Verified answer

Required Solutions:-

1. (x - 2y + 3z)²

→ We need to expand this using suitable identity.

So we know,

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

Hence,

(x - 2y + 3z)²

= (x)² + (-2y)² + (3z)² + 2[(x) × (-2y) + (-2y) × (3z) + (3z) × (x)]

= x² + 4y² + 9z² + 2[-2xy - 6yz + 3zx]

= x² + 4y² + 9z² - 4xy - 12yz + 6zx (Ans)

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2. (-5x + 2y + z)²

→ We'll expand this too by using the same identity:-

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

So,

(-5x + 2y + z)²

= (-5x)² + (2y)² + (z)² + 2[(-5x) × (2y) + (2y) × (z) + (z) × (-5x)]

= 25x² + 4y² + z² + 2[-10xy + 2yz - 5zx]

= 25x² + 4y² + z² - 20xy + 4yz - 10zx (Ans)

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3. (4x + y - 32)²

→ We'll expand this too by using the same identity:-

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

So,

(4x + y - 32)²

= (4x)² + (y)² + (-32)² + 2[(4x) × (y) + (y) × (-32) + (-32) × (4x)]

= 16x² + y² + 1024 + 2[4xy - 32y - 128x]

= 16x² + y² + 1024 + 8xy - 64y - 256x (Ans)

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4. (3a - 5b - 70)²

→ We'll expand this too by using the same identity:-

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

So,

(3a - 5b - 70)²

= (3a)² + (-5b)² + (-70)² + 2[(3a) × (-5b) + (-5b) × (-70) + (-70) × (3a)]

= 9a² + 25b² + 4900 + 2[-15ab + 350b - 210a]

= 9a² + 25b² + 4900 - 30ab + 700b - 420a (Ans)

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5. (-2a - b + 3c)²

→ We'll expand this too by using the same identity:-

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

So,

(-2a - b + 3c)²

= (-2a)² + (-b)² + (3c)² + 2[(-2a) × (-b) + (-b) × (3c) + (3c) × (-2a)]

= 4a² + b² + 9c² + 2[2ab - 3bc - 6ca]

= 4a² + b² + 9c² + 4ab - 6bc - 12ca (Ans)

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6. (-a + 6b - 2c)²

→ We'll expand this too by using the same identity:-

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

So,

(-a + 6b - 2c)²

= (-a)² + (6b)² + (-2c)² + 2[(-a) × (6b) + (6b) × (-2c) + (-2c) × (-a)]

= a² + 36b² + 4c² + 2[-6ab - 12bc + 2ca]

= a² + 36b² + 4c² - 12ab - 24bc + 4ca (Ans)

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7. (1 + 2x - 3y)²

→ We'll expand this too by using the same identity:-

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

So,

(1 + 2x - 3y)²

= (1)² + (2x)² + (-3y)² + 2[(1) × (2x) + (2x) × (-3y) + (-3y) × (1)]

= 1 + 4x² + 9y² + 2[2x - 6xy - 3y]

= 1 + 4x² + 9y² + 4x - 12xy - 6y (Ans)

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8. (2x - 4y - 12)²

→ We'll expand this too by using the same identity:-

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

So,

(2x - 4y - 12)²

= (2x)² + (-4y)² + (-12)² + 2[(2x) × (-4y) + (-4y) × (-12) + (-12) × 2x)]

= 4x² + 16y² + 144 + 2[-8xy + 48y - 24x]

= 4x² + 16y² + 144 - 16xy + 96y - 48x (Ans)

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9. (6p - 54 - 4r)²

→ We'll expand this too by using the same identity:-

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

So,

(6p - 54 - 4r)²

= (6p)² + (-54)² + (-4r)² + 2[(6p) × (-54) + (-54) × (-4r) + (-4r) × (6p)]

= 36p² + 2916 + 16r² + 2[-324p + 216r - 24pr]

= 36p² + 2916 + 16r² - 648p + 432r - 48pr (Ans)

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10. (p + 5q + 2)²

→ We'll expand this too by using the same identity:-

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

So,

(p + 59 + 2)²

= (p)² + (5q)² + (2)² + 2[(p) × (5q) + (5q) × (2) + (2) × (p)]

= p² + 25q² + 4 + 2[5pq + 10q + 2p]

= p² + 25q² + 4 + 10pq + 20q + 4p (Ans)

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11. (3x - 4y - 5)²

→ We'll expand this too by using the same identity:-

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

So,

(3x - 4y - 5)²

= (3x)² + (-4y)² + (-5)² + 2[(3x) × (-4y) + (-4y) × (-5) + (-5) × (3x)]

= 9x² + 16y² + 25 + 2[-12xy + 20y - 15x]

= 9x² + 16y² + 25 - 12xy + 20y - 20x (Ans)

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12. (-2x + 6y + 4)²

→ We'll expand this too by using the aame identity:-

(a + b + c)² = a² + b² + c² + 2(ab + bc + ca)

So,

(-2x + 6y + 4)²

= (-2x)² + (6y)² + (4)² + 2[(-2x) × (6y) + (6y) × (4) + (4) × (-2x)]

= 4x² + 36y² + 16 + 2[-12xy + 24y - 8x]

= 4x² + 36y² + 16 - 24xy + 48y - 16x (Ans)

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