Directions : Study the following questions carefully and choose the right answer.
Important for :
1
 (x2 – y2)2 + (1 – x2)2 + (y2 – 1)2 = ? (1 – x)2 (y2 – 1) (x2 – y2) (y2 – 1) (x2 – y2) (y2 – 1)
» Explain it
D
Let x2 – y2 = a

1 – x2 = b

y – 1 = c

a + b + c = 0

a + b + c = 3abc

 ∴ (x2 – y2)2 + (1 – x2)2 + (y2 – 1)2 (1 – x)2 (y2 – 1) (x2 – y2) (y2 – 1) (x2 – y2) (y2 – 1)

 = a2 + b2 + c2 bc ca ab

 = a3 + b3 + c3 abc

 = 3abc abc
= 3

Hence, option D is correct.

2
If x = √3 + √4 + √5 than x4 – 8x3 + 8x2 + 32x = ?
» Explain it
C
x = √3 + √4 + √5

x – 2 = √3 + √5

x2 – 4x + 4 = 8 + 2 √15

x2 – 4x – 4 = 2√15

x4 + 16x2 + 16 – 8x3 + 32x – 8x2 = 60

x4– 8x3 + 8x2 + 32x = 44

Hence, option C is correct.

3
 If a28 + 1 = 23 than a42 + 1 = ? a14 a21
» Explain it
A
 a28 + 1 = 23 a14

 a14 + 1 = 23 a14

 a14 + 1 + 2 = 25 a14

 a7 + 1 = 5 a7

 a21 + 1 = 53 – 3 × 5 a21

 a42 + 1 = 110 a21

Hence, option A is correct.

4
 If a2 + b2 + c2 + b2 + c2 + a2 + c2 + a2 + b2 = ? a2 – b2 – c2 b2 – c2 – a2 c2 – a2 – b2
» Explain it
A
 a2 + b2 + c2 + b2 + c2 + a2 + c2 + a2 + b2 – 3 + 3 a2 – b2 – c2 b2 – c2 – a2 c2 – a2 – b2

 = a2 + b2 + c2 + a2 – b2 – c2 + b2 + c2 + a2 + b2 – c2 – a2 + c2 + a2 + b2 + c2 – a2 – b2 – 3 a2 – b2 – c2 b2 – c2 – a2 c2 – a2 – b2

 = 2a2 + 2b2 + 2c2 – 3 a2 – b2 – c2 b2 – c2 – a2 c2 – a2 – b2

Now, a + b = c

a = c – b

a2 – b2 – c2 = – 2bc .........(i)

and a + b = c

b = c – a

b2 – c2 – a2 = – 2ac ..........(ii)

and a + b = c

c2 – a2 – b2 = 2ab .........(iii)

a + b = c

a + b – c = 0

a3 + b3 – c3 = – 3abc .........(iv)

–a3 – b3 + c3 = 3abc

 = 2a2 + 2b2 + 2c2 – 3 – 2bc –2ac 2ab

 = –a2 – b2 + c2 – 3 bc ac ab

 = –a3 – b3 + c3 – 3 abc

 = 3abc –3 abc
= 0

Hence, option A is correct.

5
 If x = 1 = √5 than √x (√x – 1) = ? x
» Explain it
C
 x2 + 1 = 7 x2

 x + 1 = 3 x

 x – 1 = √5 x

2x = 3 + √5

 x = 3 + √5 2

 x = 6 + 2√5 4

 x = ( √5 + 1 ) 2 2

 √x = √5 + 1 2

 √x (√x – 1) = ( √5 + 1 ) ( √5 + 1 – 1 ) 2 2

 = √5 + 1 × √5 – 1 = 5 – 1 = 1 2 2 4

Hence, option C is correct.