Use suitable identities to find the following products:
(i) \((x + 4)(x + 10)\)
(ii) \((x + 8)(x - 10)\)
(iii) \((3x + 4)(3x - 5)\)
(iv) \((y^2 + \dfrac{3}{2})(y^2 - \dfrac{3}{2})\)
(v) \((3 - 2x)(3 + 2x)\)
(i) \(x^2 + 14x + 40\)
(ii) \(x^2 - 2x - 80\)
(iii) \(9x^2 - 3x - 20\)
(iv) \(y^4 - \dfrac{9}{4}\)
(v) 9 - 4x^2
Evaluate the following products without multiplying directly:
(i) 103 × 107
(ii) 95 × 96
(iii) 104 × 96
(i) 11021
(ii) 9120
(iii) 9984
Factorise the following using appropriate identities:
(i) \(9x^2 + 6xy + y^2\)
(ii) \(4y^2 - 4y + 1\)
(iii) \(x^2 - \dfrac{y^2}{100}\)
(i) \((3x + y)(3x + y)\)
(ii) \((2y - 1)(2y - 1)\)
(iii) \((x + \dfrac{y}{10})(x - \dfrac{y}{10})\)
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) \([\dfrac{1}{4}a - \dfrac{1}{2}b + 1]^2\)
(i) \(x^2 + 4y^2 + 16z^2 + 4xy + 16yz + 8xz\)
(ii) \(4x^2 + y^2 + z^2 - 4xy - 2yz + 4xz\)
(iii) \(4x^2 + 9y^2 + 4z^2 - 12xy + 12yz - 8xz\)
(iv) \(9a^2 + 49b^2 + c^2 - 42ab + 14bc - 6ac\)
(v) \(4x^2 + 25y^2 + 9z^2 - 20xy - 30yz + 12xz\)
(vi) \(\dfrac{a^2}{16} + \dfrac{b^2}{4} + 1 - \dfrac{ab}{4} - b + \dfrac{a}{2}\)
Factorise:
(i) \((2x + 3y - 4z)(2x + 3y - 4z)\)
(ii) \(( -\sqrt{2}x + y + 2\sqrt{2}z)( -\sqrt{2}x + y + 2\sqrt{2}z)\)
(i) \(8x^3 + 12x^2y + 6x + 1\)
(ii) \(8a^3 - 27b^3 - 36a^2b + 54ab^2\)
Write the following cubes in expanded form:
(i) \((2x + 1)^3\)
(ii) \((2a - 3b)^3\)
(iii) \([ \dfrac{3}{2}x + 1]^3\)
(iv) \([x - \dfrac{2}{3}y]^3\)
(i) \(8x^3 + 27/8 x^3 + 27/4 x^2 + 9/2 x + 1\)
(ii) \(x^3 - \dfrac{8}{27}y^3 - 2x^2y + \dfrac{4xy^2}{3}\)
Evaluate the following using suitable identities:
(i) \((99)^3\)
(ii) \((102)^3\)
(iii) \((998)^3\)
(i) 970299
(ii) 1061208
(iii) 994011992
Factorise each of the following:
(i) \(8a^3 + b^3 + 12a^2b + 6ab^2\)
(ii) \(8a^3 - b^3 - 12a^2b + 6ab^2\)
(iii) \(27 - 125a^3 - 135a + 225a^2\)
(iv) \(64a^3 - 27b^3 - 144a^2b + 108ab^2\)
(v) \(27p^3 - \dfrac{1}{216} - \dfrac{9}{2}p^2 + \dfrac{1}{4}p\)
(i) (2a + b)(2a + b)(2a + b)
(ii) (2a - b)(2a - b)(2a - b)
(iii) (3 - 5a)(3 - 5a)(3 - 5a)
(iv) (4a - 3b)(4a - 3b)(4a - 3b)
(v) (3p - 1/6)(3p - 1/6)(3p - 1/6)
Verify:
(i) \(x^3 + y^3 = (x + y)(x^2 - xy + y^2)\)
(ii) \(x^3 - y^3 = (x - y)(x^2 + xy + y^2)\)
Simplify RHS.
Factorise the following:
(i) \(27y^3 + 125z^3\)
(ii) \(64m^3 - 343n^3\)
(i) \((3y + 5z)(9y^2 - 15yz + 25z^2)\)
(ii) \((4m - 7n)(16m^2 + 49n^2 + 28mn)\)
Factorise: \(27x^3 + y^3 + z^3 - 9xyz\)
(3x + y + z)(9x^2 + y^2 + z^2 - 3xy - yz - 3xz)
Verify that \(x^3 + y^3 + z^3 - 3xyz = \dfrac{1}{2}(x + y + z)[(x - y)^2 + (y - z)^2 + (z - x)^2]\)
Simplify RHS.
If \(x + y + z = 0\), show that \(x^3 + y^3 + z^3 = 3xyz\)
Put \(x + y + z = 0\) in the identity in Q12.
Without calculating cubes, find the value of each:
(i) \((-12)^3 + (7)^3 + (5)^3\)
(ii) \((28)^3 + (-15)^3 + (-13)^3\)
(i) -1260
(ii) 16380
Give possible expressions for the length and breadth of rectangles whose areas are:
(i) \(25a^2 - 35a + 12\)
(ii) \(35y^2 + 13y - 12\)
(i) One possible answer: Length = \(5a - 3\), Breadth = \(5a - 4\)
(ii) One possible answer: Length = \(7y - 3\), Breadth = \(5y + 4\)
Find possible expressions for the dimensions of cuboids whose volumes are:
(i) \(3x^2 - 12x\)
(ii) \(12ky^2 + 8ky - 20k\)
(i) One possible answer: \(3x, x, x - 4\)
(ii) One possible answer: \(4k, 3y + 5, y - 1\)