Question 4 of 26intermediate🔍 AnalyzeShort Answer2 marks

Restate the following statements with appropriate conditions, so that they become true.

(i)

If a2>b2a^2 > b^2, then a>ba > b.

Answer

If a2>b2a^2 > b^2 and a,b>0a, b > 0, then a>ba > b.

(ii)

If x2=y2x^2 = y^2, then x=yx = y.

Answer

If x2=y2x^2 = y^2 and x,y≥0x, y \geq 0, then x=yx = y.

(iii)

If (x+y)2=x2+y2(x + y)^2 = x^2 + y^2, then x=0x = 0.

Answer

If (x+y)2=x2+y2(x + y)^2 = x^2 + y^2 and y≠0y \neq 0, then x=0x = 0.

(iv)

The diagonals of a quadrilateral bisect each other.

Answer

The diagonals of a parallelogram bisect each other.

Explanation

This question tests the ability to identify missing conditions that make mathematical statements universally true. Each original statement has counterexamples that make it false. For (i), if a = -5 and b = 3, then a² > b² but a < b. Adding the condition that both numbers are positive makes it true. For (ii), x = 3 and y = -3 gives x² = y² but x ≠\neq y; restricting to non-negative numbers resolves this. For (iii), expanding gives 2xy = 0, so either x = 0 or y = 0; the condition y ≠\neq 0 ensures x must be 0. For (iv), not all quadrilaterals have bisecting diagonals—only parallelograms have this property.

PreviousNext