Quadratic Equations

3
and Inequalities
Polynomial Quadratic Identity
Let a0 , a1 , a2 , ... , an are real numbers, then If two quadratic expression in x are equal for all values
f ( x ) = a0 + a1x + a2x 2 + ... + an x n is known as polynomial. of x. This statement of equality between the two
expressions is called an identity.
Polynomial a0 + a1x + a2x 2 + K + an x n is known as a
polynomial of degree n.
Identical Equations
Two equations are said to be identical, if they have
Polynomial Equation same roots.
If f ( x ) is a real or complex polynomial, then f ( x ) = 0 is
known as a polynomial equation. Roots of an Equation
If f ( x )is a polynomial of degree 2, then f ( x ) = 0 is known
The values of the variables satisfying the given
as quadratic equation. General equation of quadratic
equation are called its roots i. e. , if f ( x ) = 0 is a polynomial
equation is ax 2 + bx + c = 0 ,where a , b, c ∈ R or C.
equation and f ( a ) = 0, then a is known as root of the
The quadratic equation of the form ax 2 + c = 0 is polynomial equation f ( x ) = 0.
known as pure quadratic equation.
% An equation of degree n has n roots real or imaginary.
% If an equation has a root a + b or a + ib, then another root will
Quadratic Equation be a − b or a − ib, respectively.
An equation of the form ax 2 + bx + c = 0, where a, b and % An odd degree equation has atleast one real root whose sign is
c are certain numbers and a ≠ 0 is called a quadratic opposite to that of its last term, provided that the coefficient of
the highest degree term is positive.
equation.
% Every equation of an even degree whose constant term is
The numbers a , b, and c are called the coefficients of
negative and the coefficient of the highest degree term is positive,
the quadratic equation and the number b2 − 4ac is called has atleast two real roots, one positive and one negative.
its discriminant. % If all the terms of an equation are positive and the equation
Discriminant of a quadratic equation is usually denoted involves no odd power of x, then all its roots are complex.
by D.

Quadratic Expression
Roots of a Quadratic Equation
Roots of a quadratic equation ax 2 + bx + c = 0 are
An expression of the form ax 2 + bx + c,where a , b and c
− b + b2 − 4ac − b − b2 − 4ac
are some numbers and a ≠ 0 is called a quadratic and , where b2 − 4ac is
2a 2a
expression.
known as discriminant and is denoted by D.

by hitesh kumar, (hitish.nitc@gmail.com)

 42 NDA/NA Mathematics

Example 1. The roots of the equation x2 + | x| − 6 = 0 is Example 3. If the product of the roots of the equation
(a) ± 9 (b) ± 2 mx 2 + 6 x + (2m − 1) = 0 is − 1, then the value of m is
(c) ± 3 (d) ± 4 1 1 2
(a) m = (b) m = (c) m = (d) m = 2
3 2 3
Solution (b)| x|2 + | x| − 6 = 0
⇒ (| x| + 3) (| x| − 2) = 0 ⇒| x| = 2 ⇒ x = ± 2 Solution (a) Let α and β be the roots of the equation
mx2 + 6x + (2m − 1) = 0 .
6
Relation between Coefficient ∴ α+β=−
m
...(i)

and Roots of an Equation and αβ =

2m − 1
... (ii)
m
(i) Quadratic equation Let α and β be the roots According to the question,
of the quadratic equation, ax 2 + bx + c = 0, then 2m − 1 1
−1= ⇒ − m = 2m − 1⇒ 3m = 1 ⇒ m =
b m 3
α +β = −
a
and αβ =
c Nature of Roots
a
Let the equation is ax 2 + bx + c = 0, then
(ii) Cubic equation Let α , β and γ be the roots of
1. if D = b2 − 4ac > 0, then the roots of equation are real
the cubic equation ax3 + bx 2 + cx + d = 0, then
and distinct.
b c d 2. if D = b2 − 4ac = 0, then the roots of equation are real
α + β + γ = − , αβ + βγ + γα = and αβγ = −
a a a and coincident.
% If α and β be the roots of the quadratic equation 3. if D = b2 − 4ac < 0, then the roots of equation are
ax 2 + bx + c = 0, then ax 2 + bx + c = a (x − α) (x − β). imaginary.
4. if D = b2 − 4ac > 0 and a perfect square, then the
π P Q roots of the equation are rational.
Example 2. In a ∆ PQR, ∠ R = . If tan and tan are
2 2 2 5. if D = b2 − 4ac < 0 and not a perfect square, then the
the roots of the equation ax + bx + c = 0, then which of the
2
roots of the equation are irrational.
following is correct?
(a) a + b = 2c (b) a + b = c Example 4. If the roots of the equation
(c) a + b + c = 0 (d) 2 a = b − c x 2 − 8 x + a 2 − 6 a = 0 are real and distinct, then all possible
P Q values of a is
Solution (b) Q tan and tan are the roots of the equation (a) − 3 < a < 2 (b) − 2 < a < 8 (c) − 3 < a < 4 (d) − 4 < a ≤ 1
2 2
ax2 + bx + c = 0, then Solution (b) Since, the roots of the given equation are real and
P Q b P Q c distinct, we must have
tan + tan = − and tan tan =
2 2 a 2 2 a D > 0 ⇒ 64 − 4 ( a2 − 6a) > 0 ⇒ 4 [16 − a2 + 6a] > 0
π π
Also, ∠R = ⇒ ∠P + ∠Q = ⇒ −4 ( a2 − 6a − 16) > 0 ⇒ a2 − 6a − 16 < 0
2 2
P Q ⇒ ( a − 8) ( a + 2) < 0 ⇒ −2 < a < 8
tan + tan
π  P Q 2 2 Hence, the roots of the given equation are real, if a lies
⇒ 1 = tan = tan  +  ⇒ 1 = between −2 and 8.
4 2 2 P Q
1 − tan tan
2 2
⇒ 1=
−b / a
⇒ 1=
−b
⇒ a+ b=c Symmetric Functions
1− c / a a−c
The algebraic expressions in α and β which remain
unchanged when α and β are interchanged, are known as
Equation of Given Roots symmetric functions in α and β.
(i) Quadratic equation If the roots of a An important property of such functions is that they
quadratic equation are α and β, then equation will be can always be expressed in terms of (α + β) and αβ. So, they
can be evaluated for a given quadratic equation.
x 2 − (α + β ) x + αβ = 0.
The following relations serve the purpose of useful
(ii) Cubic equation If α, β and γ are the roots of tools for the same.
the cubic equation, then the equation will be (i) (α 2 + β 2 ) = [(α + β )2 − 2αβ ]
x3 − (α + β + γ )x 2 + (αβ + βγ + γα ) x − αβγ = 0 (ii) ( 2 − β )2 = [( 2 + β )2 − 4αβ ]

by hitesh kumar, (hitish.nitc@gmail.com)

 Quadratic Equations and Inequalities 43

(iii) (α 2 − β 2 ) = (α + β ) (α − β )  b 2 2 c b 2
a 2 −  −
= (α + β ) [ (α + β ) − 4 α β ]
2 a a a
= [from Eq. (i)]
 
c  b
(iv) α3 + β3 = (α + β )3 − 3αβ (α + β ) a2  + b 2 + ab  − 
 a  a
(v) α3 − β3 = (α − β )3 + 3αβ (α − β ) 2c 2
=− =−
(vi) α 4 + β 4 = (α 2 + β 2 )2 − 2α 2β 2 ac a
= [(α + β )2 − 2αβ ]2 − 2 (αβ )2
(vii) α 4 − β 4 = (α − β ) (α + β ) (α 2 + β 2 ) Maximum and Minimum Value
= (α + β ) [(α + β )2 − 2αβ ] (α + β )2 − 4αβ of ax 2 + bx + c
Example 5. If the sum of roots of the equation  b
2
 4ac − b2 
Q ax 2 + bx + c = a  x +  + 
ax + bx + c = 0 is equal to the sum of square of their
2
 2a   4a 
reciprocal, then bc 2, ca 2 and ab 2 are in
Case I a > 0
(a) HP (b) AP (c) GP (d) None of these
4ac − b2
Q ax 2 + bx + c ≥
Solution (b) Let α and β are the roots of the euqation 4a
ax2 + bx + c = 0, then 4ac − b2
∴ Minimum value of ax 2 + bx + c is and this
b c 4a
α+β=− and αβ = b
a a value attains at x = − .
According to the question, 2a
1 1 Case II a < 0
α+β= 2+ 2
α β 4ac − b2
Q ax 2 + bx + c ≤
α 2 + β 2 (α + β) 2 − 2αβ 4a
⇒ α+β= = 4ac − b2
α 2β 2 (αβ) 2 ∴ Maximum value of ax 2 + bx + c is and this
4a
2 b
 b  c value attains at x = −
−  −2  
b  a   a b 2 − 2ac 2a
⇒ − = 2
=
a  c c2
 
 a Inequality
ab + bc
2 2
An inequation or inequality is a statement involving
⇒ − bc2 = ab 2 − 2a2c ⇒ ca2 =
2 variable or variables and anyone of the following signs of
⇒ bc2, ca2 and ab 2 are in AP.
inequalities
(i) <(less than) (ii) > (greater than) (iii) ≤ (less than or
Example 6. If α and β are the roots of ax2 + bx + c = 0, equal to) (iv) ≥ (greater than or equal to)
β α Properties of Inequalities
then the values of the following expression +
aα + b aβ + b 1. If a , b and c are real numbers such that a > b and
in terms of a, b and c is b > c, then a > c.
−2 4 More generally, if a1 , a2 , a3 ,.... , an −1 , an are real
(a) (b)
a b numbers such that a1 > a2 > a3 > a4 >... > an − 1 > an ,
3 2 then a1 > an .
(c) (d)
c 3a 2. If a and b are real numbers, then
a > b ⇒ a + c > b + c for all c ∈ R.
Solution (a) Since, α and β are the roots of ax2 + bx + c = 0.
3. If a and b are two real numbers, then
b c
∴ α+β=− ,α β = …(i) a > b ⇒ ac > bc and >
a b
for all c > 0, c ∈ R.
a a c c
β α β ( a β + b) + α ( a α + b) a b
+ = Also, a > b ⇒ ac < bc and < for all c < 0, c ∈ R.
aα + b a β + b ( a α + b) ( a β + b) c c
a (α 2 + β 2) + b (α + β) 1 1
= 4. If a > b > 0, then < .
a2αβ + b 2 + ab (α + β) a b
a{(α + β) 2 − 2αβ} + b (α + β) 5. If a1 > b1 , a2 > b2 , a3 > b3 ,... , an > bn , then
= a1 + a2 + a3 + ... + an > b1 + b2 + b3 + ....+ bn
a2αβ + b 2 + ab (α + β)

by hitesh kumar, (hitish.nitc@gmail.com)

 44 NDA/NA Mathematics

6. If a1 , a2 ,... , an and b1 , b2 ,... , bn are positive real Arithmetico-Geometric

numbers such that
a1 > b1 , a2 > b2 ,... , an > bn , Mean Inequality
then a1a2a3 ... an > b1b2b3 K bn . a+b 2
1. If a , b > 0 and a ≠ b, then > ab > .
2 1 1
7. If a and b are positive real numbers such that a < b +
and if n is any positive rational number, then a b
(a) a n < bn (b) a − n > b− n 2. If ai > 0, where i = 1, 2, 3, ... , n , then
a1 + a2 + ... + an n
(c) a1/ n < b1/ n ≥ ( a1 ⋅ a2K an )1/ n ≥
n 1 1 1
+ + ...+
8. If 0 < a < 1 and n is any positive rational number, a1 a2 an
then
(a) 0 < a n < 1 (b) a − n > 1 Example 8. If a, b and c are three distinct positive real
numbers, then
9. If a > 1 and n is any positive rational number, then
 1 1 1
(a) a n > 1 (b) 0 < a − n < 1 ( a + b + c)  + +  is
 a b c
10. If 0 < a < 1 and m , n are positive rational numbers, (a) >9 (b) <4 (c) >5 (d) <6
then
Solution (a) We have AM > GM
(a) m > n ⇒ a m < a n (b) m < n ⇒ a m > a n
a+ b+ c
∴ > ( abc)1/3
11. If a > 1 and m , n are positive rational numbers, then 3
(a) m > n ⇒ a m > a n (b) m < n ⇒ a m < a n 1 1 1
+ + 1/3
a b c  1 
and > 
Example 7. If| 2 x − 3| < | x + 5 |, then x belongs to 3  abc
(a) ( − 3, 5) (b) (5, 9) ⇒ a + b + c > 3 ( abc)1/3
− 2   2  1 1 1 3
(c)  , 8 (d)  − 8,  and  + +  >
 3   3  a b c ( abc)1/3
 1 1 1
Solution (c) Given,| 2x − 3| < | x + 5| ⇒ ( a + b + c)  + +  > 9
 a b c
⇒ | 2x − 3| −| x + 5| < 0
⇒ 3 − 2x + x + 5 < 0 , x ≤ − 5 Example 9. For real positive numbers a, b and c the
b +c c + a a+ b
⇒ 3 − 2x − x − 5 < 0 , − 5 < x ≤
3 minimum value of + + is
2 a b c
(a) 5 (b) 4 (c) 6 (d) 3
3
⇒ 2x − 3 − x − 5 < 0 , x >
Solution (c) Q  + + + + + 
2 1 a b b c c a
⇒ x > 8, x ≤ − 5 6 b a c b a c
1/ 6
−2 3 a b b c c a
⇒ x> , −5 < x≤ ≥ ⋅ ⋅ ⋅ ⋅ ⋅ 
3 2 b a c b a c
3  a b b c c a
⇒ x < 8, x > ⇒  + + + + +  ≥6
2  b a c b a c
 − 2 3 3  −2  a+ b c+ a b+ c
⇒ x ∈ , ∪  , 8 ⇒ x ∈  , 8 ⇒ Minimum value of + + =6
 3 2   2   3  c b a

by hitesh kumar, (hitish.nitc@gmail.com)

 Quadratic Equations and Inequalities 45

Comprehensive Approach
n If f(α) = 0 and f ′(α) = 0, then α is a repeated root of the quadratic (b) whose roots are Aα and Aβ, is
equation f ( x) = 0 and f ( x) = a ( x − α) 2 ax2 + Abx + A 2c = 0
In fact α = −
b α β
(c) whose roots are and , is
2a A A
For the quadratic equation ax2 + bx + c = 0 aA 2x2 + bAx + c = 0
1. One root will be reciprocal of the other, if a = c 1 1
(d) whose roots are and , is
2. One root is zero, if c = 0 α β
3. Roots are equal in magnitude but opposite in sign, if cx2 + bx + a = 0
b = 0. n If in ax2 + bx + c, a > 0 and b 2 − 4 ac < 0, then ax2 + bx + c will
4. Both roots are zero, if b = c = 0 always be positive.
5. Roots are positive, if a and c are of the same sign and b is n If in ax2 + bx + c , a < 0 and b 2 − 4 ac < 0, then ax2 + bx + c will
of the opposite sign. always be negative.
6. Roots are of opposite sign, if a and c are of opposite sign. n If the sum of coefficients of the polynomial equation
7. Roots are negative, if a, b and c are of the same sign. a0 + a1x + a2x2 + K + an xn = 0 is zero, then x = 1will be the root
of that equation.
n If the ratio of roots of the quadratic equation ax2 + bx + c = 0 be
p : q, then pqb 2 = ( p + q) 2 ac
n If the equation is ax2 + bx + c = 0 such that, a + b + c = 0, then
c
n If one root of the quadratic equation ax2 + bx + c = 0 is equal to roots of equation ax2 + bx + c = 0 will be1, and if a − b + c = 0,
a
the n th power of the other, then c
1 1 then roots of the equation will be −1, − .
( ac n ) n + 1 + ( a n c) n + 1 + b = 0 a
n If sum of roots of the equation ax2 + bx + c = 0 is equal to the sum
n If one root of the equation ax2 + bx + c = 0 be n times the other b a
root, then nb 2 = ac (n + 1) 2 of their reciprocal, then ab 2 , bc 2 and ca 2 will be in AP or , and
k+1 c b
n If the roots of the equation ax2 + bx + c = 0 are of the form c
will be in AP.
k a
k+2
and , then ( a + b + c) 2 = b 2 − 4 ac n If a1 = a2 = K = an , then AM = GM
k+1 n If a, b > 0 and a ≠ b, then
n If the roots of ax2 + bx + c = 0 are α and β, then the roots of am + b m  a + b 
m

cx2 + bx + a = 0 will be and .

1 1 (a) >  , if m < 0 or m > 1
α β 2  2 
m
The roots of the equation ax2 + bx + c = 0 are reciprocal to am + b m  a + b 
<  , if 0 < m < 1
n
(b)
a′ x2 + b ′ x + c ′ = 0, if 2  2 
( cc ′ − aa′) 2 = ( ba′ − cb ′) ( ab ′ − bc ′) n f a > 1and n is any positive rational number, then
n Let f ( x) = ax2 + bx + c, where a > 0. Then,
(a) a n > 1
1. conditions for both the roots of f ( x) = 0to be greater than a given (b) 0 < a − n < 1
−b
number K are b 2 − 4 ac ≥ 0; f (K) > 0 ; > K. n If 0 < a < 1 and m,n are positive numbers, then
2a
(a) m > n ⇒ a m < a n (b) m < n ⇒ a m > a n
2. the number K lies between the roots of f ( x) = 0, if f (K) < 0.
(a) if a > 1and x > y > 0, then log a x > log a y
3. condition for exactly one root of f ( x) = 0 to lie between d and e,
(b) if 0 < a < 1 and x > y > 0, then log a x < log a y
is f (d ) f ( e) < 0.
If a and b are positive real numbers, then
If the common root of quadratic equation a1x2 + b1x + c1 = 0 and
n
n
a+ c a
a2x2 + b2x + c2 = 0 is α, then (a) a < b ⇒ > , ∀c > 0
c a − c2a1 b c − b2c1 b+ c b
α= 1 2 or 1 2
a1b2 − a2b1 c1a2 − c2a1 a+ c a
(b) a > b ⇒ < , ∀c> 0
n If both the roots of the quadratic equations a1x2 + b1x + c1 = 0 and b+ c b
a2x2 + b2x + c2 = 0 are common, then n
1
If a is a positive real number, then a + ≥ 2 and if a is a negative
a1 b1 c1
= = a
a2 b2 c2 1
real number, then a + ≤ −2.
n Let α and β be the roots of the equation ax2 + bx + c = 0, then that a
equation n | a + b | ≤ | a | + | b | , in general
(a) whose roots are α ± A , β ± A, is | a1 + a2 + K + an |≤| a1| + | a2| + ... + | an|
a ( x m A) 2 + b ( x m A) + c = 0

by hitesh kumar, (hitish.nitc@gmail.com)

 Exercise
Level I
1. If α and β are the roots of the equation 9. What are the roots of the equation
1
4x + 3x + 7 = 0, then +
2 1
is equal to 2( y + 2)2 − 5( y + 2) = 12? (NDA 2011 II)
α β (a) − 7 / 2 , 2 (b) − 3 / 2 , 4
3 3 3 3 (c) − 5 / 3, 3
(a) − (b) (c) − (d) (d) 3 / 2, 4
7 7 5 5
10. Let α and β be the roots of the equation x 2 + x + 1 = 0.
2. If the roots of the equation 3x 2 − 5x + q = 0 are equal, The equation, whose roots are α19, and β7 is
then what is the value of q? (NDA 2011 II) (NDA 2011 II)
(a) 2 (b) 5 / 12 (a) x − x − 1 = 0
2
(b) x − x + 1 = 0
2
(c) 12 / 25 (d) 25 / 12
(c) x 2 + x − 1 = 0 (d) x 2 + x + 1 = 0
3. If the product of the roots of the equation
( a + 1) x 2 + ( 2a + 3) x + ( 3a + 4) = 0 be 2, then the ( x + 1) ( x − 3)
11. Let y = , then all real values of x for
sum of roots is ( x − 2)
(a) 1 (b) −1 which y takes real values, are
(c) 2 (d) −2 (a) −1 ≤ x < 2 or x ≥ 3 (b) −1 ≤ x < 3 or x > 2
(c) 1 ≤ x < 2 or x ≥ 3 (d) None of these
4. If the roots of the equation ax 2 + bx + c = 0 be α and β,
then the roots of the equation cx 2 + bx + a = 0 are 12. If 2 + i 3 is a root of the equation x 2 + px + q = 0,
1 where p and q are real, then ( p, q ) is equal to
(a) −α , − β (b) α, (a) ( −4, 7) (b) ( 4, − 7)
β
1 1 (c) ( 4, 7) (d) ( −4, − 7)
(c) , (d) None of these
α β 13. The coefficient of x in the equation x 2 + px + q = 0
was taken as 17 in place of 13, its roots were found to
5. If α and β are the roots of the equation be −2 and −15. The roots of the original equation are
ax 2 + bx + c = 0, then the equation, whose roots are (a) 3, 10 (b) −3, − 10
1 1 (c) −5, − 8
α + and β + , is (d) None of these
β α
14. If p, q and r are real and p ≠ q, then the roots of the
(a) acx 2 + ( a + c) bx + ( a + c)2 = 0 equation
(b) abx 2 + ( a + c) bx + ( a + c)2 = 0 ( p − q ) x 2 + 5 ( p + q ) x − 2 ( p − q ) = r are
(c) acx 2 + ( a + b) cx + ( a + c)2 = 0 (a) real and equal
(d) None of the above (b) unequal and rational
(c) unequal and irrational
6. If the equations x 2 − px + q = 0 and x 2 − ax + b = 0 (d) nothing can be said
have a common root and the roots of the second
equation are equal, then which one of the following is 15. If the roots of the equation ( p2 + q 2 )x 2 − 2q ( p + r ) x
correct? (NDA 2011 II) + ( q 2 + r 2 ) = 0 be real and equal, then p, q and r will
(a) aq = 2( b + p) (b) aq = b + p be in
(c) ap = 2( b + q ) (d) ap = b + q (a) AP (b) GP
(c) HP (d) None of these
7. The equation x 2 − 4x + 29 = 0 has one root 2 + 5i.
16. What is the value of
What is the other root? ( i = −1 ) (NDA 2011 II)
(a) 2 (b) 5 (c) 2 + 5i (d) 2 − 5i 8 + 2 8 + 2 8 + 2 8 +K∞ ?
(NDA 2011 II)
8. If one of the roots of the equation (a) 10 (b) 8
a ( b − c) x 2 + b ( c − a ) x + c ( a − b) = 0 is 1, what is the (c) 6 (d) 4
second root? (NDA 2011 II) 17. If α and β are the roots of the equation
b( c − a ) b( c − a ) x 2 − q (1 + x ) − r = 0, then what is the value of
(a) − (b)
a( b − c) a( b − c) (1 + α ) (1 + β )? (NDA 2012 I)
c( a − b) c( a − b) (a) 1 − r (b) q − r
(c) (d) −
a( b − c) a( b − c) (c) 1 + r (d) q + r

by hitesh kumar, (hitish.nitc@gmail.com)

 Quadratic Equations and Inequalities 47

18. What is the solution set for the equation (a) Only − 2 (b) Only 1
x 4 − 26x 2 + 25 = 0 ? (NDA 2011 I) (c) − 2 and 1 (d) − 2 and − 1
(a) { − 5, − 1, 1, 5} (b) { − 5, − 1} 29. If α and β are the roots of the equation
(c) { 1, 5} (d) { − 5, 0, 1, 5} 4x 2 + 3x + 7 = 0,then what is the value of (α −2 + β −2 )?
19. If the equations x 2 + kx + 64 = 0 and x 2 − 8x + k = 0 (NDA 2011 I)
have real roots, then what is the value of k? (a) 47 / 49 (b) 49 / 47
(NDA 2010 II) (c) − 47 / 49 (d) − 49 / 47
(a) 4 (b) 8 (c) 12 (d) 16 30. If the roots of the equations px 2 + 2qx + r = 0 and
20. If the product of the roots of the equation qx 2 − 2 pr x + q = 0 be real, then
x 2 − 5x + k = 15 is − 3, then what is the value of k? (a) p = q (b) q 2 = pr
(NDA 2010 I) (c) p = qr
2
(d) r 2 = pq
(a) 12 (b) 15 (c) 16 (d) 18 31. If a < b, then
α β a b a b
21. If the roots of the equation + = 1 be equal (a) < (b) >
x−α x−β ( −2) ( −2) 2 2
in magnitude but opposite in sign, then α + β is equal 1 1 a b
(c) < (d) >
to a b −2 −2
(a) 0 (b) 1 32. If 2x + 3 y = 17 and 2x + 2 − 3 y + 1 = 5, then what is the
(c) 2 (d) None of these
value of x? (NDA 2009 I)
22. If the equation x 2 − bx + 1 = 0 does not possess real (a) 3 (b) 2
roots, then which one of the following is correct? (c) 1 (d) 0
(NDA 2010 I)
33. What is the value of x satisfying the equation
(a) − 3 < b < 3 (b) − 2 < b < 2
3
(c) b > 2 (d) b < − 2  a − x a+x
16   = ? (NDA 2009 I)
23. If p and q are the roots of the equation x 2 − px + q = 0,  a + x a−x
then what are the values of p and q, respectively? a a a
(a) (b) (c) (d) 0
(a) 1, 0 (b) 0, 1 2 3 4
(c) − 2, 0 (d) − 2, 1
34. The roots of Ax 2 + Bx + C = 0 are r and s. For the
24. If α and β are the roots of ax 2 + bx + b = 0, then what roots of x 2 + px + q = 0 to be r 2 and s2 , what must be
α β b the value of p?
is + + equal to? (NDA 2009 II)
(NDA 2009 I)
β α a ( B2 − 4 AC ) ( B2 − 4 AC )
(a) (b)
(a) 0 (b) 1 (c) 2 (d) 3 A2 A2
( 2 AC − B )
2
25. If the roots of ax 2 + bx + c = 0 are sin α and cos α for (c) (d) B2 − 2 C
some α, then which one of the following is correct? A2
(NDA 2009 II) 35. If the roots of the equation x 2 − bx + c = 0 are two
(a) a + b = 2ac
2 2
(b) b − c = 2ab
2 2
consecutive integers, then what is the value of
(c) b2 − a 2 = 2ac (d) b2 + c2 = 2ab b2 − 4c ? (NDA 2008 II)
26. If x = 2 + 21/3 + 22/ 3 , then what is the value of (a) 1 (b) 2
(c) –2 (d) 3
x3 − 6x 2 + 6x? (NDA 2009 II)
36. If r and s are roots of x 2 + px + q = 0, then what is the
(a) 1 (b) 2 (c) 3 (d) −2
value of (1/ r 2 ) + (1/ s2 )? (NDA 2008 II)
27. The roots of the equation ( x − p)( x − q ) = r 2, where p,
p2 − 4q
q and r are real, are (NDA 2009 II) (a) p2 − 4q (b)
2
(a) always complex (b) always real
p2 − 4q p2 − 2q
(c) always purely imaginary (c) (d)
(d) None of the above q2 q2
28. For the two equations x 2 + mx + 1 = 0 and 37. If α and β are the roots of x 2 + 4x + 6 = 0, then what is
x 2 + x + m = 0, what is/are the value/values of m for the value of α3 + β3 ? (NDA 2008 II)
which these equations have atleast one common 2 2
root? (NDA 2009 II)
(a) − (b) (c) 4 (d) 8
3 3

by hitesh kumar, (hitish.nitc@gmail.com)

 48 NDA/NA Mathematics

38. The number of solutions of the equation 45. If the equation x 2 + k2 = 2( k + 1)x has equal roots,
x 2 − 3|x| + 2 = 0, is then what is the value of k? (NDA 2007 I)
(a) 2 (b) 3 (c) 4 (d) 5 1 1
(a) − (b) − (c) 0 (d) 1
39. What is the sum of the squares of the roots of the 3 2
equation x 2 + 2x − 143 = 0 ? (NDA 2012 I) 46. If| x|> 5, then
(a) 170 (b) 180 (c) 190 (d) 290 (a) 0 < x < 5 (b) x < − 5 or x > 5
(c) −5 < x < 5 (d) x > 5
40. If the roots of the equation λ + 8λ + µ + 6 µ = 0
2 2

are real, then µ lies between 47. What is the solution set for the equation
(a) −2 and 8 (b) −3 and 6 x 2/ 3 + x1/ 3 − 2 = 0 ?
(c) −8 and 2 (d) −6 and 3 (a) { −8, 1} (b) { 8, 1}
(c) { −8, − 1} (d) { 8, − 1}
41. What is the value of 5 5 5 K ∞ ? (NDA 2008 I) 48. What is the polynomial, whose zero is 2 ?
(a) 5 (b) 5 (a) x 2 − 2x + 2 (b) ( x 4 − 2) ( 3 − 4x + 3x3 )
(c) 1 (d) ( 5)1/ 4 (c) x − 2x + x − 4
4 3
(d) x 4 − 3x3 + 3x 2 − 3x + 2
42. One root of the equation x 2 = px + q is reciprocal of 49. If α and β are the roots of the equation
the other and p ≠ ± 1. What is the value of q? ax 2 + bx + c = 0, then what is the value of
(NDA 2008 I)
( aα + b)−1 + ( aβ + b)−1 ? (NDA 2007 I)
(a) q = − 1 (b) q = 1
1 a b
(c) q = 0 (d) q = (a) (b)
2 ( bc) ( ac)
−b −a
43. If α and β are the roots of the equation (c) (d)
( ac) ( bc)
lx 2 − mx + m = 0, l ≠ m , l ≠ 0, then which one of the
following statements is correct? (NDA 2007 II) 50. If α and β are the roots of the equation x 2 − 2x − 1 = 0,
then what is the value of α 2β −2 + α −2β 2 ? (NDA 2007 I)
α β m
(a) + − =0 (a) –2 (b) 0 (c) 30 (d) 34
β α l
a c
α β m 51. If < , then
(b) + + =0 b d
β α l a+b c+d a−b c−d
(a) < (b) <
α +β m a−b c−d a+b c+d
(c) − =0 2 2
αβ l  a  c
(c)   <   (d) None of these
 b  d
(d) The arithmetic mean of α and β is the same as
their geometric mean. 52. If a > b, then
44. For what value of k, are the roots of the quadratic (a) a + 5 > b + 5 (b) a − 5 < b − 5
equation ( k + 1)x 2 − 2( k − 1)x + 1 = 0 real and equal? (c) a + b < b + 5 (d) depends on a and b
(NDA 2007 II) 53. If one of the roots of the equation x 2 + ax − b = 0 is 1,
(a) Only k = 0 (b) Only k = − 3 then what is the value of ( a − b)? (NDA 2012 I)
(c) k = 0 or k = 3 (d) k = 0 or k = − 3 (a) −1 (b) 1 (c) 2 (d) − 2

Level II
1. Let α and β be the roots of the equation 3. If the roots of the equation
( x − a ) ( x − b) = c, c ≠ 0. Then, the roots of the ( a 2 + b2 ) x 2 − 2b ( a + c) x + ( b2 + c2 ) = 0 are equal,
equation ( x − α ) ( x − β ) + c = 0 are (NDA 2011 II) then which one of the following is correct?
(a) a, c (b) b, c (NDA 2010 II)
(c) a, b (d) a + b, a + c (a) 2b = a + c (b) b2 = ac
2. If p, q and r are rational numbers, then the roots of (c) b + c = 2a (d) b = ac
the equation x 2 − 2 px + p2 − q 2 + 2qr − r 2 = 0 are 4. Which of the following are the two roots of the
(NDA 2011 I) equation ( x 2 + 2)2 + 8x 2 = 6x ( x 2 + 2)? (NDA 2010 II)
(a) complex (b) pure imaginary (a) 1 ± i (b) 2 ± i
(c) irrational (d) rational
(c) 1 ± 2 (d) 2 ± i 2

by hitesh kumar, (hitish.nitc@gmail.com)

 Quadratic Equations and Inequalities 49

5. If α and β are the roots of the equation x 2 + x + 1 = 0, 14. The equation x − 2( x − 1)−1 = 1 − 2( x − 1)−1 has
then which of the following are the roots of the (a) no roots (NDA 2009 I)
equation x 2 − x + 1 = 0 ? (b) one root
(a) α7 and β13 (b) α13 and β7 (c) two equal roots
(d) infinite roots
(c) α and β
20 20
(d) None of these
15. If a, b and c are real numbers, then the roots of the
6. If α and β be the roots of the equation equation
2x 2 + 2 ( a + b) x + a 2 + b2 = 0, then the equation, ( x − a )( x − b) + ( x − b)( x − c) + ( x − c)( x − a ) = 0 are
whose roots are (α + β )2 and (α − β )2, is always (NDA 2009 II)
(a) x 2 − 2abx − ( a 2 − b2 )2 = 0 (a) real (b) imaginary
(b) x 2 − 4abx − ( a 2 − b2 )2 = 0 (c) positive (d) negative
2
(c) x 2 − 4abx + ( a 2 − b2 )2 = 0 16. The solution set of the equation x log x (1 − x ) = 9 is
(d) None of the above (a) { −2, 4} (b) { 4}
7. If x 2 − 3x + 2 be a factor of x 4 − px 2 + q, then ( p, q ) is (c) { 0, − 2, 4} (d) None of these
2x 1
equal to 17. If 2 > , then
(a) ( 3, 4) (b) ( 4, 5) (c) ( 4, 3) (d) ( 5, 4) 2x + 5x + 2 x + 1
8. If α and β be the roots of x 2 − px + q = 0 and α ′ , β ′ be (a) −2 > x > − 1 (b) −2 ≥ x ≥ − 1
(c) −2 < x < − 1 (d) −2 < x ≤ − 1
the roots of x 2 − p′ x + q ′ = 0, then the value of
1 1 1
(α − α ′ )2 + (β − α ′ )2 + (α − β ′ )2 + (β − β ′ )2 is 18. If the roots of the equation + = are
x+ p x+q r
(a) 2 { p2 − 2q + p′ 2 − 2q ′ − pp′ }
equal in magnitude but opposite in sign, then the
(b) 2 { p2 − 2q + p′ 2 − 2q ′ − qq ′ } product of the roots will be
(c) 2 { p2 − 2q − p′ 2 − 2q ′ − pp′ } p2 + q 2 ( p2 + q 2 )
(a) (b) −
(d) 2{ p2 − 2q − p′ 2 − 2q ′ − qq ′ } 2 2
9. If one root of the quadratic equation ax 2 + bx + c = 0 p2 − q 2 ( p2 − q 2 )
(c) (d) −
is equal to the nth power of the other root, then the 2 2
1 1 x+2
n n +1 n +1 19. If x is real, the expression takes all
value of ( ac ) + ( a c)
n
is equal to 2x 2 + 3x + 6
1 1
values in the interval
(a) b (b) −b (c) b n + 1 (d) − b n + 1
 1 1  1 1
(a)  ,  (b) − ,
10. If the difference between the roots of ax 2 + bx + c = 0  13 3  13 3
is 1, then which one of the following is correct?  1 1
(c)  − ,  (d) None of these
(NDA 2012 I)  3 13
(a) b2 = a ( a + 4c) (b) a 2 = b ( b + 4c)
20. If a < b < c < d, then the roots of the equation
(c) a 2 = c( a + 4c) (d) b2 = a ( b + 4c)
( x − a ) ( x − c) + 2 ( x − b) ( x − d ) = 0 are
1 (a) real and distinct (b) real and equal
11. If is one of the roots of ax 2 + bx + c = 0, (c) imaginary (d) None of these
2 − −2
where, a, b and c are real, then what are the values of 21. If x is an integer and satisfies 9 < 4x − 1 ≤ 19, then x is
a, b and c, respectively? (NDA 2010 I) an element of which one of the following sets?
(a) { 3, 4} (b) { 2, 3, 4} (NDA 2008 II)
(a) 6, − 4, 1 (b) 4, 6, − 1 (c) 3, − 2, 1 (d) 6, 4,1
(c) { 3, 4, 5} (d) { 2, 3, 4, 5}
12. If α and β are the roots of the quadratic equation
x 2 − x + 1 = 0, then which one of the following is 22. If x is real and x 2 − 3x + 2 > 0, x 2 − 3x − 4 ≤ 0, then
correct? (NDA 2010 I) which one of the following is correct? (NDA 2008 I)
(a) (α − β ) is real
4 4
(b) 2(α + β ) = (αβ )5
5 5 (a) −1 ≤ x ≤ 4
(b) 2 ≤ x ≤ 4
(c) (α − β ) = 0
6 6
(d) (α 8 + β 8 ) = (αβ )8
(c) −1 < x ≤ 1
13. If p and q are positive integers, then which one of the (d) −1 ≤ x < 1 or 2 < x ≤ 4
following equations has p − q as one of its roots? 23. The numerical value of the perimeter of a square
(a) x 2 − 2 px − ( p2 − q ) = 0 (NDA 2009 I) exceeds that of its area by 4. What is the side of the
square? (NDA 2008 I)
(b) x − 2 px + ( p − q ) = 0
2 2

(c) x 2 + 2 px − ( p2 − q ) = 0 (a) 1 unit (b) 2 units

(c) 3 units (d) 4 units
(d) x 2 + 2 px + ( p2 − q ) = 0

by hitesh kumar, (hitish.nitc@gmail.com)

 50 NDA/NA Mathematics

24. If the equation x 2 + kx + 1 = 0 has the roots α and β, 33. If the roots of the equations x 2 − ( a − 1)x + ( a + b) = 0
then what is the value of (α + β ) × (α −1 + β −1 )? and ax 2 − 2x + b = 0 are identical, then what are the
(NDA 2008 I) values of a and b? (NDA 2007 I)

(a) k 2
(b)
1
(c) 2k 2
(d)
1 (a) a = 2, b = 4 (b) a = 2, b = − 4
2
( 2k2 ) 1 1
k (c) a = 1, b = (d) a = − 1, b = −
2 2
25. If α and β are the roots of the equation x 2 + x + 1 = 0,
34. If − x 2 + 3x + 4 > 0, then which one of the following is
then what is the equation whose roots are α19 and β7 ?
(NDA 2007 II) correct? (NDA 2007 I)
(a) x ∈ ( −1, 4) (b) x ∈ [−1, 4]
(a) x 2 − x − 1 = 0 (b) x 2 − x + 1 = 0 (c) x ∈ ( −∞ , − 1) ∪ ( 4, ∞ ) (d) x ∈ ( −∞ , 1] ∪ [4, ∞ )
(c) x 2 + x − 1 = 0 (d) x 2 + x + 1 = 0
35. If the two quadratic equations x 2 − bx + c = 0 and
26. The number of real solution of the equation x 2 − b′ x + c′ = 0 have a common root, what is the
|x|2 − 3|x| + 2 = 0 are
value of the common root ?
(a) 1 (b) 2 b − b′ c − c′
(c) 3 (d) 4 (a) (b)
c − c′ b − b′
27. Let α and γ be the roots of Ax 2 − 4x + 1 = 0 and β , δ be b − b′ c − c′
(c) (d)
the roots of Bx 2 − 6x + 1 = 0. If α, β, γ and δ are in HP, c′ − c b′ − b
then what are the values of A and B, respectively? 36. f ( x ) = x 2 + 2ax + 1 and α is a root of the equation
(NDA 2009 I)
f ( x ) = 0, where a is real.
(a) 3, 8 (b) –3, –8
(c) 3, –8 (d) –3, 8 Which one of the following is correct ?
(a) f(α ) = 0 and f(1 / α ) ≠ 0
28. If x is real, then the maximum and minimum values
(b) f(α ) = 0 and f(1 / α ) = 0
x 2 − 3x + 4 (c) f(α ) ≠ 0 and f(1 / α ) = 0
of the expression 2 will be
x + 3x + 4 (d) f(α ) ≠ 0 and f(1 / α ) ≠ 0
1
(a) 2, 1 (b) 5, 37. The roots of the quadratic x 2 + 4a = 8x − 12a 2 are real
5 and unequal. Which one of the following is correct?
1
(c) 7, (d) None of these (a) 4 / 3 < a < 2 (b) − 4 / 3 < a < − 1
7
(c) −4 / 3 < a < 2 (d) −4 / 3 < a < 1
29. If l , m and n are real and l ≠ m, then the roots of the
equation ( l − m ) x 2 − 5 ( l + m ) x − 2 ( l − m ) = 0 are 38. The sum of the two roots of a quadratic equation is
3
λ and the sum of their squares is 5 µ 2 . Which one of
(a) complex (b) real and distinct
(c) real and equal (d) None of these the following is that equation ?
3 5
30. The number of rows in a lecture hall equals to the (a) x 2 − 3 λ x + ( λ 2 − λ 2 ) = 0
number of seats in a row. If the number of rows is 3
(b) x 2 − 3 λ x + ( λ 2 + 5 µ 2 ) = 0
doubled and the number of seats in every row is 3
reduced by 10, the number of seats is increased by (c) 2x 2 − 2 3 λ x + ( λ 2 − 5 µ 2 ) = 0
3
300. If x denotes the number of rows in the lecture (d) 2x 2 − 2 3 λ x + ( λ 2 + 5 µ 2 ) = 0
hall, then what is the value of x? (NDA 2007 II)
(a) 10 (b) 15 (c) 20 (d) 30 39. The roots of the equation x 2 + px + q = 0 both real
and greater than 1. If r = p + q + 1, then which one of
31. If α and β are the roots of ax 2 + 2bx + c = 0 and α + δ , the following is correct?
β + δ are the roots of Ax 2 + 2Bx + C = 0, then what is (a) r must be greater than 0
( b2 − ac)/( B2 − AC ) equal to (NDA 2007 I) (b) r must be less than 0
2 2 (c) r must be equal to 0
 b  a (d) r may be equal to 0
(a)   (b)  
 B  A
40. Which one of the following is correct?
( a 2b2 ) ( ab)  7   7 
(c) (d) The equation x − 
( A2B2 ) ( AB)  = 3−  
 x − 3  x − 3
32. How many real values of x satisfy the equation (a) has only one integral root
| x| +| x − 1| = 1 ? (NDA 2007 I) (b) has no roots
(a) 1 (b) 2 (c) has two equal integral roots
(c) Infinite (d) No value of x (d) has two unequal integral roots

by hitesh kumar, (hitish.nitc@gmail.com)

 Quadratic Equations and Inequalities 51

41. One of the roots of a quadratic equation with real 46. Match List I (equations ) with List II (their roots )
1 and select the correct answer using the codes given
coefficients is . Which of the following
(2 − 3 i) below the lists.
implications is/are true? List I List II
1 (Equations) (Their roots)
I. The second root of the equation will be .
(3 − 2 i)
A. 6 x2 + 7 x − 10 = 0 1. 6, 40
II. The equation has no real root. 13
III. The equation is 13x 2 − 4x + 1 = 0. B. x + 1 = 5 2. − 5 + i 11 −5 − i 11
,
Which of the above is/are correct ? x 2 6 6
(a) Only I and II C. 3 x2 + 5 x + 3 = 0 3. −2, 5
(b) Only III 6
(c) Only II and III
4. 2, 1
(d) I, II and III
2
42. Given, 4a − 2 b + c = 0, where a , b, c ∈ R, which of the
following statements is/are not true in general? Codes
I. ( x + 2) will always be a factor of the expression A B C A B C
ax 2 + bx + c. (a) 3 4 2 (b) 3 1 2
II. ( x − 2) will always be a factor of the expression (c) 1 4 2 (d) 1 3 2
ax 2 + bx + c.
47. Let a , b ∈ {1, 2, 3}. What is the number of equations of
III. There will be a factor of the expression the form ax 2 + bx + 1 = 0 having real roots?
ax 2 + bx + c different from ( x + 2).
(a) 1 (b) 2
Select the correct answer using the code given below (c) 5 (d) 3
(a) Only I and II (b) I, II and III 48. If 3 <| x|< 6, then x belongs to
(c) Only II (d) Only I (a) ( −6, − 3) ∪ ( 3, 6) (b) ( −6, 6)
43. Which one of the following sets has all elements as (c) ( −3, − 3) ∪ ( 3, 6) (d) None of these
odd positive integers? 1 1
49. If < , then
(a) S = { x ∈ R x3 − 8x 2 + 19x − 12 = 0} a b
(a) | a|>| b| (b) a < b
(b) S = { x ∈ R x3 − 9x 2 + 23x − 15 = 0} (c) a > b (d) None of these

50. x + < 3, then x belongs to

(c) S = { x ∈ R x3 − 7x 2 + 14x − 8 = 0} 2
 x
(d) S = { x ∈ R x3 − 12x 2 + 44x − 48 = 0} (a) ( −2, − 1) ∪ (1, 2)
(b) ( −∞ , − 2) ∪ ( −1, 1) ∪ ( 2, ∞ )
44. What is the number of real solutions of (c) ( −2, 2)
|x 2 − x − 6| = x + 2 ? (d) ( −3, 3)
(a) 4 (b) 3 51. x 2 − 3|x| + 2 < 0, then x belongs to
(c) 2 (d) 1
(a) (1, 2)
45. Consider the following statements (b) ( −2, − 1)
I. If the quadratic equation is ax 2 + bx + c = 0 such (c) ( −2, − 1) ∪ (1, 2)
that a + b + c = 0, then roots of the equation (d) ( −3, 5)
c
ax 2 + bx + c = 0 will be 1, . 52. The set of values of x satisfying the inequalities
a ( x − 1) ( x − 2) < 0 and ( 3x − 7) ( 2x − 3) > 0 is
II. If ax 2 + bx + c = 0 is a quadratic equation such  7
that a − b + c = 0, then roots of the equation will (a) (1, 2) (b)  2 , 
 3
c
be −1, .  7  3
a (c) 1,  (d) 1, 
 3  2
Which of the statements given above is/are correct?
(a) Only I 53. log2 x > 4, then x belongs to
(b) Only II (a) x > 4 (b) x > 16
(c) Both I and II (c) x > 8 (d) None of these
(d) Neither I nor II

by hitesh kumar, (hitish.nitc@gmail.com)

 52 NDA/NA Mathematics

Directions (Q. Nos. 54-58) Each of these Directions (Q. Nos. 59-61) Consider the
questions contain two statements, one is Assertion (A) quadratic equation 2x 2 − 8x + 3 = 0, whose roots are α
and other is Reason (R). Each of these questions also has and β, then
four alternative choices, only one of which is the correct 59. The sum and product of the roots are, respectively is
answer. You have to select one of the codes (a), (b), (c) and 3
(d) given below. (a) 3, 4 (b) 4,
2
Codes 3
(a) Both A and R are individually true and R is the (c) 5, (d) 4, 7
2
correct explanation of A. 60. The value of α3 + β3 is
(b) Both A and R are individually true and R is not
the correct explanation of A. (a) 46 (b) 40
(c) A is true but R is false. (c) 30 (d) − 18
(d) A is false but R is true. 1 1
61. The quadratic equation, if roots are and , is
54. Assertion (A) One root of the equation α β
7x 2 + 5x + 7 = 0 is reciprocal of the other. (a) 3x 2 − 8x + 2 = 0 (b) x 2 − 4x + 3 = 0
Reason (R) For the quadratic equation (c) 3x 2 − 4x + 1 = 0 (d) None of these
ax 2 + bx + c = 0 one root will be reciprocal of the
other, if a = c. Directions (Q. Nos. 62-64) Consider the
55. Assertion (A) The common root of the quadratic equation ax 2 + bx + c = 0, then condition that
equation 2x 2 − 3x − 5 = 0 and x 2 − 3x − 4 = 0 is −1. 62. One roots is the reciprocal of the other roots is
Reason (R) If the common root of the quadratic c
(a) a = c (b) a = −
equation a1x 2 + b1x + c1 = 0 and a2x 2 + b2x + c2 = 0 is 2
α, then (c) 2b = a (d) b = a
c a − c2a1 63. One roots is n times the other root is
α= 1 2 . (a) ac ( n + 1)2 = b2n (b) ab2 ( n + 1)2
a1b2 − a2b1
(c) ac ( n + 2) = b
2 2
(d) 4a 2 = b2
56. Assertion (A) The real quadratic equation whose 64. One root is the square of the other root
one root is 2 − 3 is x 2 − 4x + 1 = 0. (a) ca 2 + c2a − 3abc = − b3 (b) a + b + c = 0
Reason (R) If an equation has a root 2 − 3, then (c) a3 + b3 + c3 = 0 (d) None of these
another root will be 2 + 3.
57. Assertion (A) For b = − 5, x + 3 is a factor of Directions (Q. Nos. 65-66) The equation formed
x3 + 2x 2 + bx − 6 . by multiplying each root of ax 2 + bx + c = 0 by 2 is
Reason (R) If f ( x ) is a polynomial and f ( a ) = 0, x 2 + 36x + 24 = 0. (NDA 2012 I)
then x − a is a factor of f ( x ). 65. What is the value of b : c?
58. Assertion (A) The equation x 2 + 2ax − b2 = 0 can (a) 3 : 1 (b) 1 : 2 (c) 1 : 3 (d) 3 : 2
have repeated roots, where a and b are real numbers.
66. Which one of the following is correct?
Reason (R) The equation Ax 2 + Bx + C = 0 will (a) bc = a 2 (b) bc = 36a 2
have repeated roots when the discriminant becomes (c) bc = 72a 2
(d) bc = 108a 2
zero.

Answers
Level I
1. (a) 2. (d) 3. (b) 4. (c) 5. (a) 6. (c) 7. (d) 8. (c) 9. (a) 10. (d)
11. (a) 12. (a) 13. (b) 14. (d) 15. (b) 16. (d) 17. (a) 18. (a) 19. (d) 20. (a)
21. (a) 22. (b) 23. (a) 24. (a) 25. (c) 26. (b) 27. (b) 28. (c) 29. (c) 30. (b)
31. (d) 32. (a) 33. (b) 34. (c) 35. (a) 36. (d) 37. (d) 38. (c) 39. (d) 40. (c)
41. (a) 42. (a) 43. (a) 44. (c) 45. (b) 46. (b) 47. (a) 48. (d) 49. (b) 50. (d)
51. (d) 52. (a) 53. (a)
Level II
1. (c) 2. (d) 3. (b) 4. (a) 5. (d) 6. (b) 7. (d) 8. (a) 9. (b) 10. (a)
11. (a) 12. (c) 13. (b) 14. (a) 15. (a) 16. (b) 17. (c) 18. (b) 19. (b) 20. (a)
21. (c) 22. (d) 23. (b) 24. (a) 25. (d) 26. (d) 27. (a) 28. (c) 29. (b) 30. (d)
31. (b) 32. (b) 33. (b) 34. (a) 35. (b) 36. (b) 37. (d) 38. (c) 39. (a) 40. (b)
41. (c) 42. (c) 43. (b) 44. (b) 45. (c) 46. (a) 47. (d) 48. (a) 49. (d) 50. (a)
51. (c) 52. (d) 53. (b) 54. (a) 55. (a) 56. (a) 57. (a) 58. (a) 59. (b) 60. (a)
61. (a) 62. (a) 63. (a) 64. (a) 65. (a) 66. (d)

by hitesh kumar, (hitish.nitc@gmail.com)