Prove- -4X-7Y- 2-4X-7Y- 2-112Xy - Maths Q-A

Determine the expression of (4X+7Y)^2 (4X 7Y)^2=112 XYUse formula:(a+b)^2=a^2+b^2+2ab 0ex0ex(a b)^2=a^2+b^2 2abNow, considerL.H.S=(4X+7Y)^2 (4X 7Y)^20ex0ex ...

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Standard XII

Mathematics

Definition of a Determinant

Prove: (4X+7Y...

Question

Prove: ${(4 X + 7 Y)}^{2} - {(4 X - 7 Y)}^{2} = 112 X Y$

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Solution

Determine the expression of ${(4 X + 7 Y)}^{2} - {(4 X - 7 Y)}^{2} = 112 X Y$
Use formula:
${(a + b)}^{2} = a^{2} + b^{2} + 2 a b {(a - b)}^{2} = a^{2} + b^{2} - 2 a b$
Now, consider
$L . H . S = {(4 X + 7 Y)}^{2} - {(4 X - 7 Y)}^{2} = {{(4 X)}^{2} + (7 Y)}^{2} + 2 . 4 X . 7 Y - ({{(4 X)}^{2} + (7 Y)}^{2} - 2 . 4 X . 7 Y) {= {16 X}^{2} + 49 Y}^{2} + 56 X Y - ({{16 X}^{2} + 49 Y}^{2} - 56 X Y) {{= 16 X}^{2} + 49 Y}^{2} + 56 X Y - {{16 X}^{2} - 49 Y}^{2} + 56 X Y = 112 X Y = R . H . S$
Hence, proved.

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{(4 x + 7 y)}^{2} - {(4 x - 7 y)}^{2} = 112 x y

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Prove: (4X+7Y)2−(4X−7Y)2=112XY

Determine the expression of (4X+7Y)2−(4X−7Y)2=112XYUse formula:(a+b)2=a2+b2+2ab(a-b)2=a2+b2-2abNow, considerL.H.S=(4X+7Y)2−(4X−7Y)2=(4X)2+(7Y)2+2.4X.7Y−(4X)2+(7Y)2-2.4X.7Y=16X2+49Y2+56XY−16X2+49Y2-56XY=16X2+49Y2+56XY−16X2-49Y2+56XY=112XY=R.H.S Hence, proved.

Prove: ${(4 X + 7 Y)}^{2} - {(4 X - 7 Y)}^{2} = 112 X Y$