ax + by = 5,ax2 + by2 = 10,ax3 + by3 = 50,ax4 + by4 = 130,13 (x +

1.	ax + by = 5,ax2 + by2 = 10,ax3 + by3 = 50,ax4 + by4 = 130,13 (x + y - xy) - 120(a + b) = ?
Answer» ax + by = 5---(1) ax² + by² = 10---(2) ax³ + by³ = 50---(3) ax⁴ + by⁴ = 130---(4) From (2), ax² + by² = 10 ⇒ (ax² + by²) (x + y) = 10(x + y) ⇒ ax³ + by³ + ax²y + bxy² = 10(x + y) ⇒ 50 + xy(ax + by) = 10(x + y) (From (3)) ⇒ 50 + 5xy = 10(x + y)---(5) From (1) From (3), ax² + by³ = 50 ⇒ (ax³ + by³)(x + y) = 50 (x + y) ⇒ 130 + (ax² + by²)xy = 50 (x + y) From (4) ⇒ 130 + 10xy = 50(x + y)----(6) Now, (6) - 2 x (5), we get 130 + 10xy - (100 + 10xy) = 50(x + y) - 20(x + y) ⇒ 130 - 100 = 30(x + y) ⇒ x + y = 30/30 = 1 ⇒ x + y = 1 Put x + y = 1 in equation (5) we get 50 + 5xy = 10 ⇒ 5xy = 10 - 50 = -40 ⇒ xy = -40/5 = -8 From (1), ax + by = 5 ⇒ (ax + by)(x + y) = 5(x + y) ⇒ ax² + by² + axy + bxy = 5 (\(\because\) x + y = 1) ⇒ 10 + xy (a + b) = 5 (From (2)) ⇒ 10 - 8(a + b) = 5 (\(\because\) xy = -8) ⇒ a + b = \(\frac{5-10}{-8}\) = \(\frac{-5}{-8}\) = \(\frac58\) Now, 13 (x + y - xy) - 120(a + b) = 13(1 - (-8)) -120 x \(\frac58\) = 13 x 9 - 15 x 5 = 117 - 75 = 42

ax + by = 5,ax2 + by2 = 10,ax3 + by3 = 50,ax4 + by4 = 130,13 (x + y - xy) - 120(a + b) = ?

Answer»

ax + by = 5---(1)

ax² + by² = 10---(2)

ax³ + by³ = 50---(3)

ax⁴ + by⁴ = 130---(4)

From (2),

ax² + by² = 10

⇒ (ax² + by²) (x + y) = 10(x + y)

⇒ ax³ + by³ + ax²y + bxy² = 10(x + y)

⇒ 50 + xy(ax + by) = 10(x + y) (From (3))

⇒ 50 + 5xy = 10(x + y)---(5) From (1)

From (3),

ax² + by³ = 50

⇒ (ax³ + by³)(x + y) = 50 (x + y)

⇒ 130 + (ax² + by²)xy = 50 (x + y) From (4)

⇒ 130 + 10xy = 50(x + y)----(6)

Now, (6) - 2 x (5), we get

130 + 10xy - (100 + 10xy) = 50(x + y) - 20(x + y)

⇒ 130 - 100 = 30(x + y)

⇒ x + y = 30/30 = 1

⇒ x + y = 1

Put x + y = 1 in equation (5) we get

50 + 5xy = 10

⇒ 5xy = 10 - 50 = -40

⇒ xy = -40/5 = -8

From (1), ax + by = 5

⇒ (ax + by)(x + y) = 5(x + y)

⇒ ax² + by² + axy + bxy = 5 (\(\because\) x + y = 1)

⇒ 10 + xy (a + b) = 5 (From (2))

⇒ 10 - 8(a + b) = 5 (\(\because\) xy = -8)

⇒ a + b = \(\frac{5-10}{-8}\) = \(\frac{-5}{-8}\) = \(\frac58\)

Now, 13 (x + y - xy) - 120(a + b) = 13(1 - (-8)) -120 x \(\frac58\)

= 13 x 9 - 15 x 5 = 117 - 75 = 42