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Directions : Study the following questions carefully and choose the right answer.
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.