Which one of the following statement is correct? (1) If x^6 + 1 is divided by x + 1 reminder is -2. (2) If x^6 + 1 is divided by x - 1 reminder is 2. (3) If x^6 + 1 is divided by x + 1 reminder is 1. (4) If x^6 + 1 is divided by x - 1 reminder is -1.
Answers
Answer: (2)
To understand the true statement each condition has to be considered
Let us consider the case (1)
When f(x) = x6 + 1 is divided by x + 1 then the remainder is -2
Here x = -1 then
f(-1) = (-1)6 + 1
f(-1) = 2
A statement (1) is wrong.
Let us consider the case (2)
When f(x) = x6 + 1 is divided by x – 1 then the remainder is 2
Here x = +1 then
f(1) = (1)6 + 1
f(1) = 2
A statement (2) is correct.
Let us consider the case (3)
When f(x) = x6 + 1 is divided by x + 1 then the remainder is 1
Here x = -1 then
f(-1) = (-1)6 + 1
f(1) = -2
A statement (3) is wrong.
Let us consider the case (4)
When f(x) = x6 + 1 is divided by x – 1 then the remainder is -1
Here x = +1 then
f(-1) = (1)6 + 1
f(1) = 2
A statement (4) is wrong.
Given equation
3/x-1+1/x-3=4/x-2
We need to find the value of x
3/x-1+1/x-3=4/x-2
On taking LCM in LHS we get
3(x-3) + 1 (x-1) / (x-1)(x-3)= 4/ x-2
=>3x-9 + x-1 / (x-1)(x-3) = 4/ x-2
=> (4x -10) (x-2) = 4 (x-1) (x-3)
= >4×2 – 8x -10x+20= 4 (x2 -3x -x+3)
=> 4x2 -18x +20 = 4x2 – 12x -4x +12
=> 4x2 -18x +20 = 4x2 – 16x +12
4x2 gets cancelled on both sides
=> -18x +20 = – 16x +12
=> -18x+16x = 12 – 20
=> -2x = -8
Negative sign gets cancelled on both sides
=> 2x= 8
=> x= 8/2
=>x=4
Hence x=4