CBSE Class 09 Mathematics – Exploring Algebraic Identities (NCERT Chapter 4)
1. The identity \((a + b)^2 = a^2 + 2ab + b^2\) is true for
(a) only positive integers
(b) only rational numbers
(c) all real numbers
(d) only natural numbers
2. Which of the following is not an identity?
(a) \((x + y)^2 = x^2 + 2xy + y^2\)
(b) \((x - y)^2 = x^2 - 2xy + y^2\)
(c) \(x^2 - 1 = 24\)
(d) \((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca\)
3. Using the identity \((a - b)^2 = a^2 - 2ab + b^2\), the value of \(79^2\) is
(a) 6241
(b) 6400
(c) 6084
(d) 6240
4. The factors of \(x^2 + 7x + 12\) obtained by splitting the middle term are
(a) \((x + 3)(x + 4)\)
(b) \((x + 2)(x + 5)\)
(c) \((x + 6)(x + 1)\)
(d) \((x - 3)(x - 4)\)
5. Assertion (A): \((a + b)^2 = a^2 + 2ab + b^2\) holds for all real values of \(a\) and \(b\).
Reason (R): An algebraic identity is true for every value of the variables involved.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
6. Assertion (A): \(x^3 - y^3 = (x - y)(x^2 + xy + y^2)\) is an identity.
Reason (R): The identity \(x^3 - y^3 = (x - y)(x^2 + xy + y^2)\) can be verified by expanding the right-hand side using the distributive property.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
7. The expression \(36x^2 + 12x + 1\) factors as
(a) \((6x + 1)^2\)
(b) \((6x - 1)^2\)
(c) \((9x + 1)^2\)
(d) \((3x + 1)^2\)
8. Which identity is used to evaluate \(119^2\) as \((100 + 10 + 9)^2\)?
(a) \((a + b)^2 = a^2 + 2ab + b^2\)
(b) \((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca\)
(c) \((a - b)^2 = a^2 - 2ab + b^2\)
(d) \(a^2 - b^2 = (a + b)(a - b)\)
9. The simplified form of \(\frac{x^2 - 7x + 12}{5x^2 + 5x - 100}\) (assuming the denominator ≠ 0) is
(a) \(\frac{x - 3}{5(x + 5)}\)
(b) \(\frac{x - 4}{5(x + 5)}\)
(c) \(\frac{x - 3}{5(x - 4)}\)
(d) \(\frac{x - 4}{5(x - 5)}\)
10. The identity \((a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3\) is obtained by
(a) multiplying \((a + b)\) by \((a^2 + 2ab + b^2)\)
(b) adding \((a + b)^2\) and \((a + b)\)
(c) subtracting \((a - b)^3\) from \((a + b)^3\)
(d) using only the distributive property on three linear factors
11. Expand \((7x + 4y)^2\) using a suitable identity.
12. Find the value of \(105^2\) using the identity \((a + b)^2 = a^2 + 2ab + b^2\).
13. Factorise completely: \(9x^2 + 24xy + 16y^2\).
14. Using the identity \((a - b)^2 = a^2 - 2ab + b^2\), evaluate \(193^2\).
15. Factorise: \(x^2 + 11x + 30\) by splitting the middle term.
16. Verify the identity \((x + y)^2 = x^2 + 2xy + y^2\) for \(x = -2\), \(y = -3\).
17. Expand \((p + 3q + 7r)^2\) using the identity \((a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca\).
18. Factorise completely: \(16s^2 + 25t^2 - 40st\).
19. Evaluate \(117^2\) using a suitable identity and state which identity is used.
20. Factorise: \(50p^2 + 60pq + 18q^2\) (take out the common factor first).
21. Simplify the rational expression \(\frac{x^2 - 7x + 12}{5x^2 + 5x - 100}\) (assuming the denominator ≠ 0).
22. (Multi-step) A rectangular pool has length \(x\) metres and breadth \(x - 4\) metres. Its area is 96 m².
(i) Form the quadratic equation.
(ii) Factorise the equation using the splitting-the-middle-term method.
(iii) Find the possible values of \(x\) and state which value is valid.
(iv) Hence find the length and breadth of the pool.
23. Prove that if three consecutive square numbers are taken as \((n-1)^2\), \(n^2\) and \((n+1)^2\), then the sum of the smallest and largest minus twice the middle square always equals 2. Use algebraic identities to justify each step.
24. Factorise completely using suitable identities:
\(9a^2 + b^2 + 4c^2 - 6ab + 12ac - 4bc\).
25. Saira arranged one square tile of side \(x\) units, eight rectangular strips of size \(x \times 1\) and fifteen unit squares of side 1 to form a larger rectangle.
(i) Write the total area covered by all the pieces.
(ii) Factorise the expression obtained in (i).
(iii) State the length and breadth of the larger rectangle in terms of \(x\).
(iv) Verify that the area of the larger rectangle equals the total area of the pieces.
26. In a village, a square playground of side 40 m has a path of uniform width \(s\) metres constructed all around it for walking.
(i) Express the outer length of the larger square formed by the playground plus the path.
(ii) Write an expression for the area of the larger square.
(iii) Write an expression for the area of the original playground.
(iv) Hence obtain an algebraic expression (in terms of \(s\)) for the area of the path only.
1. (c)
2. (c)
3. (a)
4. (a)
5. (a)
6. (a)
7. (a)
8. (b)
9. (a)
10. (a)
11.
\((7x + 4y)^2 = (7x)^2 + 2(7x)(4y) + (4y)^2\)
= \(49x^2 + 56xy + 16y^2\)
(correct identity — 1 mark; expansion — 1 mark)
12.
\(105^2 = (100 + 5)^2 = 100^2 + 2 \times 100 \times 5 + 5^2 = 10000 + 1000 + 25 = 11025\)
(correct identity — 1 mark; substitution & answer — 1 mark)
13.
\(9x^2 + 24xy + 16y^2 = (3x)^2 + 2(3x)(4y) + (4y)^2 = (3x + 4y)^2\)
(correct identity & factorisation — 2 marks)
14.
\(193^2 = (200 - 7)^2 = 200^2 - 2 \times 200 \times 7 + 7^2 = 40000 - 2800 + 49 = 37249\)
(correct identity — 1 mark; steps & answer — 1 mark)
15.
\(x^2 + 11x + 30 = x^2 + (5x + 6x) + 30 = (x + 5)(x + 6)\)
(a + b = 11, ab = 30 — 1 mark; correct splitting & factors — 1 mark)
16.
LHS = \((-2 - 3)^2 = (-5)^2 = 25\)
RHS = \((-2)^2 + 2(-2)(-3) + (-3)^2 = 4 + 12 + 9 = 25\)
LHS = RHS (identity verified) — 2 marks
17.
\((p + 3q + 7r)^2 = p^2 + (3q)^2 + (7r)^2 + 2p(3q) + 2(3q)(7r) + 2(7r)p\)
= \(p^2 + 9q^2 + 49r^2 + 6pq + 42qr + 14pr\)
(correct identity — 1 mark; all six terms — 2 marks)
18.
\(16s^2 + 25t^2 - 40st = (4s)^2 + (5t)^2 - 2(4s)(5t) = (4s - 5t)^2\)
(correct identity — 2 marks; factorisation — 1 mark)
19.
\(117^2 = (100 + 17)^2\) or better \((120 - 3)^2 = 120^2 - 2 \times 120 \times 3 + 3^2 = 14400 - 720 + 9 = 13689\)
(identity stated — 1 mark; correct steps — 2 marks)
20.
\(50p^2 + 60pq + 18q^2 = 2(25p^2 + 30pq + 9q^2) = 2(5p + 3q)^2\)
(common factor — 1 mark; identity application — 2 marks)
21.
Numerator: \(x^2 - 7x + 12 = (x - 3)(x - 4)\)
Denominator: \(5x^2 + 5x - 100 = 5(x^2 + x - 20) = 5(x - 4)(x + 5)\)
Expression simplifies to \(\frac{x - 3}{5(x + 5)}\) (x ≠ 4)
(factorisation of num. & den. — 2 marks; cancellation — 1 mark)
22.
(i) \(x(x - 4) = 96 \Rightarrow x^2 - 4x - 96 = 0\) (1 mark)
(ii) Split –4x as –12x + 8x → \(x(x - 12) + 8(x - 12) = (x - 12)(x + 8) = 0\) (2 marks)
(iii) x = 12 or x = –8; length = 12 m (discard negative) (1 mark)
(iv) Breadth = 12 – 4 = 8 m (1 mark)
23.
\((n-1)^2 + (n+1)^2 - 2n^2 = (n^2 - 2n + 1 + n^2 + 2n + 1) - 2n^2 = 2n^2 + 2 - 2n^2 = 2\)
(each expansion using identity — 2 marks; simplification — 3 marks)
24.
\(9a^2 + b^2 + 4c^2 - 6ab + 12ac - 4bc = (3a - b + 2c)^2\)
(recognise as perfect square — 2 marks; verification by expansion — 3 marks)
25.
(i) Area = \(x^2 + 8x + 15\) (1 mark)
(ii) \(x^2 + 8x + 15 = (x + 3)(x + 5)\) (1 mark)
(iii) Length = x + 5, breadth = x + 3 (1 mark)
(iv) Area of rectangle = (x + 5)(x + 3) = x² + 8x + 15 (matches) (1 mark)
26.
(i) Outer side = 40 + 2s (1 mark)
(ii) Outer area = (40 + 2s)² (1 mark)
(iii) Playground area = 40² = 1600 (1 mark)
(iv) Path area = (40 + 2s)² – 1600 = 1600 + 160s + 4s² – 1600 = 160s + 4s² (1 mark)
All questions are answerable from the NCERT chapter text. Reviewed by GFIS faculty.