Question

Hola Maths Aryabhatta's Here are some Interesting Questions.Try to solve all :-

$x {}^{2011} - x {}^{2010} + 2x {}^{100} - 3x {}^{97} + 5x {}^{3} - x {}^{2} + 4$
is divided by 2x⁴ -x³ + 3x² + x -5 get the remainder ax³+bx²+cx+D then
a + b + c + D is ??
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x belongs to Real numbers x² -15x +1 =0 then x⁴+1/x⁴ is

1) 49279
2) 49727
3) 49772
4) 49227
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Answer 1

Answer:

As per the reminder theory for any

polynomial equation

p(x)=g(x).q(x)+r(x)→(i)

where p(x),g(x)= polynomials

q(x)= quotient

r(x)= reminder

Assuming r(x)=Ax+B

here p(x)=x

100 ,g(x)=x 2 −3x+2

Simplifying g(x)

g(x)=x 2 −3x+2=x 2 −2x−x+2=(x−2)(x−1)

Substituting in eqn (i)

p(x)=(x−2)(x−1).q(x)+Ax+B

taking (x)=1

p(1)=(1−2)(1−1).q(1)+A(1)+B

100 =0+A+B

A+B=1→(i)

taking x=2

p(2)=(2−2)(2−1).q(x)+A(2)+B2 100 =0+2A+B2A+B=2

100 →(ii)

from eqn

(i) B=1−A putting it in eqn

(i)2A+1−A=2 100

A=2 100 −1

now B=1−2 100 +1=2−2 100

∴B(2−2 100 )

∴ Reminder =Ax+B=(2 100 −1)−x+(2−2 100 )

Step-by-step explanation:

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Answer 2

Answer:

1)7

2) 49727

Step-by-step explanation:

1) Given,

Dividend = x²⁰¹¹- x²⁰¹⁰ + 2x¹⁰⁰ - 3x⁹⁷ + 5x³ - x² + 4

Let,

f(x) = x²⁰¹¹- x²⁰¹⁰ + 2x¹⁰⁰ - 3x⁹⁷ + 5x³ - x² + 4

Divisor = 2x⁴ -x³ + 3x² + x -5

Let,

g(x) = 2x⁴ -x³ + 3x² + x -5

Remainder = ax³ + bx² + cx + d

Let,

r(x) = ax³ + bx² + cx + d

Solution :-

As they not given any value of 'x' :-

Let us take the value of 'x = 1' :-

We know that :-

Dividend = Divisor × Quotient + Remainder

→ f(x) = g(x) × q(x) + r(x)

x²⁰¹¹- x²⁰¹⁰ + 2x¹⁰⁰ - 3x⁹⁷ + 5x³ - x² + 4 = (2x⁴ -x³ + 3x² + x -5) q(x) + ax³ + bx² + cx + d

Substituting x = 1 :-

(1)²⁰¹¹ - (1)²⁰¹⁰ + 2(1)¹⁰⁰ - 3(1)⁹⁷ + 5(1)³ - (1)² + 4 = [2(1)⁴ - (1)³ + 3(1)² + 1 - 5)] q(x) + a(1)³ + b(1)² + c(1) + d

1 - 1 + 2 - 3 + 5 - 1 + 4 = [2 - 1 + 3 + 1 - 5]q(x) + a + b + c + d

7 = 0 [q(x) ] + a + b + c + d

→ a + b + c + d = 7

2) Given,

x² - 15x + 1 = 0

Dividing the above equation with 'x' :-

(x²-15x+1)/x = 0/x

x - 15 + (1/x) = 0

x + 1/x = 15

Squaring on both sides :-

(x + 1/x)² = (15)²

x² + 1/x² + 2x²(1/x²) = 225

x² + 1/x² = 223

Squaring on both sides :-

(x² + 1/x²)² = (223)²

x⁴ + 1/x⁴ + 2x²(1/x²) = 49729

x⁴ + 1/x⁴ = 49727