometrically visualize karna inki understanding ko strong karta hai.

Identity 1: \((a+b)^2 = a^2 + 2ab + b^2\)

Geometrical Proof:

Ek square banao jiski side length \((a+b)\) ho.
Is square ko do chote squares aur do rectangles mein divide karo.
Ek square ki side \(a\) hogi, area \(a^2\).
Dusre square ki side \(b\) hogi, area \(b^2\).
Do rectangles honge, har ek ki sides \(a\) aur \(b\), so area \(ab\) each.
Total area of big square \((a+b)^2\) = Sum of areas of all smaller parts \( = a^2 + b^2 + ab + ab = a^2 + 2ab + b^2\).

Application: Numerical calculations ko simplify karna. Jaise \((103)^2 = (100+3)^2\).

Identity 2: \((a-b)^2 = a^2 - 2ab + b^2\)

Derivation: Identity 1 mein \(b\) ko \(-b\) se replace karke ya geometrical method se bhi derive kar sakte hain.
\((a+(-b))^2 = a^2 + 2a(-b) + (-b)^2 = a^2 - 2ab + b^2\).
Geometrical Visualisation: Ek square of side \(a\) lo. Usmein se ek strip of width \(b\) hata do. Bacha hua area \((a-b)^2\) ko represent karta hai. [IMAGE: TODO: Geometrical representation of (a-b)^2]

Identity 3: \(a^2 - b^2 = (a-b)(a+b)\)

Derivation: RHS ko expand karke:

\((a-b)(a+b) = a(a+b) - b(a+b) = a^2+ab - ba-b^2 = a^2-b^2\).

Geometrical Visualisation: Ek square of side \(a\) mein se ek square of side \(b\) remove karo. Remaining area \(a^2-b^2\) ko do rectangles mein rearrange karke \((a-b)(a+b)\) show kar sakte hain. [IMAGE: TODO: Geometrical representation of a^2-b^2]

Application: Factorisation aur numerical simplification. Jaise \(99^2 - 1^2 = (99-1)(99+1) = 98 \times 100 = 9800\).

🧮Formula

Basic Identities:

\((a+b)^2 = a^2 + 2ab + b^2\)
\((a-b)^2 = a^2 - 2ab + b^2\)
\(a^2 - b^2 = (a-b)(a+b)\)

💡Tip

Geometrical proofs direct exam mein kam aate hain, but conceptual clarity ke liye bahut important hain. Inko samajh ke rakhna.

Identity: \((x+a)(x+b)\)

Yeh identity quadratic expressions ko factorise karne aur products ko expand karne mein use hoti hai.

Derivation: Distributive property se:

\((x+a)(x+b) = x(x+b) + a(x+b)\) \( = x^2 + xb + ax + ab\) \( = x^2 + (a+b)x + ab\)

Formula: \((x+a)(x+b) = x^2 + (a+b)x + ab\)

Application:
Expansion: \((x+3)(x+4) = x^2 + (3+4)x + (3)(4) = x^2 + 7x + 12\).
Factorisation: \(x^2 + 7x + 12\) ko factorise karne ke liye, humein aise do numbers \(a\) aur \(b\) dhundhne hain jinka sum \(7\) ho aur product \(12\) ho. (Numbers are 3 and 4). So, \(x^2 + 7x + 12 = (x+3)(x+4)\).

Splitting the Middle Term: Quadratic expressions \(ax^2+bx+c\) ko factorise karne ka yeh ek common method hai. \(bx\) ko do terms mein split karte hain, jinka sum \(b\) ho aur product \(ac\) ho.
Steps for \(x^2+bx+c\):

Do numbers \(p\) aur \(q\) dhundo such that \(p+q=b\) aur \(pq=c\).
Middle term \(bx\) ko \(px+qx\) mein split karo.
Common factors leke factorise karo.

Example: Factorise \(x^2 + 5x + 6\)

\(p+q=5\), \(pq=6\). Numbers are 2 and 3.
\(x^2 + 2x + 3x + 6\)
\(x(x+2) + 3(x+2)\)
\((x+2)(x+3)\)

🧮Formula

Identity: \((x+a)(x+b) = x^2 + (a+b)x + ab\)

🚧Misconception

Students aksar \((a+b)\) aur \(ab\) ko confuse kar dete hain. Hamesha yaad rakho, middle term \(x\) ka coefficient sum hota hai, aur constant term product hota hai.

Identity: \((a+b+c)^2\)

Jab teen terms ka square find karna ho, toh yeh identity use hoti hai.

Derivation:

\((a+b+c)^2 = ((a+b)+c)^2\) Let \((a+b) = x\). Then \((x+c)^2 = x^2 + 2xc + c^2\) Substitute \(x = (a+b)\) back: \((a+b)^2 + 2(a+b)c + c^2\) \( = (a^2+2ab+b^2) + 2ac+2bc + c^2\) \( = a^2+b^2+c^2+2ab+2bc+2ca\)

Formula: \((a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca\)

Application: Expressions ko expand karna aur kabhi-kabhi factorise karna (jab expression is form mein ho).
Example: \((x+2y+3z)^2\)

\( = x^2 + (2y)^2 + (3z)^2 + 2(x)(2y) + 2(2y)(3z) + 2(3z)(x)\) \( = x^2 + 4y^2 + 9z^2 + 4xy + 12yz + 6zx\)

🧮Formula

Identity: \((a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca\)

⭐Remember

Is identity mein har term ka square aur har pair ka twice product aata hai.

Cubic Identities: \((a+b)^3\) aur \((a-b)^3\)

Yeh identities cubes of binomials ko expand karne ke liye use hoti hain.

Identity 1: \((a+b)^3 = a^3 + b^3 + 3ab(a+b)\)

Derivation:

\((a+b)^3 = (a+b)^2 (a+b)\) \( = (a^2+2ab+b^2)(a+b)\) \( = a(a^2+2ab+b^2) + b(a^2+2ab+b^2)\) \( = a^3+2a^2b+ab^2 + a^2b+2ab^2+b^3\) \( = a^3 + 3a^2b + 3ab^2 + b^3\) \( = a^3 + b^3 + 3ab(a+b)\) (Common term \(3ab\) leke)

Formula: \((a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\) or \(a^3 + b^3 + 3ab(a+b)\)

Identity 2: \((a-b)^3 = a^3 - b^3 - 3ab(a-b)\)

Derivation: \((a+b)^3\) mein \(b\) ko \(-b\) se replace karke:

\((a+(-b))^3 = a^3 + (-b)^3 + 3a(-b)(a+(-b))\) \( = a^3 - b^3 - 3ab(a-b)\) \( = a^3 - b^3 - 3a^2b + 3ab^2\)

Formula: \((a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\) or \(a^3 - b^3 - 3ab(a-b)\)

Application:
Expansion: \((2x+3y)^3 = (2x)^3 + (3y)^3 + 3(2x)(3y)(2x+3y)\)

\( = 8x^3 + 27y^3 + 18xy(2x+3y)\) \( = 8x^3 + 27y^3 + 36x^2y + 54xy^2\)

Numerical Calculation: \((99)^3 = (100-1)^3 = 100^3 - 1^3 - 3(100)(1)(100-1)\)

\( = 1000000 - 1 - 300(99)\) \( = 999999 - 29700 = 970299\)

🧮Formula

Cubic Identities:

\((a+b)^3 = a^3 + b^3 + 3ab(a+b)\) or \(a^3 + 3a^2b + 3ab^2 + b^3\)
\((a-b)^3 = a^3 - b^3 - 3ab(a-b)\) or \(a^3 - 3a^2b + 3ab^2 - b^3\)

💡Tip

In identities ko expand form aur factorised form dono mein yaad rakho, depending on question ki requirement.

Factorisation Identities: \((a^3+b^3)\) aur \((a^3-b^3)\)

Yeh identities cubic expressions ko factorise karne ke liye use hoti hain. Inko cubic binomial identities se confuse mat karna.

Identity 1: \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)

Derivation: Hum jante hain \((a+b)^3 = a^3 + b^3 + 3ab(a+b)\)

Rearranging terms: \(a^3 + b^3 = (a+b)^3 - 3ab(a+b)\) Common factor \((a+b)\) leke: \(a^3 + b^3 = (a+b)[(a+b)^2 - 3ab]\) \( = (a+b)[a^2+2ab+b^2 - 3ab]\) \( = (a+b)(a^2 - ab + b^2)\)

Formula: \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)

Identity 2: \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)

Derivation: Hum jante hain \((a-b)^3 = a^3 - b^3 - 3ab(a-b)\)

Rearranging terms: \(a^3 - b^3 = (a-b)^3 + 3ab(a-b)\) Common factor \((a-b)\) leke: \(a^3 - b^3 = (a-b)[(a-b)^2 + 3ab]\) \( = (a-b)[a^2-2ab+b^2 + 3ab]\) \( = (a-b)(a^2 + ab + b^2)\)

Formula: \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)

Application:
Factorise: \(8x^3 + 27y^3 = (2x)^3 + (3y)^3\)

\( = (2x+3y)((2x)^2 - (2x)(3y) + (3y)^2)\) \( = (2x+3y)(4x^2 - 6xy + 9y^2)\)

Factorise: \(x^3 - 64 = x^3 - 4^3\)

\( = (x-4)(x^2 + x(4) + 4^2)\) \( = (x-4)(x^2 + 4x + 16)\)

🧮Formula

Factorisation Identities (Cubic):

\(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)
\(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)

🚧Misconception

Sign errors! \(a^3+b^3\) mein \((a+b)\) aur \(-ab\) hota hai. \(a^3-b^3\) mein \((a-b)\) aur \(+ab\) hota hai. Signs ko carefully yaad rakho.

Special Identity: \((a^3+b^3+c^3-3abc)\)

Yeh identity thodi complex hai but bahut important hai, especially for factorisation of cubic trinomials.

Formula: \(a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca)\)

Derivation (Brief Idea): Yeh derivation thoda lamba hai, but isko \((a+b)^3\) identity aur \(a^3+b^3\) identity ko combine karke derive kiya ja sakta hai. Basically, \(a^3+b^3+c^3-3abc\) ko \((a+b)^3+c^3-3ab(a+b)-3abc\) likh kar, phir \(X^3+Y^3\) form mein factorise kiya jata hai.

Special Case:
Agar \(a+b+c=0\) ho, toh identity ka RHS \(0\) ho jayega.
Iska matlab, agar \(a+b+c=0\), then \(a^3+b^3+c^3-3abc = 0\).
So, if \(a+b+c=0\), then \(a^3+b^3+c^3 = 3abc\).
Yeh special case numerical problems mein bahut use hota hai.

Application:
Factorise: \(x^3+y^3+z^3-3xyz\) is already in the form.
Evaluate: \((10)^3 + (-7)^3 + (-3)^3\)

Yahan \(a=10, b=-7, c=-3\). Check \(a+b+c = 10 + (-7) + (-3) = 10 - 7 - 3 = 0\). Since \(a+b+c=0\), then \(a^3+b^3+c^3 = 3abc\). \( = 3(10)(-7)(-3)\) \( = 3(10)(21) = 30 \times 21 = 630\).

🧮Formula

Special Identity: \(a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca)\)

Important Special Case: If \(a+b+c=0\), then \(a^3+b^3+c^3 = 3abc\)

💡Tip

Is identity ka special case \(a+b+c=0 \implies a^3+b^3+c^3 = 3abc\) bahut frequent board exam question hai. Isko hamesha check karna.

Identities ka Application: Factorisation aur Simplification

Identities sirf expansion ke liye nahi, balki factorisation aur complex expressions ko simplify karne ke liye bhi use hoti hain.

Factorisation Steps (General):

Common Factor: Sabse pehle common factor dhundo.
Identity Recognition: Dekho kya expression kisi standard identity ke form mein hai.

\(A^2+2AB+B^2 \implies (A+B)^2\)
\(A^2-2AB+B^2 \implies (A-B)^2\)
\(A^2-B^2 \implies (A-B)(A+B)\)
\(x^2+(A+B)x+AB \implies (x+A)(x+B)\) (Middle term splitting)
\(A^3+B^3 \implies (A+B)(A^2-AB+B^2)\)
\(A^3-B^3 \implies (A-B)(A^2+AB+B^2)\)
\(A^3+B^3+C^3-3ABC \implies (A+B+C)(A^2+B^2+C^2-AB-BC-CA)\)

Grouping: Agar identity directly apply nahi ho rahi, toh terms ko group karke common factors ya identities banao.

Simplification of Rational Expressions:
Rational expression \(\frac{P(x)}{Q(x)}\) ko simplify karne ke liye, numerator aur denominator ko factorise karte hain.
Phir common factors ko cancel out karte hain (provided common factor zero na ho).
Example: Simplify \(\frac{x^2-4}{x^2+4x+4}\)

Numerator: \(x^2-4 = (x-2)(x+2)\) (using \(a^2-b^2\)) Denominator: \(x^2+4x+4 = (x+2)^2\) (using \((a+b)^2\)) So, \(\frac{(x-2)(x+2)}{(x+2)(x+2)} = \frac{x-2}{x+2}\) (provided \(x+2 \neq 0\) i.e., \(x \neq -2\))

Numerical Simplification: Identities ko numbers ke calculations mein bhi use karte hain.
\(105 \times 95 = (100+5)(100-5) = 100^2 - 5^2 = 10000 - 25 = 9975\)
\(101^2 = (100+1)^2 = 100^2 + 2(100)(1) + 1^2 = 10000 + 200 + 1 = 10201\)

💡Tip

Factorisation aur simplification ke questions mein step-by-step approach follow karna. Pehle common factor, phir identities, phir grouping. Signs ka dhyaan rakhna.

❗Important

Rational expressions ko simplify karte waqt, jo factor cancel karte ho, usko zero ke equal nahi hona chahiye. For example, \(\frac{x-2}{x+2}\) is valid only if \(x \neq -2\).

Easy (15)

An algebraic identity is an equation that is true for all values of the variables occurring in it. Is this statement true or false?
Why: An algebraic identity is defined as an equation true for all variable values.
The expansion of (a + b)² is a² + 2ab + b². True or False?
Why: This is the fundamental identity for the square of a binomial sum.
The identity (a - b)² is equal to a² - 2ab - b². True or False?
Why: The correct identity is (a - b)² = a² - 2ab + b².
The area of a square with side (a + b) units can be represented by the identity (a + b)² = a² + 2ab + b². What is the area of the square with side (a + b) units?
Why: The area of a square is side squared, so for side (a+b), the area is (a+b)².
The identity (a + b + c)² expands to a² + b² + c² + 2ab + 2bc + 2ca. What is the coefficient of the 'ab' term in this expansion?
Why: In the expansion of (a+b+c)², the term 2ab appears.
The product of (x + a)(x + b) is x² + (a + b)x + ab. What is the constant term in this expansion?
Why: The constant term in the expansion of (x+a)(x+b) is the product of 'a' and 'b'.
The expansion of (a + b)³ is a³ + 3a²b + 3ab² + b³. What is the coefficient of the 'a³' term?
Why: The term a³ has an implicit coefficient of 1.
Which of the following is the correct expansion of (x + 5)²?
- x² + 25
- x² + 10x + 25 (correct)
- x² + 5x + 25
- x² + 10x + 5
Why: Using (a+b)² = a² + 2ab + b², we get x² + 2(x)(5) + 5² = x² + 10x + 25.
Which identity is used to expand (2x - 3y)²?
- (a + b)² = a² + 2ab + b²
- (a - b)² = a² - 2ab + b² (correct)
- (a + b)(a - b) = a² - b²
- (x + a)(x + b) = x² + (a + b)x + ab
Why: The expression is a difference of two terms squared, matching (a-b)².
What is the result of (x + 3)(x + 4) using the identity (x + a)(x + b) = x² + (a + b)x + ab?
- x² + 7x + 12 (correct)
- x² + 12x + 7
- x² + 7x + 7
- x² + 12
Why: Substitute a=3, b=4 into the identity: x² + (3+4)x + (3)(4) = x² + 7x + 12.
Which of the following is a factor of x² - y²?
- x + y (correct)
- x² + y²
- x - y²
- x² - y
Why: The difference of squares identity states x² - y² = (x - y)(x + y).
What is the value of (100 + 1)² using the identity (a + b)²?
- 10001
- 10101
- 10201 (correct)
- 1001
Why: (100+1)² = 100² + 2(100)(1) + 1² = 10000 + 200 + 1 = 10201.
Which of the following is NOT an algebraic identity?
- (a + b)² = a² + 2ab + b²
- (a - b)² = a² - 2ab + b²
- a² - b² = (a - b)(a + b)
- a + b = 5 (correct)
Why: a + b = 5 is an equation, not an identity, as it's only true for specific values.
When visualising identities using geometrical models, what shapes are typically used to represent terms like a² and b²?
- Circles
- Triangles
- Squares (correct)
- Cylinders
Why: a² and b² represent areas of squares with sides 'a' and 'b' respectively.
The expression x² + 7x + 12 can be factorised into (x + 3)(x + 4). This process is known as factorisation. True or False?
Why: Breaking down an expression into a product of simpler expressions is factorisation.

Medium (15)

Expand (3x + 2y)² using a suitable algebraic identity.
- 9x² + 4y²
- 9x² + 6xy + 4y²
- 9x² + 12xy + 4y² (correct)
- 3x² + 12xy + 2y²
Why: Using (a+b)² = a² + 2ab + b², with a=3x and b=2y, gives 9x² + 12xy + 4y².
Factorise the expression 4x² - 9y².
- (2x - 3y)²
- (2x + 3y)²
- (2x - 3y)(2x + 3y) (correct)
- (4x - 9y)(4x + 9y)
Why: This is a difference of squares: (2x)² - (3y)² = (2x - 3y)(2x + 3y).
What is the expanded form of (x + y + z)²?
- x² + y² + z² + 2xy + 2yz + 2zx (correct)
- x² + y² + z² + xy + yz + zx
- x² + y² + z² + 2xyz
- x³ + y³ + z³ + 3xyz
Why: The identity for the square of a trinomial is x² + y² + z² + 2xy + 2yz + 2zx.
If (x + a)(x + b) = x² + 9x + 20, what are the values of 'a' and 'b'?
- a=1, b=20
- a=2, b=10
- a=4, b=5 (correct)
- a=3, b=6
Why: Comparing with x² + (a+b)x + ab, we need a+b=9 and ab=20. So, a=4, b=5 (or vice versa).
Which of the following is the correct expansion of (2x + 1)³?
- 8x³ + 1
- 8x³ + 12x² + 6x + 1 (correct)
- 8x³ + 6x² + 12x + 1
- 2x³ + 6x² + 6x + 1
Why: Using (a+b)³ = a³ + 3a²b + 3ab² + b³ with a=2x, b=1, we get 8x³ + 12x² + 6x + 1.
Match the algebraic expressions with their correct expanded forms.
Why: Each pair matches an identity with its expanded form: (a+b)², (a-b)², (a+b)(a-b), and (x+a)(x+b).
Match the factorised expressions with their original forms.
Why: Each expression is a perfect square trinomial, difference of squares, or general quadratic factorisation.
Match the identities with their corresponding geometric interpretations (assuming positive values for variables).
Why: Identities relate to geometric areas of squares/rectangles or volumes of cubes/cuboids.
What is the value of 99² using an algebraic identity?
Why: Use (100-1)² = 100² - 2(100)(1) + 1² = 10000 - 200 + 1 = 9801.
What is the common factor of x² - y² and x³ - y³?
Why: x² - y² = (x-y)(x+y) and x³ - y³ = (x-y)(x² + xy + y²). Both have (x-y) as a factor.
Arrange the steps to expand (a + b)³ using the distributive property.
Why: The sequence follows breaking down the cubic term, expanding the square, distributing, and finally combining like terms.
Arrange the steps to factorise x² + 7x + 12 by splitting the middle term.
Why: The steps involve finding two numbers, splitting the middle term, grouping, factoring common terms, and finally factoring the common binomial.
Identify the algebraic identity represented by the given diagram.
- (a + b)² = a² + b²
- (a + b)² = a² + 2ab + b² (correct)
- (a - b)² = a² - 2ab + b²
- a² - b² = (a - b)(a + b)
Why: The diagram shows a square of side (a+b) divided into a², b², and two ab rectangles.
The given diagram represents the factorisation of x² + 7x + 12 using algebra tiles. Which of the following statements is true about the arrangement?
- The 7x term is split into 2x and 5x.
- The total area of the tiles forms a rectangle with sides (x+3) and (x+4). (correct)
- The diagram represents the expansion of (x+7)(x+12).
- The diagram shows a perfect square trinomial.
Why: Fig 4.7 shows a rectangle formed by x² + 7x + 12 tiles, with side lengths (x+3) and (x+4).
Observe the given figure which represents the identity (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca. What does the central square with side 'c' represent?
- The term ab
- The term c² (correct)
- The term 2bc
- The term (a+b)²
Why: In Fig 4.4, the square with side 'c' represents the area c².

Hard (17)

If x + 1/x = 5, find the value of x² + 1/x².
Why: Square both sides of x + 1/x = 5 to get x² + 2(x)(1/x) + 1/x² = 25, which simplifies to x² + 2 + 1/x² = 25. So x² + 1/x² = 23.
If a - b = 7 and ab = 12, find the value of a² + b².
Why: Use (a-b)² = a² - 2ab + b². Substitute given values: 7² = a² + b² - 2(12), so 49 = a² + b² - 24. Thus, a² + b² = 73.
If x + y + z = 10 and xy + yz + zx = 30, find the value of x² + y² + z².
Why: Use (x+y+z)² = x²+y²+z² + 2(xy+yz+zx). Substitute values: 10² = x²+y²+z² + 2(30), so 100 = x²+y²+z² + 60. Thus, x²+y²+z² = 40.
Evaluate 103³ using an algebraic identity.
Why: Use (100+3)³ = 100³ + 3(100)²(3) + 3(100)(3)² + 3³ = 1000000 + 90000 + 2700 + 27 = 1092727.
Assertion (A): The expression (x - y)² is always greater than or equal to 0 for any real numbers x and y. Reason (R): The square of any real number is always non-negative.
Why: The assertion is true because (x-y) is a real number, and its square must be non-negative, as stated by the reason.
Assertion (A): The expression x² + 5x + 6 can be factorised into (x + 2)(x + 3). Reason (R): For a quadratic trinomial x² + (a+b)x + ab, its factors are (x+a) and (x+b).
Why: The assertion is true because 2+3=5 and 2*3=6, fitting the pattern described in the reason.
A rectangular plot has a length of (2x + 5) meters and a width of (2x - 5) meters. If the area of the plot is 100 square meters, what is the value of x?
- 5
- 10 (correct)
- 15
- 20
Why: Area = (x+5)(x-5) = x² - 25. Set x² - 25 = 75, solve for x.
A company manufactures square tiles. They decide to increase the side length of their tiles from 'a' units to '(a + 2)' units. If the increase in the area of the tile is 40 square units, what was the original side length 'a'?
- 8 units (correct)
- 9 units
- 10 units
- 12 units
Why: New Area - Old Area = (a+2)² - a² = 40. Expand (a+2)² and solve for 'a'.
nThis is good.
stem": "A company manufactures square tiles. They decide to increase the side length of their tiles from 'a' units to '(a + 2)' units. If the increase in the area of the tile is 36 square units
what was the original side length 'a'?
- 7 units
- 8 units (correct)
- 9 units
- 10 units
Why: New Area - Old Area = (a+2)² - a² = 36. Expand (a+2)² and solve for 'a'.
The given diagram shows a large square of side 'a' from which a smaller square of side 'b' is removed. The remaining area is then rearranged to form a rectangle. Which identity does this diagram visually represent?
- (a + b)² = a² + 2ab + b²
- (a - b)² = a² - 2ab + b²
- a² - b² = (a - b)(a + b) (correct)
- (a + b)³ = a³ + 3a²b + 3ab² + b³
Why: Fig 4.5 shows a²-b² rearranged into a rectangle with sides (a-b) and (a+b).
The diagram shows algebra tiles representing the product of (2x + 3) and (3x + 1). Based on the arrangement, what is the expanded form of this product?
- 5x² + 7x + 3
- 6x² + 11x + 3 (correct)
- 6x² + 9x + 3
- 6x² + 13x + 3
Why: The diagram shows 6 x² tiles, 11 x tiles, and 3 unit tiles, representing 6x² + 11x + 3.