Question 1:Use suitable identities to find the following products: (i) (x + 4) (x + 10) (ii) (x + 8) (x – 10) (iii) (3x + 4) (3x – 5) (iv) (y2 + 3 2 ) (y2 – 3 2 ) (v) (3 – 2x) (3 + 2x)
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Question 2:Evaluate the following products without multiplying directly: (i) 103 × 107 (ii) 95 × 96 (iii) 104 × 96
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Question 3:Factorise the following using appropriate identities: (i) 9x2 + 6xy + y2 (ii) 4y2 – 4y + 1 (iii) x2 – 2 100
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Question 4:Expand each of the following, using suitable identities: (i) (x + 2y + 4z)2 (ii) (2x – y + z)2 (iii) (–2x + 3y + 2z)2 (iv) (3a – 7b – c)2 (v) (–2x + 5y – 3z)2 (vi) 2 1 1 1 4 2 a b − +
Question 5:Factorise: (i) 4x2 + 9y2 + 16z2 + 12xy – 24yz – 16xz (ii) 2x2 + y2 + 8z2 – 2 2 xy + 4 2 yz – 8xz
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Question 6:Write the following cubes in expanded form: (i) (2x + 1)3 (ii) (2a – 3b)3 (iii) 3 3 1 2 x + (iv) 3 2 3 x y −
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Question 7:Evaluate the following using suitable identities: (i) (99)3 (ii) (102)3 (iii) (998)3
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Question 8:Factorise each of the following: (i) 8a3 + b3 + 12a2b + 6ab2 (ii) 8a3 – b3 – 12a2b + 6ab2 (iii) 27 – 125a3 – 135a + 225a2 (iv) 64a3 – 27b3 – 144a2b + 108ab2 (v) 27p3 – 1 216 – 2 9 1 2 4 p + p
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Question 9:Verify : (i) x3 + y3 = (x + y) (x2 – xy + y2) (ii) x3 – y3 = (x – y) (x2 + xy + y2)
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Question 10:Factorise each of the following: i ) (27y3 + 125z3 (ii) 64m3 – 343n3 [Hint : See Question 9.]
EXERCISE 2.51. Use suitable identities to find the
Question 11:Factorise : 27x3 + y3 + z3 – 9xyz
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Question 12:Verify that x3 + y3 + z3 – 3xyz = 2 2 2 1 ( ) ( ) ( ) ( ) 2 x + y + z x − y + y − z + z − x
Question 13:If x + y + z = 0, show that x3 + y3 + z3 = 3xyz.
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Question 14:Without actually calculating the cubes, find the value of each of the following: (i) (–12)3 + (7)3 + (5)3 (ii) (28)3 + (–15)3 + (–13)3
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Question 15:Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given: Area : 25a2 – 35a + 12 A rea : 35y2 + 13y –12
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Question 16:What are the possible expressions for the dimensions of the cuboids whose volumes are given below? Volume : 3x2 – 12x Volume : 12ky2 + 8ky – 20k