Q. 155.0( 6 Votes )

i. Let’s write for which value of t, will be a whole square form.

ii. Let’s write the expression which when added to gives a whole square.

iii. If a and b are positive integers and lets write the value of a & b.

iv. an identity or an equation? Write with reason.

v. For each positive or negative value of x and y except zero the value of is always (positive or negative)

Answer :

i. For making a complete square, we need to reduce the expression in x2± 2xy + y2form, so if we put t = ±1, then we get expression as





Hence, for t = ±1, the given equation will be a whole square form.


ii. For making a complete square, we need to reduce the expression in x2 + 2xy + y2form, so if we add ±4x to given equation, we have


a2 + 4 ± 4x


= a2± 4x + 4


= a2± 2(2)x + 22


= (a ± 2)2


Hence, if we add 4x to the given expression, the expression becomes a whole square.


iii. Given,


a2 – b2 = 9 × 11


we know, a2 – b2 = (a - b)(a + b)


(a – b)(a + b) = 9 × 11


On comparison, we get


a – b = 9 [1]


a + b = 11 [2]


Adding equation [1] and [2]


a + b + a – b = 9 + 11


2a = 20


a = 10


Putting value of ‘a’ in equation [1], we get


10 – b = 9


b = 10 – 9


b = 1


iv. Identity, as


Taking LHS


Now, we know


(x + y)2 = x2 + 2xy + y2


and (x – y)2 = x2 – 2xy + y2


Applying above identities in LHS, we get


= (x + y)2 – (x – y)2


= x2 + 2xy + y2 – (x2 – 2xy + y2)


= x2 + 2xy + y2 – x2 + 2xy – y2


= 4xy


= RHS


As, LHS = RHS the given expression is an identity!


v. Square of any number is always positive


[Explanation: Suppose -a (a > 0) is a negative number, then


(-a)2 = -a × (-a) = a2]


Therefore, a2and b2 both are positive numbers


a2 + b2 is a positive number, as sum of two positive numbers is positive!


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