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Page 1
Pinnacle Day: 39th - 48th Algebra
Algebra
Important Formulas:-
· = ( ?? + ?? )
2
??
2
+ ??
2
+ 2 ????
· = ( ?? - ?? )
2
??
2
+ ??
2
- 2 ????
· = ( ?? + ?? )
2
+ ( ?? - ?? )
2
2 ( ??
2
+ ??
2
)
· ( ?? + ?? )
2
- ( ?? - ?? )
2
= 4 ????
· ( ?? + ?? )
2
- 4 ???? = ( ?? - ?? )
2
· ??
2
- ??
2
= ( ?? + ?? )( ?? - ?? )
· ??
4
- ??
4
= ( ??
2
+ ??
2
)( ?? + ?? )( ?? - ?? )
· = ( )
2
2 ??
2
+
1
??
2
?? +
1
??
-
= ( )
2
?? -
1
??
+ 2
· = ??
2
+ ??
2
+ ??
2
- ???? - ???? - ????
+
1
2
{ ?? - ?? ( )
2
( ?? - ?? )
2
+ ?? - ?? ( )
2
}
If then, ?? = ?? = ?? = ??
, ??
2
+ ??
2
+ ??
2
= ???? + ???? + ????
· ( ?? + ?? )
3
= ??
3
+ ??
3
+ 3 ???? ( ?? + ?? )
· ( ?? - ?? )
3
= ??
3
- ??
3
- 3 ???? ( ?? - ?? )
· ??
3
+ ??
3
= ( ?? + ?? )( ??
2
+ ??
2
- ???? )
· ??
3
- ??
3
= ( ?? - ?? )( ??
2
+ ??
2
+ ???? )
· + = ??
3 1
??
3
?? +
1
??
( )
3
- 3 ?? +
1
??
( )
· = ??
3
-
1
??
3
?? -
1
??
( )
3
+ 3 ?? -
1
??
( )
· (a + b + c) ??
3
+ ??
3
+ ??
3
- 3 ?????? =
( ??
2
+ ??
2
+ ??
2
- ???? - ???? - ???? )
Or ,
1
2
( ?? + ?? + ?? ){ ?? - ?? ( )
2
+ ?? - ?? ( )
2
Or, + ?? - ?? ( )
2
}
(a + b + c)[ - 3(ab + bc + ca)] ( ?? + ?? + ?? )
2
Or , (a + b + c)
1
2
3 ( ??
2
+ ??
2
+ ??
2
) - ( ?? + ?? + ?? )
2
[ ]
· If , then ?? + ?? + ?? ( ) = 0
??
3
+ ??
3
+ ??
3
= 3 ??????
· If ( ) = 0, then ?? - ?? - ??
??
3
- ??
3
- ??
3
= 3 ??????
· If ( ) = 0, then ?? + ?? - ??
??
3
+ ??
3
- ??
3
=- 3 ??????
· If , ??
3
+ ??
3
+ ??
3
= 3 ??????
then either a = b = c or a + b + c = 0
· = ( ?? + ?? + ?? )
2
??
2
+ ??
2
+ ??
2
+ 2 ( ???? + ???? + ???? )
· = ( ?? + ?? + ?? )
3
??
3
+ ??
3
+ ??
3
+
3 ( ?? + ?? )( ?? + ?? )( ?? + ?? )
· = ??
4
+ ??
2
??
2
+ ??
4
( ??
2
+ ???? + ??
2
)
( ??
2
- ???? + ??
2
)
· If = 0, ( ??
2
+ ??
2
+ ??
2
)
then = 0, = 0, = 0 ?? ?? ??
· If + = , then + = ??
1
??
?? ??
3 1
??
3
??
3
- 3 ??
· If - = , then - = ??
1
??
?? ??
3 1
??
3
??
3
+ 3 ??
· If + = , then + = ??
1
??
?? ??
2 1
??
2
??
2
- 2
· If - = , then + = ??
1
??
?? ??
2 1
??
2
??
2
+ 2
· If + = , then - = ??
1
??
?? ??
1
??
??
2
- 4
· If - = , then + = ??
1
??
?? ??
1
??
??
2
+ 4
· If + = , then ??
1
??
??
- = ??
2 1
??
2
?? ??
2
- 4
· If - = , then ??
1
??
??
- = ??
2 1
??
2
?? ??
2
+ 4
· If + = , then ??
1
??
??
+ = ??
4 1
??
4
( ??
2
- 2 )
2
- 2
· If - = , then ??
1
??
??
+ = ??
4 1
??
4
( ??
2
+ 2 )
2
- 2
· If + = a , then ??
1
??
+ = ( + ) ( + ) - ( x + ) ??
5 1
??
5
??
2 1
??
2
??
3 1
??
3
1
??
If - = a, then • ??
1
??
- = ( + ) ( - ) - ( - ) ??
5 1
??
5
??
2 1
??
2
??
3 1
??
3
??
1
??
· If + = , then = -1 ??
1
??
± 3 ??
6
· If + = 1, then = -1 ??
1
??
??
3
· If + = -1, then = 1 ??
1
??
??
3
· If + = 2, then = 1, similarly if ??
1
??
??
+ = -2, then = -1 ??
1
??
??
Some Special Case :-
? if - ab + = 0 then + = 0 ??
2
??
2
??
3
??
3
? if - = then + = 0
1
??
1
??
1
?? - ??
??
3
??
3
? if + = then - = 0
1
??
1
??
1
?? + ??
??
3
??
3
? if + = -1 then - = 0
??
??
??
??
??
3
??
3
? if + = 1 then + = 0
??
??
??
??
??
3
??
3
? if + a + 1 = 0 then - 1 = 0 or = 1 ??
2
??
3
??
3
? if ab(a + b) = 1 then - - = 3
1
??
3
??
3
??
3
??
3
· Dividendo – Componendo rule:
If = then, =
??
??
??
??
?? + ??
?? - ??
?? + ??
?? - ??
If = then =
?? + ??
?? - ??
??
??
??
??
?? + ??
?? - ??
· If 0 ?? - ?? ( )
2
+ ?? - ?? ( )
2
+ ?? - ?? ( )
2
=
then by putting in ?? = ?? , ?? = ?? , ?? = ??
an expression we can ?nd the value of
the expression.
· If , then will be ?? + ?? = ?? ????
maximum when = . ?? ??
· If = a, then ( will be ???? ?? + ?? )
minimum when = = ?? ?? ??
Law of Indices :
(1) = (2) = ??
??
× ??
??
??
?? + ?? ??
??
??
??
??
?? - ??
(3) = (4) = ( ??
??
)
??
??
????
(
??
??
)
??
??
??
??
??
(5) = 1 (6) = ??
0
??
1
??
??
??
If a is negative number and n is an odd
number then ( )
n
=
n
- ?? - ??
Laws of surds :
(1). =
??
?? ??
1
??
(2). =
??
?? ×
??
??
??
????
(3). =
??
??
??
??
??
??
??
(4). =
??
??
??
????
??
(5). = (
??
?? )
??
??
??
??
Ex.1. If 3 is given, ?nd out the 5
rationalizing factor.
Sol : 3 = 3 5 = 15, so the 5 × 5 ×
rationalizing factor of 3 is 5 5 .
Rules for making a perfect square :
1. Middle term =
± ?????????? ???????? × ??h?????? ????????
2. First term =
( ???????????? ???????? )
2
4 × ??h?????? ????????
3. Third term =
( ???????????? ???????? )
2
4 × ?????????? ????????
Quadratic and cubic equation :
? For quadratic equation :
= 0, then ?? ??
2
+ ???? + ??
Sum of its zeros ( + ) = - a ß
??
??
www.ssccglpinnacle.com Download Pinnacle Exam Preparation App 262
Page 2
Pinnacle Day: 39th - 48th Algebra
Algebra
Important Formulas:-
· = ( ?? + ?? )
2
??
2
+ ??
2
+ 2 ????
· = ( ?? - ?? )
2
??
2
+ ??
2
- 2 ????
· = ( ?? + ?? )
2
+ ( ?? - ?? )
2
2 ( ??
2
+ ??
2
)
· ( ?? + ?? )
2
- ( ?? - ?? )
2
= 4 ????
· ( ?? + ?? )
2
- 4 ???? = ( ?? - ?? )
2
· ??
2
- ??
2
= ( ?? + ?? )( ?? - ?? )
· ??
4
- ??
4
= ( ??
2
+ ??
2
)( ?? + ?? )( ?? - ?? )
· = ( )
2
2 ??
2
+
1
??
2
?? +
1
??
-
= ( )
2
?? -
1
??
+ 2
· = ??
2
+ ??
2
+ ??
2
- ???? - ???? - ????
+
1
2
{ ?? - ?? ( )
2
( ?? - ?? )
2
+ ?? - ?? ( )
2
}
If then, ?? = ?? = ?? = ??
, ??
2
+ ??
2
+ ??
2
= ???? + ???? + ????
· ( ?? + ?? )
3
= ??
3
+ ??
3
+ 3 ???? ( ?? + ?? )
· ( ?? - ?? )
3
= ??
3
- ??
3
- 3 ???? ( ?? - ?? )
· ??
3
+ ??
3
= ( ?? + ?? )( ??
2
+ ??
2
- ???? )
· ??
3
- ??
3
= ( ?? - ?? )( ??
2
+ ??
2
+ ???? )
· + = ??
3 1
??
3
?? +
1
??
( )
3
- 3 ?? +
1
??
( )
· = ??
3
-
1
??
3
?? -
1
??
( )
3
+ 3 ?? -
1
??
( )
· (a + b + c) ??
3
+ ??
3
+ ??
3
- 3 ?????? =
( ??
2
+ ??
2
+ ??
2
- ???? - ???? - ???? )
Or ,
1
2
( ?? + ?? + ?? ){ ?? - ?? ( )
2
+ ?? - ?? ( )
2
Or, + ?? - ?? ( )
2
}
(a + b + c)[ - 3(ab + bc + ca)] ( ?? + ?? + ?? )
2
Or , (a + b + c)
1
2
3 ( ??
2
+ ??
2
+ ??
2
) - ( ?? + ?? + ?? )
2
[ ]
· If , then ?? + ?? + ?? ( ) = 0
??
3
+ ??
3
+ ??
3
= 3 ??????
· If ( ) = 0, then ?? - ?? - ??
??
3
- ??
3
- ??
3
= 3 ??????
· If ( ) = 0, then ?? + ?? - ??
??
3
+ ??
3
- ??
3
=- 3 ??????
· If , ??
3
+ ??
3
+ ??
3
= 3 ??????
then either a = b = c or a + b + c = 0
· = ( ?? + ?? + ?? )
2
??
2
+ ??
2
+ ??
2
+ 2 ( ???? + ???? + ???? )
· = ( ?? + ?? + ?? )
3
??
3
+ ??
3
+ ??
3
+
3 ( ?? + ?? )( ?? + ?? )( ?? + ?? )
· = ??
4
+ ??
2
??
2
+ ??
4
( ??
2
+ ???? + ??
2
)
( ??
2
- ???? + ??
2
)
· If = 0, ( ??
2
+ ??
2
+ ??
2
)
then = 0, = 0, = 0 ?? ?? ??
· If + = , then + = ??
1
??
?? ??
3 1
??
3
??
3
- 3 ??
· If - = , then - = ??
1
??
?? ??
3 1
??
3
??
3
+ 3 ??
· If + = , then + = ??
1
??
?? ??
2 1
??
2
??
2
- 2
· If - = , then + = ??
1
??
?? ??
2 1
??
2
??
2
+ 2
· If + = , then - = ??
1
??
?? ??
1
??
??
2
- 4
· If - = , then + = ??
1
??
?? ??
1
??
??
2
+ 4
· If + = , then ??
1
??
??
- = ??
2 1
??
2
?? ??
2
- 4
· If - = , then ??
1
??
??
- = ??
2 1
??
2
?? ??
2
+ 4
· If + = , then ??
1
??
??
+ = ??
4 1
??
4
( ??
2
- 2 )
2
- 2
· If - = , then ??
1
??
??
+ = ??
4 1
??
4
( ??
2
+ 2 )
2
- 2
· If + = a , then ??
1
??
+ = ( + ) ( + ) - ( x + ) ??
5 1
??
5
??
2 1
??
2
??
3 1
??
3
1
??
If - = a, then • ??
1
??
- = ( + ) ( - ) - ( - ) ??
5 1
??
5
??
2 1
??
2
??
3 1
??
3
??
1
??
· If + = , then = -1 ??
1
??
± 3 ??
6
· If + = 1, then = -1 ??
1
??
??
3
· If + = -1, then = 1 ??
1
??
??
3
· If + = 2, then = 1, similarly if ??
1
??
??
+ = -2, then = -1 ??
1
??
??
Some Special Case :-
? if - ab + = 0 then + = 0 ??
2
??
2
??
3
??
3
? if - = then + = 0
1
??
1
??
1
?? - ??
??
3
??
3
? if + = then - = 0
1
??
1
??
1
?? + ??
??
3
??
3
? if + = -1 then - = 0
??
??
??
??
??
3
??
3
? if + = 1 then + = 0
??
??
??
??
??
3
??
3
? if + a + 1 = 0 then - 1 = 0 or = 1 ??
2
??
3
??
3
? if ab(a + b) = 1 then - - = 3
1
??
3
??
3
??
3
??
3
· Dividendo – Componendo rule:
If = then, =
??
??
??
??
?? + ??
?? - ??
?? + ??
?? - ??
If = then =
?? + ??
?? - ??
??
??
??
??
?? + ??
?? - ??
· If 0 ?? - ?? ( )
2
+ ?? - ?? ( )
2
+ ?? - ?? ( )
2
=
then by putting in ?? = ?? , ?? = ?? , ?? = ??
an expression we can ?nd the value of
the expression.
· If , then will be ?? + ?? = ?? ????
maximum when = . ?? ??
· If = a, then ( will be ???? ?? + ?? )
minimum when = = ?? ?? ??
Law of Indices :
(1) = (2) = ??
??
× ??
??
??
?? + ?? ??
??
??
??
??
?? - ??
(3) = (4) = ( ??
??
)
??
??
????
(
??
??
)
??
??
??
??
??
(5) = 1 (6) = ??
0
??
1
??
??
??
If a is negative number and n is an odd
number then ( )
n
=
n
- ?? - ??
Laws of surds :
(1). =
??
?? ??
1
??
(2). =
??
?? ×
??
??
??
????
(3). =
??
??
??
??
??
??
??
(4). =
??
??
??
????
??
(5). = (
??
?? )
??
??
??
??
Ex.1. If 3 is given, ?nd out the 5
rationalizing factor.
Sol : 3 = 3 5 = 15, so the 5 × 5 ×
rationalizing factor of 3 is 5 5 .
Rules for making a perfect square :
1. Middle term =
± ?????????? ???????? × ??h?????? ????????
2. First term =
( ???????????? ???????? )
2
4 × ??h?????? ????????
3. Third term =
( ???????????? ???????? )
2
4 × ?????????? ????????
Quadratic and cubic equation :
? For quadratic equation :
= 0, then ?? ??
2
+ ???? + ??
Sum of its zeros ( + ) = - a ß
??
??
www.ssccglpinnacle.com Download Pinnacle Exam Preparation App 262
Pinnacle Day: 39th - 48th Algebra
Product of its zeros ( ) = aß
??
??
where, amd are roots of a a ß
given quadratic equation.
? For cubic equation :
= 0, then ?? ??
3
+ ?? ??
2
+ ???? + ??
Sum of its zeros ( + + )= - a ß ?
??
??
Product of its zeros ( ) = - aß?
??
??
Sum of its zeros taken two at a
time ( + + ) = aß ß? ?a
??
??
where , and are roots of a a ß ?
given cubic equation.
Variety Questions
Q.1. If , then the value of ??
4
+ ??
4
= ??
2
??
2
+ is : ??
6
??
6
SSC CPO 04/10/2023 (1st Shift)
(a) 1 (b) 3 (c) 2 (d) 0
Q.2. If and ???? + ???? = 1 ???? + ???? =
then the value of x (in terms of
2 ????
??
2
+ ??
2
,
a and b) is:
SSC CHSL 11/08/2023 (4th Shift)
(a) (b)
2 ??
??
2
+ ??
2
??
??
2
+ ??
2
(c) (d)
??
??
2
+ ??
2
2 ??
??
2
+ ??
2
Q.3. If x = (7 + 3 ) then ?nd the value 5
of x
2
+
1
??
2
SSC CHSL 09/08/2023 (4th Shift)
(a) (b)
580 + 315 5
8
799 + 328 5
8
(c) (d)
799 + 315 5
12
799 + 315 5
8
Q.4. If x
2
- 11x + 1 = 0 what is the value
of x
8
- 14159x
4
+ 11?
SSC CHSL 08/08/2023 (1st Shift)
(a) 9 (b) 10 (c) 12 (d) 11
Q.5. Simplify:
= ?
( 1 - ?? )
3
+ ( ?? - ?? )
3
+ ( ?? - 1 )
3
12 ( 1 - ?? ) ( ?? - ?? ) ( ?? - 1 )
SSC CHSL 07/08/2023 (3rd Shift)
(a) (b) (c) (d)
1
2
2
3
3
2
1
4
Q.6. For what positive value of k, the
equation 3x² + 2kx + 3 = 0 has real and
equal roots?
SSC CHSL 07/08/2023 (2nd Shift)
(a) 12 (b) 1 (c) 3 (d) 9
Q.7. Simplify the following expression.
??
2
+ ??
2
- ??
2
( )
2
- ??
2
- ??
2
+ ??
2
( )
2
??
2
- ??
2
SSC CHSL 04/08/2023 (4th Shift)
(a) 3a
2
(b) 4a
2
(c) 5a
2
(d) 2a
2
Q.8. Find the value of (a
3
+ b
3
+ c
3
- 3abc),
where a = 335, b = 215 and c = 180.
SSC CHSL 04/08/2023 (3rd Shift)
(a) 14472250 (b) 15421320
(c) 14502230 (d) 15452630
Q.9. Simplify the expression :
( ?? - ?? )
3
+ ( ?? - ?? )
3
+ ( ?? - ?? )
3
( ??
2
- ??
2
)
3
+ ( ??
2
- ??
2
)
3
+ ( ??
2
- ??
2
)
3
SSC CHSL 04/08/2023 (1st Shift)
(a) (b) 1
1
( ?? + ?? ) ( ?? + ?? ) ( ?? + ?? )
(c) (d) 0
3
( ?? + ?? ) ( ?? + ?? ) ( ?? + ?? )
Q.10. If x
3
= 270 + y
3
and x = (6 + y), then
what is the value of (x + y)?
(given that x > 0 andy > 0)
SSC CHSL 03/08/2023 (4th Shift)
(a) 2 (b) (c) 4 (d) 4 3 3 3 2
Q.11. If p + q + r = 0 , then what is the
simpli?ed value of the expression
( + + )
??
2
??
2
- ????
??
2
??
2
- ????
??
2
??
2
- ????
SSC CHSL 03/08/2023 (2nd Shift)
(a) 0 (b) 2 (c) 1 (d) - 1
Q.12. If ( ) = , and x > 1, what is ?? +
1
??
5 2
the value of ( ) ? ??
6
-
1
??
6
SSC CHSL 02/08/2023 (2nd Shift)
(a) 22970 (b) 23030 23 23
(c) 23060 (d) 22960 23 23
Q.13. The difference between the cubes
of two given natural numbers is 6272,
while the positive difference between the
two given numbers is 8. What is the sum
of the cubes of the two given numbers?
SSC CGL 27/07/2023 (1st shift)
(a) 9728 (b) 9684 (c) 8000 (d) 9600
Q.14. If a = 101, b = 102 and c = 103,
then a² + b² + c² - ab - bc - ca = ______.
SSC CGL 24/07/2023 (1st shift)
(a) 2 (b) 4 (c) 3 (d) 6
Q.15. If ) , and x > 1, what is ( ?? +
1
??
= 5
the value of ) ? ( ??
8
-
1
??
8
SSC CGL 18/07/2023 (3rd shift)
(a) 60605 (b) 60615 21 21
(c) 60705 (d) 60725 21 21
Q.16. If What will be the ?? -
1
??
= - 6 ,
value of ??
5
-
1
??
5
?
SSC CGL 18/07/2023 (2nd shift)
(a) 8898 (b) 8896 - -
(c) 8886 (d) 8892 - -
Q.17. If ( + ) = 7, and 0 < x < 1, ?nd ?? ²
1
?? ²
the value of - . ?? ²
1
?? ²
SSC CGL 17/07/2023 (1st shift)
(a) (b) (c) (d) 3 5 4 3 - 4 3 - 3 5
Q.18. If x > 0, and + = 254, what is ??
4 1
??
4
the value of + ? ??
5 1
??
5
SSC CGL 14/07/2023 (4th shift)
(a) 717 (b) 723 (c) 720 (d) 726 2 2 2 2
Q.19. If + -
2 ??
?? + 2
1
2 ?? + 4
12 - ??
2 ??
2
+ 4 ??
= a + , where a and b are integers, then
??
??
the value of ab = ___________ .
Graduate Level 27/06/2023 (Shift - 4)
(a) 3 (b) - 6 (c) 6 (d) - 3
Q.20. If 6
X
= 3
Y
= 2
z
, then what is the
value of + - ?
1
??
1
??
1
??
SSC CHSL 20/03/2023 (2nd Shift)
(a) 1 (b) 0 (c) 3 (d) 6
Q.21. For what value of m will the system
of equations 17x + my + 102 = 0 and 23x
+ 299y + 138 = 0 have an in?nite number
of solutions?
SSC CGL Tier II (02/03/2023)
(a) 221 (b) 223 (c) 220 (d) 219
Q.22. If a + b + c = 5, a
3
+ b
3
+ c
3
= 85 and
abc = 25, then ?nd the value of a
2
+ b
2
+
c
2
- ab - bc - ca.
SSC CGL 13/12/2022 (4th Shift)
(a) 2 (b) 4 (c) 6 (d) 8
Q.23. If , are the roots of 6x² + 13x + a ß
7 = 0, then the equation whose roots are
2
,
2
is: a ß
SSC CGL 12/12/2022 (4th Shift)
(a) 36 - 87x + 49 = 0 ??
2
(b) 36 - 85x + 49 = 0 ??
2
(c) 36 - 85x - 49 = 0 ??
2
(d) 36 + 87x - 49 = 0 ??
2
Q.24. If x = 222, y = 223 and z = 224, then
?nd the value of + + - 3xyz. ??
3
??
3
??
3
SSC CGL 12/12/2022 (4th Shift)
(a) 2007 (b) 2004 (c) 2006 (d) 2005
Q.25. What is the value of
( )( )( ) ?? -
1
??
??
2
+
1
??
2
??
4
+
1
??
4
( )( )( ) ? ??
8
+
1
??
8
??
16
+
1
??
16
??
32
+
1
??
32
www.ssccglpinnacle.com Download Pinnacle Exam Preparation App 263
Page 3
Pinnacle Day: 39th - 48th Algebra
Algebra
Important Formulas:-
· = ( ?? + ?? )
2
??
2
+ ??
2
+ 2 ????
· = ( ?? - ?? )
2
??
2
+ ??
2
- 2 ????
· = ( ?? + ?? )
2
+ ( ?? - ?? )
2
2 ( ??
2
+ ??
2
)
· ( ?? + ?? )
2
- ( ?? - ?? )
2
= 4 ????
· ( ?? + ?? )
2
- 4 ???? = ( ?? - ?? )
2
· ??
2
- ??
2
= ( ?? + ?? )( ?? - ?? )
· ??
4
- ??
4
= ( ??
2
+ ??
2
)( ?? + ?? )( ?? - ?? )
· = ( )
2
2 ??
2
+
1
??
2
?? +
1
??
-
= ( )
2
?? -
1
??
+ 2
· = ??
2
+ ??
2
+ ??
2
- ???? - ???? - ????
+
1
2
{ ?? - ?? ( )
2
( ?? - ?? )
2
+ ?? - ?? ( )
2
}
If then, ?? = ?? = ?? = ??
, ??
2
+ ??
2
+ ??
2
= ???? + ???? + ????
· ( ?? + ?? )
3
= ??
3
+ ??
3
+ 3 ???? ( ?? + ?? )
· ( ?? - ?? )
3
= ??
3
- ??
3
- 3 ???? ( ?? - ?? )
· ??
3
+ ??
3
= ( ?? + ?? )( ??
2
+ ??
2
- ???? )
· ??
3
- ??
3
= ( ?? - ?? )( ??
2
+ ??
2
+ ???? )
· + = ??
3 1
??
3
?? +
1
??
( )
3
- 3 ?? +
1
??
( )
· = ??
3
-
1
??
3
?? -
1
??
( )
3
+ 3 ?? -
1
??
( )
· (a + b + c) ??
3
+ ??
3
+ ??
3
- 3 ?????? =
( ??
2
+ ??
2
+ ??
2
- ???? - ???? - ???? )
Or ,
1
2
( ?? + ?? + ?? ){ ?? - ?? ( )
2
+ ?? - ?? ( )
2
Or, + ?? - ?? ( )
2
}
(a + b + c)[ - 3(ab + bc + ca)] ( ?? + ?? + ?? )
2
Or , (a + b + c)
1
2
3 ( ??
2
+ ??
2
+ ??
2
) - ( ?? + ?? + ?? )
2
[ ]
· If , then ?? + ?? + ?? ( ) = 0
??
3
+ ??
3
+ ??
3
= 3 ??????
· If ( ) = 0, then ?? - ?? - ??
??
3
- ??
3
- ??
3
= 3 ??????
· If ( ) = 0, then ?? + ?? - ??
??
3
+ ??
3
- ??
3
=- 3 ??????
· If , ??
3
+ ??
3
+ ??
3
= 3 ??????
then either a = b = c or a + b + c = 0
· = ( ?? + ?? + ?? )
2
??
2
+ ??
2
+ ??
2
+ 2 ( ???? + ???? + ???? )
· = ( ?? + ?? + ?? )
3
??
3
+ ??
3
+ ??
3
+
3 ( ?? + ?? )( ?? + ?? )( ?? + ?? )
· = ??
4
+ ??
2
??
2
+ ??
4
( ??
2
+ ???? + ??
2
)
( ??
2
- ???? + ??
2
)
· If = 0, ( ??
2
+ ??
2
+ ??
2
)
then = 0, = 0, = 0 ?? ?? ??
· If + = , then + = ??
1
??
?? ??
3 1
??
3
??
3
- 3 ??
· If - = , then - = ??
1
??
?? ??
3 1
??
3
??
3
+ 3 ??
· If + = , then + = ??
1
??
?? ??
2 1
??
2
??
2
- 2
· If - = , then + = ??
1
??
?? ??
2 1
??
2
??
2
+ 2
· If + = , then - = ??
1
??
?? ??
1
??
??
2
- 4
· If - = , then + = ??
1
??
?? ??
1
??
??
2
+ 4
· If + = , then ??
1
??
??
- = ??
2 1
??
2
?? ??
2
- 4
· If - = , then ??
1
??
??
- = ??
2 1
??
2
?? ??
2
+ 4
· If + = , then ??
1
??
??
+ = ??
4 1
??
4
( ??
2
- 2 )
2
- 2
· If - = , then ??
1
??
??
+ = ??
4 1
??
4
( ??
2
+ 2 )
2
- 2
· If + = a , then ??
1
??
+ = ( + ) ( + ) - ( x + ) ??
5 1
??
5
??
2 1
??
2
??
3 1
??
3
1
??
If - = a, then • ??
1
??
- = ( + ) ( - ) - ( - ) ??
5 1
??
5
??
2 1
??
2
??
3 1
??
3
??
1
??
· If + = , then = -1 ??
1
??
± 3 ??
6
· If + = 1, then = -1 ??
1
??
??
3
· If + = -1, then = 1 ??
1
??
??
3
· If + = 2, then = 1, similarly if ??
1
??
??
+ = -2, then = -1 ??
1
??
??
Some Special Case :-
? if - ab + = 0 then + = 0 ??
2
??
2
??
3
??
3
? if - = then + = 0
1
??
1
??
1
?? - ??
??
3
??
3
? if + = then - = 0
1
??
1
??
1
?? + ??
??
3
??
3
? if + = -1 then - = 0
??
??
??
??
??
3
??
3
? if + = 1 then + = 0
??
??
??
??
??
3
??
3
? if + a + 1 = 0 then - 1 = 0 or = 1 ??
2
??
3
??
3
? if ab(a + b) = 1 then - - = 3
1
??
3
??
3
??
3
??
3
· Dividendo – Componendo rule:
If = then, =
??
??
??
??
?? + ??
?? - ??
?? + ??
?? - ??
If = then =
?? + ??
?? - ??
??
??
??
??
?? + ??
?? - ??
· If 0 ?? - ?? ( )
2
+ ?? - ?? ( )
2
+ ?? - ?? ( )
2
=
then by putting in ?? = ?? , ?? = ?? , ?? = ??
an expression we can ?nd the value of
the expression.
· If , then will be ?? + ?? = ?? ????
maximum when = . ?? ??
· If = a, then ( will be ???? ?? + ?? )
minimum when = = ?? ?? ??
Law of Indices :
(1) = (2) = ??
??
× ??
??
??
?? + ?? ??
??
??
??
??
?? - ??
(3) = (4) = ( ??
??
)
??
??
????
(
??
??
)
??
??
??
??
??
(5) = 1 (6) = ??
0
??
1
??
??
??
If a is negative number and n is an odd
number then ( )
n
=
n
- ?? - ??
Laws of surds :
(1). =
??
?? ??
1
??
(2). =
??
?? ×
??
??
??
????
(3). =
??
??
??
??
??
??
??
(4). =
??
??
??
????
??
(5). = (
??
?? )
??
??
??
??
Ex.1. If 3 is given, ?nd out the 5
rationalizing factor.
Sol : 3 = 3 5 = 15, so the 5 × 5 ×
rationalizing factor of 3 is 5 5 .
Rules for making a perfect square :
1. Middle term =
± ?????????? ???????? × ??h?????? ????????
2. First term =
( ???????????? ???????? )
2
4 × ??h?????? ????????
3. Third term =
( ???????????? ???????? )
2
4 × ?????????? ????????
Quadratic and cubic equation :
? For quadratic equation :
= 0, then ?? ??
2
+ ???? + ??
Sum of its zeros ( + ) = - a ß
??
??
www.ssccglpinnacle.com Download Pinnacle Exam Preparation App 262
Pinnacle Day: 39th - 48th Algebra
Product of its zeros ( ) = aß
??
??
where, amd are roots of a a ß
given quadratic equation.
? For cubic equation :
= 0, then ?? ??
3
+ ?? ??
2
+ ???? + ??
Sum of its zeros ( + + )= - a ß ?
??
??
Product of its zeros ( ) = - aß?
??
??
Sum of its zeros taken two at a
time ( + + ) = aß ß? ?a
??
??
where , and are roots of a a ß ?
given cubic equation.
Variety Questions
Q.1. If , then the value of ??
4
+ ??
4
= ??
2
??
2
+ is : ??
6
??
6
SSC CPO 04/10/2023 (1st Shift)
(a) 1 (b) 3 (c) 2 (d) 0
Q.2. If and ???? + ???? = 1 ???? + ???? =
then the value of x (in terms of
2 ????
??
2
+ ??
2
,
a and b) is:
SSC CHSL 11/08/2023 (4th Shift)
(a) (b)
2 ??
??
2
+ ??
2
??
??
2
+ ??
2
(c) (d)
??
??
2
+ ??
2
2 ??
??
2
+ ??
2
Q.3. If x = (7 + 3 ) then ?nd the value 5
of x
2
+
1
??
2
SSC CHSL 09/08/2023 (4th Shift)
(a) (b)
580 + 315 5
8
799 + 328 5
8
(c) (d)
799 + 315 5
12
799 + 315 5
8
Q.4. If x
2
- 11x + 1 = 0 what is the value
of x
8
- 14159x
4
+ 11?
SSC CHSL 08/08/2023 (1st Shift)
(a) 9 (b) 10 (c) 12 (d) 11
Q.5. Simplify:
= ?
( 1 - ?? )
3
+ ( ?? - ?? )
3
+ ( ?? - 1 )
3
12 ( 1 - ?? ) ( ?? - ?? ) ( ?? - 1 )
SSC CHSL 07/08/2023 (3rd Shift)
(a) (b) (c) (d)
1
2
2
3
3
2
1
4
Q.6. For what positive value of k, the
equation 3x² + 2kx + 3 = 0 has real and
equal roots?
SSC CHSL 07/08/2023 (2nd Shift)
(a) 12 (b) 1 (c) 3 (d) 9
Q.7. Simplify the following expression.
??
2
+ ??
2
- ??
2
( )
2
- ??
2
- ??
2
+ ??
2
( )
2
??
2
- ??
2
SSC CHSL 04/08/2023 (4th Shift)
(a) 3a
2
(b) 4a
2
(c) 5a
2
(d) 2a
2
Q.8. Find the value of (a
3
+ b
3
+ c
3
- 3abc),
where a = 335, b = 215 and c = 180.
SSC CHSL 04/08/2023 (3rd Shift)
(a) 14472250 (b) 15421320
(c) 14502230 (d) 15452630
Q.9. Simplify the expression :
( ?? - ?? )
3
+ ( ?? - ?? )
3
+ ( ?? - ?? )
3
( ??
2
- ??
2
)
3
+ ( ??
2
- ??
2
)
3
+ ( ??
2
- ??
2
)
3
SSC CHSL 04/08/2023 (1st Shift)
(a) (b) 1
1
( ?? + ?? ) ( ?? + ?? ) ( ?? + ?? )
(c) (d) 0
3
( ?? + ?? ) ( ?? + ?? ) ( ?? + ?? )
Q.10. If x
3
= 270 + y
3
and x = (6 + y), then
what is the value of (x + y)?
(given that x > 0 andy > 0)
SSC CHSL 03/08/2023 (4th Shift)
(a) 2 (b) (c) 4 (d) 4 3 3 3 2
Q.11. If p + q + r = 0 , then what is the
simpli?ed value of the expression
( + + )
??
2
??
2
- ????
??
2
??
2
- ????
??
2
??
2
- ????
SSC CHSL 03/08/2023 (2nd Shift)
(a) 0 (b) 2 (c) 1 (d) - 1
Q.12. If ( ) = , and x > 1, what is ?? +
1
??
5 2
the value of ( ) ? ??
6
-
1
??
6
SSC CHSL 02/08/2023 (2nd Shift)
(a) 22970 (b) 23030 23 23
(c) 23060 (d) 22960 23 23
Q.13. The difference between the cubes
of two given natural numbers is 6272,
while the positive difference between the
two given numbers is 8. What is the sum
of the cubes of the two given numbers?
SSC CGL 27/07/2023 (1st shift)
(a) 9728 (b) 9684 (c) 8000 (d) 9600
Q.14. If a = 101, b = 102 and c = 103,
then a² + b² + c² - ab - bc - ca = ______.
SSC CGL 24/07/2023 (1st shift)
(a) 2 (b) 4 (c) 3 (d) 6
Q.15. If ) , and x > 1, what is ( ?? +
1
??
= 5
the value of ) ? ( ??
8
-
1
??
8
SSC CGL 18/07/2023 (3rd shift)
(a) 60605 (b) 60615 21 21
(c) 60705 (d) 60725 21 21
Q.16. If What will be the ?? -
1
??
= - 6 ,
value of ??
5
-
1
??
5
?
SSC CGL 18/07/2023 (2nd shift)
(a) 8898 (b) 8896 - -
(c) 8886 (d) 8892 - -
Q.17. If ( + ) = 7, and 0 < x < 1, ?nd ?? ²
1
?? ²
the value of - . ?? ²
1
?? ²
SSC CGL 17/07/2023 (1st shift)
(a) (b) (c) (d) 3 5 4 3 - 4 3 - 3 5
Q.18. If x > 0, and + = 254, what is ??
4 1
??
4
the value of + ? ??
5 1
??
5
SSC CGL 14/07/2023 (4th shift)
(a) 717 (b) 723 (c) 720 (d) 726 2 2 2 2
Q.19. If + -
2 ??
?? + 2
1
2 ?? + 4
12 - ??
2 ??
2
+ 4 ??
= a + , where a and b are integers, then
??
??
the value of ab = ___________ .
Graduate Level 27/06/2023 (Shift - 4)
(a) 3 (b) - 6 (c) 6 (d) - 3
Q.20. If 6
X
= 3
Y
= 2
z
, then what is the
value of + - ?
1
??
1
??
1
??
SSC CHSL 20/03/2023 (2nd Shift)
(a) 1 (b) 0 (c) 3 (d) 6
Q.21. For what value of m will the system
of equations 17x + my + 102 = 0 and 23x
+ 299y + 138 = 0 have an in?nite number
of solutions?
SSC CGL Tier II (02/03/2023)
(a) 221 (b) 223 (c) 220 (d) 219
Q.22. If a + b + c = 5, a
3
+ b
3
+ c
3
= 85 and
abc = 25, then ?nd the value of a
2
+ b
2
+
c
2
- ab - bc - ca.
SSC CGL 13/12/2022 (4th Shift)
(a) 2 (b) 4 (c) 6 (d) 8
Q.23. If , are the roots of 6x² + 13x + a ß
7 = 0, then the equation whose roots are
2
,
2
is: a ß
SSC CGL 12/12/2022 (4th Shift)
(a) 36 - 87x + 49 = 0 ??
2
(b) 36 - 85x + 49 = 0 ??
2
(c) 36 - 85x - 49 = 0 ??
2
(d) 36 + 87x - 49 = 0 ??
2
Q.24. If x = 222, y = 223 and z = 224, then
?nd the value of + + - 3xyz. ??
3
??
3
??
3
SSC CGL 12/12/2022 (4th Shift)
(a) 2007 (b) 2004 (c) 2006 (d) 2005
Q.25. What is the value of
( )( )( ) ?? -
1
??
??
2
+
1
??
2
??
4
+
1
??
4
( )( )( ) ? ??
8
+
1
??
8
??
16
+
1
??
16
??
32
+
1
??
32
www.ssccglpinnacle.com Download Pinnacle Exam Preparation App 263
Pinnacle Day: 39th - 48th Algebra
SSC CGL 06/12/2022 (3rd Shift)
(a) (b)
??
64
-
1
??
64
?? +
1
??
??
32
-
1
??
32
?? -
1
??
(c) (d)
??
32
-
1
??
32
?? +
1
??
??
32
+
1
??
32
?? +
1
??
Q.26. Which of the following statements
is correct ?
I. The value of 100
2
- 99
2
+ 98
2
- 97
2
+ 96
2
- 95
2
+ 94
2
- 93
2
+ ...... + 2
2
- 1
2
is 5050.
II. If 8x + = -16 and x < 0, then the
8
??
value of x
197
+ x
- 197
is 2.
SSC CGL 02/12/2022 (4th Shift)
(a) Only I (b) Only II
(c) Both I and II (d) Neither I nor II
Q.27. If m – n = 0 then what is the ??
??
??
??
value of + in terms of
1
??
??
+ ??
??
1
??
??
- ??
??
? ??
??
SSC CHSL 24/05/2022 (Morning)
(a) (b)
2 ????
??
??
( ??
2
+ ??
2
)
2 ??
??
??
( ??
2
- ??
2
)
(c) (d)
2 ????
??
??
( ??
2
- ??
2
)
2 ????
??
??
( ??
2
- ??
2
)
Q.28. , and ??
2
+ ??
2
= 25 ??
2
+ ??
2
= 17
ax + by = 8 then ?nd the value of (ay - bx)
SSC CHSL 13/4/2021 (Afternoon)
(a) 33 (b) 25 (c) 21 (d) 19
Q.29. If 2 = x + , t hen the
1
1 +
1
5 +
1
2
value of x is equal to:
SSC CGL Tier II (18/11/2020)
(a) (b) 1 (c) (d)
14
13
15
13
13
15
Q.30. If x
3
+ 2 x
2
ax b is exactly - -
divisible by ( x
2
1), then the values of a -
and b are:
SSC CHSL 21/10/2020 (Afternoon)
(a) a = -1, and b = 2 (b) a = 1 and b = -2
(c) a = 1 and b = 2 (d) a = 2 and b = 2
Q.31. If = + 2, then the value of ??
3
5
is equal to: ??
3
- 6 ??
2
+ 12 ?? - 12
SSC CHSL 21/10/2020 (Morning)
(a) 0 (b) 2 (c) 1 (d) - 1
Q.32. (ax + by) is a factor of:
SSC CHSL 16/10/2020 (Afternoon)
(a) a
2
x
2
+ 2ab - b
2
y
2
(b) a
2
x
2
+ 2abxy + b
2
y
2
(c) a
2
x
3
+ 2abx + b
2
y
2
x
(d) a
2
x
2
+ 2ab + b
2
y
2
Q.33. If ‘a’ is a natural number, then (7a
2
+ 7a) is always divisible by:
SSC CHSL 16/10/2020 (Morning shift)
(a) 7 and 14 both (b) 7 only
(c) 14 only (d) 21 only
Q.34. The value of (a + 2a + 3a + 2a
2
3
1
2
1
3
1
6
+ 1)(a - 2a + 1) - a (a - 2),
1
3
1
6
1
2
1
2
when a = 7, is:
SSC CHSL 13/10/2020 (Morning)
(a) 7 (b) 0 (c) 1 (d) 7
Q.35. If a + b + c + d = 2, then the
maximum value of —
(1 + a)(1 + b)(1 + c) (1 + d)
SSC CHSL 18/03/2020 (Afternoon)
(a) (b) (c) (d)
91
9
81
16
63
22
54
13
Q.36. The coe?cient of x
2
in ( 2 ?? + ?? )
3
is:
SSC CGL 05/03/2020 (Evening)
(a) 12y
2
(b) 12 (c) 12y (d) 8
Q.37. If x
2a
= y
2b
= z
2c
0 and x
2
= yz, ?
then the value of is:
???? + ???? + ????
????
SSC CGL 03/03/2020 (Morning)
(a) 3 (b) 3ac (c) 3bc (d) 3ab
Q.38. If x = 5.51, y = 5.52 and z = 5.57,
then what is the value of
? ??
3
+ ??
3
+ ??
3
- 3 ??????
SSC CPO 11/12/2019 (Evening)
(a) 5.146 (b) 51.46
(c) 0.05146 (d) 0.5146
Q.39. The value of is closest
1
7 - 4 3
to:
SSC CPO 09/12/2019 (Evening)
(a) 4.1 (b) 4.2 (c) 1.2 (d) 3.7
Q.40. If (2x + 3y + 4)(2x +3y - 5) is
equivalent to (ax
2
+ by
2
+ 2hxy + 2gx + 2fy
+ c ), then what is the value of {(g + f - c
)/abh}
SSC CPO 09/12/2019 (Morning)
(a) (b) (c) (d)
35
432
19
108
19
316
37
216
Q.41. (4 y - 6 + 4x - ) can be ??
3
??
2
??
2
??
3
??
4
expressed as:
SSC CPO 09/12/2019 (Morning)
(a) (x - y)
4
- x
4
(b) (x + y)
4
- y
4
(c) (x + y)
4
- x
4
(d) x
4
- (x - y)
4
Q.42. Let x = and
6
27 - 6
3
4
y = , then the value
45 + 605 + 245
80 + 125
of is : ??
2
+ ??
2
SSC CGL TIER II (13/09/2019)
(a) (b) (c) (d)
223
36
221
36
221
9
227
9
Q.43. If = a - b , then what 86 - 60 2 2
will be the value of , correct to ??
2
+ ??
2
one decimal place ?
SSC CGL TIER II (11/09/2019)
(a) 8.4 (b) 8.2 (c) 7.8 (d) 7.2
Q.44. If a = and a = , then 8 - 7
1
??
is equal to :
??
2
+ ??
2
- 3 ????
??
2
+ ???? + ??
2
SSC CHSL 11/07/2019 (Evening)
(a) (b) (c) (d)
27
31
27
32
29
33
29
31
Q.45. If (a + b + 4){ab + 4(a + b)} - 4ab = 0,
a - 4, b - 4, then , ? ?
{ - } is equal to :
1
( ?? + ?? + 4 )
117
2
- 234
SSC CHSL 11/07/ 2019 (Afternoon)
(a) (b) (c) - (d) 0
1
4
117
1
2
117
1
2
234
Q.46. ( x + y ) + ( z + y ) = - ( x + z ) then
1
3
1
3
1
3
( can be expressed as : ??
3
+ ??
3
+ ??
3
)
SSC CHSL 10/07/2019 (Morning)
(a) xyz
1
8
(b) (x + y)(y + z)(z + x)
(c) (x + y)(y + z)(z + x)
3
8
(d) 3xyz
Q.47. The value of the expression
{( a + )
2
( a )
2
} is :
1
4
1
??
- -
1
??
SSC CHSL 09/07/2019 (Evening)
(a) (b) (c) 1 (d) 4
1
2
1
4
Q.48. If 3 + 4 = 5 , then the value
4
??
4
??
4
??
of x is :
SSC CHSL 08/07/2019 (Morning)
(a) 4 (b) 2 (c) 8 (d) 16
Q.49. If x = 2 - then the value of 3
is : ??
3
- ??
- 3
SSC CHSL 05/07/2019 (Evening)
(a) -30 (b) 30 (c) -30 (d) 30 3 3 2 2
Q.50. If x - 1, 2 and 5, then the ?
simpli?ed value of {
2 ( ??
3
- 8 )
??
2
- ?? - 2
×
} is equal to :
??
2
+ 2 ?? + 1
??
2
- 4 ?? - 5
÷
??
2
+ 2 ?? + 4
3 ?? - 15
SSC CHSL 03/07/2019 (Afternoon)
(a) (b) 6 (c) (d)
1
6
3
2
2
3
Q.51. If x is real, and ??
4
- 5 ??
2
- 1 = 0 ,
www.ssccglpinnacle.com Download Pinnacle Exam Preparation App 264
Page 4
Pinnacle Day: 39th - 48th Algebra
Algebra
Important Formulas:-
· = ( ?? + ?? )
2
??
2
+ ??
2
+ 2 ????
· = ( ?? - ?? )
2
??
2
+ ??
2
- 2 ????
· = ( ?? + ?? )
2
+ ( ?? - ?? )
2
2 ( ??
2
+ ??
2
)
· ( ?? + ?? )
2
- ( ?? - ?? )
2
= 4 ????
· ( ?? + ?? )
2
- 4 ???? = ( ?? - ?? )
2
· ??
2
- ??
2
= ( ?? + ?? )( ?? - ?? )
· ??
4
- ??
4
= ( ??
2
+ ??
2
)( ?? + ?? )( ?? - ?? )
· = ( )
2
2 ??
2
+
1
??
2
?? +
1
??
-
= ( )
2
?? -
1
??
+ 2
· = ??
2
+ ??
2
+ ??
2
- ???? - ???? - ????
+
1
2
{ ?? - ?? ( )
2
( ?? - ?? )
2
+ ?? - ?? ( )
2
}
If then, ?? = ?? = ?? = ??
, ??
2
+ ??
2
+ ??
2
= ???? + ???? + ????
· ( ?? + ?? )
3
= ??
3
+ ??
3
+ 3 ???? ( ?? + ?? )
· ( ?? - ?? )
3
= ??
3
- ??
3
- 3 ???? ( ?? - ?? )
· ??
3
+ ??
3
= ( ?? + ?? )( ??
2
+ ??
2
- ???? )
· ??
3
- ??
3
= ( ?? - ?? )( ??
2
+ ??
2
+ ???? )
· + = ??
3 1
??
3
?? +
1
??
( )
3
- 3 ?? +
1
??
( )
· = ??
3
-
1
??
3
?? -
1
??
( )
3
+ 3 ?? -
1
??
( )
· (a + b + c) ??
3
+ ??
3
+ ??
3
- 3 ?????? =
( ??
2
+ ??
2
+ ??
2
- ???? - ???? - ???? )
Or ,
1
2
( ?? + ?? + ?? ){ ?? - ?? ( )
2
+ ?? - ?? ( )
2
Or, + ?? - ?? ( )
2
}
(a + b + c)[ - 3(ab + bc + ca)] ( ?? + ?? + ?? )
2
Or , (a + b + c)
1
2
3 ( ??
2
+ ??
2
+ ??
2
) - ( ?? + ?? + ?? )
2
[ ]
· If , then ?? + ?? + ?? ( ) = 0
??
3
+ ??
3
+ ??
3
= 3 ??????
· If ( ) = 0, then ?? - ?? - ??
??
3
- ??
3
- ??
3
= 3 ??????
· If ( ) = 0, then ?? + ?? - ??
??
3
+ ??
3
- ??
3
=- 3 ??????
· If , ??
3
+ ??
3
+ ??
3
= 3 ??????
then either a = b = c or a + b + c = 0
· = ( ?? + ?? + ?? )
2
??
2
+ ??
2
+ ??
2
+ 2 ( ???? + ???? + ???? )
· = ( ?? + ?? + ?? )
3
??
3
+ ??
3
+ ??
3
+
3 ( ?? + ?? )( ?? + ?? )( ?? + ?? )
· = ??
4
+ ??
2
??
2
+ ??
4
( ??
2
+ ???? + ??
2
)
( ??
2
- ???? + ??
2
)
· If = 0, ( ??
2
+ ??
2
+ ??
2
)
then = 0, = 0, = 0 ?? ?? ??
· If + = , then + = ??
1
??
?? ??
3 1
??
3
??
3
- 3 ??
· If - = , then - = ??
1
??
?? ??
3 1
??
3
??
3
+ 3 ??
· If + = , then + = ??
1
??
?? ??
2 1
??
2
??
2
- 2
· If - = , then + = ??
1
??
?? ??
2 1
??
2
??
2
+ 2
· If + = , then - = ??
1
??
?? ??
1
??
??
2
- 4
· If - = , then + = ??
1
??
?? ??
1
??
??
2
+ 4
· If + = , then ??
1
??
??
- = ??
2 1
??
2
?? ??
2
- 4
· If - = , then ??
1
??
??
- = ??
2 1
??
2
?? ??
2
+ 4
· If + = , then ??
1
??
??
+ = ??
4 1
??
4
( ??
2
- 2 )
2
- 2
· If - = , then ??
1
??
??
+ = ??
4 1
??
4
( ??
2
+ 2 )
2
- 2
· If + = a , then ??
1
??
+ = ( + ) ( + ) - ( x + ) ??
5 1
??
5
??
2 1
??
2
??
3 1
??
3
1
??
If - = a, then • ??
1
??
- = ( + ) ( - ) - ( - ) ??
5 1
??
5
??
2 1
??
2
??
3 1
??
3
??
1
??
· If + = , then = -1 ??
1
??
± 3 ??
6
· If + = 1, then = -1 ??
1
??
??
3
· If + = -1, then = 1 ??
1
??
??
3
· If + = 2, then = 1, similarly if ??
1
??
??
+ = -2, then = -1 ??
1
??
??
Some Special Case :-
? if - ab + = 0 then + = 0 ??
2
??
2
??
3
??
3
? if - = then + = 0
1
??
1
??
1
?? - ??
??
3
??
3
? if + = then - = 0
1
??
1
??
1
?? + ??
??
3
??
3
? if + = -1 then - = 0
??
??
??
??
??
3
??
3
? if + = 1 then + = 0
??
??
??
??
??
3
??
3
? if + a + 1 = 0 then - 1 = 0 or = 1 ??
2
??
3
??
3
? if ab(a + b) = 1 then - - = 3
1
??
3
??
3
??
3
??
3
· Dividendo – Componendo rule:
If = then, =
??
??
??
??
?? + ??
?? - ??
?? + ??
?? - ??
If = then =
?? + ??
?? - ??
??
??
??
??
?? + ??
?? - ??
· If 0 ?? - ?? ( )
2
+ ?? - ?? ( )
2
+ ?? - ?? ( )
2
=
then by putting in ?? = ?? , ?? = ?? , ?? = ??
an expression we can ?nd the value of
the expression.
· If , then will be ?? + ?? = ?? ????
maximum when = . ?? ??
· If = a, then ( will be ???? ?? + ?? )
minimum when = = ?? ?? ??
Law of Indices :
(1) = (2) = ??
??
× ??
??
??
?? + ?? ??
??
??
??
??
?? - ??
(3) = (4) = ( ??
??
)
??
??
????
(
??
??
)
??
??
??
??
??
(5) = 1 (6) = ??
0
??
1
??
??
??
If a is negative number and n is an odd
number then ( )
n
=
n
- ?? - ??
Laws of surds :
(1). =
??
?? ??
1
??
(2). =
??
?? ×
??
??
??
????
(3). =
??
??
??
??
??
??
??
(4). =
??
??
??
????
??
(5). = (
??
?? )
??
??
??
??
Ex.1. If 3 is given, ?nd out the 5
rationalizing factor.
Sol : 3 = 3 5 = 15, so the 5 × 5 ×
rationalizing factor of 3 is 5 5 .
Rules for making a perfect square :
1. Middle term =
± ?????????? ???????? × ??h?????? ????????
2. First term =
( ???????????? ???????? )
2
4 × ??h?????? ????????
3. Third term =
( ???????????? ???????? )
2
4 × ?????????? ????????
Quadratic and cubic equation :
? For quadratic equation :
= 0, then ?? ??
2
+ ???? + ??
Sum of its zeros ( + ) = - a ß
??
??
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Pinnacle Day: 39th - 48th Algebra
Product of its zeros ( ) = aß
??
??
where, amd are roots of a a ß
given quadratic equation.
? For cubic equation :
= 0, then ?? ??
3
+ ?? ??
2
+ ???? + ??
Sum of its zeros ( + + )= - a ß ?
??
??
Product of its zeros ( ) = - aß?
??
??
Sum of its zeros taken two at a
time ( + + ) = aß ß? ?a
??
??
where , and are roots of a a ß ?
given cubic equation.
Variety Questions
Q.1. If , then the value of ??
4
+ ??
4
= ??
2
??
2
+ is : ??
6
??
6
SSC CPO 04/10/2023 (1st Shift)
(a) 1 (b) 3 (c) 2 (d) 0
Q.2. If and ???? + ???? = 1 ???? + ???? =
then the value of x (in terms of
2 ????
??
2
+ ??
2
,
a and b) is:
SSC CHSL 11/08/2023 (4th Shift)
(a) (b)
2 ??
??
2
+ ??
2
??
??
2
+ ??
2
(c) (d)
??
??
2
+ ??
2
2 ??
??
2
+ ??
2
Q.3. If x = (7 + 3 ) then ?nd the value 5
of x
2
+
1
??
2
SSC CHSL 09/08/2023 (4th Shift)
(a) (b)
580 + 315 5
8
799 + 328 5
8
(c) (d)
799 + 315 5
12
799 + 315 5
8
Q.4. If x
2
- 11x + 1 = 0 what is the value
of x
8
- 14159x
4
+ 11?
SSC CHSL 08/08/2023 (1st Shift)
(a) 9 (b) 10 (c) 12 (d) 11
Q.5. Simplify:
= ?
( 1 - ?? )
3
+ ( ?? - ?? )
3
+ ( ?? - 1 )
3
12 ( 1 - ?? ) ( ?? - ?? ) ( ?? - 1 )
SSC CHSL 07/08/2023 (3rd Shift)
(a) (b) (c) (d)
1
2
2
3
3
2
1
4
Q.6. For what positive value of k, the
equation 3x² + 2kx + 3 = 0 has real and
equal roots?
SSC CHSL 07/08/2023 (2nd Shift)
(a) 12 (b) 1 (c) 3 (d) 9
Q.7. Simplify the following expression.
??
2
+ ??
2
- ??
2
( )
2
- ??
2
- ??
2
+ ??
2
( )
2
??
2
- ??
2
SSC CHSL 04/08/2023 (4th Shift)
(a) 3a
2
(b) 4a
2
(c) 5a
2
(d) 2a
2
Q.8. Find the value of (a
3
+ b
3
+ c
3
- 3abc),
where a = 335, b = 215 and c = 180.
SSC CHSL 04/08/2023 (3rd Shift)
(a) 14472250 (b) 15421320
(c) 14502230 (d) 15452630
Q.9. Simplify the expression :
( ?? - ?? )
3
+ ( ?? - ?? )
3
+ ( ?? - ?? )
3
( ??
2
- ??
2
)
3
+ ( ??
2
- ??
2
)
3
+ ( ??
2
- ??
2
)
3
SSC CHSL 04/08/2023 (1st Shift)
(a) (b) 1
1
( ?? + ?? ) ( ?? + ?? ) ( ?? + ?? )
(c) (d) 0
3
( ?? + ?? ) ( ?? + ?? ) ( ?? + ?? )
Q.10. If x
3
= 270 + y
3
and x = (6 + y), then
what is the value of (x + y)?
(given that x > 0 andy > 0)
SSC CHSL 03/08/2023 (4th Shift)
(a) 2 (b) (c) 4 (d) 4 3 3 3 2
Q.11. If p + q + r = 0 , then what is the
simpli?ed value of the expression
( + + )
??
2
??
2
- ????
??
2
??
2
- ????
??
2
??
2
- ????
SSC CHSL 03/08/2023 (2nd Shift)
(a) 0 (b) 2 (c) 1 (d) - 1
Q.12. If ( ) = , and x > 1, what is ?? +
1
??
5 2
the value of ( ) ? ??
6
-
1
??
6
SSC CHSL 02/08/2023 (2nd Shift)
(a) 22970 (b) 23030 23 23
(c) 23060 (d) 22960 23 23
Q.13. The difference between the cubes
of two given natural numbers is 6272,
while the positive difference between the
two given numbers is 8. What is the sum
of the cubes of the two given numbers?
SSC CGL 27/07/2023 (1st shift)
(a) 9728 (b) 9684 (c) 8000 (d) 9600
Q.14. If a = 101, b = 102 and c = 103,
then a² + b² + c² - ab - bc - ca = ______.
SSC CGL 24/07/2023 (1st shift)
(a) 2 (b) 4 (c) 3 (d) 6
Q.15. If ) , and x > 1, what is ( ?? +
1
??
= 5
the value of ) ? ( ??
8
-
1
??
8
SSC CGL 18/07/2023 (3rd shift)
(a) 60605 (b) 60615 21 21
(c) 60705 (d) 60725 21 21
Q.16. If What will be the ?? -
1
??
= - 6 ,
value of ??
5
-
1
??
5
?
SSC CGL 18/07/2023 (2nd shift)
(a) 8898 (b) 8896 - -
(c) 8886 (d) 8892 - -
Q.17. If ( + ) = 7, and 0 < x < 1, ?nd ?? ²
1
?? ²
the value of - . ?? ²
1
?? ²
SSC CGL 17/07/2023 (1st shift)
(a) (b) (c) (d) 3 5 4 3 - 4 3 - 3 5
Q.18. If x > 0, and + = 254, what is ??
4 1
??
4
the value of + ? ??
5 1
??
5
SSC CGL 14/07/2023 (4th shift)
(a) 717 (b) 723 (c) 720 (d) 726 2 2 2 2
Q.19. If + -
2 ??
?? + 2
1
2 ?? + 4
12 - ??
2 ??
2
+ 4 ??
= a + , where a and b are integers, then
??
??
the value of ab = ___________ .
Graduate Level 27/06/2023 (Shift - 4)
(a) 3 (b) - 6 (c) 6 (d) - 3
Q.20. If 6
X
= 3
Y
= 2
z
, then what is the
value of + - ?
1
??
1
??
1
??
SSC CHSL 20/03/2023 (2nd Shift)
(a) 1 (b) 0 (c) 3 (d) 6
Q.21. For what value of m will the system
of equations 17x + my + 102 = 0 and 23x
+ 299y + 138 = 0 have an in?nite number
of solutions?
SSC CGL Tier II (02/03/2023)
(a) 221 (b) 223 (c) 220 (d) 219
Q.22. If a + b + c = 5, a
3
+ b
3
+ c
3
= 85 and
abc = 25, then ?nd the value of a
2
+ b
2
+
c
2
- ab - bc - ca.
SSC CGL 13/12/2022 (4th Shift)
(a) 2 (b) 4 (c) 6 (d) 8
Q.23. If , are the roots of 6x² + 13x + a ß
7 = 0, then the equation whose roots are
2
,
2
is: a ß
SSC CGL 12/12/2022 (4th Shift)
(a) 36 - 87x + 49 = 0 ??
2
(b) 36 - 85x + 49 = 0 ??
2
(c) 36 - 85x - 49 = 0 ??
2
(d) 36 + 87x - 49 = 0 ??
2
Q.24. If x = 222, y = 223 and z = 224, then
?nd the value of + + - 3xyz. ??
3
??
3
??
3
SSC CGL 12/12/2022 (4th Shift)
(a) 2007 (b) 2004 (c) 2006 (d) 2005
Q.25. What is the value of
( )( )( ) ?? -
1
??
??
2
+
1
??
2
??
4
+
1
??
4
( )( )( ) ? ??
8
+
1
??
8
??
16
+
1
??
16
??
32
+
1
??
32
www.ssccglpinnacle.com Download Pinnacle Exam Preparation App 263
Pinnacle Day: 39th - 48th Algebra
SSC CGL 06/12/2022 (3rd Shift)
(a) (b)
??
64
-
1
??
64
?? +
1
??
??
32
-
1
??
32
?? -
1
??
(c) (d)
??
32
-
1
??
32
?? +
1
??
??
32
+
1
??
32
?? +
1
??
Q.26. Which of the following statements
is correct ?
I. The value of 100
2
- 99
2
+ 98
2
- 97
2
+ 96
2
- 95
2
+ 94
2
- 93
2
+ ...... + 2
2
- 1
2
is 5050.
II. If 8x + = -16 and x < 0, then the
8
??
value of x
197
+ x
- 197
is 2.
SSC CGL 02/12/2022 (4th Shift)
(a) Only I (b) Only II
(c) Both I and II (d) Neither I nor II
Q.27. If m – n = 0 then what is the ??
??
??
??
value of + in terms of
1
??
??
+ ??
??
1
??
??
- ??
??
? ??
??
SSC CHSL 24/05/2022 (Morning)
(a) (b)
2 ????
??
??
( ??
2
+ ??
2
)
2 ??
??
??
( ??
2
- ??
2
)
(c) (d)
2 ????
??
??
( ??
2
- ??
2
)
2 ????
??
??
( ??
2
- ??
2
)
Q.28. , and ??
2
+ ??
2
= 25 ??
2
+ ??
2
= 17
ax + by = 8 then ?nd the value of (ay - bx)
SSC CHSL 13/4/2021 (Afternoon)
(a) 33 (b) 25 (c) 21 (d) 19
Q.29. If 2 = x + , t hen the
1
1 +
1
5 +
1
2
value of x is equal to:
SSC CGL Tier II (18/11/2020)
(a) (b) 1 (c) (d)
14
13
15
13
13
15
Q.30. If x
3
+ 2 x
2
ax b is exactly - -
divisible by ( x
2
1), then the values of a -
and b are:
SSC CHSL 21/10/2020 (Afternoon)
(a) a = -1, and b = 2 (b) a = 1 and b = -2
(c) a = 1 and b = 2 (d) a = 2 and b = 2
Q.31. If = + 2, then the value of ??
3
5
is equal to: ??
3
- 6 ??
2
+ 12 ?? - 12
SSC CHSL 21/10/2020 (Morning)
(a) 0 (b) 2 (c) 1 (d) - 1
Q.32. (ax + by) is a factor of:
SSC CHSL 16/10/2020 (Afternoon)
(a) a
2
x
2
+ 2ab - b
2
y
2
(b) a
2
x
2
+ 2abxy + b
2
y
2
(c) a
2
x
3
+ 2abx + b
2
y
2
x
(d) a
2
x
2
+ 2ab + b
2
y
2
Q.33. If ‘a’ is a natural number, then (7a
2
+ 7a) is always divisible by:
SSC CHSL 16/10/2020 (Morning shift)
(a) 7 and 14 both (b) 7 only
(c) 14 only (d) 21 only
Q.34. The value of (a + 2a + 3a + 2a
2
3
1
2
1
3
1
6
+ 1)(a - 2a + 1) - a (a - 2),
1
3
1
6
1
2
1
2
when a = 7, is:
SSC CHSL 13/10/2020 (Morning)
(a) 7 (b) 0 (c) 1 (d) 7
Q.35. If a + b + c + d = 2, then the
maximum value of —
(1 + a)(1 + b)(1 + c) (1 + d)
SSC CHSL 18/03/2020 (Afternoon)
(a) (b) (c) (d)
91
9
81
16
63
22
54
13
Q.36. The coe?cient of x
2
in ( 2 ?? + ?? )
3
is:
SSC CGL 05/03/2020 (Evening)
(a) 12y
2
(b) 12 (c) 12y (d) 8
Q.37. If x
2a
= y
2b
= z
2c
0 and x
2
= yz, ?
then the value of is:
???? + ???? + ????
????
SSC CGL 03/03/2020 (Morning)
(a) 3 (b) 3ac (c) 3bc (d) 3ab
Q.38. If x = 5.51, y = 5.52 and z = 5.57,
then what is the value of
? ??
3
+ ??
3
+ ??
3
- 3 ??????
SSC CPO 11/12/2019 (Evening)
(a) 5.146 (b) 51.46
(c) 0.05146 (d) 0.5146
Q.39. The value of is closest
1
7 - 4 3
to:
SSC CPO 09/12/2019 (Evening)
(a) 4.1 (b) 4.2 (c) 1.2 (d) 3.7
Q.40. If (2x + 3y + 4)(2x +3y - 5) is
equivalent to (ax
2
+ by
2
+ 2hxy + 2gx + 2fy
+ c ), then what is the value of {(g + f - c
)/abh}
SSC CPO 09/12/2019 (Morning)
(a) (b) (c) (d)
35
432
19
108
19
316
37
216
Q.41. (4 y - 6 + 4x - ) can be ??
3
??
2
??
2
??
3
??
4
expressed as:
SSC CPO 09/12/2019 (Morning)
(a) (x - y)
4
- x
4
(b) (x + y)
4
- y
4
(c) (x + y)
4
- x
4
(d) x
4
- (x - y)
4
Q.42. Let x = and
6
27 - 6
3
4
y = , then the value
45 + 605 + 245
80 + 125
of is : ??
2
+ ??
2
SSC CGL TIER II (13/09/2019)
(a) (b) (c) (d)
223
36
221
36
221
9
227
9
Q.43. If = a - b , then what 86 - 60 2 2
will be the value of , correct to ??
2
+ ??
2
one decimal place ?
SSC CGL TIER II (11/09/2019)
(a) 8.4 (b) 8.2 (c) 7.8 (d) 7.2
Q.44. If a = and a = , then 8 - 7
1
??
is equal to :
??
2
+ ??
2
- 3 ????
??
2
+ ???? + ??
2
SSC CHSL 11/07/2019 (Evening)
(a) (b) (c) (d)
27
31
27
32
29
33
29
31
Q.45. If (a + b + 4){ab + 4(a + b)} - 4ab = 0,
a - 4, b - 4, then , ? ?
{ - } is equal to :
1
( ?? + ?? + 4 )
117
2
- 234
SSC CHSL 11/07/ 2019 (Afternoon)
(a) (b) (c) - (d) 0
1
4
117
1
2
117
1
2
234
Q.46. ( x + y ) + ( z + y ) = - ( x + z ) then
1
3
1
3
1
3
( can be expressed as : ??
3
+ ??
3
+ ??
3
)
SSC CHSL 10/07/2019 (Morning)
(a) xyz
1
8
(b) (x + y)(y + z)(z + x)
(c) (x + y)(y + z)(z + x)
3
8
(d) 3xyz
Q.47. The value of the expression
{( a + )
2
( a )
2
} is :
1
4
1
??
- -
1
??
SSC CHSL 09/07/2019 (Evening)
(a) (b) (c) 1 (d) 4
1
2
1
4
Q.48. If 3 + 4 = 5 , then the value
4
??
4
??
4
??
of x is :
SSC CHSL 08/07/2019 (Morning)
(a) 4 (b) 2 (c) 8 (d) 16
Q.49. If x = 2 - then the value of 3
is : ??
3
- ??
- 3
SSC CHSL 05/07/2019 (Evening)
(a) -30 (b) 30 (c) -30 (d) 30 3 3 2 2
Q.50. If x - 1, 2 and 5, then the ?
simpli?ed value of {
2 ( ??
3
- 8 )
??
2
- ?? - 2
×
} is equal to :
??
2
+ 2 ?? + 1
??
2
- 4 ?? - 5
÷
??
2
+ 2 ?? + 4
3 ?? - 15
SSC CHSL 03/07/2019 (Afternoon)
(a) (b) 6 (c) (d)
1
6
3
2
2
3
Q.51. If x is real, and ??
4
- 5 ??
2
- 1 = 0 ,
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Pinnacle Day: 39th - 48th Algebra
then the value of
( + + 1) is : ??
6
- 3 ??
2 3
??
2
-
1
??
6
SSC CHSL 03/07/2019 (Morning)
(a) 126 (b) 110 (c) 116 (d) 96
Q.52. If x + y = 7 and xy = 10, then the
value of + is:
1
??
3
1
??
3
SSC CHSL 01/07/2019 (Evening)
(a) 0.543 (b) 0.131 (c) 0.133 (d) 0.453
Q.53. If ( 3 ?? - 1 )
3
+ ( 4 ?? - 3 )
3
+
= 3(3x - 1)(4x - 3)(2x + 1) and ( 2 ?? + 1 )
3
x then x = ? ?
1
3
SSC CGL 10/06/2019 (Evening)
(a) (b) 2 (c) (d) 1
1
2
1
4
Q.54. If x + y + z = 0 , then the value of
( - xy) is : ( ??
2
+ ??
2
+ ??
2
) ÷ ??
2
SSC CGL 10/06/2019 (Morning)
(a) 1 (b) 2 (c) - 2 (d) - 1
Q.55. If - 3x - 1 = 0 , then the value of ??
2
( + 8x - 1 ) is : ??
2
( ??
3
+ ??
- 1
)
- 1
SSC CGL 07/06/2019 (Afternoon)
(a) (b) 8 (c) 1 (d) 3
3
8
Q.56. If x = 2 - p , then , is ??
3
+ 6 ???? + ??
3
equal to :
SSC CGL 07/06/2019 (Morning)
(a) 12 (b) 6 (c) 8 (d) 4
Q.57. If + 6x - 2xy + 9 = 0, then 2 ??
2
+ ??
2
the value of is : 4 ??
3
- ??
3
+ ??
2
??
2
SSC CGL 06/06/2019 (Afternoon)
(a) 0 (b) 9 (c) - 3 (d) - 9
Q.58. If ab + bc + ca = 8 and + + ??
2
??
2
??
2
= 20, then a possible value of (a + b +
1
2
c) [(a - b + (b - c + (c - a ] is )
2
)
2
)
2
SSC CGL 06/06/2019 (Morning)
(a) 72 (b) 56 (c) 84 (d) 80
Q.59. If x + y = 1 and xy(xy - 2) = 12, then
the value of is: ??
4
+ ??
4
SSC CGL 04/06/2019 (Evening)
(a) 19 (b) 25 (c) 20 (d) 23
Q.60. If 24(x - y + z) 16 ??
2
+ 9 ??
2
+ 4 ??
2
=
, then the value of (xy + 2z) is : - 61
SSC CGL 04/06/2019 (Afternoon)
(a) 1 (b) 2 (c) 3 (d) 5
Q.61. If ( ) 5 5 ??
3
- 81 3 ??
3
÷ 5 ?? -
= , then the value 3 3 ?? ?? ??
2
+ ?? ??
2
+ ??????
of (6A + B - ) is : 15 ??
SSC CGL 04/06/2019 (Morning)
(a) 10 (b) 9 (c) 15 (d) 12
Q.62. If x + y + z = 19,
and xz = , then the ??
2
+ ??
2
+ ??
2
= 133 ??
2
difference between z and x is :
SSC CGL 04/06/2019 (Morning)
(a) 5 (b) 3 (c) 6 (d) 4
Q.63. If , then the ??
4
+ ??
- 4
= 194 ?? ? 0
value of is : ( ?? - 2 )
2
SSC CGL 04/06/2019 (Morning)
(a) 1 (b) 6 (c) 2 (d) 3
Q.64. If roots of - 4x + a = 0 are equal, ??
2
then a = ?
SSC CPO 16/03/2019 (Evening)
(a) - 4 (b) 4 (c) 8 (d) -8
Practice Questions
SSC CHSL 2023 Tier - 2
Q.65. If (a - 2b) = 8 and 2ab = 9, then ?nd
the value of (a + 2b).
SSC CHSL Tier II (02/11/2023)
(a) 12 (b) 6 (c) 10 (d) 8
Q.66. A and B each have a certain
number of oranges. A says to B, “If you
give me 10 of your oranges, I will have
twice the number of oranges left with
you.” B replies, “If you give me 10 of your
oranges, I will have the same number of
oranges left with you.” Find the number
of oranges that A has.
SSC CHSL Tier II (02/11/2023)
(a) 65 (b) 50 (c) 55 (d) 70
SSC CPO 2023 Tier - 1
Q.67. If + = 1 and = 2, then
??
??
??
??
?? + ??
the value of is: ??
3
+ ??
3
SSC CPO 03/10/2023 (1st Shift)
(a) 0 (b) 1 (c) 3 (d) 2
Q.68. If + = 1 and + = 1,
??
??
??
??
??
??
??
??
then the value of + is :
??
??
??
??
SSC CPO 03/10/2023 (2nd Shift)
(a) 1 (b) (c) (d) 0
??
??
??
??
Q.69. If a ? b, b ? and c ? d, then the
1
??
relation between a and b is :
SSC CPO 03/10/2023 (2nd Shift)
(a) a ? (b) a ? d (c) a ? (d) a ? ??
1
??
??
2
Q.70. If A + = , then the
1
1 +
1
2 +
1
3
9
10
value of A is:
SSC CPO 03/10/2023 (3rd Shift)
(a) (b) (c) (d)
3
10
2
5
1
10
1
5
Q.71. If (5 - ) = , then the value of ??
2
??
5
??
x² + is:
1
??
2
SSC CPO 03/10/2023 (3rd Shift)
(a) (b) (c) (d)
54
23
53
28
53
27
54
25
Q.72. The value of
157 × 157 ( ) + 157 × 133 ( ) + 133 × 133 ( )
157 × 157 × 157 ( ) - 133 × 133 × 133 ( )
is :
SSC CPO 04/10/2023 (2nd Shift)
(a) 24 (b) (c) 290 (d)
1
290
1
24
Q.73. Simplify the given expression.
3 . 254 × 3 . 254 × 3 . 254 ( ) - 0 . 746 × 0 . 746 × 0 . 746 ( )
( 3 . 254 × 3 . 254 ) + ( 0 . 746 × 0 . 746 ) + ( 3 . 254 × 0 . 746 )
SSC CPO 04/10/2023 (2nd Shift)
(a) 2.508 (b) 4.508 (c) 2.146 (d) 3.156
Q.74. The value of +
??
2
- ( ?? - ?? )
2
( ?? + ?? )
2
- ??
2
is :
??
2
- ( ?? - ?? )
2
( ?? + ?? )
2
- ??
2
+
??
2
- ( ?? - ?? )
2
( ?? + ?? )
2
- ??
2
SSC CPO 04/10/2023 (3rd Shift)
(a) 3 (b) -1 (c) 1 (d) 2
Q.75. Simplify the given expression.
( 1 + ?? )
3
+ ( 1 - ?? )
3
+ (- 2 )
3
SSC CPO 04/10/2023 (3rd Shift)
(a) 3(1 - x
2
) (b) 6(1 - x
2
)
(c) - 6(1 - x
2
) (d) - 3(1 - x
2
)
Q.76. Simplify the given expression.
( 1 - 2 ?? )
2
- ( 1 + 2 ?? )
2
SSC CPO 05/10/2023 (1st Shift)
(a) (b) - 8 ?? 8 ??
(c) (d) - 2 + 8 ??
2
( 2 + 8 ??
2
)
Q.77. If a + b + c = 5 and ab + bc + ca = 7,
then the value of is: ??
3
+ ??
3
+ ??
3
- 3 ??????
SSC CPO 05/10/2023 (1st Shift)
(a) 20 (b) 25 (c) 15 (d) 30
Q.78. Simplify the given expression.
(x - 2y)(y - 3x) + (x + y)(x - y) + (x - 3y)(2x
+ y)
SSC CPO 05/10/2023 (3rd Shift)
(a) 2x(x - 3y) (b) 2y(x + 3y)
(c) 2x(x + 3y) (d) 2y(x - 3y)
SSC MTS 2023 Tier - 1
Q.79. If (x - 2) is a factor of (2x
2
+ 12kx -
25k), then what is the value of k?
SSC MTS 01/09/2023 (2nd Shift)
(a) 2 (b) 8 (c) 4 (d) 6
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Page 5
Pinnacle Day: 39th - 48th Algebra
Algebra
Important Formulas:-
· = ( ?? + ?? )
2
??
2
+ ??
2
+ 2 ????
· = ( ?? - ?? )
2
??
2
+ ??
2
- 2 ????
· = ( ?? + ?? )
2
+ ( ?? - ?? )
2
2 ( ??
2
+ ??
2
)
· ( ?? + ?? )
2
- ( ?? - ?? )
2
= 4 ????
· ( ?? + ?? )
2
- 4 ???? = ( ?? - ?? )
2
· ??
2
- ??
2
= ( ?? + ?? )( ?? - ?? )
· ??
4
- ??
4
= ( ??
2
+ ??
2
)( ?? + ?? )( ?? - ?? )
· = ( )
2
2 ??
2
+
1
??
2
?? +
1
??
-
= ( )
2
?? -
1
??
+ 2
· = ??
2
+ ??
2
+ ??
2
- ???? - ???? - ????
+
1
2
{ ?? - ?? ( )
2
( ?? - ?? )
2
+ ?? - ?? ( )
2
}
If then, ?? = ?? = ?? = ??
, ??
2
+ ??
2
+ ??
2
= ???? + ???? + ????
· ( ?? + ?? )
3
= ??
3
+ ??
3
+ 3 ???? ( ?? + ?? )
· ( ?? - ?? )
3
= ??
3
- ??
3
- 3 ???? ( ?? - ?? )
· ??
3
+ ??
3
= ( ?? + ?? )( ??
2
+ ??
2
- ???? )
· ??
3
- ??
3
= ( ?? - ?? )( ??
2
+ ??
2
+ ???? )
· + = ??
3 1
??
3
?? +
1
??
( )
3
- 3 ?? +
1
??
( )
· = ??
3
-
1
??
3
?? -
1
??
( )
3
+ 3 ?? -
1
??
( )
· (a + b + c) ??
3
+ ??
3
+ ??
3
- 3 ?????? =
( ??
2
+ ??
2
+ ??
2
- ???? - ???? - ???? )
Or ,
1
2
( ?? + ?? + ?? ){ ?? - ?? ( )
2
+ ?? - ?? ( )
2
Or, + ?? - ?? ( )
2
}
(a + b + c)[ - 3(ab + bc + ca)] ( ?? + ?? + ?? )
2
Or , (a + b + c)
1
2
3 ( ??
2
+ ??
2
+ ??
2
) - ( ?? + ?? + ?? )
2
[ ]
· If , then ?? + ?? + ?? ( ) = 0
??
3
+ ??
3
+ ??
3
= 3 ??????
· If ( ) = 0, then ?? - ?? - ??
??
3
- ??
3
- ??
3
= 3 ??????
· If ( ) = 0, then ?? + ?? - ??
??
3
+ ??
3
- ??
3
=- 3 ??????
· If , ??
3
+ ??
3
+ ??
3
= 3 ??????
then either a = b = c or a + b + c = 0
· = ( ?? + ?? + ?? )
2
??
2
+ ??
2
+ ??
2
+ 2 ( ???? + ???? + ???? )
· = ( ?? + ?? + ?? )
3
??
3
+ ??
3
+ ??
3
+
3 ( ?? + ?? )( ?? + ?? )( ?? + ?? )
· = ??
4
+ ??
2
??
2
+ ??
4
( ??
2
+ ???? + ??
2
)
( ??
2
- ???? + ??
2
)
· If = 0, ( ??
2
+ ??
2
+ ??
2
)
then = 0, = 0, = 0 ?? ?? ??
· If + = , then + = ??
1
??
?? ??
3 1
??
3
??
3
- 3 ??
· If - = , then - = ??
1
??
?? ??
3 1
??
3
??
3
+ 3 ??
· If + = , then + = ??
1
??
?? ??
2 1
??
2
??
2
- 2
· If - = , then + = ??
1
??
?? ??
2 1
??
2
??
2
+ 2
· If + = , then - = ??
1
??
?? ??
1
??
??
2
- 4
· If - = , then + = ??
1
??
?? ??
1
??
??
2
+ 4
· If + = , then ??
1
??
??
- = ??
2 1
??
2
?? ??
2
- 4
· If - = , then ??
1
??
??
- = ??
2 1
??
2
?? ??
2
+ 4
· If + = , then ??
1
??
??
+ = ??
4 1
??
4
( ??
2
- 2 )
2
- 2
· If - = , then ??
1
??
??
+ = ??
4 1
??
4
( ??
2
+ 2 )
2
- 2
· If + = a , then ??
1
??
+ = ( + ) ( + ) - ( x + ) ??
5 1
??
5
??
2 1
??
2
??
3 1
??
3
1
??
If - = a, then • ??
1
??
- = ( + ) ( - ) - ( - ) ??
5 1
??
5
??
2 1
??
2
??
3 1
??
3
??
1
??
· If + = , then = -1 ??
1
??
± 3 ??
6
· If + = 1, then = -1 ??
1
??
??
3
· If + = -1, then = 1 ??
1
??
??
3
· If + = 2, then = 1, similarly if ??
1
??
??
+ = -2, then = -1 ??
1
??
??
Some Special Case :-
? if - ab + = 0 then + = 0 ??
2
??
2
??
3
??
3
? if - = then + = 0
1
??
1
??
1
?? - ??
??
3
??
3
? if + = then - = 0
1
??
1
??
1
?? + ??
??
3
??
3
? if + = -1 then - = 0
??
??
??
??
??
3
??
3
? if + = 1 then + = 0
??
??
??
??
??
3
??
3
? if + a + 1 = 0 then - 1 = 0 or = 1 ??
2
??
3
??
3
? if ab(a + b) = 1 then - - = 3
1
??
3
??
3
??
3
??
3
· Dividendo – Componendo rule:
If = then, =
??
??
??
??
?? + ??
?? - ??
?? + ??
?? - ??
If = then =
?? + ??
?? - ??
??
??
??
??
?? + ??
?? - ??
· If 0 ?? - ?? ( )
2
+ ?? - ?? ( )
2
+ ?? - ?? ( )
2
=
then by putting in ?? = ?? , ?? = ?? , ?? = ??
an expression we can ?nd the value of
the expression.
· If , then will be ?? + ?? = ?? ????
maximum when = . ?? ??
· If = a, then ( will be ???? ?? + ?? )
minimum when = = ?? ?? ??
Law of Indices :
(1) = (2) = ??
??
× ??
??
??
?? + ?? ??
??
??
??
??
?? - ??
(3) = (4) = ( ??
??
)
??
??
????
(
??
??
)
??
??
??
??
??
(5) = 1 (6) = ??
0
??
1
??
??
??
If a is negative number and n is an odd
number then ( )
n
=
n
- ?? - ??
Laws of surds :
(1). =
??
?? ??
1
??
(2). =
??
?? ×
??
??
??
????
(3). =
??
??
??
??
??
??
??
(4). =
??
??
??
????
??
(5). = (
??
?? )
??
??
??
??
Ex.1. If 3 is given, ?nd out the 5
rationalizing factor.
Sol : 3 = 3 5 = 15, so the 5 × 5 ×
rationalizing factor of 3 is 5 5 .
Rules for making a perfect square :
1. Middle term =
± ?????????? ???????? × ??h?????? ????????
2. First term =
( ???????????? ???????? )
2
4 × ??h?????? ????????
3. Third term =
( ???????????? ???????? )
2
4 × ?????????? ????????
Quadratic and cubic equation :
? For quadratic equation :
= 0, then ?? ??
2
+ ???? + ??
Sum of its zeros ( + ) = - a ß
??
??
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Pinnacle Day: 39th - 48th Algebra
Product of its zeros ( ) = aß
??
??
where, amd are roots of a a ß
given quadratic equation.
? For cubic equation :
= 0, then ?? ??
3
+ ?? ??
2
+ ???? + ??
Sum of its zeros ( + + )= - a ß ?
??
??
Product of its zeros ( ) = - aß?
??
??
Sum of its zeros taken two at a
time ( + + ) = aß ß? ?a
??
??
where , and are roots of a a ß ?
given cubic equation.
Variety Questions
Q.1. If , then the value of ??
4
+ ??
4
= ??
2
??
2
+ is : ??
6
??
6
SSC CPO 04/10/2023 (1st Shift)
(a) 1 (b) 3 (c) 2 (d) 0
Q.2. If and ???? + ???? = 1 ???? + ???? =
then the value of x (in terms of
2 ????
??
2
+ ??
2
,
a and b) is:
SSC CHSL 11/08/2023 (4th Shift)
(a) (b)
2 ??
??
2
+ ??
2
??
??
2
+ ??
2
(c) (d)
??
??
2
+ ??
2
2 ??
??
2
+ ??
2
Q.3. If x = (7 + 3 ) then ?nd the value 5
of x
2
+
1
??
2
SSC CHSL 09/08/2023 (4th Shift)
(a) (b)
580 + 315 5
8
799 + 328 5
8
(c) (d)
799 + 315 5
12
799 + 315 5
8
Q.4. If x
2
- 11x + 1 = 0 what is the value
of x
8
- 14159x
4
+ 11?
SSC CHSL 08/08/2023 (1st Shift)
(a) 9 (b) 10 (c) 12 (d) 11
Q.5. Simplify:
= ?
( 1 - ?? )
3
+ ( ?? - ?? )
3
+ ( ?? - 1 )
3
12 ( 1 - ?? ) ( ?? - ?? ) ( ?? - 1 )
SSC CHSL 07/08/2023 (3rd Shift)
(a) (b) (c) (d)
1
2
2
3
3
2
1
4
Q.6. For what positive value of k, the
equation 3x² + 2kx + 3 = 0 has real and
equal roots?
SSC CHSL 07/08/2023 (2nd Shift)
(a) 12 (b) 1 (c) 3 (d) 9
Q.7. Simplify the following expression.
??
2
+ ??
2
- ??
2
( )
2
- ??
2
- ??
2
+ ??
2
( )
2
??
2
- ??
2
SSC CHSL 04/08/2023 (4th Shift)
(a) 3a
2
(b) 4a
2
(c) 5a
2
(d) 2a
2
Q.8. Find the value of (a
3
+ b
3
+ c
3
- 3abc),
where a = 335, b = 215 and c = 180.
SSC CHSL 04/08/2023 (3rd Shift)
(a) 14472250 (b) 15421320
(c) 14502230 (d) 15452630
Q.9. Simplify the expression :
( ?? - ?? )
3
+ ( ?? - ?? )
3
+ ( ?? - ?? )
3
( ??
2
- ??
2
)
3
+ ( ??
2
- ??
2
)
3
+ ( ??
2
- ??
2
)
3
SSC CHSL 04/08/2023 (1st Shift)
(a) (b) 1
1
( ?? + ?? ) ( ?? + ?? ) ( ?? + ?? )
(c) (d) 0
3
( ?? + ?? ) ( ?? + ?? ) ( ?? + ?? )
Q.10. If x
3
= 270 + y
3
and x = (6 + y), then
what is the value of (x + y)?
(given that x > 0 andy > 0)
SSC CHSL 03/08/2023 (4th Shift)
(a) 2 (b) (c) 4 (d) 4 3 3 3 2
Q.11. If p + q + r = 0 , then what is the
simpli?ed value of the expression
( + + )
??
2
??
2
- ????
??
2
??
2
- ????
??
2
??
2
- ????
SSC CHSL 03/08/2023 (2nd Shift)
(a) 0 (b) 2 (c) 1 (d) - 1
Q.12. If ( ) = , and x > 1, what is ?? +
1
??
5 2
the value of ( ) ? ??
6
-
1
??
6
SSC CHSL 02/08/2023 (2nd Shift)
(a) 22970 (b) 23030 23 23
(c) 23060 (d) 22960 23 23
Q.13. The difference between the cubes
of two given natural numbers is 6272,
while the positive difference between the
two given numbers is 8. What is the sum
of the cubes of the two given numbers?
SSC CGL 27/07/2023 (1st shift)
(a) 9728 (b) 9684 (c) 8000 (d) 9600
Q.14. If a = 101, b = 102 and c = 103,
then a² + b² + c² - ab - bc - ca = ______.
SSC CGL 24/07/2023 (1st shift)
(a) 2 (b) 4 (c) 3 (d) 6
Q.15. If ) , and x > 1, what is ( ?? +
1
??
= 5
the value of ) ? ( ??
8
-
1
??
8
SSC CGL 18/07/2023 (3rd shift)
(a) 60605 (b) 60615 21 21
(c) 60705 (d) 60725 21 21
Q.16. If What will be the ?? -
1
??
= - 6 ,
value of ??
5
-
1
??
5
?
SSC CGL 18/07/2023 (2nd shift)
(a) 8898 (b) 8896 - -
(c) 8886 (d) 8892 - -
Q.17. If ( + ) = 7, and 0 < x < 1, ?nd ?? ²
1
?? ²
the value of - . ?? ²
1
?? ²
SSC CGL 17/07/2023 (1st shift)
(a) (b) (c) (d) 3 5 4 3 - 4 3 - 3 5
Q.18. If x > 0, and + = 254, what is ??
4 1
??
4
the value of + ? ??
5 1
??
5
SSC CGL 14/07/2023 (4th shift)
(a) 717 (b) 723 (c) 720 (d) 726 2 2 2 2
Q.19. If + -
2 ??
?? + 2
1
2 ?? + 4
12 - ??
2 ??
2
+ 4 ??
= a + , where a and b are integers, then
??
??
the value of ab = ___________ .
Graduate Level 27/06/2023 (Shift - 4)
(a) 3 (b) - 6 (c) 6 (d) - 3
Q.20. If 6
X
= 3
Y
= 2
z
, then what is the
value of + - ?
1
??
1
??
1
??
SSC CHSL 20/03/2023 (2nd Shift)
(a) 1 (b) 0 (c) 3 (d) 6
Q.21. For what value of m will the system
of equations 17x + my + 102 = 0 and 23x
+ 299y + 138 = 0 have an in?nite number
of solutions?
SSC CGL Tier II (02/03/2023)
(a) 221 (b) 223 (c) 220 (d) 219
Q.22. If a + b + c = 5, a
3
+ b
3
+ c
3
= 85 and
abc = 25, then ?nd the value of a
2
+ b
2
+
c
2
- ab - bc - ca.
SSC CGL 13/12/2022 (4th Shift)
(a) 2 (b) 4 (c) 6 (d) 8
Q.23. If , are the roots of 6x² + 13x + a ß
7 = 0, then the equation whose roots are
2
,
2
is: a ß
SSC CGL 12/12/2022 (4th Shift)
(a) 36 - 87x + 49 = 0 ??
2
(b) 36 - 85x + 49 = 0 ??
2
(c) 36 - 85x - 49 = 0 ??
2
(d) 36 + 87x - 49 = 0 ??
2
Q.24. If x = 222, y = 223 and z = 224, then
?nd the value of + + - 3xyz. ??
3
??
3
??
3
SSC CGL 12/12/2022 (4th Shift)
(a) 2007 (b) 2004 (c) 2006 (d) 2005
Q.25. What is the value of
( )( )( ) ?? -
1
??
??
2
+
1
??
2
??
4
+
1
??
4
( )( )( ) ? ??
8
+
1
??
8
??
16
+
1
??
16
??
32
+
1
??
32
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Pinnacle Day: 39th - 48th Algebra
SSC CGL 06/12/2022 (3rd Shift)
(a) (b)
??
64
-
1
??
64
?? +
1
??
??
32
-
1
??
32
?? -
1
??
(c) (d)
??
32
-
1
??
32
?? +
1
??
??
32
+
1
??
32
?? +
1
??
Q.26. Which of the following statements
is correct ?
I. The value of 100
2
- 99
2
+ 98
2
- 97
2
+ 96
2
- 95
2
+ 94
2
- 93
2
+ ...... + 2
2
- 1
2
is 5050.
II. If 8x + = -16 and x < 0, then the
8
??
value of x
197
+ x
- 197
is 2.
SSC CGL 02/12/2022 (4th Shift)
(a) Only I (b) Only II
(c) Both I and II (d) Neither I nor II
Q.27. If m – n = 0 then what is the ??
??
??
??
value of + in terms of
1
??
??
+ ??
??
1
??
??
- ??
??
? ??
??
SSC CHSL 24/05/2022 (Morning)
(a) (b)
2 ????
??
??
( ??
2
+ ??
2
)
2 ??
??
??
( ??
2
- ??
2
)
(c) (d)
2 ????
??
??
( ??
2
- ??
2
)
2 ????
??
??
( ??
2
- ??
2
)
Q.28. , and ??
2
+ ??
2
= 25 ??
2
+ ??
2
= 17
ax + by = 8 then ?nd the value of (ay - bx)
SSC CHSL 13/4/2021 (Afternoon)
(a) 33 (b) 25 (c) 21 (d) 19
Q.29. If 2 = x + , t hen the
1
1 +
1
5 +
1
2
value of x is equal to:
SSC CGL Tier II (18/11/2020)
(a) (b) 1 (c) (d)
14
13
15
13
13
15
Q.30. If x
3
+ 2 x
2
ax b is exactly - -
divisible by ( x
2
1), then the values of a -
and b are:
SSC CHSL 21/10/2020 (Afternoon)
(a) a = -1, and b = 2 (b) a = 1 and b = -2
(c) a = 1 and b = 2 (d) a = 2 and b = 2
Q.31. If = + 2, then the value of ??
3
5
is equal to: ??
3
- 6 ??
2
+ 12 ?? - 12
SSC CHSL 21/10/2020 (Morning)
(a) 0 (b) 2 (c) 1 (d) - 1
Q.32. (ax + by) is a factor of:
SSC CHSL 16/10/2020 (Afternoon)
(a) a
2
x
2
+ 2ab - b
2
y
2
(b) a
2
x
2
+ 2abxy + b
2
y
2
(c) a
2
x
3
+ 2abx + b
2
y
2
x
(d) a
2
x
2
+ 2ab + b
2
y
2
Q.33. If ‘a’ is a natural number, then (7a
2
+ 7a) is always divisible by:
SSC CHSL 16/10/2020 (Morning shift)
(a) 7 and 14 both (b) 7 only
(c) 14 only (d) 21 only
Q.34. The value of (a + 2a + 3a + 2a
2
3
1
2
1
3
1
6
+ 1)(a - 2a + 1) - a (a - 2),
1
3
1
6
1
2
1
2
when a = 7, is:
SSC CHSL 13/10/2020 (Morning)
(a) 7 (b) 0 (c) 1 (d) 7
Q.35. If a + b + c + d = 2, then the
maximum value of —
(1 + a)(1 + b)(1 + c) (1 + d)
SSC CHSL 18/03/2020 (Afternoon)
(a) (b) (c) (d)
91
9
81
16
63
22
54
13
Q.36. The coe?cient of x
2
in ( 2 ?? + ?? )
3
is:
SSC CGL 05/03/2020 (Evening)
(a) 12y
2
(b) 12 (c) 12y (d) 8
Q.37. If x
2a
= y
2b
= z
2c
0 and x
2
= yz, ?
then the value of is:
???? + ???? + ????
????
SSC CGL 03/03/2020 (Morning)
(a) 3 (b) 3ac (c) 3bc (d) 3ab
Q.38. If x = 5.51, y = 5.52 and z = 5.57,
then what is the value of
? ??
3
+ ??
3
+ ??
3
- 3 ??????
SSC CPO 11/12/2019 (Evening)
(a) 5.146 (b) 51.46
(c) 0.05146 (d) 0.5146
Q.39. The value of is closest
1
7 - 4 3
to:
SSC CPO 09/12/2019 (Evening)
(a) 4.1 (b) 4.2 (c) 1.2 (d) 3.7
Q.40. If (2x + 3y + 4)(2x +3y - 5) is
equivalent to (ax
2
+ by
2
+ 2hxy + 2gx + 2fy
+ c ), then what is the value of {(g + f - c
)/abh}
SSC CPO 09/12/2019 (Morning)
(a) (b) (c) (d)
35
432
19
108
19
316
37
216
Q.41. (4 y - 6 + 4x - ) can be ??
3
??
2
??
2
??
3
??
4
expressed as:
SSC CPO 09/12/2019 (Morning)
(a) (x - y)
4
- x
4
(b) (x + y)
4
- y
4
(c) (x + y)
4
- x
4
(d) x
4
- (x - y)
4
Q.42. Let x = and
6
27 - 6
3
4
y = , then the value
45 + 605 + 245
80 + 125
of is : ??
2
+ ??
2
SSC CGL TIER II (13/09/2019)
(a) (b) (c) (d)
223
36
221
36
221
9
227
9
Q.43. If = a - b , then what 86 - 60 2 2
will be the value of , correct to ??
2
+ ??
2
one decimal place ?
SSC CGL TIER II (11/09/2019)
(a) 8.4 (b) 8.2 (c) 7.8 (d) 7.2
Q.44. If a = and a = , then 8 - 7
1
??
is equal to :
??
2
+ ??
2
- 3 ????
??
2
+ ???? + ??
2
SSC CHSL 11/07/2019 (Evening)
(a) (b) (c) (d)
27
31
27
32
29
33
29
31
Q.45. If (a + b + 4){ab + 4(a + b)} - 4ab = 0,
a - 4, b - 4, then , ? ?
{ - } is equal to :
1
( ?? + ?? + 4 )
117
2
- 234
SSC CHSL 11/07/ 2019 (Afternoon)
(a) (b) (c) - (d) 0
1
4
117
1
2
117
1
2
234
Q.46. ( x + y ) + ( z + y ) = - ( x + z ) then
1
3
1
3
1
3
( can be expressed as : ??
3
+ ??
3
+ ??
3
)
SSC CHSL 10/07/2019 (Morning)
(a) xyz
1
8
(b) (x + y)(y + z)(z + x)
(c) (x + y)(y + z)(z + x)
3
8
(d) 3xyz
Q.47. The value of the expression
{( a + )
2
( a )
2
} is :
1
4
1
??
- -
1
??
SSC CHSL 09/07/2019 (Evening)
(a) (b) (c) 1 (d) 4
1
2
1
4
Q.48. If 3 + 4 = 5 , then the value
4
??
4
??
4
??
of x is :
SSC CHSL 08/07/2019 (Morning)
(a) 4 (b) 2 (c) 8 (d) 16
Q.49. If x = 2 - then the value of 3
is : ??
3
- ??
- 3
SSC CHSL 05/07/2019 (Evening)
(a) -30 (b) 30 (c) -30 (d) 30 3 3 2 2
Q.50. If x - 1, 2 and 5, then the ?
simpli?ed value of {
2 ( ??
3
- 8 )
??
2
- ?? - 2
×
} is equal to :
??
2
+ 2 ?? + 1
??
2
- 4 ?? - 5
÷
??
2
+ 2 ?? + 4
3 ?? - 15
SSC CHSL 03/07/2019 (Afternoon)
(a) (b) 6 (c) (d)
1
6
3
2
2
3
Q.51. If x is real, and ??
4
- 5 ??
2
- 1 = 0 ,
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Pinnacle Day: 39th - 48th Algebra
then the value of
( + + 1) is : ??
6
- 3 ??
2 3
??
2
-
1
??
6
SSC CHSL 03/07/2019 (Morning)
(a) 126 (b) 110 (c) 116 (d) 96
Q.52. If x + y = 7 and xy = 10, then the
value of + is:
1
??
3
1
??
3
SSC CHSL 01/07/2019 (Evening)
(a) 0.543 (b) 0.131 (c) 0.133 (d) 0.453
Q.53. If ( 3 ?? - 1 )
3
+ ( 4 ?? - 3 )
3
+
= 3(3x - 1)(4x - 3)(2x + 1) and ( 2 ?? + 1 )
3
x then x = ? ?
1
3
SSC CGL 10/06/2019 (Evening)
(a) (b) 2 (c) (d) 1
1
2
1
4
Q.54. If x + y + z = 0 , then the value of
( - xy) is : ( ??
2
+ ??
2
+ ??
2
) ÷ ??
2
SSC CGL 10/06/2019 (Morning)
(a) 1 (b) 2 (c) - 2 (d) - 1
Q.55. If - 3x - 1 = 0 , then the value of ??
2
( + 8x - 1 ) is : ??
2
( ??
3
+ ??
- 1
)
- 1
SSC CGL 07/06/2019 (Afternoon)
(a) (b) 8 (c) 1 (d) 3
3
8
Q.56. If x = 2 - p , then , is ??
3
+ 6 ???? + ??
3
equal to :
SSC CGL 07/06/2019 (Morning)
(a) 12 (b) 6 (c) 8 (d) 4
Q.57. If + 6x - 2xy + 9 = 0, then 2 ??
2
+ ??
2
the value of is : 4 ??
3
- ??
3
+ ??
2
??
2
SSC CGL 06/06/2019 (Afternoon)
(a) 0 (b) 9 (c) - 3 (d) - 9
Q.58. If ab + bc + ca = 8 and + + ??
2
??
2
??
2
= 20, then a possible value of (a + b +
1
2
c) [(a - b + (b - c + (c - a ] is )
2
)
2
)
2
SSC CGL 06/06/2019 (Morning)
(a) 72 (b) 56 (c) 84 (d) 80
Q.59. If x + y = 1 and xy(xy - 2) = 12, then
the value of is: ??
4
+ ??
4
SSC CGL 04/06/2019 (Evening)
(a) 19 (b) 25 (c) 20 (d) 23
Q.60. If 24(x - y + z) 16 ??
2
+ 9 ??
2
+ 4 ??
2
=
, then the value of (xy + 2z) is : - 61
SSC CGL 04/06/2019 (Afternoon)
(a) 1 (b) 2 (c) 3 (d) 5
Q.61. If ( ) 5 5 ??
3
- 81 3 ??
3
÷ 5 ?? -
= , then the value 3 3 ?? ?? ??
2
+ ?? ??
2
+ ??????
of (6A + B - ) is : 15 ??
SSC CGL 04/06/2019 (Morning)
(a) 10 (b) 9 (c) 15 (d) 12
Q.62. If x + y + z = 19,
and xz = , then the ??
2
+ ??
2
+ ??
2
= 133 ??
2
difference between z and x is :
SSC CGL 04/06/2019 (Morning)
(a) 5 (b) 3 (c) 6 (d) 4
Q.63. If , then the ??
4
+ ??
- 4
= 194 ?? ? 0
value of is : ( ?? - 2 )
2
SSC CGL 04/06/2019 (Morning)
(a) 1 (b) 6 (c) 2 (d) 3
Q.64. If roots of - 4x + a = 0 are equal, ??
2
then a = ?
SSC CPO 16/03/2019 (Evening)
(a) - 4 (b) 4 (c) 8 (d) -8
Practice Questions
SSC CHSL 2023 Tier - 2
Q.65. If (a - 2b) = 8 and 2ab = 9, then ?nd
the value of (a + 2b).
SSC CHSL Tier II (02/11/2023)
(a) 12 (b) 6 (c) 10 (d) 8
Q.66. A and B each have a certain
number of oranges. A says to B, “If you
give me 10 of your oranges, I will have
twice the number of oranges left with
you.” B replies, “If you give me 10 of your
oranges, I will have the same number of
oranges left with you.” Find the number
of oranges that A has.
SSC CHSL Tier II (02/11/2023)
(a) 65 (b) 50 (c) 55 (d) 70
SSC CPO 2023 Tier - 1
Q.67. If + = 1 and = 2, then
??
??
??
??
?? + ??
the value of is: ??
3
+ ??
3
SSC CPO 03/10/2023 (1st Shift)
(a) 0 (b) 1 (c) 3 (d) 2
Q.68. If + = 1 and + = 1,
??
??
??
??
??
??
??
??
then the value of + is :
??
??
??
??
SSC CPO 03/10/2023 (2nd Shift)
(a) 1 (b) (c) (d) 0
??
??
??
??
Q.69. If a ? b, b ? and c ? d, then the
1
??
relation between a and b is :
SSC CPO 03/10/2023 (2nd Shift)
(a) a ? (b) a ? d (c) a ? (d) a ? ??
1
??
??
2
Q.70. If A + = , then the
1
1 +
1
2 +
1
3
9
10
value of A is:
SSC CPO 03/10/2023 (3rd Shift)
(a) (b) (c) (d)
3
10
2
5
1
10
1
5
Q.71. If (5 - ) = , then the value of ??
2
??
5
??
x² + is:
1
??
2
SSC CPO 03/10/2023 (3rd Shift)
(a) (b) (c) (d)
54
23
53
28
53
27
54
25
Q.72. The value of
157 × 157 ( ) + 157 × 133 ( ) + 133 × 133 ( )
157 × 157 × 157 ( ) - 133 × 133 × 133 ( )
is :
SSC CPO 04/10/2023 (2nd Shift)
(a) 24 (b) (c) 290 (d)
1
290
1
24
Q.73. Simplify the given expression.
3 . 254 × 3 . 254 × 3 . 254 ( ) - 0 . 746 × 0 . 746 × 0 . 746 ( )
( 3 . 254 × 3 . 254 ) + ( 0 . 746 × 0 . 746 ) + ( 3 . 254 × 0 . 746 )
SSC CPO 04/10/2023 (2nd Shift)
(a) 2.508 (b) 4.508 (c) 2.146 (d) 3.156
Q.74. The value of +
??
2
- ( ?? - ?? )
2
( ?? + ?? )
2
- ??
2
is :
??
2
- ( ?? - ?? )
2
( ?? + ?? )
2
- ??
2
+
??
2
- ( ?? - ?? )
2
( ?? + ?? )
2
- ??
2
SSC CPO 04/10/2023 (3rd Shift)
(a) 3 (b) -1 (c) 1 (d) 2
Q.75. Simplify the given expression.
( 1 + ?? )
3
+ ( 1 - ?? )
3
+ (- 2 )
3
SSC CPO 04/10/2023 (3rd Shift)
(a) 3(1 - x
2
) (b) 6(1 - x
2
)
(c) - 6(1 - x
2
) (d) - 3(1 - x
2
)
Q.76. Simplify the given expression.
( 1 - 2 ?? )
2
- ( 1 + 2 ?? )
2
SSC CPO 05/10/2023 (1st Shift)
(a) (b) - 8 ?? 8 ??
(c) (d) - 2 + 8 ??
2
( 2 + 8 ??
2
)
Q.77. If a + b + c = 5 and ab + bc + ca = 7,
then the value of is: ??
3
+ ??
3
+ ??
3
- 3 ??????
SSC CPO 05/10/2023 (1st Shift)
(a) 20 (b) 25 (c) 15 (d) 30
Q.78. Simplify the given expression.
(x - 2y)(y - 3x) + (x + y)(x - y) + (x - 3y)(2x
+ y)
SSC CPO 05/10/2023 (3rd Shift)
(a) 2x(x - 3y) (b) 2y(x + 3y)
(c) 2x(x + 3y) (d) 2y(x - 3y)
SSC MTS 2023 Tier - 1
Q.79. If (x - 2) is a factor of (2x
2
+ 12kx -
25k), then what is the value of k?
SSC MTS 01/09/2023 (2nd Shift)
(a) 2 (b) 8 (c) 4 (d) 6
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SSC CHSL 2023 Tier - 1
Q.80. If , then the value of 2 ?? +
2
??
= 5
( ) will be : ??
3
+
1
??
3
+ 2
SSC CHSL 02/08/2023 (1st Shift)
(a) (b) (c) (d)
81
11
81
7
71
8
81
8
Q.81. If (a - b) = 9 and ( = 4401, ??
3
- ??
3
)
?nd the value of ab.
SSC CHSL 02/08/2023 (2nd Shift)
(a) 190 (b) 112 (c) 162 (d) 136
Q.82. If (a + ) , then what is the
1
??
= 6
value of ( + ) ?
3
4
??
2 1
??
2
SSC CHSL 02/08/2023 (2nd Shift)
(a) 22.5 (b) 34 (c) 25.5 (d) 36
Q.83. If , then the ?? =
1
?? - 6
( ?? > 0 )
value of ( ) is : ?? +
1
??
SSC CHSL 02/08/2023 (3rd Shift)
(a) (b) (c) (d) 6 10 15 7
Q.84. If = +
??
??
8
3
??
??
, then what is the ?????? ( ?? + ?? ) = 30
value of ab ?
SSC CHSL 03/08/2023 (1st Shift)
(a) 64 (b) 28 (c) 81 (d) 36
Q.85. If , and , 2 ?? + 3 ?? = 9 ???? = 3
what is ? 8 ??
3
+ 27 ??
3
SSC CHSL 03/08/2023 (1st Shift)
(a) 235 (b) 244 (c) 243 (d) 234
Q.86. If , then what is the ??
2
+
1
??
2
=
7
3
value of ( - )? ??
3 1
??
3
SSC CHSL 03/08/2023 (3rd Shift)
(a) (b) (c) (d)
5
3 3
10
3 3
7
3 3
8
3 3
Q.87. If a + b = 10 and , ?nd ??
2
+ ??
2
= 58
the value of ab.
SSC CHSL 03/08/2023 (3rd Shift)
(a) 24 (b) 27 (c) 30 (d) 21
Q.88. If (a + b + c) ? 0, then (a + b + c)
(a² + b² + c²- ab - bc - ca) is equal to:
SSC CHSL 03/08/2023 (4th Shift)
(a) a
3
+ b
3
- c
3
- 3abc (b) a
3
- b
3
+ c
3
- 3abc
(c) a
3
+ b
3
+ c
3
- 3abc (d) a
3
+ b
3
+ c
3
+ 3abc
Q.89. If 2a + 3b = 14 and 2a - 3b = 10,
then ?nd the value of 'ab'.
SSC CHSL 03/08/2023 (4th Shift)
(a) 5 (b) 3 (c) 4 (d) 2
Q.90. If (w + ) = 6, then what will be
1
??
the value of ( w – ) ?
1
??
SSC CHSL 04/08/2023 (1st Shift)
(a) (b) (c) (d) 4 2 2 3 2 2 2
Q.91. If - and a - b
??
??
=
8
3
??
??
= 10 then the value of ab is :
SSC CHSL 04/08/2023 (2nd Shift)
(a) 32 (b) 32 (c) 32 (d) 32
1
7
3
7
4
7
2
7
Q.92. Which of the following is an
identity?
SSC CHSL 04/08/2023 (2nd Shift)
(a) (a - b)
2
= a
2
- b
2
(b) (a + b)
2
= a
2
+ b
2
+ 2ab
(c) (a + b)
2
= a
2
+ b
2
(d) (a - b)
2
= a
2
- b
2
- 2ab
Q.93. If + = 1 then the value of ??
2
??
is:
??
2
+ 7 ?? + 2
??
2
+ 13 ?? + 2
SSC CHSL 04/08/2023 (2nd Shift)
(a) (b) (c) (d)
5
7
3
7
4
7
2
7
Q.94. If ( + ) = 7 and > 1, what is ??
1
??
??
the value of ( ³ - )? ??
1
?? ³
SSC CHSL 04/08/2023 (3rd Shift)
(a) 174 (b) 164 (c) 154 (d) 144 5 5 5 5
Q.95. If (y - ) = 9, ?nd the value of
1
??
(y³ - )
1
?? ³
SSC CHSL 04/08/2023 (3rd Shift)
(a) 766 (b) 756 (c) 702 (d) 729
Q.96. If 2x + 3y = 4 and 4x² + 9y
2
= 64,
then what is the value of 8x
3
+ 27y³.
SSC CHSL 04/08/2023 (4th Shift)
(a) 253 (b) 235 (c) 352 (d) 325
Q.97. If a + b + c = 15 and ab + bc + ca =
22, then ?nd the value of . ??
2
+ ??
2
+ ??
2
SSC CHSL 07/08/2023 (1st Shift)
(a) 131 (b) 141 (c) 161 (d) 181
Q.98. Simplify the following expression.
{[( - )( - 1)] - [(9 - 5)(9 - 1)]} ?? 5 ?? ?? ?? ÷ 16 ??
SSC CHSL 07/08/2023 (1st Shift)
(a) (b) 2 ?? ( 5 ?? - 3 ) - ( 5 ?? - 3 )
(c) (d) ?? ( 5 ?? - 3 ) - 6 ?? ( 5 ?? - 3 )
Q.99. If , then ?nd the value ?? +
1
??
= 2
of ??
1823
+
1
??
1929
SSC CHSL 07/08/2023 (1st Shift)
(a) 2 (b) 1 (c) 0 (d) -1
Q.100. The algebraic expression 2x
2
- 32
is equal to______.
SSC CHSL 07/08/2023 (2nd Shift)
(a) 2(x - 4)
2
(b) (x + 4)(x - 4)
(c) 2(x + 4)² (d) 2(x + 4)(x - 4)
Q.101. If (x - ) = 0.4, and x>0, what is
1
??
the value of (x² + )?
1
?? ²
SSC CHSL 07/08/2023 (2nd Shift)
(a) 4.16 (b) 3.32 (c) 1.84 (d) 2.16
Q.102. If (x + ) = 5, then the value of
1
??
(x
5
+ ) is:
1
??
5
SSC CHSL 07/08/2023 (2nd Shift)
(a) 2535 (b) 2540 (c) 2515 (d) 2525
Q.103. If x² + y ² + z
2
= 628 and xy + yz +
zx = 24, then ?nd the value of (x + y + z).
SSC CHSL 07/08/2023 (3rd Shift)
(a) 22 (b) 28 (c) 26 (d) 24
Q.104. if x + = - 1, then the value of
1
??
x
15
+ will be:
1
??
15
SSC CHSL 07/08/2023 (3rd Shift)
(a) 3 (b) 2 (c) -1 (d) 0
Q.105. Simplify the following expression.
??
3
+ ??
3
+ ??
3
- 3 ??????
( ?? - ?? )
2
+ ( ?? - ?? )
2
+( ?? - ?? )
2
SSC CHSL 07/08/2023 (4th Shift)
(a) (x +y + z) (b) (x + y + z)
1
3
(c) (x + y + z) (d) (x + y + z)
1
4
1
2
Q.106. If (x - ) = 7, then the value of
1
?? - 2
(x - 2)³ — is:
1
( ?? - 2 )
3
SSC CHSL 07/08/2023 (4th Shift)
(a) 130 (b) 115 (c) 140 (d) 125
Q.107. If a + b + c = 16, ab + bc + ac = 81,
what is the value of a
2
+ b
2
+ c
2
SSC CHSL 08/08/2023 (1st Shift)
(a) 94 (b) 98 (c) 89 (d) 87
Q.108. Simplify the following expression
(a + b + c)
2
- (a - b + c)
2
+ 4ac
SSC CHSL 08/08/2023 (1st Shift)
(a) 4(bc + ac) (b) 2(ab + bc + ac)
(c) 4(ab + bc + ac) (d) 4(bc + ab)
Q.109. If , and x > 0, ??
2
- 5 5 ?? + 1 = 0
what is the value of ( )? ??
3
-
1
??
3
SSC CHSL 08/08/2023 (2nd Shift)
(a) 1331 (b) 1364 (c) 1296 (d) 1244
Q.110. If + = 194, > 0 , then ?nd ??
4 1
??
4
??
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