Wipro Sample Algebraic Problems

Below are three model problems dealing with the formulae (x+y)² = x² + y² + 2xy and (x-y)² = x² + y² - 2xy
Question 1
Find X when X - Y = 3 and X² + Y² = 89 where X and Y are integers.
a)10 b)-5 c)-10 d)-3
Answer : b)-5.
Solution :
We know that (x - y)² = x² + y² - 2xy
Sub. the given values,
3² = 89 - 2XY
9 - 89 = -2XY
80 = 2XY
XY = 40
Since X and Y are integers and XY = 40,the possibilities of X and Y are as follows:
(1,40), (2,20), (4,10), (5,8), (-1,-40), (-2,-20), (-4,-10) and (-5,-8)
X - Y is a positive integer(3), so X will be greater than Y.
By checking the given condition X - Y = 3, we have X = 8, Y = 5 or X = -5, Y = -8.
i.e., X is either 8 or -5.
From the given options we can conclude that X = -5.
Question 2
If the summation and multiplication of two integer is 24 and 143 respectively then the difference of them is:
a)2 b)1 c)12 d)4
Answer : a)2
Solution :
Let A and B be two integers.
Then A + B = 24 and AB = 143.
We know that (x+y)² = x² + y² + 2xy
Here, 24² = A² + B² + 2(143)
576 - 286 = A² + B²
A² + B² = 290.
i.e., Summation of the square of two integers is 290.
i.e., A² or B² is < or = 290.
Since A and B are integers, the possibilities are {1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289}.
From the above set of square numbers, the values may be
A² + B² = 1 + 289 or 121 + 169 = 290
A² + B² = 1² + 17² or 11² + 13² = 290
The condition A + B = 24 and AB = 143, is satisfied by A = 11, B = 13 or A = 13, B = 11
We have to find the difference of A and B.
Hence the difference is 2.
Question 3
If X - Y = 9, X² + Y² = 257 and X,Y are integers then what will be the value of X and Y ?
a)21,12 b)15,6 c)none of these d)cannot be determined
Answer : d)cannot be determined
Solution :
Here we use the formula, (x-y)² = x² + y² - 2xy
9² = 257 - 2XY
XY = (257-81)/2 = 176/2 = 88
XY = 88
Since X and Y are integers and XY = 88 then the possibilities of X and Y are (1,88), (2,44), (4,22), (8,11), (-1,-88), (-2,-44), (-4,-22) and (-8,-11).
By observing, none of the above possibility of X and Y satisfies X - Y = 9.
Hence, the values of X and Y cannot be determined by given conditions.