Exploring Algebraic Identities: CBSE Class 9 Mathematics practice questions

Practise 50 questions on Exploring Algebraic Identities, free, with an explanation for each answer.

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Sample questions

Use the area model to expand $(3x + 4y)^2$.

1. $9x^2 + 16y^2$
2. $9x^2 + 12xy + 16y^2$
3. $3x^2 + 24xy + 4y^2$
4. $9x^2 + 24xy + 16y^2$

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D. $9x^2 + 24xy + 16y^2$

$(3x)^2 + 2(3x)(4y) + (4y)^2 = 9x^2 + 24xy + 16y^2$. Leaving out the two $12xy$ rectangles gives $9x^2 + 16y^2$; forgetting the 2 gives $12xy$; and $3x^2$ comes from not squaring the coefficient 3.

Find $117^2$ by writing $117 = 100 + 10 + 7$ and using $(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$.

1. 10149
2. 11919
3. 10289
4. 13689

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D. 13689

$10000 + 100 + 49 + 2(100)(10) + 2(10)(7) + 2(7)(100) = 10149 + 2000 + 140 + 1400 = 13689$. $10149$ drops all the rectangles, $11919$ forgets to double them, and $10289 = 100^2 + 17^2$ uses the trap $(a+b)^2 = a^2 + b^2$.

Expand $\left(\dfrac{2}{3}m + 6n\right)^2$.

1. $\dfrac{4}{9}m^2 + 36n^2$
2. $\dfrac{4}{9}m^2 + 8mn + 36n^2$
3. $\dfrac{4}{9}m^2 + 4mn + 36n^2$
4. $\dfrac{2}{3}m^2 + 8mn + 6n^2$

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B. $\dfrac{4}{9}m^2 + 8mn + 36n^2$

$\left(\dfrac23 m\right)^2 + 2\left(\dfrac23 m\right)(6n) + (6n)^2 = \dfrac49 m^2 + 8mn + 36n^2$. Dropping the middle term or forgetting its 2 gives the wrong options, and $\dfrac23 m^2 + 6n^2$ comes from not squaring the coefficients.

Topics covered

• Area model of (a + b)²
• Area model of (a − b)²
• Why (a − b)² ends in + b²
• Area model of (a + b + c)²
• Algebra tiles: (x + 3)(x + 4)
• Algebra tiles: (2x + 3)(3x + 1)
• Splitting the middle term
• Cube of edge (a + b)
• Area of a path
• Standard identities
• Pieces of the cube (a + b)³
• Factorising x² + bx + c
• Mental maths with identities
• Three consecutive squares
• Visualising (a − b)²
• Comparing (a + b)², a² + b², (a − b)² and a² − b²
• Simplifying rational expressions
• Square of a trinomial
• Identity or equation?
• Factors of x⁴ − y⁴
• n³ − n is divisible by 6
• Identities hold for all numbers
• Equivalent forms of (a − b)²
• Number plus its reciprocal
• When is (a + b)² smaller than a² + b²?
• Expanding (a + b)²
• Expanding with fractions
• Combining identities
• Squaring numbers with (a − b)²
• Squaring numbers with (a + b + c)²
• Products with (a + b)(a − b)
• Four consecutive squares
• Śrīdharāchārya's method
• Cubing numbers with (a − b)³
• Factorising with a common factor
• Factorising a square of three terms
• Using (a + b)³
• Sum of two cubes
• x³ + y³ + z³ − 3xyz
• Sum of cubes of three numbers
• Length and breadth from area
• Algebra tiles: factorising

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More CBSE Class 9 Mathematics chapters

• Orienting Yourself: The Use of Coordinates
• Introduction to Linear Polynomials
• The World of Numbers
• I’m Up and Down, and Round and Round
• Measuring Space: Perimeter and Area
• The Mathematics of Maybe: Introduction to Probability
• Predicting What Comes Next: Exploring Sequences and Progressions
• Propositions and Their Converses
• How Quantities Combine: Understanding Data
• The World of Algorithms
• Quadrilaterals
• Two Variables, One Line
• Math of Space: Surface Area and Volume