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ALGEBRA
Importance : Algebra based 2-3 questions are essentially asked in almost all competitive exams obviously this chapter should be given sufficient time and practice done. Scope of questions : Questions based on different algebraic expressions, equations (e.g. quadratic or higher order, square root, cube root and inverse) or based on graphic representation of equations and the value of a variable is asked or an equation is required to be validated. Way to success : Solution of questions of this chapter can be ensured by memorising the concerved formulae/rules and by regular practice.
Polynomials : An algebraic expression in which the variables involved have only non-negative integral powers is called a polynomial. General Form : p ( x ) = a 0 + a 1 x + a 2 x^2 + ... + anxn^ is a polynomial in variable x , where a 0 , a 1 , a 2 , a 3 ... an are real numbers and n is non-negative integer. Remainder Theorem : Let f ( x ) be a polynomial of degree n > 1, and let a be any real number. When f ( x ) is divided by ( x – a ), then the remainder is f ( a ). Proof : Suppose that when f ( x ) is divided by ( x – a ), the quotient is g ( x ) and the remainder is r ( x ). Then,degree r ( x ) < degree ( x – a )
Þ degree r ( x ) < 1
Þ degree r ( x ) = 0 [Q degree of ( x – a ) = 1]
Þ r ( x ) is constant, equal to r (say).
Thus, when f ( x ) is divided by ( x – a ), then the quotient is g ( x ) and the remainder is r.
\ f ( x ) = ( x – a ). g ( x ) + r ... (i)
Putting x = a in (i), we get r = f ( a ). Thus, when f ( x ) is divided by ( x – a ), then the remainder is f ( a ). Remarks (i) If a polynomial p ( x ) is divided by ( x + a ), the remainder is the value of p ( x ) at x = – a i.e. p (– a )
[Q x + a = 0 Þ x = – a ]
(ii) If a polynomial p ( x ) is divided by ( ax – b ), the remainder
is the value of p ( x ) at x =
b
a
i.e. p
b
a
F
HG
I
KJ^.
[Q ax – b = 0 Þ x =
b
a ]
(iii) If a polynomial p ( x ) is divided by ( ax + b ), then
remainder is the value of p ( x ) at x = -^
b
a i.e. p^
F-
HG^
I
KJ
b
a
[Q ax + b = 0 Þ x = –
b
a
]
(iv) If a polynomial p ( x ) is divided by b – ax , the remainder
is the value of p ( x ) at x =
b
a i.e.^
p b
a
F
H
G
I
K
J
[Q b – ax = 0 Þ x =
b
a
]
Factor Theorem Let p ( x ) be a polynomial of degree greater than or equal to 1 and a be a real number such that p ( a ) = 0, then ( x – a ) is a factor of p (x). Conversely, if ( x – a ) is a factor of p ( x ), then p ( a ) = 0
Þ p ( x ), when divided by ( x – a ) gives remainder zero.
But by Remainder theorem, p ( x ) when divided by ( x – a ) gives the remainder equal to p ( a ).
\ p ( a ) = 0
Remarks (i) ( x + a ) is a factor of a polynomial iff (if and only if) p (– a ) = 0
(ii) ( ax – b ) is a factor of a polynomial if p
b
a
F
HG
I
KJ^ = 0
(iii) ( ax + b ) is a factor of a polynomial p ( x ) if p^
b
a
F -
HG^
I
KJ^ = 0
(iv) ( x – a ) ( x – b ) are factors of a polynomial p ( x ) if p ( a ) = 0 and p ( b ) = 0
ALGEBRAIC IDENTITIES
An algebraic identity is an algebraic equation which is true for all values of the variable (s).
IMPORTANT FORMULAE
- ( a + b )^2 = a^2 + 2 ab + b^2
- ( a – b )^2 = a^2 – 2 ab + b^2
- ( a + b )^2 = ( a – b )^2 + 4 ab
- ( a – b )^2 = ( a + b )^2 – 4 ab
- a^2 – b^2 = ( a + b ) ( a – b )
- a^3 + b^3 = ( a + b ) ( a^2 – ab + b^2 )
- a^3 – b^3 = ( a – b ) ( a^2 + ab + b^2 )
- ( a + b )^3 = a^3 + b^3 + 3 ab ( a + b )
- ( a – b )^3 = a^3 – b^3 – 3 ab ( a – b )
- a^3 + b^3 = ( a + b )^3 – 3 ab ( a + b )
- a^3 – b^3 = ( a – b )^3 + 3 ab ( a – b )
- a^3 + b^3 + c^3 – 3 abc = ( a + b + c ) ( a^2 + b^2 + c^2 – ab – bc – ac )
= ( a + b + c )
1 2
(2 a^2 + 2 b^2 + 2 c^2 – 2 ab – 2 bc – 2 ac )
1 2
( a + b + c ) [( a – b )^2 + ( b – c )^2 + ( c – a )^2 ]
- If a + b + c = 0, then a^3 + b^3 + c^3 = 3 abc
- ( a + b + c )^3 = a^3 + b^3 + c^3 + 3 ( b + c ) ( c + a ) ( a + b )
- a^2 + b^2 = ( a + b )^2 – 2 ab
- a^2 + b^2 = ( a – b )^2 + 2 ab
- ( a + b + c )^2 = a^2 + b^2 + c^2 + 2 ab + 2 ac + 2 bc
- a^4 + b^4 + a^2 b^2 = ( a^2 – ab + b^2 ) ( a^2 + ab + b^2 )
GRAPHIC REPRESENTATION OF STRAIGHT LINES
Ordered Pair : A pair of numbers a and b listed in a specific order with a at the first place and b at the second place is called an ordered pair ( a , b ).
Note that ( a , b ) ¹ ( b , a ).
Thus, (2, 3) is one ordered pair and (3, 2) is another ordered pair. CO-ORDINATE SYSTEM Co-ordinate Axes : The position of a point in a plane is determined with reference to two fixed mutually perpendicular lines, called the coordinate axes. Let us draw two lines X’OX and YOY’, which are perpendicular to each other and intersect at the point O. These lines are called the coordinate axes or the axes of reference. The horizontal line X’OX is called the x-axis. The vertical line YOY’ is called the y-axis. The point O is called the origin. The distance of a point from y-axis is called its x-co- ordinate or abscissa and the distance of the point from x-axis is called its y-co ordinate or ordinate. If x and y, denote respectively the abscissa and ordinate of a point P, then (x, y) are called the coordinates of the point P. The y–co-ordinate of every point on x - axis is zero. i.e. when a straight line intersects at x-axis, its y-co-ordinate is zero. So, the co-ordinates of any point on the x-axis are of the form (x, 0). The x- co-ordinate of every point on y-axis is zero. So, the co-ordinates of any point on y-axis are of the form (0, y). The co-ordinates of the origin are (0, 0). y = a where a is constant denotes a straight line parallel to x-axis. x = a where a is constant, denotes a straight line parallel to y-axis. x = 0 denotes y-axis. y = 0 denotes x-axis.
X' X
Y
Y'
–5 –4–3 –2–1^0 1 2 3 4
II I
III IV
We can fix a convenient unit of length and taking the origin as zero, mark equal distances on the x-axis as well as on the y-axis. Convention of Signs : The distances measured along OX and OY are taken as positive and those along OX’ and OY’ are taken as negative, as shown in the figure given above. CO-ORDINATES OF A POINT IN A PLANE Let P be a point in a plane. Let the distance of P from the y-axis = a units. And, the distance of P from the x-axis = b units. Then, we say that the co-ordinates of P are (a, b). a is called the x-co-ordinate, or abscissa of P. b is called the y co-ordinate, or ordinate of P.
X' O X
Y'
Y
a
M
P
b
( a , b )
Quadrants : Let X’ OX and YOY’ be the co-ordinate axes. These axes divide the plane of the paper into four regions, called quadrants. The regions XOY, YOX’,X’OY’ and Y’OX are respectively known as the first, second, third and fourth quadrants.
Rule 22. If a^3 + b^3 + c^3 = 3abc, then a + b + c = 0 or a = b = c.
Proof Q a^3 + b^3 + c^3 = 3abc
Þ a^3 + b^3 + c^3 – 3abc = 0
Now, a^3 + b^3 + c^3 – 3abc =
(a + b + c) [(a – b)^2 +
(b – c)^2 + (c – a)^2 ]
Þ 0 =
2 (a + b + c) [(a – b)
(^2) + (b – c) (^2) + (c – a) (^2) ]
\ Either a + b + c = 0 or, (a – b)^2 + (b – c)^2 + (c – a)^2
= 0, i.e., a – b = 0
Þ a = b, b – c = 0
Þ b = c, c – a = 0
Þ c = a
\ a = b = c
Rule 23. If a^2 + b^2 + c^2 = ab + bc + ca, then a = b = c. Rule 24. Componendo and Dividendo Rule, If
a
b
c
d
then
a b
a b
c d
c d
Rule 25. If
a b
a b
c
d
then
a
b
c d
c d
Rule 26. If x + x + x+... ¥where x = n(n + 1)
then (^) x + x + x + ...¥ = (^) bn + (^1) g
Rule 27. If x - x - x-... ¥ where x = n(n + 1) then,
x - x - x-... ¥ = n.
Rule 28. (a + b + c)^3 = a^3 + b^3 + c^3 – 3(a + b) (b + c) (c + a) Rule 29. a^4 + a^2 b^2 + b^4 = (a^2 + ab + b^2 ) (a^2 – ab + b^2 )
Rule 30. If a^
a
+ 1 =x, then a
a
33 x 3 x
Rule 31. If a^
a
a
3 - 13 = x 3 + 3 x.
Rule 32. Binomial theorem : (a + b)n^ = nC 0 anb^0 + nC 1 an – 1b^1 + nC 2 an – 2b^2 + ... + nC n – 1a
(^1) bn – 1 (^) + nC na
(^0) bn, where, n is a positive number and
n
Cr
n
r n r
! (^) b g! Permutation and Combination Permutation : It is used where we have to arrange things. Out of total n things, r things (taken at a time) can be arranged as npr or P ( n , r )
P ( n , r ) = n Pr = (^) ( n n - r^ !)! where n > r
Combination : It is used where we have to select things. It is written as nCr or C( n , r )
C ( n , r ) =
n
r r
(n – )!! n^ >^ r
Some important results.
n P o =1; n Pn = n!
nCo = nCn =1 ; nCr = nCn – r = nC 1 = nCn –1 = n.
Ex. (^7 )
P =^210
b g!
C 2 =^10
b g! n! (is called as n factorial) 5! = 5.4! = 5.4.3! = 5.4.3.2! = 5.4.3.2.1!
Also 0!^ =^1
Importance : Coordinate geometry is separate and important filled in mathematics but very rarely asked in competitive exams. However in two-dimensional (2–D) geometry introductory/easy questions should be practised for improving marks. Scope of questions : Mostly questions are related to distance between two points, linear/non-linear these coplaner points, cutting a line a specific ratio by a given point. Way to success : The concept of coordinate geometry and practice of above mentioned questions is very important to solve questions.
Important Points : x–coordinate is called the abscissa of P, where (x, y) are co-ordinates of any point P. y–co-ordinate is called the ordinate of P, where (x, y) are co–ordinates of any point P. Quadrants :
IInd quadrant (–x, y) x¢ x
y¢
y Ist quadrant (x, y)
IIIrd quadrant (–x, –y) IVth quadrant (x, –y)
COORDINATE GEOMETRY
Cartesian Co–ordinate System : y
x¢ x
y¢
P (x, y)
y-Co-ordinate (ordinate)
x-Co-ordinate (abscissa)
C0-ordinate
Polar Coordinate System : y
x¢ x
y¢
P (r, q)
x
q
r (^) y ^ r = x + y^2 2
RULE 1 : The distance between any two points in the plane is the length of the line segment joining them. The distance between two points P (x 1 , y 1 ) and Q (x 2 , y 2 ) is
PQ = ( x 1 - x 2 ) 2 + ( y 1 - y 2 )^2 or,,
PQ = (^) (differenc e of abscissa ) 2 +(differenc e of ordinates)^2
RULE 2 : The area of a triangle, the Co-ordinates of whose vertices are (x 1 , y 1 ), (x 2 , y 2 ) and (x 3 , y 3 ) is
Area D =
F HG
I KJ^ |x^1 (y^2 – y^3 ) + x^2 (y^3 – y^1 ) + x^3 (y^1 – y^2 )|
F H
G
I K
J
x y
x y
x y
1 1 2 2 3 3
If all three points are collinear,
then area of D = 0
RULE 3 : The Co-ordinates of the point which divides the line segment joining the points (x 1 , y 1 ) and (x 2 , y 2 ) internally in the ratio m : n are given by
x =
mx nx
m n
+ y =
my ny
m n
RULE 4 : If P is the mid-point of AB, such that it divides AB in the ratio 1 : 1, then its Co-ordinates are (x,y) =
x 1 x 2 y 1 y 2
F^ +^ + H
G
I K
, (^) Jalso called mid point formula.
RULE 5 : The Co–ordinates of the point which divides the line segment joining the points (x 1 , y 1 ) and (x 2 , y 2 ) externally in the ratio m : n, are
mx nx
m n
my ny
m n
F HG^
I KJ
RULE 6 : The Co-ordinates of the centroid of a triangle whose vertices are (x 1 , y 1 ), (x 2 , y 2 ) and (x 3 , y 3 ) is given by
x 1 x 2 x 3 y 1 y 2 y 3
F^ +^ +^ +^ + HG^
I KJ
RULE 7 : The Co–ordinates of the in–centre of a triangle whose vertices are A (x 1 , y 1 ), B(x 2 , y 2 ), C(x 3 , y 3 ) are given by
ax bx cx
a b c
ay by cy
a b c
1 +^2 + 3 1 2 3
F HG^
I KJ
, where a = BC,
b = CA and c = AB. Equation of straight line. A straight line is a curve such that every point on the line segment joining any two points on it lies on it. RULE 8 : If (x 1 , y 1 ) and (x 2 , y 2 ) are the Co-ordinates of any two points on a line, then its slope is
(tan q ) = m =
y y
x x
2 1 2 1
difference of ordinates
difference of abscissa
RULE 9 : The angle q between the lines having slopes
m 1 and m 2 is given by tan q = ±^
m m
m m
2 1
RULE 10 : If two lines having slopes m 1 and m 2 are (i) parallel if m 1 = m 2 (ii) Perpendicular if m 1 x m 2 = – RULE 11 : (Slope–Intercept) The equation of a line with slope m and making an intercept c on y-axis is y = mx + c. RULE 12 : (Point-Slope form) The equation of a line which passes through the point (x 1 , y 1 ) and has the slope ‘m’ is (y – y 1 ) = m(x – x 1 ) RULE 13 : (Two-point form) The equation of a line passing through two points (x 1 , y 1 ) and (x 2 , y 2 ) is given by
x x
x x
1 2 1
y y y y
1 2 2 RULE 14 : (Intercept form) The equation of a line which cuts off intercepts a and b respectively on the x and y–axes is
x
a
y
b
RULE 15 : (i) The slope of a line whose general quation
is given by Ax + By + C = 0 is
B
(ii) The intercepts of a line on x and y axes respectively whose general equation is Ax + By + C = 0 is given by :-
x-intercept =
A
and y-intercept =
B
RULE 16 : General equation of straight line is ax+ by + c = 0
\ Now the area of the triangle made by the given straight
line and its intercepts is
D =
2 ×^
F- HG^
I KJ
c
a ×^
F- HG^
I KJ
c
b sq. units
qq q
- If
2 x^
, the value of x is
(1) 31 (2) 32 (3) 36 (4) 37 (SSC Section Officer (Commercial Audit) Exam.16.11.2003)
- The value of
5 2 1 1
b g
n (^) n
n n
is
(1) 1 (2) 9 (3) 3 (4) 3n (SSC CGL Prelim Exam. 08.02. (First Sitting)
- If x = 0.5 and y = 0.2, then
value of (^) 0 6. × (3 y ) x^ is equal to (1) 1.0 (2) 0. (3) 0.6 (4) 1. (SSC CGL Prelim Exam. 08.02. (Second Sitting)
24. If x x^ x^ x x
x = (^) e j , then x equals
(SSC CPO S.I. Exam. 05.09.2004)
- If a = 7, b = 5 and c = 3, then the value of a^2 + b^2 + c^2 – ab – bc – ca is (1) 12 (2) – (3) 0 (4) 8 (SSC CPO S.I. Exam.05.09.2004)
- If 7
x (^) = (^) , then the value of x
is (1) 3 (2) –
(3)
(SSC CPO S.I. Exam. 05.09.2004)
- If
a
b
c
,then a^ b^ c
c
= = +^ +^ is
equal to (1) 2 (2) 4 (3) 5 (4) 6 (SSC Data Entry Operator Exam. 31.08.2008)
- If 0.13 ÷ p^2 = 13, then p is equal to (1) 10 (2) 0. (3) 0.1 (4) 100 (SSC CGL Prelim Exam. 13.11. (Second Sitting)
- If
a b
= , then value of
a b
a b
is
(SSC CHSL DEO & LDC Exam. 04.12.2011 (Ist Sitting (East Zone)
- For what value(s) of a is
x + x + a
(^2) a perfect square?
(1) ±^
18 (2)^
(3) -^
5 (4)^
(SSC CPO S.I. Exam. 03.09.2006)
- If a ¹ b, then which of the follow- ing statements is true?
a + b = ab
a + b < ab
a + b > ab
(4) All of the above (SSC CPO S.I. Exam. 03.09.2006)
- If
a
a
b
b
c
1 – 1 1 c
= 1, then
the value of
1 – a
1 - b
1 - c
is
(1) 1 (2) 2 (3) 3 (4) 4 (SSC CHSL DEO & LDC Exam. 04.12.2011 (IInd Sitting (East Zone) & (SSC GL Tier-I Exam. 19.05.2013)
- If x , y are two positive real num- bers and x 1/3^ = y1/4, then which of the following relations is true? (1) x^3 = y^4 (2) x^3 = y (3) x = y^4 (4) x^20 = y^15 (SSC Section Officer (Commercial Audit) Exam. 26.11. (Second Sitting) 34. If a^2 x +2^ = 1, where a is a positive real number other than 1, then x is equal to (1) –2 (2) – (3) 0 (4) 1 (SSC CGL Prelim Exam. 04.02. (First Sitting)
- If x is real, then the minimum value of ( x^2 – x + 1) is
(1)
(SSC CGL Prelim Exam. 04.02. (Second Sitting)
- If
= a + b , then the
value of a is
3 (2)^
3 (4)^
(SSC CPO S.I. Exam. 16.12.2007)
- If (125) x^ = 3125, then the value of x is
(1)
5 (2)^
3 (4)^
(SSC CGL Prelim Exam. 27.07. (First Sitting)
- If 5 x^ + 12 x^ = 13 x , then x is equal to
(SSC CGL Prelim Exam. 27.07. (First Sitting)
- If 2^2 x^ –^ y^ = 16 and 2 x^ +^ y^ = 32, the value of xy is (1) 2 (2) 4 (3) 6 (4) 8 (SSC CPO S.I. Exam. 06.09.2009)
- If
F^3 6 2 HG
I KJ
F HG
I KJ^
= F HG
I KJ
is equal to (1) –2 (2) 2 (3) –1 (4) 1 (SSC CGL Tier-I Exam. 16.05. (First Sitting)
- If
x y
x y
= , then value of
x y
x y
+ is :
(SSC CHSL DEO & LDC Exam. 11.12.2011 (IInd Sitting (Delhi Zone)
- If a and b be positive integers such that a^2 – b^2 = 19, then the value of a is (1) 19 (2) 20 (3) 9 (4) 10 (SSC CGL Tier-I Exam. 16.05. (First Sitting)
x x = x x
then x is
equal to
12 (2)^
(SSC CGL Tier-I Exam. 16.05. (First Sitting)
- If x +
x
= 5, then
x x - x + is equal to
(SSC CHSL DEO & LDC Exam. 11.12.2011 (Ist Sitting (East Zone)
45. If x = 3
, then the value of
F
H
G
I
K
J
x x
x x is
(SSC SAS Exam. 26.06. (Paper-1)
- If x^ =^
and y^ =^
, then
value of x^2 + y^2 is : (1) 14 (2) 13 (3) 15 (4) 10 (SSC CGL Prelim Exam. 11.05. (First Sitting)
- If 4^4 x^ + 1^ =
, then the value of
x is
(3) -^
2 (4)^
(SSC CISF ASI Exam. 29.08. (Paper-1)
- If
x x x x
(^2) then x is
equal to (1) 2.4 (2) 3. (3) 4 (4) 5 (SSC (South Zone) Investigator Exam.12.09.2010)
49. If 2 x^ = 256 , then the value of
x is (1) 14 (2) 16 (3) 18 (4) 20 (SSC CPO S.I. Exam. 12.12.2010 (Paper-I)
50. If 5 5 5
7 5 e j ¸^ e j =^
p (^) , then the
value of p is (1) 5 (2) 2
(SSC CPO S.I.
Exam. 12.12.2010 (Paper-I)
51. If 1
3
-^ x^ = , then x equals
(1) 2 (2) 4
1 3 b g
/
(SSC CGL Tier-1 Exam. 19.06. (First Sitting)
- If a b = 2 a + 3 b – ab , then the value of (3 5 + 5 3) is (1) 10 (2) 6 (3) 4 (4) 2 (SSC CGL Tier-1 Exam. 19.06. (First Sitting) 53. If 1 9
+^ x^ = , (^) then the value of
x is
(1)
(SSC CGL Tier-1 Exam. 19.06. (Second Sitting)
- If
= a + b , (^) then
the values of a and b are respectively
(1)
(SSC CGL Tier-1 Exam. 19.06. (Second Sitting)
- If x + y = 2 z then the value of x x z
z
(3)
(SSC Delhi Police S.I.(SI) Exam. 19.08.2012)
- If a * b = ab , then the value of 5 * 3 is (1) 125 (2) 243 (3) 53 (4) 15 (SSC CGL Tier-1 Exam. 19.06. (Second Sitting)
57. If 0 03. ´ 0 3. a = 0 3. ´ 0 3. ´ b ,
value of
a
b
is
(1) 0.009 (2) 0. (3) 0.9 (4) 0. (SSC CGL Tier-1 Exam 19.06. (Second Sitting)
- If x * y = ( x + 3)^2 ( y –1), then the value of 5 * 4 is (1) 192 (2) 182 (3) 180 (4) 172 (SSC CGL Tier-1 Exam 26.06. (First Sitting)
59. If 9 x = 12 + 147 , then
x =? (1) 2 (2) 3 (3) 4 (4) 5 (SSC CGL Tier-1 Exam 26.06. (First Sitting)
- If a , b , c are real and a^2 + b^2 + c^2 = 2 ( a – b – c ) – 3, then the value of 2 a – 3 b + 4 c is (1) –1 (2) 0 (3) 1 (4) 2 (SSC CHSL DEO & LDC Exam. 11.12.2011 (IInd Sitting (East Zone) & (SSC GL Tier-I Exam. 21.04.2013) & (SSC CHSL DEO & LDC Exam. 20.10.2013)
- If (3 a + 1)^2 + ( b – 1)^2 + (2 c – 3)^2 = 0, then the value of (3 a + b + 2 c ) is equal to : (1) 3 (2) – (3) 2 (4) 5 (SSC CHSL DEO & LDC Exam. 11.12.2011 (IInd Sitting (Delhi Zone)
- The value of the expression
a b
b c c a
b g b gb g
2
b c
a b c a
b g b gb g
2
c a
a b b c
b g b (^) gb (^) g
2 is :
(SSC CHSL DEO & LDC Exam. 11.12.2011 (IInd Sitting (Delhi Zone) & (SSC CHSL DEO & LDC Exam. 27.10.2013)
- If ( a –3)^2 + ( b – 4)^2 + ( c – 9)^2 = 0,
then the value of a + b + c is :
(SSC CHSL DEO & LDC Exam. 11.12.2011 (IInd Sitting (East Zone)
- If a^3 b = abc = 180, a , b , c are positive integers, then the value of c is (1) 110 (2) 1 (3) 4 (4) 25 (SSC Graduate Level Tier-II Exam. 16.09.2012)
- If ( x – 3)^2 + ( y – 5)^2 + ( z – 4)^2 = 0, then the value of
x^2 y^2 z^2 9 25 16
(1) 12 (2) 9 (3) 3 (4) 1 (SSC Graduate Level Tier-I Exam. 19.05.2013)
- If a , b are rational numbers and ( a - 1 ) 2 + 3 = b 2 + a , the value of ( a + b ) is (1) –5 (2) 3 (3) –3 (4) 5 (SSC Graduate Level Tier-II Exam. 16.09.2012)
- If a =
and b =
then the value of
a ab b
a ab b
2 2 2 2
is
5 (4)^
(SSC CGL Prelim Exam. 13.11. (Second Sitting)
- If 64 x +1^ =
4 x^
, then the value of
x is (1) 1 (2) 0
(3)
(SSC Assistant Grade-III Exam. 11.11.2012 (IInd Sitting)
- If ax^2 + bx + c = a ( x – p )^2 , then the relation among a , b , c would be (1) abc = 1 (2) b^2 = ac (3) b^2 = 4 ac (4) 2 b = a + c (SSC Delhi Police S.I. (SI) Exam. 19.08.2012)
- If a + b + c + d = 1, then the maximum value of (1 + a ) (1 + b ) (1 + c ) (1 + d ) is
3 F H
G
I K
J
F^3 HG
I KJ^
F^4 HG
I KJ (SSC Graduate Level Tier-I Exam. 11.11.2012, Ist Sitting)
- x varies inversely as square of y. Given that y = 2 for x = 1, the value of x for y = 6 will be equal to (1) 3 (2) 9
3 (4)^
(SSC Multi-Tasking Staff Exam. 17.03.2013, Kolkata Region)
- If x =
, then
x x
x x
is equal to
(SSC CPO S.I. Exam. 03.09.2006)
- If a^2 + b^2 + c^2 + 3 = 2 ( a – b – c ), then the value of 2 a – b + c is : (1) 3 (2) 4 (3) 0 (4) 2 (SSC Graduate Level Tier-I Exam. 21.04.2013, Ist Sitting)
94. If x^2 - y^2 = 80 and x – y = 8,
then the average of x and y is (1) 2 (2) 3 (3) 4 (4) 5 (SSC Graduate Level Tier-I Exam. 21.04.2013 IInd Sitting)
- If for non-zero, x , x^2 – 4 x – 1
= 0, the value of x^
x
2 2
+ is
(SSC Section Officer (Commercial Audit) Exam. 26.11. (Second Sitting)
- The third proportional to
x
y
y
x
F H
G
I K
J (^) and (^) x^2 + y^2 is
(1) xy (2) xy
(3) 3 xy (4) 4 xy
(SSC Graduate Level Tier-I Exam. 21.04.2013)
- If
x
of P, x = 6? (1) 6 (2) 4 (3) 2 (4) 1 (SSC Graduate Level Tier-I Exam. 19.05.2013)
- The value of
is
(SSC Graduate Level Tier-I Exam. 19.05.2013)
- Let
a = 6 – 5 , b = 5 – 2,
c = 2 – 3
Then point out the correct alternative among the fou r alternatives given below. (1) b < a < c (2) a < c < b (3) b < c < a (4) a < b < c (SSC CHSL DEO & LDC Exam. 20.10.2013)
- If x =
, the value of
x
x
x
x
is
(1) 1 (2) 2
(SSC CHSL DEO & LDC Exam. 27.10.2013 IInd Sitting)
101. If x = 5 – 21 , then the value of
x
32 - 2 x - 21
is
e 3 -^7 j
e 7 -^3 j
e 7 +^3 j
e^7 -^3 j
(SSC CHSL DEO & LDC Exam. 10.11.2013, Ist Sitting)
- If 6 x – 5y = 13, 7 x + 2 y = 23
then 11 x + 18 y = (1) –15 (2) 51 (3) 33 (4) 15 (SSC CHSL DEO & LDC Exam. 10.11.2013, IInd Sitting)
- The value of
x b^ c^ x x
- b^ - c^^ c + a c^ -^ a^ a + b a^ - b e j e j e j ,
( x ¹ 0) is
(SSC CHSL DEO & LDC Exam. 10.11.2013, IInd Sitting)
- If
x
a a x
, then the value of
x – x^2 is :
(1) – a (2)
a
a
(4) a
(SSC Graduate Level Tier-I Exam. 21.04.2013, Ist Sitting)
- If x +
x = 99, find the value of
x
x + x +
6 (2)^
(SSC Graduate Level Tier-I Exam. 19.05.2013 Ist Sitting)
- I f
4 x (^3 4 3 4 3 ) x
y y
z z
then the value of
x y z
+ + is
(SSC Graduate Level Tier-I Exam. 19.05.2013 Ist Sitting)
- If
xy
x y
a
xz
x z
b
, = and
yz
y z
c
= , where a , b , c are all
non-zero numbers, then x equals to
2 abc
ab + bc - ac
2 abc
ab + ac - bc
2 abc
ac + bc - ab
2 abc
ab + bc + ac
(SSC CHSL DEO & LDC Exam. 10.11.2013, IInd Sitting)
108. If x = 3 + 8 , then x^2 +
x^2 is
equal to (1) 38 (2) 36 (3) 34 (4) 30 (SSC CGL Prelim Exam. 04.02. (Ist Sitting) & (SSC CGL Prelim Exam. 27.07.2008 (IInd Sitting) & (SSC Investigator Exam. 12.09.2010) (South Zone)
- If x and y are positive real num- bers and xy = 8, then the mini- mum value of 2 x + y is (1) 9 (2) 17 (3) 10 (4) 8 (SSC Graduate Level Tier-I Exam. 19.05.2013 Ist Sitting)
- If the expression x^2 + x + 1 is written in the form
x + q
F HG^
I KJ^
2
2 , then the possi-
ble values of q are
(1) ±^
3 (2)^
(3) ±^
3 (4)^
(SSC Graduate Level Tier-I Exam. 21.04.2013 IInd Sitting)
- If a^2 – 4 a – 1 = 0, then value of
a
a
a
a
2 2
+ + - is
(SSC Graduate Level Tier-I Exam. 21.04.2013 IInd Sitting)
- If a +
b
= 1 and b +
c
then c + 1
a
is equal to
(SSC CGL Prelim Exam. 04.02. (First Sitting)
- The minimum value of ( x – 2) ( x – 9) is
(1) -^
(3) 0 (4) -^
(SSC Graduate Level Tier-I Exam. 21.04.2013)
9 (2)^
9 (4)^
(SSC Graduate Level Tier-I Exam. 19.05.2013)
146. If x^2 - 3 x + 1 = 0 , then the val-
ue of x x
x x
2 2
+ + + is
(SSC Graduate Level Tier-I Exam. 19.05.2013 Ist Sitting)
- If a^2 + b^2 = 5 ab , then the value
of
a
b
b
a
2 2
2
F
H
G
I
K
J is :
(SSC CAPFs SI & CISF ASI Exam. 23.06.2013)
- If xy + yz + zx = 0, then
x^2 - yz +^ y^2 - zx +^ z^2 - xy
F HG^
I KJ
( x , y , z ¹ 0) is equal to
(3) x + y + z (4) 0 (SSC CHSL DEO & LDC Exam. 20.10.2013)
- If a + b + c = 9 (where a , b , c are real numbers), then the min- imum value of a^2 + b^2 + c^2 is (1) 100 (2) 9 (3) 27 (4) 81 (SSC CHSL DEO & LDC Exam. 20.10.2013)
- If x + y + z = 13 and x^2 + y^2 + z^2 = 69, then xy + z ( x + y ) is equal to (1) 70 (2) 40 (3) 50 (4) 60 (SSC CHSL DEO & LDC Exam. 10.11.2013, IInd Sitting)
- If a = 0.1039, then the value of
4 a 2 – 4 a + 1 + 3 a is
(SSC CPO S.I. Exam.12.01.2003)
- If a + b + c = 0, then the value of 1 ( a + b )( b + c )+^
( a + c )( b + a )
( c + a )( c + b )is: (1) 1 (2) 0 (3) –1 (4) – (SSC CHSL DEO & LDC Exam. 11.12.2011 (IInd Sitting (East Zone)
- If a + b + c = 0, then the value of
a b c
a bc
2 2 2 2
is
(1) 0 (2) 1 (3) 2 (4) 3 (SSC Graduate Level Tier-II Exam. 16.09.2012)
136. If n = 7 + 4 3 , then the value
of n^^ + n
F HG^
I KJ
is
(SSC Graduate Level Tier-II Exam. 16.09.2012)
137. If x = 3 + 2 , then the value of
x
x
F + HG^
I KJ
is
(SSC CHSL DEO & LDC Exam. 21.10.2012 (Ist Sitting)
- If p + q = 10 and pq = 5, then the
numerical value of
p q
q p
(1) 16 (2) 20 (3) 22 (4) 18 (SSC CHSL DEO & LDC Exam. 21.10.2012 (Ist Sitting)
139. If x = 3 + 2 2 and xy = 1, then
the value of
x xy y
x xy y
2 2 2 2
is
31 (2)^
31 (4)^
(SSC CHSL DEO & LDC Exam. 21.10.2012 (IInd Sitting)
140. If
x
b c
y
c a
z
+ a b
, then
x y
b a
y z
c b
z x
a c
x
a
y
b
z
c
x y
c
y z
b
z x
c
(4) None of the above is true (SSC CHSL DEO & LDC Exam. 04.11.2012, Ist Sitting)
- If a + b + c = 0, then the value of
a b
c
b c
a
c a
b
F + H
G
I K
J
a
b c
b
c a
c
+ a b
F HG^
I KJ^ is : (1) 8 (2) – (3) 9 (4) 0 (SSC Graduate Level Tier-I Exam. 21.04.2013)
- If a , b , c are non-zero,
a +
b =1 and^ b^ +^
c =1, then the
value of abc is : (1) – 1 (2) 3 (3) – 3 (4) 1 (SSC Graduate Level Tier-I Exam. 21.04.2013)
- If a + b + c = 2 s , then
s a s b s c s a b c
b g b g b g
(^2 2 2 ) 2 2 2
is equal to (1) a^2 + b^2 + c^2 (2) 0 (3) 1 (4) 2 (SSC Graduate Level Tier-I Exam. 21.04.2013)
144. If x = 3 + 2 2 , the value
of x^
x
2
is
(1) 36 (2) 30 (3) 32 (4) 34 (SSC Graduate Level Tier-I Exam. 19.05.2013 Ist Sitting)
145. If x^
x x
F – HG^
I KJ^
= , then the val-
ue of x^
x
2 2
+ is
- If a = 0.25, b = – 0.05, c = 0.5, then the value of
a b c bc
a b ab c
2 2 2 2 2 2
is
(SSC CPO S.I. Exam. 12.01.2003)
- If a = 23 and b = –29 then the value of 25 a^2 + 40 ab + 16 b^2 is : (1) 1 (2) – (3) 0 (4) 2 FCI Assistant Grade-III Exam.05.02.2012 (Paper-I) East Zone (IInd Sitting)
- If x – y = 2 and x^2 + y^2 = 20, then value of ( x + y )^2 is (1) 38 (2) 36 (3) 16 (4) 12 (SSC CHSL DEO & LDC Exam. 28.11.2010 (IInd Sitting)
- If x^2 + y^2 – 4 x – 4 y + 8 = 0, then the value of x – y is (1) 4 (2) – (3) 0 (4) 8 (SSC CHSL DEO & LDC Exam. 04.12.2011 (Ist Sitting (North Zone)
- If x = b + c – 2 a , y = c + a – 2 b , z = a + b – 2 c , then the value of x^2 + y^2 – z^2 + 2 xy is (1) 0 (2) a + b + c (3) a – b + c (3) a + b – c (SSC CHSL DEO & LDC Exam. 04.12.2011 (Ist Sitting (East Zone)
- For real a , b , c if a^2 + b^2 + c^2 = ab
a c
b
is (1) 1 (2) 2 (3) 3 (4) 0 (SSC CHSL DEO & LDC Exam. 11.12.2011 (Ist Sitting (Delhi Zone) & (SSC CHSL DEO & LDC Exam. 10.11.2013)
- If x – y = 2, xy = 24, then the value of ( x^2 + y^2 ) is : (1) 25 (2) 36 (3) 63 (4) 52 (SSC CHSL DEO & LDC Exam. 21.10.2012 (IInd Sitting)
- If the expression
x y
tx y
2 2
2 4
a perfect square, then the val- ues of t is (1) + 1 (2) + 2 (3) 0 (4) + 3 (SSC CHSL DEO & LDC Exam. 28.10.2012 (Ist Sitting)
- If a = x + y , b = x – y , c = x + 2 y , then a^2 + b^2 + c^2 – ab – bc – ca is (1) 4 y^2 (2) 5 y^2 (3) 6 y^2 (4)7 y^2 (SSC CHSL DEO & LDC Exam. 04.11.2012 (IInd Sitting)
- If a^2 + b^2 + c^2 = ab + bc + ca, where a, b, c are non zero real numbers, then the valu e of
a b
c
is
(1) 2 (2) 1 (3) 0 (4) – (SSC CHSL DEO & LDC Exam. 28.10.2012, Ist Sitting)
- If a^2 + b^2 + 4 c^2 = 2 ( a + b – 2 c ) - 3 and a , b , c are real, then the value of ( a^2 + b^2 + c^2 ) is
(SSC Graduate Level Tier-I Exam. 19.05.2013 Ist Sitting)
- If
x a b c
x b c a
x c a b
2 2 2
= 4( a + b + c ), then x is equal to (1) ( a + b + c )^2 (2) a^2 + b^2 + c^2 (3) ab + bc + ca (4) a^2 + b^2 + c^2 – ab – bc – ca (SSC Graduate Level Tier-II Exam. 29.09.2013)
- Number of solutions of the two equations 4 x – y = 2 and 2 y – 8 x + 4 = 0 is (1) zero (2) one (3) two (4) infinitely many (SSC CHSL DEO & LDC Exam. 20.10.2013)
- If
a
b
and
b
c
, then
2 2 2 2
c a
c a
is equal to
3 (2)^
4 (4)^
(SSC Graduate Level Tier-I Exam. 21.04.2013 IInd Sitting)
- If x ¹ 0, y ¹ 0 and z ¹ 0 and
x^2 +^ y^2 +^ z^2 =^ xy +^ yz +^ zx ,
then the relation among x , y , z is (1) x + y + z = 0 (2) x + y = z
x y z
(4) x = y = z (SSC Graduate Level Tier-I Exam. 21.04.2013)
- The term to be added to 121 a^2 + 64 b^2 to make a perfect square is (1) 176 ab (2) 276 a^2 b (3) 178 ab (4) 188 b^2 a (SSC CGL Tier-I Re-Exam. (2013) 27.04.2014)
168. If a = 2 + 3 , then the value of
a
a
2 2
F HG^
I KJ^ is (1) 12 (2) 14 (3) 16 (4) 10 (SSC CGL Tier-I Re-Exam. (2013) 27.04.2014)
- For what valsue (s) of k the ex-
pression p^ +^ p^ + k
(^2) is a
perfect square?
(1) ±^
3 (2)^
(3) ±^
(4) ±^
(SSC CGL Tier-I Re-Exam. (2013) 27.04.2014)
- If
b c
a
a c
b
a b
c
= 1 and
a – b + c ¹ 0 then which one of
the following relations is true?
(1)
c a b
a b c
b a c
b a c
(SSC CGL Tier-I Re-Exam. (2013) 27.04.2014)
- If a + b = 1, c + d = 1 and
a – b =
d
c
, then the value of c^2 – d^2 is
(1)
a
b
b
a
(SSC CGL Tier-I Re-Exam. (2013) 20.07.2014 (Ist Sitting)
- If a^2 + b^2 + c^2 = 2 a - 2 b - 2,then the value of 3 a - 2 b + c is (1) 0 (2) 3 (3) 5 (4) 2 (SSC CGL Tier-I Exam. 19.10. TF No. 022 MH 3)
- If a + b + c = 3, a^2 + b^2 + c^2 = 6
and
a +^
b +^
c = 1, where^ a ,
b , c are all non-zero, then ‘ abc ’ is equal to
(1)
(SSC CGL Tier-I Exam. 19.10. TF No. 022 MH 3)
193. If a^2 – 4 a – 1 = 0, a ¹ 0, then
the value of a^2 + 3 a +
a^2 –^
a
is
(1) 24 (2) 26 (3) 28 (4) 30 (SSC CGL Tier-I Exam. 19.10. TF No. 022 MH 3)
194. If x = 2 + 3 , then x^2 +
x^2 is
equal to (1) 10 (2) 12 (3) –12 (4) 14 (SSC CGL Tier-I Exam. 19.10. TF No. 022 MH 3)
- If a^2 + b^2 + c^2 = 2 ( a – b – c ) – 3, then the value of ( a + b + c ) is (1) 0 (2) 1 (3) – 1 (4) 2 (SSC CHSL (10+2) DEO & LDC Exam. 16.11.2014 , Ist Sitting TF No. 333 LO 2)
- If x is a prime number and
–1 <
x –
< 1 then the num- ber of values of x is (1) 4 (2) 3 (3) 2 (4) 5 (SSC CHSL (10+2) DEO & LDC Exam. 16.11.2014, IInd Sitting TF No. 545 QP 6)
- If
x
x +
3 – 5y
2y +^
3 – 5z
2z =
0, the value of
x +
y +^
z is
(SSC CGL Tier-II Exam. 12.04. TF No. 567 TL 9)
- If 2s = a + b + c, then the value of s(s – c) + (s – a) (s – b) is (1) ab (2) abc
a + b + c
(SSC CGL Tier-II Exam. 12.04. TF No. 567 TL 9)
- If
p
p r^ p +
4 ]^ then the val-
ue of p^
p
F HG^
I KJ^ is
(SSC CGL Tier-II Exam. 12.04. TF No. 567 TL 9)
200. If 1
x
13, then^ x
equals (1) 1 (2) 27
(SSC CGL Tier-II Exam, 2014 12.04.2015 (Kolkata Region) TF No. 789 TH 7)
201. If 2 x = a +
a ' a^ > 0, then
the value of
x
x x
2 2
is
(1) a + 1 (2) 1
( a + 1)
( a – 1) (4) a – 1
(SSC CGL Tier-II Exam, 2014 12.04.2015 (Kolkata Region) TF No. 789 TH 7)
- If a , b , c are real numbers and a^2
+ _b_^2 + _c_^2 = 2 ( _a_ – _b_ – _c_ ) – 3, then the value of _a + b + c_ is (1) – 1 (2) 1 (3) 3 (4) 0 (SSC CGL Tier-II Exam,
2014 12.04.2015 (Kolkata Region) TF No. 789 TH 7)
- If
a b c
a b
b c a
b c
c a b
c a
and a + b + c ¹ 0, then
(1) a ¹ b ¹ c (2) a = b = c
(3) a = b ¹ c (4) a ¹ b = c
(SSC CGL Tier-II Exam, 2014 12.04.2015 (Kolkata Region) TF No. 789 TH 7)
- If bc + ab + ca = abc , then the
value of
b c
bc a
( – 1 ) +^
a c
ac b
a b
ab c
( – 1 ) is
(SSC CGL Tier-II Exam, 2014 12.04.2015 (Kolkata Region) TF No. 789 TH 7)
- If
a bc
a bc
2 2
b ca
b ca
2 2
c ab
c ab
2 2
= 1, then the value of
a
a bc
2
2 +^ +^
b
b ac
2
2 +^ +^
c
c ab
2
2 +^ is
(SSC CGL Tier-II Exam, 2014 12.04.2015 (Kolkata Region) TF No. 789 TH 7)
- If 999 x + 888 y = 1332 888 x + 999 y = 555, then the value of x + y is (1) 888 (2) 555 (3) 1 (4) 999 (SSC CAPFs SI, CISF ASI & Delhi Police SI Exam, 21.06. (Ist Sitting) TF No. 8037731)
- If a =
x x
x x
, then
the value of ( a^2 – ax ) is (1) 1 (2) 2 (3) – 1 (4) 0 (SSC CAPFs SI, CISF ASI & Delhi Police SI Exam, 21.06. IInd Sitting)
- If x =
2 + 3 ,^ y^ =
then the value of 8 xy ( x^2 + y^2 ) is (1) 196 (2) 290 (3) 112 (4) 194 (SSC CAPFs SI, CISF ASI & Delhi Police SI Exam, 21.06. IInd Sitting)
- If a^2 + b^2 + c^2 = ab + bc + ca ,
then the value of
a c
b
is
(1) 3 (2) 2 (3) 0 (4) 1 (SSC CAPFs SI, CISF ASI & Delhi Police SI Exam, 21.06. IInd Sitting)
- If
m a
b c
m b
c a
2
+ +^2 +^2
m c
a b
2 2 3 , then the value of
m is (1) a^2 + b^2 – c^2 (2) a^2 + b^2 (3) a^2 + b^2 + c^2 (4) a^2 – b^2 – c^2 (SSC CGL Tier-I Exam, 09.08. (Ist Sitting) TF No. 1443088)
- If x +
x
= then the value of
x x
x x
2 2
is
(SSC CGL Tier-I Exam, 09.08. (IInd Sitting) TF No. 4239378)
- If p = 99 then, the value of
p ( p^2 + 3 p + 3) is : (1) 989898 (2) 988899 (3) 999999 (4) 998889 (SSC CGL Tier-I Exam, 16.08. (Ist Sitting) TF No. 3196279)
- If x =
and y
then the value of
x xy y
x xy y
2 2 2 2
61 (2)^
63 (4)^
(SSC CGL Tier-I Exam, 16.08. (IInd Sitting) TF No. 2176783)
- If x +
x
= 1 then the value of
x^2 – x + 2
(SSC CGL Tier-I Exam, 16.08. (IInd Sitting) TF No. 2176783)
- If x =
a b
a b
+ ,^ y^ =^
b c
b c
+ ,^ z^ =
c a
c a
, then
x y z
+ x + y + z
is equal to (1) 1 (2) 0
(SSC CGL Tier-I Re-Exam, 30.08.2015)
- Let x =
and y =
x
then the value of 3 x^2 – 5 xy + 3 y^2 is (1) 1717 (2) 1177 (3) 1771 (4) 1171 (SSC CGL Tier-II Exam, 25.10.2015, TF No. 1099685)
217. If a +
b
= b +
c
= c +
a
where a ¹ b ¹ c ¹ 0, then the value of a^2 b^2 c^2 is (1) –1 (2) abc (3) 0 (4) 1 (SSC CGL Tier-II Exam, 25.10.2015, TF No. 1099685)
- For real a , b , c if a^2 + b^2 + c^2 =
ab + bc + ca , the value of
a c
b
is (1) 3 (2) 1 (3) 2 (4) 0 (SSC CHSL (10+2) LDC, DEO & PA/SA Exam, 01.11.2015, IInd Sitting)
- 9 x^2 + 25 – 30 x can be expressed as the square of (1) –3 x – 5 (2) 3 x + 5 (3) 3 x – 5 (4) 3 x^2 – 25 (SSC CHSL (10+2) LDC, DEO & PA/SA Exam, 01.11.2015, IInd Sitting)
- If
x
3 +^
x = 1 then the value of
x^3 is (1) 1 (2) 27 (3) 0 (4) – (SSC CHSL (10+2) LDC, DEO & PA/SA Exam, 01.11.2015, IInd Sitting)
- If x + y = 2 a, then the value of
a
x – a +
a
y – a is
(SSC CHSL (10+2) LDC, DEO & PA/SA
Exam, 01.11.2015, IInd Sitting)
- If
x
x
a
b
and
+ y =^
b
a
then the value of
x y
xy
1 + is
2 2
ab
a – b
a b
ab
2 2
a b
ab
2 2
a b
ab
(SSC CHSL (10+2) LDC, DEO
& PA/SA Exam, 15.11. (Ist Sitting) TF No. 6636838)
- If
a
b
b
a
= 2, then the value of
( a – b ) is : (1) 1 (2) 2 (3) –1 (4) 0 (SSC CHSL (10+2) LDC, DEO & PA/SA Exam, 15.11. (IInd Sitting) TF No. 7203752)
224. If y = 4 x , then
x
y
2 is :
(SSC CHSL (10+2) LDC, DEO
& PA/SA Exam, 15.11. (IInd Sitting) TF No. 7203752)
243. If 2
x
x
+ =1, then the value
of x^
x
2 2
+ is
(SSC CHSL (10+2) Tier-I (CBE) Exam. 08.09.2016) (Ist Sitting)
- The value of
a
a – b
b
b – a
is
a b
a b
a + f a – f (2) – (3) 2 ab (4) 1 (SSC CGL Tier-I (CBE) Exam. 09.09.2016) (Ist Sitting)
- If a +
b = 1 and^ b^ +^
c = 1
then c +
a is equal to
(SSC CAPFs (CPO) SI & ASI, Delhi Police Exam. 05.06.2016) (Ist Sitting)
- If
a
b =^
2 , find the value of the
expression
a b
a b
b – g b + g
(1) –32 (2) 11
(SSC CAPFs (CPO) SI & ASI, Delhi Police Exam. 05.06.2016) (Ist Sitting)
- If
2
2
x
+ x represents the redi-
us of circle P and
x
+ x = ,
which of the following best ap- proximates the circumference of circle P?
(1) 287 p (2) 547 p
(3) 574 p (4) 278 p
(SSC CPO SI & ASI, Online Exam. 06.06.2016) (IInd Sitting)
- What is the value of m in the quadratic equation x^2 + mx + 24
= 0 if one of its roots is
2 (4)^
(SSC CPO SI & ASI, Online Exam. 06.06.2016) (IInd Sitting)
- If ab = 21 and
a b
a b
2 2
then the value of a^2 + b^2 + 3 ab is (1) 115 (2) 121 (3) 125 (4) 127 (SSC CGL Tier-I (CBE) Exam. 27.08.2016) (Ist Sitting)
250. If a^^ +^ a - =
4, then the value
of ( a – 2)^2 +
2
a -
F HG^
I KJ^ is : (1) 0 (2) 2 (3) –2 (4) 4 (SSC CGL Tier-I (CBE) Exam. 27.08.2016) (IInd Sitting)
251. If x^
pq
p q
, then the value of
x p
x p
x q
x q
is
(1) 6 (2) 8 (3) 2 (4) 3 (SSC CGL Tier-I (CBE) Exam. 27.08.2016) (IInd Sitting)
252. If x^
x
4 , then the value
2
x + x 2 is
(SSC CGL Tier-I (CBE) Exam. 28.08.2016) (IInd Sitting)
253. If x^^3 x x
F – HG^
I KJ^
= , then the value
of x^
x
2 2
+ will be
9 (2)^
9 (4)^
(SSC CGL Tier-I (CBE) Exam. 28.08.2016) (IInd Sitting)
254. If x^
x
2 2
+ = 2 , then the value
of x^^ – x
is
(1) –2 (2) 0 (3) 1 (4) – (SSC CGL Tier-I (CBE) Exam. 29.08.2016) (IInd Sitting)
- If 9 x^2 + 16 y^2 = 60 and 3 x + 4 y = 6, then the value of xy is (1) –1 (2) 1 (3) –2 (4) 2 (SSC CGL Tier-I (CBE) Exam. 29.08.2016) (IInd Sitting)
256. If p^2 + q^2 = 7 pq, then the value
of
p
q
q
p
+ is equal to
(SSC CGL Tier-I (CBE) Exam. 30.08.2016) (Ist Sitting)
- If x = 99, then the value of 2( x^2 + 3 x + 3 ) is equal to (1) 1000001 (2) 1000000 (3) 999999 (4) 9999999 (SSC CGL Tier-I (CBE) Exam. 30.08.2016) (Ist Sitting)
- If
p
p – p +
= , then the val-
ue of p^^ +^ p
will be
(1) 8 (2) 10 (3) 12 (4) None of these (SSC CGL Tier-I (CBE) Exam. 31.08.2016) (Ist Sitting)
- If ( a – b ) = 3 and ( a^2 + b^2 ) = 25, then the value of (ab) is (1) 16 (2) 8 (3) 10 (4) 15 (SSC CGL Tier-I (CBE) Exam. 31.08.2016) (Ist Sitting)
- If a +
a = 1, then the value of
a a
a a
2 2
is ( a ¹ 0)
(1) 1 (2) – (3) 0 (4) 2 (SSC CGL Tier-I (CBE) Exam. 02.09.2016) (Ist Sitting)
- If x –
x = 2, then what is the
value of x^2 +
x^2?
(SSC CGL Tier-I (CBE) Exam. 02.09.2016) (IInd Sitting)
- If a + b = 2 c , then the value of
a
a c
c
+ is equal to (where
a ¹ b ¹ c )
(SSC CGL Tier-I (CBE) Exam. 04.09.2016) (Ist Sitting)
263. If x^^ +^ x
= 5, then the value of
x
1 + x + x^2
is
5 (2)^
(SSC CGL Tier-I (CBE) Exam. 04.09.2016) (Ist Sitting)
- If
a
b c
2
b
c a
2
c
a b
2
= 1 then find the value of
a b + c
(SSC CGL Tier-I (CBE) Exam. 04.09.2016) (Ist Sitting)
265. If 5
x
x
+ = 10, then x^2 +
25 x^2
is equal to
5 (2)^
5 (4)^
(SSC CGL Tier-I (CBE) Exam. 06.09.2016) (Ist Sitting)
266. If 4 r = h + r^2 + h^2 then r : h is
? ( r ¹ 0)
(SSC CGL Tier-I (CBE) Exam. 06.09.2016) (Ist Sitting)
- If p = 99, then the value of p ( p^2 + 3 p + 3) will be (1) 999999 (2) 1000000 (3) 1000001 (4) 999998 (SSC CGL Tier-I (CBE) Exam. 07.09.2016) (Ist Sitting)
- If
x
a + b
x
a b
a b
then x is equal to (1) 2 a – b (2) a + b (3) a – b (4) 2 a + b (SSC CGL Tier-I (CBE) Exam. 07.09.2016) (Ist Sitting)
- If x^2 + y^2 = 29 and xy = 10, where x > 0, y > 0, x > y then
the value of
x y
x y
(SSC CGL Tier-I (CBE) Exam. 30.08.2016) (IInd Sitting)
- If 4 x^2 – 12 x + k is a perfect square, then the value of k is (1) 2 (2) 9 (3) 12 (4) 10 (SSC CGL Tier-I (CBE) Exam. 31.08.2016) (IInd Sitting)
- The value of 1 1 1 a p – n fa n – q f a n – q fa q – p f a q – p fa p – n f
F HG^
I KJ is (1) 1 (2) 0
(3) p + q + n (4)
2 n
p + q
(SSC CGL Tier-I (CBE) Exam. 01.09.2016) (IInd Sitting)
- If
a
b c
2
b
c a
2
c
a b
2
then
1 + a
1 + b
1 + c
is
(1) 1 (2) 2 (3) 3 (4) 4 (SSC CGL Tier-I (CBE) Exam. 01.09.2016) (IInd Sitting)
273. If a^2 + 1 = 9 a , ( a ¹ 0) then the
value of ( a )^2 +
a f a^2 is (1) 81 (2) 18 (3) 79 (4) 83 (SSC CGL Tier-I (CBE) Exam. 02.09.2016) (IInd Sitting)
- If p = 99, then the value of p ( p^2
+ 3 _p_ + 3) is (1) 9999 (2) 999999 (3) 99999 (4) 9999999 (SSC CGL Tier-II (CBE) Exam. 30.11.2016)
275. If x^
x
c
c
then the value of x is
(1) c
c
(2) c , c^2 (3) c , 2 c (4) 0, 1 (SSC CGL Tier-II (CBE) Exam. 30.11.2016)
- If x^2 + y^2 + 6 x + 5 = 4 ( x – y ) then ( x – y ) is (1) 1 (2) – (3) 0 (4) 4 (SSC CGL Tier-II (CBE) Exam. 30.11.2016)
277. If x^
x
F HG^
I KJ^ =^
, the value of 3
x
x
F H
G I K
J (^) is :
(1) –1 (2) 1 (3) –2 (4) 2 (SSC CGL Tier-I (CBE) Exam. 28.08.2016 (IST Sitting)
- If
a
q r
b
r p
c
, find
the value of ( pa + qb + rc ). (1) 0 (2) 1 (3) 2 (4) – (SSC CGL Tier-I (CBE) Exam. 29.08.2016 (IST Sitting)
- If
a b
c d
a b
c d
- , then (1) ab = cd (2) ad = bc
(3) ac = bd (4) a = b = c ¹ d
(SSC CGL Tier-I (CBE) Exam. 30.08.2016 (IIIrd Sitting)
280. If x^^ + x
F HG^
I KJ
= 2 then x^
x
2 2
F + HG^
I KJ is equal to (1) 0 (2) 2 (3) 4 (4) 8 (SSC CGL Tier-I (CBE) Exam. 31.08.2016 (IIIrd Sitting)
- If a + b = 17 and a – b = 9, then the value of (4 a^2 + 4 b^2 ) is : (1) 710 (2) 720 (3) 730 (4) 740 (SSC CGL Tier-I (CBE) Exam. 31.08.2016 (IIIrd Sitting)
282. If x + y = 3 and x – y = 2 ,
then the value of 8 xy ( x^2 + y^2 ) is :
(SSC CGL Tier-I (CBE) Exam. 31.08.2016 (IIIrd Sitting)
- If a^2 + 1 = a , then the value of a^3 is (1) 0 (2) 1 (3) –1 (4) 2 (SSC CGL Tier-I (CBE) Exam. 01.09.2016 (IIIrd Sitting)
