Direction: Study the following questions carefully and choose the right answer.
Important for :
1
If x + y = 15, then (x – 10)3 + (y – 5)3 is
» Explain it
D
Given, x + y = 15

∴     (x – 10)3 + (y – 5)3

We can express it in the form of

a3 + b3 = (a + b)3 – 3ab(a + b)

(x – 10)3 + (y – 5)3 = (x – 10 + y – 5)3 – 3(x – 10)(y – 5)(x – 10 + y – 5)

⇒    (x + y – 15)3 – 3(x – 10)(y – 5)(x + y – 15) = 0

Putting the value of x + y, we get

⇒ (15 – 15)3 – 3(x – 10)(y – 5)(15 – 15)

⇒ (0)3 – 3(x – 10)(y – 5)(0) = 0.

Hence, option D is correct.

2
 If x + 1 = 5, then the value of x6 + 1 is x x6
» Explain it
A
 x + 1 = 5 x

On cubing both sides, we get

 ( x + 1 ) 3 = (5)3 x

 ⇒  x3 + 1 + 3x . 1 ( x + 1 ) = 125 x3 x x

 ⇒  x3 + 1 + 3 × 5 = 125 x3

 ⇒   x3 + 1 = 125 – 15 = 110 x3

On squaring both sides, we get

 ⇒   x6 + 1 + 2 . x3 . 1 = 12100 x6 x3

 ⇒   x6 + 1 = 12100 – 2 = 12098 x6

Hence, option A is correct.

3
 If x3 + 3 = 4(a3 + b3) and 3x + 1 = 4(a3 – b3), then a2 – b2 is equal to x x3
» Explain it
C
 x3 + 3 = 4(a3 + b3)         .....(i) x

 3x + 1 = 4(a3 – b3)       .....(ii) x3

On adding both equations, we get

 x3 + 3x + 3 + 1 = 8a3 x x3

 ⇒ ( x + 1 ) 3 = (2a)3 x

⇒
 x + 1 x
= 2a   ⇒   a =  1 ( x +  1 )
2 x

Similarly, on subtracting

 x3 + 3 – 3x – 1 = 8b3 x x3

 ⇒ ( x – 1 ) 3 = (2b)3 x

 ⇒ b = 1 ( x – 1 ) 2 x

∴   a2 – b2

 = 1 [( x + 1 ) 2 – ( x – 1 ) 2 ] 4 x x

 = 1 [( x + 1 + x – 1 )( x + 1 – x + 1 )] 4 x x x x

 = 1 [ 2x × 2 ] 4 x

 = 1 × 4 = 1. 4

Hence, option C is correct.

4
 If x2 – 3x + 1 = 0, then the value of x6 + x4 + x2 + 1 will be x3
» Explain it
C
Given,

x2 – 3x + 1 = 0

 ⇒  x2 + 1 = 3x   ⇒ x2 + 1 = 3 x

 ⇒ ( x + 1 ) = 3     ....(i) x

Cubing both sides here, we get

 x3 + 1 + 3 ( x + 1 ) = 27 x3 x

 ⇒ ( x3 + 1 ) + 3 × 3 = 27 x3

 ⇒ ( x3 + 1 ) = 27 – 9 = 18    ...(ii) x3

 Now, x6 + x4 + x2 + 1 = x6 + x4 + x2 + 1 x3 x3 x3 x3 x3

 = x3+ x + 1 + 1 x x3

 = ( x3 + 1 ) + ( x + 1 ) x3 x

From eqn. (i) and (ii), we get

= 18 + 3 = 21

Hence, option C is correct.

5
 If x = 6 + 1 , then the value of x4 + 1 is x x4
» Explain it
B
 x = 6 + 1 ⇒  x – 1 = 6 x x

On squaring both sides, we get

 ⇒   x2 + 1 – 2 = 36 x2

 ⇒  x2 + 1 = 36 + 2 = 38 x2

On squaring again,

 ⇒   x4 + 1 + 2 = 1444 x4

 ⇒   x4 + 1 = 1444 – 2 = 1442. x4

Hence, option B is correct.