⇒ (a – b)2 + ( b – c)2 + ( c – a)2 = 0
⇒ a = b, b = c, c = a, since no square can be a negative integer
⇒ a = b = c
7. If a = 2.234, b = 3.121 and c = -5.355, then the value of a3 + b3 + c3 – 3abc is
a. – 1
b. 0
c. 1
d. 2
Answer (b).
a + b + c = 2.234 + 3.121 – 5.355 = 0
If a + b + c = 0, then a3 + b3 + c3 – 3abc = 0, which can be proved as under
a + b = – c
⇒ (a + b)3 = (– c)3
⇒ a3 + b3 + 3ab(a + b) = – c3
⇒ a3 + b3 + 3ab(– c) = – c3
⇒ a3 + b3 – 3abc = – c3
⇒ a3 + b3 + c3 – 3abc = 0
8. The equations 3x + 4y = 10 and – x + 2y = 0 have the solution (a, b). The value of a + b is
⇒ x = 2
Substituting in equation 2
⇒ – 2 + 2y = 0
⇒ 2y = 2 or y = 1
Therefore a + b = 3
9. If x2 + 4x + 3 = 0, then the value of
x3
is
x6 + 27x3 + 27
a. –
1
2
b. 1
c.
1
2
d. –
1
Answer (d).
x2 + 4x + 3 = 0 ⇒ x2 + 3 = – 4 ⇒ x +
3
= – 4 ....... (1)
x
⇒ (x +
3
)3 = (– 4)3
x
⇒ x3 +
27
+ 3(x)
3
(x +
3
) = – 64
x3
x
x
⇒ x3 +
27
+ 9 x – 4 = – 64 [see (1) above]
x3
⇒ x3 +
27
– 36 = – 64
x3
⇒ x3 +
27
= – 28 ...... (2)
x3
x3
=
x3 / x3
=
1
x6 + 27x3 + 27
(x6 + 27x3 + 27)/x3
x3 + 27 + 27/x3
Substituting the value of (2) in the above
1
= – 1
27 – 28
10. If 999x + 888y = 1332 and 888x + 999y = 555, then x2 – y2 is equal to
a. 7
b. 8
c. 9
d. 5
Answer (a).
Adding the given equations we get 1887x + 1887y = 1887 or x + y = 1
Subtracting the second equation from the first we get 111x – 111y = 777 or x – y = 7
x2 + y2 = (x + y)(x – y) = 7
11. If a – b = 4, and a2 + b2 = 40 where a and b are positive integers, then a3 + b6 is equal to
(a + b)2 = a2 + b2 + 2ab
(a + b)2 = 40 + 24
a + b = √64 = 8
a + b = 8
a – b = 4
2a = 12
a = 6
a = 6; b = 2
a3 + b6 = 63 + 26
216 + 64 = 280
12. If (x2 +
1
) =
17
, then what is (x3 –
1
) equal to?
x2
4
x3
a.
75
16
b.
63
8
c.
95
8
d.
None
of these
Answer (b).
x2 +
1
– 2 =
17
– 2
x2
4
(x –
1
)2 =
9
x
4
x –
1
=
3
x
2
x3 –
1
= (x –
1
) ( x2 +
1
+ 1)
x3
x
x2
x3 –
1
= (
3
) x (
17
+ 1) =
63
x3
2
4
8
13. If x + y = 5, y + 3 = 10 and z + x = 15, then which of the following is correct?
a. z > x > y
b. z > y > x
c. x > y > z
d. x > z > y
Answer (a).
2x + 2y + 2z = 5 + 10 + 15 = 30
x + y + z = 15
x + y = 5 z = 10 [x + y + z - (x + y)]
y + z = 10 x = 5 [x + y + z - (y + z)]
z + x = 15 y = 0 [x + y + z - (z + x)]
z > x > y
14. What is
1
–
1
–
2b
–
4b3
–
8b7
equal to
a – b
a + b
a2 + b2
a4 + b4
a8 – b8
a. a + b
b. a – b
c. 1
d. 0
Answer (d).
Taking 2 expressions at a time
1
–
1
=
a + b – a + b
=
2b
a – b
a + b
a2 – b2
a2 – b2
2b
–
2b
=
2b(a2 + b2) – 2b(a2 – b2)
=
4b3
a2 – b2
a2 + b2
a4 – b4
a4 – b4
4b3
–
4b3
=
4b3(a4 + b4) – 4b3(a4 – b4)
=
8b7
a4 – b4
a4 + b4
a8 – b8
a8 – b8
8b7 – 8b7
= 0
a8 – b8
15. If x + y – 7 = 0 and 3x + y – 13 = 0, then what is 4x2 + y2 + 4xy equal to
a. 75
b. – 85
c. 91
d. 100
Answer (d).
x + y = 7 ......... (1)
3y + y = 13 ...........(2)
Subtracting (1) from (2) we get x = 3
Substituting the value of x in (1) we get y = 4
4x2 + y2 + 4xy = (2x + y)2
⇒ (6 + 4)2 = 100
z = 10 [x + y + z - (x + y)] 
a2 + b2 = ab]