# PRACTICE TEST USING ALGEBRAIC IDENTITIES

(1)  Expand following

(i)  (2x + 3y + 4z)2

(ii) (−p +2q + 3r)2

(iii) (2p + 3)(2p −4)(2p −5)

(iv) (3a +1)(3a −2)(3a + 4)           Solution

(1)  Using algebraic identity, find the coefficients of x2 , x and constant term without actual expansion.

(i) (x + 5)(x + 6)(x + 7)

(ii) (2x + 3)(2x −5)(2x −6)            Solution

(2)  If (x + a)(x + b)(x + c) = x3 + 14x2 + 59x + 70 , find the value of

(i)  a + b + c

(ii)  (1/a) + (1/b) + (1/c)

(iii)  a2 + b2 + c2

(iv)  (a/bc) + (b/ac) + (c/ab)          Solution

(3)  Expand :

(i) (3a - 4b)3

(ii)  (x + (1/y))3                  Solution

(2)  Evaluate the following by using identities:

(i) 983

(ii)  10013                 Solution

(3)  If (x + y + z) = 9 and (xy + yz + zx) = 26, then find the value of x2 + y2 + z2                 Solution

(4)  Find 27a3 + 64b3 , if 3a + 4b = 10 and ab = 2

Solution

(5)  Find x3 - y3, if  x - y  =  5 and xy  =  14   Solution

(6)  If a + (1/a)  =  6, then find the value of a3 + 1/a3

(7)  If x2 + 1/x2  =  23, then find the value of x + (1/x) and x3 + (1/x3)           Solution

(8)  If (y - (1/y))3  =  27, then find the value of y3 - (1/y)3

Solution

(9)  Simplify:

(i)  (2a + 3b + 4c)(4a2 + 9b2 + 16c2 - 6ab - 12bc - 8ca)

(ii) (x −2y + 3z)(x2 + 4y2 + 9z2 + 2xy + 6yz −3xz)

Solution

(10)  By using identity evaluate the following:

(i) 73 - 103 + 33

(ii)  1 + (1/8) - (27/8)            Solution

(11)  If 2x −3y −4z = 0 , then find 8x3 - 27y3 - 64z3

Solution

(12)  If 2x + (2/x)  =  3, what is the value of x2 + 1/x2 ?

(13)  If a+b+c  =  6 and a2+b2+c2  =  14, what is the value of (a-b)2 + (b-c)2 + (c-a)2 ?        Solution

(14)  If x-y  =  8 and xy  =  5, what is the value of

x3-y3+8(x+y)2    Solution

(15)  If x + y  =  5 and xy  =  6 and x > y, then find 2(x2+y2)          Solution

(16)  If a3-b3  =  513 and a-b  =  3, what is the value of ab ?             Solution

(17)  If a2+b2+c2  =  9 and ab+bc+ca  =  8, what is the vale of (a+b+c)2         Solution

(18)  a+b  =  √7 and a-b  =  √5, then find the value of 8ab(a2+b2)           Solution

(19) If a-b  =  4 and ab  =  60, what is the value of a+b ?

(20)  If a+b  =  9m and ab  =  18m2, what is the value of a-b ?  Solution

(21)  Simplify :

(2345 x 2345 - 759x759)/(2345-759)

(22)  Factorize 9x² - 24xy + 16y²

Solution

(23)  Factorize 4x² + 12xy + 9y²

Solution

(24)  Factorize 16a² - 8a + 1

Solution

(25)  Factorize 16a² - 9b²

Solution

(26)  Factorize (a + b)² - (a - b)²

Solution

(27)  Factorize 8x³ - 125y³

Solution

(28)  Factorize 27x³ + 64y³

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