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Direction: Study the following questions carefully and choose the right answer.
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 . x 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.