RowQ
The Vault
RowQ
The Vault
CBSE Class 9 Maths · 11 questions · 26 marks
Identities are shortcuts that hold for every value of the variable, which is exactly what makes them so powerful in an exam. Instead of grinding through the multiplication of (2a - 3b)³, you recognise the pattern and write the answer down in one line. This chapter collects the square, cube and three-variable identities and drills you on spotting which one fits.
The expansion of (3x - 4y)² is:
Answer
The expansion is 9x² - 24xy + 16y². Apply (a - b)² = a² - 2ab + b² with a = 3x and b = 4y. Then a² = 9x², b² = 16y², and 2ab = 2 × 3x × 4y = 24xy, which is subtracted. The option 9x² - 12xy + 16y² comes from forgetting to double the product term.
Using a suitable identity, the value of 103 × 97 is:
Answer
The value is 9991. Write the product as (100 + 3)(100 - 3), which matches (a + b)(a - b) = a² - b². So the product is 100² - 3² = 10000 - 9 = 9991. Recognising the difference-of-squares pattern avoids long multiplication entirely.
If a + 1/a = 6, then the value of a² + 1/a² is:
Answer
The value is 34. Squaring both sides gives (a + 1/a)² = a² + 2(a)(1/a) + 1/a² = a² + 1/a² + 2. So 36 = a² + 1/a² + 2, giving a² + 1/a² = 34. The answer 36 is what you get by wrongly squaring term by term and ignoring the middle term.
The factorisation of 27p³ - 64q³ is:
Answer
It factorises as (3p - 4q)(9p² + 12pq + 16q²). Note 27p³ = (3p)³ and 64q³ = (4q)³, so use a³ - b³ = (a - b)(a² + ab + b²) with a = 3p and b = 4q. Then a² = 9p², ab = 12pq and b² = 16q², all added inside the second bracket. The sign in the quadratic factor is plus for a difference of cubes, which is what distinguishes it from the second option.
Assertion (A): If x + y + z = 0, then x³ + y³ + z³ = 3xyz. Reason (R): x³ + y³ + z³ - 3xyz factorises as (x + y + z)(x² + y² + z² - xy - yz - zx).
Answer
Both A and R are true and R is the correct explanation of A. The factorisation in R is a standard identity and is correct. Substituting x + y + z = 0 into it makes the whole right-hand side zero, so x³ + y³ + z³ - 3xyz = 0, which is exactly the statement in A. Since A follows directly from R, R is the correct explanation.
Expand (2m + 5n + 3p)² using a suitable identity.
Answer
Use (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca with a = 2m, b = 5n and c = 3p. Squares: (2m)² = 4m², (5n)² = 25n², (3p)² = 9p². Cross terms: 2ab = 2(2m)(5n) = 20mn, 2bc = 2(5n)(3p) = 30np, 2ca = 2(3p)(2m) = 12mp. So (2m + 5n + 3p)² = 4m² + 25n² + 9p² + 20mn + 30np + 12mp.
Evaluate 98³ without direct multiplication, using an algebraic identity.
Answer
Write 98 = 100 - 2 and use (a - b)³ = a³ - b³ - 3ab(a - b) with a = 100, b = 2. a³ = 1000000 and b³ = 8. 3ab(a - b) = 3 × 100 × 2 × 98 = 600 × 98 = 58800. So 98³ = 1000000 - 8 - 58800 = 941192. Hence 98³ = 941192.
Factorise 49x² - 84xy + 36y².
Answer
Notice 49x² = (7x)² and 36y² = (6y)², while the middle term 84xy = 2 × 7x × 6y. This fits the pattern a² - 2ab + b² = (a - b)² with a = 7x and b = 6y. Therefore 49x² - 84xy + 36y² = (7x - 6y)².
(a) If a - b = 5 and ab = 14, find a² + b² and a³ - b³. (b) Factorise 8x³ + 125y³ + 1 - 30xy.
Answer
(a) From a² + b² = (a - b)² + 2ab, we get a² + b² = 5² + 2(14) = 25 + 28 = 53. For the cube, use a³ - b³ = (a - b)³ + 3ab(a - b) = 5³ + 3(14)(5) = 125 + 210 = 335. So a² + b² = 53 and a³ - b³ = 335. (b) Write the expression as (2x)³ + (5y)³ + (1)³ - 3(2x)(5y)(1), since 3 × 2x × 5y × 1 = 30xy. This matches a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca) with a = 2x, b = 5y, c = 1. So the factorisation is (2x + 5y + 1)(4x² + 25y² + 1 - 10xy - 5y - 2x).
If x + 1/x = 4, find (a) x² + 1/x², (b) x³ + 1/x³, and (c) x⁴ + 1/x⁴, showing every step.
Answer
(a) Square both sides: (x + 1/x)² = x² + 2 + 1/x². So 16 = x² + 1/x² + 2, giving x² + 1/x² = 14. (b) Use the identity x³ + 1/x³ = (x + 1/x)³ - 3(x + 1/x). So x³ + 1/x³ = 4³ - 3(4) = 64 - 12 = 52. (c) Square the result of part (a): (x² + 1/x²)² = x⁴ + 2 + 1/x⁴. So 14² = x⁴ + 1/x⁴ + 2, that is 196 = x⁴ + 1/x⁴ + 2, giving x⁴ + 1/x⁴ = 194. Hence the three values are 14, 52 and 194.
Read the following and answer the questions that follow: A gardener has a square lawn of side (x + 6) metres. She decides to remove a smaller square patch of side (x - 6) metres from one corner to build a shed, and lays turf over the remaining area. (a) Write an expression for the area of turf laid, using a suitable identity, and simplify it fully. (b) Find the turf area when x = 10 metres. (c) The gardener also fences the larger square. Write its perimeter in terms of x. (d) If the turf costs Rs 45 per square metre, what is the cost of turfing when x = 10?
Answer
(a) Turf area = (x + 6)² - (x - 6)². Using a² - b² = (a + b)(a - b) with a = (x + 6) and b = (x - 6): a + b = 2x and a - b = 12, so the area is 2x × 12 = 24x square metres. (b) When x = 10, area = 24 × 10 = 240 square metres. (Check directly: 16² - 4² = 256 - 16 = 240, which agrees.) (c) The larger square has side (x + 6), so its perimeter is 4(x + 6) = 4x + 24 metres. (d) Cost = 240 × 45 = Rs 10800.
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