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NCERT Solutions · Maths Ch 4
Exploring Algebraic Identities
98 questions · 6 exercises
Closed
Open all
Ex 4.1
Ex 4.2
Ex 4.3
Ex 4.4
Ex 4.5
Ex 4.6
NCERT answers with full exam-style working, step by step. Tap a question to open its solution.
1
9 QUESTIONS
Exercise 4.1
Q1(i)
Using the identity (a + b)² = a² + 2ab + b², expand: (7x + 4y)²
(2 marks)
Q1(ii)
Using the identity (a + b)² = a² + 2ab + b², expand: (7x/5 + 3y/2)²
(2 marks)
Q1(iii)
Using the identity (a + b)² = a² + 2ab + b², expand: (2.5p + 1.5q)²
(2 marks)
Q1(iv)
Using the identity (a + b)² = a² + 2ab + b², expand: (3s/4 + 8t)²
(2 marks)
Q1(v)
Using the identity (a + b)² = a² + 2ab + b², expand: (x + 1/(2y))²
(2 marks)
Q1(vi)
Using the identity (a + b)² = a² + 2ab + b², expand: (1/x + 1/y)²
(2 marks)
Q2(i)
Using the identity (a + b)² = a² + 2ab + b², find the value of: (64)²
(2 marks)
Q2(ii)
Using the identity (a + b)² = a² + 2ab + b², find the value of: (105)²
(2 marks)
Q2(iii)
Using the identity (a + b)² = a² + 2ab + b², find the value of: (205)²
(2 marks)
2
9 QUESTIONS
Exercise 4.2
Q1(i)
Factor completely: 9x² + 24xy + 16y²
(2 marks)
Q1(ii)
Factor completely: 4s² + 20st + 25t²
(2 marks)
Q1(iii)
Factor completely: 49x² + 28xy + 4y²
(2 marks)
Q1(iv)
Factor completely: 64p² + (32/3)pq + (4/9)q²
(2 marks)
Q1(v)
Factor completely: 3a² + 4ab + (4/3)b²
(2 marks)
Q1(vi)
Factor completely: (9/5)s² + 6sv + 5v²
(2 marks)
Q2(i)
Find the value using the identity (a − b)² = a² − 2ab + b²: (79)²
(2 marks)
Q2(ii)
Find the value using the identity (a − b)² = a² − 2ab + b²: (193)²
(2 marks)
Q2(iii)
Find the value using the identity (a − b)² = a² − 2ab + b²: (299)²
(2 marks)
3
14 QUESTIONS
Exercise 4.3
Q1(i)
Find the following square using a suitable identity: 117²
(2 marks)
Q1(ii)
Find the following square using a suitable identity: 78²
(2 marks)
Q1(iii)
Find the following square using a suitable identity: 198²
(2 marks)
Q1(iv)
Find the following square using a suitable identity: 214²
(2 marks)
Q1(v)
Find the following square using a suitable identity: 1104²
(2 marks)
Q1(vi)
Find the following square using a suitable identity: 1120²
(2 marks)
Q2(i)
Factor using a suitable identity: 16y² – 24y + 9
(2 marks)
Q2(ii)
Factor using a suitable identity: (9/4)s² + 6st + 4t²
(2 marks)
Q2(iii)
Factor using a suitable identity: m²/9 + mk/3 + k²/4 + 3nk + 2mn + 9n²
(2 marks)
Q2(iv)
Factor using a suitable identity: p²/16 – 2 + 16/p²
(2 marks)
Q2(v)
Factor using a suitable identity: 9a² + 4b² + c² – 12ab + 6ac – 4bc
(2 marks)
Q3(i)
Expand using the identity (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca: (p + 3q + 7r)²
(2 marks)
Q3(ii)
Expand using the identity (a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca: (3x – 2y + 4z)²
(2 marks)
Q4
Is this an identity? (a + b – c)² + (a – b + c)² + (a – b – c)² = 2a² + 2b² + 2c².
(3 marks)
4
17 QUESTIONS
Exercise 4.4
Q1(i)
Fill in the blanks to complete the identity: s² – 11s + 24 = (____)(____)
(1 mark)
Q1(ii)
Fill in the blank to complete the identity: (____)(x + 1) = 3x² – 4x – 7
(1 mark)
Q1(iii)
Fill in the blanks to complete the identity: 10x² – 11x – 6 = (2x – ___)(___ + 2)
(1 mark)
Q1(iv)
Fill in the blanks to complete the identity: 6x² + 7x + 2 = (____)(____)
(1 mark)
Q2(i)
Select and use the identity that helps you find the product without multiplying directly: (41)²
(1 mark)
Q2(ii)
Select and use the identity that helps you find the product without multiplying directly: (27)²
(1 mark)
Q2(iii)
Select and use the identity that helps you find the product without multiplying directly: (23 × 17)
(1 mark)
Q2(iv)
Select and use the identity that helps you find the product without multiplying directly: (135)²
(1 mark)
Q2(v)
Select and use the identity that helps you find the product without multiplying directly: (97)²
(1 mark)
Q2(vi)
Select and use the identity that helps you find the product without multiplying directly: (18 × 29)
(1 mark)
Q2(vii)
Select and use the identity that helps you find the product without multiplying directly: (34 × 43)
(1 mark)
Q2(viii)
Select and use the identity that helps you find the product without multiplying directly: (205)²
(1 mark)
Q3(i)
Factor the following: 9a² + b² + 4c² – 6ab + 12ac – 4bc
(2 marks)
Q3(ii)
Factor the following: 16s² + 25t² – 40st
(2 marks)
Q3(iii)
Factor the following: r² – r – 42
(2 marks)
Q3(iv)
Factor the following: 49g² + 14gh + h²
(2 marks)
Q3(v)
Factor the following: 64u² + 121v² + 4w² – 176uv – 32uw + 44vw
(2 marks)
5
6 QUESTIONS
Exercise 4.5
Q1(i)
Simplify the rational expression (denominators are not zero): (3p² – 3pq – 18q²)/(p² + 3pq – 10q²)
(3 marks)
Q1(ii)
Simplify the rational expression (denominators are not zero): (n³ – 3n²m + 3nm² – m³)/(5m² – 10mn + 5n²)
(3 marks)
Q1(iii)
Simplify the rational expression (denominators are not zero): (w³ – v³ + x³ + 3wvx)/(w² + v² + x² – 2wv – 2vx + 2wx)
(3 marks)
Q1(iv)
Simplify the rational expression (denominators are not zero): (4y² – 20yz + 25z²)/(25z² – 4y²)
(2 marks)
Q1(v)
Simplify the rational expression (denominators are not zero): (x² + x – 6)(x² – 7x + 12)/[(x² – 6x + 8)(x² – 9)]
(3 marks)
Q1(vi)
Simplify the rational expression (denominators are not zero): (p⁴ – 16)/(p² – 4p + 4)
(2 marks)
6
EXERCISE 4.6 · 43 QUESTIONS
End-of-Chapter Exercises
Q1(i)
Use suitable identities to find the product: (–3x + 4)²
(1 mark)
Q1(ii)
Use suitable identities to find the product: (2s + 7)(2s – 7)
(1 mark)
Q1(iii)
Use suitable identities to find the product: (p² + 1/2)(p² – 1/2)
(1 mark)
Q1(iv)
Use suitable identities to find the product: (2n + 7)(2n – 7)
(1 mark)
Q1(v)
Use suitable identities to find the product: (s – 2t)(s² + 2st + 4t²)
(1 mark)
Q1(vi)
Use suitable identities to find the product: (1/(2r) – 4r)²
(1 mark)
Q1(vii)
Use suitable identities to find the product: (–3m + 4k – l)²
(2 marks)
Q1(viii)
Use suitable identities to find the product: (x – y/3)³
(2 marks)
Q1(ix)
Use suitable identities to find the product: (7k/2 – 2m/3)³
(2 marks)
Q2(i)
Find the value using a suitable identity: 17 × 21
(1 mark)
Q2(ii)
Find the value using a suitable identity: 104 × 96
(1 mark)
Q2(iii)
Find the value using a suitable identity: 24 × 16
(1 mark)
Q2(iv)
Find the value using a suitable identity: 147³
(2 marks)
Q2(v)
Find the value using a suitable identity: 199³
(2 marks)
Q2(vi)
Find the value using a suitable identity: 127³
(2 marks)
Q2(vii)
Find the value using a suitable identity: (–107)³
(2 marks)
Q2(viii)
Find the value using a suitable identity: (–299)³
(2 marks)
Q3(i)
Factor the following algebraic expression: 4y² + 1 + 1/(16y²)
(2 marks)
Q3(ii)
Factor the following algebraic expression: 9m² – 1/(25n²)
(2 marks)
Q3(iii)
Factor the following algebraic expression: 27b³ – 1/(64b³)
(2 marks)
Q3(iv)
Factor the following algebraic expression: x² + 5x/6 + 1/6
(2 marks)
Q3(v)
Factor the following algebraic expression: 27u³ – 1/125 – 27u²/5 + 9u/25
(2 marks)
Q3(vi)
Factor the following algebraic expression: 64y³ + z³/125
(2 marks)
Q3(vii)
Factor the following algebraic expression: p³ + 27q³ + r³ – 9pqr
(2 marks)
Q3(viii)
Factor the following algebraic expression: 9m² – 12m + 4
(2 marks)
Q3(ix)
Factor the following algebraic expression: 9x³ – (8/3)y³ + z³/3 + 6xyz
(2 marks)
Q3(x)
Factor the following algebraic expression: 4x² + 9y² + 36z² + 12xz + 36yz + 24xy
(2 marks)
Q3(xi)
Factor the following algebraic expression: 27u³ – 1/216 – 9u²/2 + u/4
(2 marks)
Q4(i)
Simplify (denominator ≠ 0): (4x² + 4x + 1)/(4x² – 1)
(2 marks)
Q4(ii)
Simplify (denominator ≠ 0): 9(3a³ – 24b³)/(9a² – 36b²)
(3 marks)
Q4(iii)
Simplify (denominator ≠ 0): (s³ + 125t³)/(s² – 2st – 35t²)
(3 marks)
Q5(i)
Find possible expressions for the length and breadth of the rectangle whose area is 25a² – 30ab + 9b² square units.
(2 marks)
Q5(ii)
Find possible expressions for the length and breadth of the rectangle whose area is 36s² – 49t² square units.
(2 marks)
Q6(i)
Find possible expressions for the length, breadth and height of the cuboid whose volume is 6a² – 24b² cubic units.
(2 marks)
Q6(ii)
Find possible expressions for the length, breadth and height of the cuboid whose volume is 3ps² – 15ps + 12p cubic units.
(2 marks)
Q7
The village playground is shaped as a square of side 40 metres. A path of width s metres is created around the playground for people to walk. Find an expression for the area of the path in terms of s.
(3 marks)
Q8
If a number plus its reciprocal equals 10/3, find the number.
(2 marks)
Q9
A rectangular pool has area 2x² + 7x + 3 square hastas. If its width is 2x + 1 hastas, find its length.
(2 marks)
Q10
If both x – 2 and x – 1/2 are factors of px² + 5x + r, show that p = r.
(3 marks)
Q11
If a + b + c = 5 and ab + bc + ca = 10, then prove that a³ + b³ + c³ – 3abc = –25.
(3 marks)
Q12
By factoring the expression, check that n³ – n is always divisible by 6 for all natural numbers n. Give reasons.
(3 marks)
Q13(i)
Find the value of x³ + y³ – 12xy + 64, when x + y = –4.
(3 marks)
Q13(ii)
Find the value of x³ – 8y³ – 36xy – 216, when x = 2y + 6.
(3 marks)