1-A (OBJECTIVE TYPE QUESTIONS) Single choice 1.
Number of values of ' p ' for which the equation (p2 3p + 2) x 2 (p2 5p + 4) x + p p2 = 0 possess more than two roots, is: (A) 0 (B ) 1 (C) 2 (D) none
2.
The roots of the equation (b – c) x2 + (c – a) x + (a – b) = 0 are (A)
3.
c a ,1 bc
(B )
a b ,1 bc
(C )
bc ,1 ab
If , are the roots roots of quadratic quadratic equation x2 + p x + q = 0 and then ( ) . ( ) is equal equal to : ( A) q + r (B) q r (C) (q (q + r)
(D )
,
ca ,1 ab
are the roots roots of x2 + p x r = 0, (D )
(p + q + r)
4.
Two real numbers & are such that + = 3 & = 4, then & are the roots of the quadratic equation: (A) 4x 2 12x 7 = 0 (B) 4x2 12x + 7 = 0 (C) 4x 2 12 1 2x + 25 = 0 (D) none o f thes e
5.
If a, b, c are integers and b 2 = 4(ac + 5d 2), d (A) Ir-rational (C) Com plex conjugate
N, then roots of the equation ax2 + bx + c = 0 are (B) Rational & dif ferent (D) Rational & equal
6.
Let a, b and c be real numbers such that 4a + 2b + c = 0 and ab > 0. Then the equation ax 2 + bx + c = 0 has ( A) r eal r oo t s (B ) i m a gi na ry r oo t s (C ) e x ac t l y o ne r oo t (D) none o f thes e
7.
Consider the equation x 2 + 2x – n = 0, where n N and n [5, 100]. Total Total number of diff erent values of 'n' so that the given equation has integral roots, is (A) 4 (B) 6 (C) 8 (D) 3
8.
If the equations k (6x 2 + 3) + rx + 2x 2 – 1 = 0 and 6k (2x 2 + 1) + px + 4x 2 – 2 = 0 have both roots common, then the value of (2r – p) is (A) 0 (B ) 1 / 2 (C) 1 (D) none o f thes e
9.
If a, b, p, q are non zero real numbers, then two equations 2a 2 x 2 2 ab x + b 2 = 0 and p2 x 2 + 2 pq x + q2 = 0 have: ( A) n o c o m m o n ro ot (B) one com m on root if 2 a2 + b 2 = p 2 + q2 (C) t wo com m on on roots if 3 pq = 2 ab (D) two com mo mon roots if 3 qb = 2 ap
10.
Which of the following graph represents the expression f(x) = a x 2 + b x + c (a a > 0, b < 0 & c < 0 ?
(A)
(B)
(C)
(D)
20
0)
when
11.
The expression y = ax 2 + bx + c has always the same sign as of 'a' if : (A) 4ac < b 2 (B) 4ac > b 2 (C) ac < b 2 (D) ac < b 2
12.
The entire graph of the expression y = x2 + kx – x + 9 is strictly above the x-axis if and only if (A) k < 7 (B) –5 < k < 7 (C) k > – 5 (D) none
13.
If a, b R, a 0 and the quadratic equation ax2 bx + 1 = 0 has im aginary roots then a + b + 1 is: (A) positive (B) negative (C) zero (D) depends on the sign of b
14.
If a and b are the non-zero distinct roots of x 2 + ax + b = 0, then t he least value of x2 + ax + b is (A)
3 2
(B)
9 4
(C) –
9 4
15.
If y = – 2x2 – 6x + 9, then (A) maximum value of y is –11 and it occurs at x = 2 (B) minimum value of y is –11 and it occurs at x = 2 (C) maximum value of y is 13.5 and it occurs at x = –1.5 (D) minimum value of y is 13.5 and it occurs at x = –1.5
16.
Let f(x) = x 2 + 4x + 1. Then (A) f(x) > 0 for all x (C) f(x) 1 when x – 4
17.
19.
(B) f(x) > 1 when x 0 (D) f(x) = f(– x) for all x
The complete set of values of x which satisfy the inequations : 5x + 2 < 3x + 8 and (A) (–
18.
(D) 1
, 1)
(C) (– , 3)
(B) (2, 3)
The number of the integer solutions of x 2 + 9 < (x + 3) 2 < 8x + 25 is : (A) 1 (B) 2 (C) 3
The complete solution set of the inequality (A) ( , 5) (1, 2) (6, ) (C) ( , 5] [1, 2] [6, )
{0} {0}
(D) (– , 1)
x2 < 4 is x 1
(2, 3)
(D) none
x4
3 x 3 2x 2 0 is: x 2 x 30 (B) ( , 5) [1, 2] (6, ) {0} (D) none of these
20.
If the inequality (m 2)x 2 + 8x + m + 4 > 0 is satisfied for all x R, then the least integral m is: (A) 4 (B) 5 (C) 6 (D) none
21.
The complete set of real ' x ' satisfying (A) [0, 2]
22.
If log1/3
3x 1 x2
(B) [ 1, 3]
1 1 1 is: (C) [ 1, 1]
(D) [1, 3]
is less than unity, then x must lie in the interval :
(A) ( 2) (5/8, ) (C) ( 2) (1/3, 5/8) 23.
x
(B) ( 2, 5/8) (D) ( 2, 1/3)
Solution set of the inequality 2 log 2 (x 2 + 3x) 0 is : (A) [ 4, 1] (B) [ 4, 3) (0, 1] (C) ( 3) (1, ) (D) ( 4) [1, ) 21
24.
25.
26.
The set of all the solutions of the inequality log1 – x (x – 2) – 1 is (A) (– , 0) (B) (2, ) (C) (– , 1)
(D)
If log0.3 (x 1) < log 0.09 (x 1), then x lies in the interval (A) (2, ) (B) (1, 2) (C) ( 2, 1)
(D) none of these
If log0.5 log 5 (x 2 – 4) > log 0.5 1, then ‘x’ lies in the interval (A) (– 3, –
5)
(
5 , 3)
(B) (– 3, –
(C) ( 5 , 3 5 )
27.
If x is real and k =
(A)
1 3
k 3
(D)
(
5,3 5)
2
x 1 , x2 x 1 x
5)
then:
(B) k 5
(C) k 0
(D) none
28.
The real values of 'a' for which the quadratic equation 2x2 (a3 + 8a 1) x + a 2 4a = 0 possesses roots of opposite sign is given by: (A) a > 5 (B) 0 < a < 4 (C) a > 0 (D) a > 7
29.
If , are the roots of the quadratic equation x2 2p(x 4) 15 = 0, then the set of values of p for which one root is less than 1 & the other root is greater than 2 is: (A) (7/3, ) (B) ( , 7/3) (C) x R (D) none
30.
If , be the roots of 4x 2 – 16x + = 0, where R, such that 1 < < 2 and 2 < < 3, then the number of integral solutions of is (A) 5 (B) 6 (C) 2 (D) 3
31.
If two roots of the equation x3 px 2 + qx r = 0 are equal in magnitude but opposite in sign, then: (A) pr = q (B) qr = p (C) pq = r (D) none
32.
If
, & are the roots of the equation x3 x 1 = 0 then, (B) 1
(A) zero 33.
1 1
+
1 1
(C) 7
+
1 1
has the value equal to:
(D) 1
Let , , be the roots of (x – a) (x – b) (x – c) = d, d 0 then the roots of the equation (x – ) (x – ) (x – ) + d = 0 are : (A) a + 1, b + 1, c + 1 (B) a, b, c
(C) a – 1, b – 1, c – 1
(D)
a b c , , b c a
Multiple choice 34.
For the equation |x| 2 + |x| – 6 = 0, the correct statement (s) is (are) : (A) sum of roots is 0 (B) product of roots is – 4 (C) there are 4 roots (D) there are only 2 roots
35.
If
, are the roots of ax 2 + bx + c = 0, and + h, + h are the roots of px 2 + qx + r = 0, then
(A)
a p
=
b q
=
c r
(B) h =
1 2
b q a p
(C) h =
1 2
b q a p
(D)
b2
4ac a
2
22
=
q2
4pr p2
36.
If a, b are non-zero real numbers and , the roots of x2 + ax + b = 0, then 2, 2 are the roots of x 2 – (2b – a2) x + a2 = 0 (A) (B)
(C) (D) 37.
1 1 , are the roots of bx 2 + ax + 1 = 0
2 2 , are the roots of bx + (2b – a ) x + b = 0 ( – 1), ( – 1) are the roots of the equation x2 + x (a + 2) +
The graph of the quadratic polynomial y = ax 2 + bx + c is as shown in the figure. Then :
(A) b2 4ac > 0 38.
(B) b < 0
(C) a > 0
(B)
(C)
(D)
The adjoining figure shows the graph of y = ax 2 + bx + c. Then
(B) b2 < 4ac (D) a and b are of opposite sign
(A) a < 0 (C) c > 0 40.
(D) c < 0
For which of the f ollowing graphs of the quadratic expression y = a x 2 + b x + c, the product a b c is negative.
(A)
39.
1+a+b=0
x 2 + x + 1 is a factor of a x 3 + b x 2 + c x + d = 0, then the real root of above equation is (a, b, c, d R) (A) d/a (B) d/a (C) (b – a)/a (D) (a – b)/a
1-B (SUBJECTIVE TYPE QUESTIONS)
1.
For what value of a, the equation (a2 – a – 2)x2 + (a2 – 4)x + (a 2 – 3a + 2) = 0, will have more than two solutions ? Does there exist a real value of x for which the above equation will be an identity in a ?
2.
If (i)
3.
and are the roots of the 2 + 2
equation 2x2 + 3x + 4 = 0, then find the values of (ii)
+
If and are the roots of the equation ax2 + bx + c = 0, then find the equation whose roots are given by (i)
1
1
+ , +
(ii)
2 + 2, 2 + 2
23
4.
The length of a rectangle is 2 metre more than its width. If the length is increased by 6 metre and the width is decreased by 2 metre, the area becomes 119 square metre. Find the dimensions of the original rectangle.
5.
If
6.
In copying a quadratic equation of the form x2 + px + q = 0, the coefficient of x was wrongly written as – 10 in place of – 11 and the roots were found to be 4 and 6. Find the roots of t he correct equation.
7.
If 2 + i 3 is a root of the equation x2 + px + q = 0, where p, q R, then f ind the ordered pair (p, q)
8.
If one root of equation ( – m) x2 + x + 1 = 0 be double of the other and if be real, show that m
9.
Solve the equation x4 + 4x 3 + 5x2 + 2x – 2 = 0, one root being – 1 +
10.
If the roots of the equation x2 – 2cx + ab = 0 are real and unequal, then prove that the roots of x 2 – 2(a + b) x + a2 + b2 + 2c 2 = 0 will be imaginary.
11.
For what values of k the expression kx2 + (k + 1)x + 2 will be a perfect square of linear factor.
12.
Show that the roots of the equation (a 2 – bc) x 2 + 2(b 2 – ac) x + c 2 – ab = 0 are equal if either b = 0 or a 3 + b 3 + c 3 = 3abc
13.
Prove that the roots of the equation
but 2 = 5 – 3, 2 = 5 – 3, then
1 x a
find the equation whose roots are
+
1 x b
+
1 x c
and .
9 8
.
1 .
= 0 are always real and cannot have roots
if a = b = c.
14.
If the roots of the equation
1 ( x , p)
+
1 ( x , q)
=
1 are equal in magnitude but opposite in sign, show that r
p + q = 2 r & that the product of t he roots is equal to ( 1/2) (p² + q²).
3, 5
.
1
15.
Find the value of the expression 2x 3 + 2x 2 – 7x + 72 when x =
16.
Find the value of 'a' so that x2 – 11 x + a = 0 an d x2 – 14x + 2a = 0 have a common root.
17.
If ax2 + bx + c = 0 and bx 2 + cx + a = 0 have a common root and a, b, c are non-zero real numbers, then find the value of
a3
b3 c 3 abc
2
.
18.
If x 2 + px + q = 0 and x 2 + qx + p = 0 have a common root, show that 1 + p + q = 0 ; show that their other roots are the roots of the equation x2 + x + pq = 0.
19.
If one of the roots of the equation ax 2 + b x + c = 0 be reciprocal of one of the roots of a1 x 2 + b 1 x + c 1 = 0, then prove that (a a1 c c 1) 2 = (b c 1 a b1) (b 1c a1b).
24
20.
21.
22.
Draw the graph of the following expressions : (i) y = x 2 + 4x + 3 (ii) y = 9x2 + 6x + 1
(i)
f(x) = –x 2 + 2x + 3
(ii)
f(x) = x2 – 2x + 3
(iii)
f(x) = x2 – 4x + 6
x R x [0, 2] x [0, 1]
If x be real, then find the range of the following rational expressions :
x 1 x2 1
x2
y=
Solve the following inequalities : (i) x 2 – 7x + 10 > 0 2x (iii)
2x
2
5x 2
2x 9 2x 9
y=
(ii)
x2 – 4x + 3 < 0
(iv)
x2 2x 3 > . x2 4x 1
1 x 1.
x2
(ii)
x
2
24.
Find all real values of x which satisfy x2 – 3x + 2 > 0 and x2 – 3x – 4 0.
25.
Solve the following inequalities :
26.
27.
28.
y = – 2x2 + x – 1
Find the range of following quadratic expressions :
(i)
23.
(iii)
(i)
(x – 1) (x + 1) (x – 4) 0
(ii)
(iii)
(x 2 – x – 1) (x 2 – x – 7) < – 5
(iv)
2
3
x 4 ( x 1)2 ( x 2) ( x 3) 3 ( x 4)
0
( x 2) ( x 2
2x 1) 0 4 3x x 2
Solve the following inequalities : 3
(i)
1
(iv)
|x2 + 3x| + x2 – 2 0
x
> 2
(ii)
3x x
2
4
1
(iii)
(v)
|x + 3| > |2x – 1|
| x 3 | x > 1 x2
Solve the following inequalities :
4x 6 x
(i)
log1/5
0
(ii)
log2(4 x – 2.2 x + 17) > 5.
(iii)
log2x log x + 2
(iv)
log1/2 (x + 1) > log2 (2 x)
Solve the following inequalities :
(i)
log0.5 (x + 5) 2 > log1/2 (3x – 1) 2. (ii)
(iii)
log(
3 x2 1
)
2<
1 2
(iv)
log1/2 log3
x 1 x 1
0
log x² (2 + x) < 1
29.
If both roots of the equation x2 – 6ax + 2 – 2a + 9a 2 = 0 exceed 3, then show that a > 11/9.
30.
Find all the values of 'K' for which one root of the equation x² (K + 1) x + K² + K 8 = 0, exceeds 2 & the other root is smaller than 2. 25
31.
Find all the real values of 'a', so that the roots of the equation (a 2 – a + 2) x 2 + 2(a – 3) x + 9 (a 4 – 16) = 0 are of opposite sign.
32.
Find all the values of 'a', so that exactly one root of the equation x2 – 2ax + a2 – 1 = 0, lies between the numbers 2 and 4, and no root of the equation is either equal to 2 nor equal to 4.
33.
If and be two real roots of the equation x 3 + px2 + qx + r = 0 satisfying the relation + 1 = 0, then prove that r 2 + pr + q + 1 = 0
34.
If
35.
If
1 , , are the roots of the equation x3 + px 2 + qx + r = 0, then f ind the value of 1 . , and
are roots of 2x3 + x 2 – 7 = 0, then find the value of
1
.
2-A (OBJECTIVE TYPE QUESTIONS) Single choice 1.
If the quadratic equations 3x2 + ax + 1 = 0 & 2x2 + bx + 1 = 0 have a common root, then the value of the expression 5ab 2a2 3b2 is: (A) 0 (B) 1 (C) 1 (D) none
2.
If both roots of the quadratic equation (2 x) (x + 1) = p are distinct & positive, then p must lie in the interval: (A) (2, ) (B ) (2, 9/4) (C) (– , – 2) (D) (– , )
3.
The equation, x = (A ) no solution
4.
5.
2x 2 + 6x 9 has: (B) one solution
(C) two solutions
(D) infinite solutions
Let a > 0, b > 0 & c > 0. Then both the roots of the equation ax2 + bx + c = 0 (A) are real & negative (B) have negative real parts (C) are rational numbers (D) have positive real parts 2 The set of all solutions of the inequality (1/ 2) x 2 x < 1/4 contains the set
(A) (–
, 0)
(B) (– , 1)
(C) (1, )
(D) (3, )
6.
The value of 'a' for which the sum of the squares of the roots of the equation x2 (a 2) x a 1 = 0 assume the least value is: (A) 0 (B) 1 (C) 2 (D) 3
7.
The set of values of x satisfying simultaneously the inequalities
2x 3 31 > 0 is : (A) a unit set (C) an infinite set
( x . 8) (2 . x )
10 log0.3 7
(log2 5 . 1)
0 and
(B) an empty set (D) a set consisting of exactly two elements. 26
2x 8.
Consider y = (A) [ 1, 1]
9.
1 x2
, where x is real , then the range of expression y2 + y 2 is: (B) [0, 1]
(C)
9 / 4 , 0
The least value of expression x 2 + 2 xy + 2 y2 + 4 y + 7 is: (A) 1 (B) 1 (C) 3
(D)
9 / 4 , 1
(D) 7
10.
The values of k, for which the equation x2 + 2 (k 1) x + k + 5 = 0 possess atleast one positive root, a re: (A) [4, ) (B) ( , 1] [4, ) (C) [ 1, 4] (D) ( , 1]
11.
If b > a, then the equation (x a) (x b) + 1 = 0, has: (A ) both roots in (a, b) (B) both roots in ( , a) (C) both roots in (b, ) (D) one root in ( , a) & other in (b,
12.
If
)
6x 2
5x 3 4, then least and the highest values of 4 x2 are: x 2 2x 6
(A) 0 & 81
(B) 9 & 81
(C) 36 & 81
(D) none of these
13.
If , , , are the roots of the equation x4 Kx3 + Kx2 + Lx + M = 0, where K, L & M are real numbers, then the minimum value of 2 + 2 + 2 + 2 is: (A) 0 (B ) 1 (C) 1 (D) 2
14.
If the roots of the equation x3 + Px 2 + Qx 19 = 0 are each one more than the roots of the equaton x 3 Ax2 + Bx C = 0, where A, B, C, P & Q are constants, then the value of A + B + C is equal to : (A ) 18 (B) 19 (C) 20 (D) none
15.
The equations x3 + 5x2 + px + q = 0 an d x3 + 7x2 + px + r = 0 have two roots in common. If the third root of each equation is represented by x1 and x 2 respectively, then the ordered pair (x 1, x 2) is: (A) ( 5, 7) (B) (1, 1) (C) ( 1, 1) (D) (5, 7)
16.
If coefficients of the equation ax2 + bx + c = 0, a 0 are real and roots of the equation are non-real complex and a + c < b, then (A) 4a + c > 2b (B) 4a + c < 2b (C) 4a + c = 2b (D) none of these
17.
If (2 + – 2)x 2 + ( + 2) x < 1 for all x R, then belongs to the interval (A) (–2, 1)
18.
2, 2 5
(C)
2 , 1 5
(D) none of these
The set of possible values of for which x2 – ( 2 – 5 + 5)x + (2 2 – 3 – 4) = 0 has roots, whose sum and product are both less than 1, is (A)
19.
(B)
.
1,
5
2
(B) (1, 4)
(C) 1 ,
5
2
1, (D)
5
2
Let conditions C1 and C2 be defined as follows : C1 : b 2 – 4ac 0, C 2 : a, –b, c are of same s ign. The roots of ax 2 + bx + c = 0 are real and positive, if (A) both C1 and C2 are satisfied (B) only C2 is satisfied (C) only C1 is satisfied (D) none of these
27
20.
If the roots of the equation x2 + 2ax + b = 0 are real and distinct and they differ by at most 2m , then b lies in the interval (A) (a 2 – m2, a2) (B) [a2 – m2, a2) (C) (a 2, a2 + m2) (D) none of these
Multiple choice
21.
If
1 2
log 0.1 x 2,
then
(A) maximum value of x is
(C) minimum value of x is
22.
If
x 1) > 1, log2 ( x 2 1)
log2 ( 4x 2
(A)
, 2 3
1 10 1 10
(B) x lies between
1 and 100
(D) minimum value of x is
1 10
1 100
then x lies in the interval
(B) (1,
)
(C)
2 , 0 3
(D) none of these
23.
If the quadratic equations x2 + abx + c = 0 a nd x2 + acx + b = 0 have a com mon root, then the equation containing their other roots is/are: (A) x 2 + a (b + c) x a 2bc = 0 (B) x 2 a (b + c) x + a2bc = 0 (C) a (b + c) x 2 (b + c) x + abc = 0 (D) a (b + c) x 2 + (b + c) x abc = 0
24.
If the quadratic equations ax2 + bx + c = 0 (a, b, c R, a 0) and x 2 + 4x + 5 = 0 have a common root, then a, b, c must satisfy the relations: (A) a > b > c (B) a < b < c (C) a = k; b = 4k; c = 5k (k R, k 0) (D) b 2 4ac is negative.
25.
If , are the real and distinct roots of x2 + px + q = 0 and 4, 4 are the roots of x 2 – rx + s = 0, then the equation x 2 – 4qx + 2q2 – r = 0 has always (A) two real roots (B) two negative roots (C) two positive roots (D) one positive root and one negative root
2-B (SUBJECTIVE TYPE QUESTIONS) 1.
Solve the following equations : (i) 2 2x + 2x+2 – 32 = 0 (ii) x (x + 1) (x + 2) (x + 3) = 120.
2.
If one root of the equation ax2 + bx + c = 0 is equal to nth power of the other root, show that (ac n) 1/(n + 1) + (a nc) 1/(n + 1) + b = 0.
3.
If a, b are the roots of x 2 + px + 1 = 0 and c, d are the roots of x2 + qx + 1 = 0. Show that q2 p 2 = (a c) (b c) (a + d) (b + d).
4.
If a, b, c are non–zero, unequal rational numbers then prove that the roots of the equation (abc2)x 2 + 3a 2 cx + b2 cx – 6a 2 – ab + 2b 2 = 0 are rational
28
5.
If p, q, r, s
R and pr = 2 (q + s), then show that atleast one of the
equations
x 2 + px + q = 0, x2 + rx + s = 0 has real roots. 6.
If one root of the equation 4x 2 + 2x – 1 = 0 b e then prove that its second root is 43 – 3 .
7.
Find all values of the parameter ' a ' such that the roots , of the equation 2 x 2 + 6 x + a = 0 satisfy the inequality
8.
< 2.
Solve for real values of x (i)
2 (5 2 6 ) x 3
(5 2
2 6 ) x 3 = 10
x2 – 2a |x – a| – 3a 2 = 0, a 0
(ii)
x2
kx 1 x 1 < 2 is valid x R.
9.
If ' x ' is real, find values of ' k ' for which
10.
The equations x 2 ax + b = 0 & x 3 px 2 + qx = 0, where b
x
2
0, q 0 have one common root & the
second equation has two equal roots. Prove that 2 (q + b) = ap. 11.
Obtain real solutions of the simultaneous equations xy + 3 y² x + 4 y 7 = 0, 2 xy + y² 2 x 2 y + 1 = 0.
12.
If the equations x2 + a x + 12 = 0, x2 + b x + 15 = 0 & x2 + (a + b) x + 36 = 0 have a comm on positive root, then find a, b and the roots of the equations.
& are the , ( 6, 1).
two distinct roots of x² + 2 (K
3) x +
13.
If
9 = 0, then find the values of K such that
14.
Find all values of a for which atleast one of the roots of the equat ion x2 – (a – 3) x + a = 0 is greater than 2.
15.
If x1 is a root of ax 2 + bx + c = 0, x 2 is a root of
ax 2 + bx + c = 0 where 0 < x 1 <
x2, show that the
equation ax2 + 2 bx + 2c = 0 has a root x 3 satisfying 0 < x 1 < x3 < x2.
+ i ; , R, be a root of the equation x3 + qx + r = 0; q, r R. Find a real cubic equation, independent of and , whose one root is 2.
16.
Let
17.
Find the set of values of 'a' if (x2 + x) 2 + a (x2 + x) + 4 = 0 has (i)
all four real & distinct roots.
(ii)
O nl y t wo root s ar e r ea l and dis ti nc t.
(iii)
a ll fou r r oo ts are im ag in ar y.
(iv)
four real roots in which only two are equal.
29
3-A (MATCH THE COLUMN)
1.
Column –
Column –
(A)
If the roots of x2 – bx + c = 0 are two consecutive integers, then b2 – 4c =
(B)
(p)
1
If x2 + ax + b = 0 and x 2 + bx + a = 0 (a 0) have a common root, then a + b =
(q)
7
(C)
If are roots of x2 – x + 3 = 0 then
(r)
17
(D)
If are the roots of x3 – 7x2 + 16 x – 12 = 0 then 2 + 2 +
(s)
–1
4 = ?
=
2.
Column –
Column –
(A)
If + 4 are two roots of x2 – 8 x + k = 0, then possible value of k is
(p)
2
(B)
Number of real roots of equation x2 – 5|x| + 6 = 0
(q)
3
are 'n', then value of
n is 2
(C)
If 3 – i is a root of x2 + ax + b = 0 (a, b R), then b is
(r)
12
(D)
If both roots of x2 – 2 kx + k 2 + k – 5 = 0 are less than 5, then 'k' m ay be equal to
(s)
10
3-B (ASSERTION/REASONING)
3.
Statement - 1 : Maximum value of log1/3 (x2 – 4x + 5) is '0'. Statement - 2 : loga x 0 for x 1 and 0 < a < 1. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True
4.
Statement - 1 : The nearest point from x - axis on the curve f(x) = x2 – 6x + 11 is (3, 2) Statement - 2 : If a > 0 and D < 0, then ax2 + bx + c > 0 x R. (A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True
30
5.
Let , be the roots of f(x) = 3x2 – 4x + 5 = 0. Statement-1 : The equation whose roots are 2, 2 is given by 3x 2 + 8x – 20 = 0. Statement-2 : To obtain, from the equation f(x) = 0, having roots and , the equation having roots 2, 2 one needs to change x to
x in f(x) = 0. 2
(A) Statement-1 is True, Statement-2 is True; Statement-2 is a correct explanation for Statement-1. (B) Statement-1 is True, Statement-2 is True; Statement-2 is NOT a correct explanation for Statement-1 (C) Statement-1 is True, Statement-2 is False (D) Statement-1 is False, Statement-2 is True 3-C (COMPREHENSION) 6.
Comprehension In the given figure
Y
OBC is an isosceles right triangle
y = x2 + bx + c
C
in which AC is a median, then answer the following O
questions : 6.1
Roots of y = 0 are (A) {2, 1}
(B) {4, 2}
(C) {1, 1/2}
A
B
X
(D) {8, 4}
6.2
The equation whose roots are ( + ) & ( – ), where , ( > ) are roots obtained in previous question, is (A) x2 – 4x + 3 = 0 (B) x2 – 8x + 12 = 0 (C) 4x2 – 8x + 3 = 0 (D) x2 – 16x + 48 = 0
6.3
Minimum value of the quadratic expression correspoinding to the quadratic equation obtained in Q.No. 6.2 occurs at x = (A) 8 (B) 1 (C) 4 (D) 2
7.
Comprehension Consider the equation x 4 – x 2 + 9 = 0.
7.1
If the equation has four real and distinct solutions, then lies in the interval (A) (– , –6) (6, ) (B) (0, ) (C) (6, ) (D) (– , –6)
7.2
If the equation has no real solution, then lies in the interval (A) (– , 0) (B) (–, 6) (C) (6, )
(D) (0, )
If the equation has two real solutions, then set of values of is (A) (– , –6) (B) (–6, 6) (C) {6}
(D) none of these
7.3
3-D (TRUE/FALSE)
8.
The equation 2x2 + 3x + 1 = 0 has an irrational root.
9.
If and re roots of equation x2 + 2x + 4 = 0, then
10.
If a < b < c < d, then the roots of t he equation (x – a) (x – c) + 2 (x – b) (x – d) = 0 are real and distinct.
1
3
1
3
=
1 4
31
11.
If a, b, c, d R , then the roots of the quadratic equation (a4 + b4) x 2 + (4abcd)x + c 4 + d 4 = 0 can not be distinct, if real.
12.
If x 2 + 3x + 5 = 0 and ax2 + bx + c = 0 have a common root and a, b, c N, then the minimum value of (a + b + c) is 10.
3-E (FILL IN THE BLANKS)
13.
The value of the biquadratic expression x4 8 x 3 + 18 x 2 8 x + 2, when x = 2 +
14.
If the product of the roots of the equation x2 – 3kx + 2 e 2lnk – 1 = 0 is 7, then the roots are real for k = ...............
15.
If (x + 1)2 is greater than 5x 1 & less than 7x 3 then the integral value of x is equal to ........ ..
x 1 x
3
+
3 , is ...........
x 1 = 0 is ............ x
16.
The number of real roots of
17.
If x – y and y – 2x are two factors of the expression x3 – 3x 2y + xy 2 + y3, then = ......... and = .......
4-A (PREVIOUS JEE QUESTIONS) IIT-JEE - 2008 1.
Let a, b, c, p, q be real numbers. Suppose , are the roots of the equation x2 + 2px + q = 0 and ,
1
are
the roots of the equation ax2 + 2bx + c = 0, where 2 {–1, 0, 1} STATEMENT -1 : (p2 – q) (b2 – ac) 0 and STATEMENT-2 : b pa or c qa (A)
STATEMENT-1 is True, STATEMENT-2 is True ; STATEMENT-2 is a correct explanation for STATEMENT-1 STATEMENT-1 is True, STATEMENT-2 is True ; STATEMENT-2 is NOT a correct explanation for STATEMENT-1 STATEMENT-1 is True, STATEMENT-2 is False STATEMENT-1 is False, STATEMENT-2 is True
(B) (C) (D) IIT-JEE - 2007 2.
Let
, be the roots of the equation x2 – px + r = 0 and
2
, 2 be the roots of the equation x2 – qx + r = 0.
Then the value of r is (A)
(C)
2 (p – q) (2q – p) 9 2 9
(q – 2p) (2q – p)
(B)
(D)
2 (q – p) (2p – q) 9 2 9
(2p – q) (2q – p)
32
3.
Let f(x) =
x2 x
2
6x 5 5x 6
Column –
Column –
(A)
If – 1 < x < 1, then f(x) satisfies
(p)
0 < f(x) < 1
(B)
If 1 < x < 2, then f(x) satisfies
(q)
f(x) < 0
(C)
If 3 < x < 5, then f(x) satisfies
(r)
f(x) > 0
(D)
If x > 5, then f(x) satisfies
(s)
f(x) < 1
IIT-JEE - 2006 4.
Let a, b, c be the sides of a triangle (No two of them are equal) and x2 + 2(a + b + c)x + 3 (ab + bc + ca) = 0 are real, then (A) <
5.
4 3
(B) >
5 3
(C)
1 , 5 3 3
R. If the roots of the equation (D)
4 , 5 3 3
If roots of the equation x2 – 10ax – 11b = 0 are c and d and those of x 2 – 10cx – 11d = 0 are a and b, then find the value of a + b + c + d. (where a, b, c, d are all distinct numbers)
IIT-JEE - 2004 6.
If the quadratic expression x 2 + 2ax – 3a + 10 > 0 (A) a > 5
7.
(B) | a | < 5
x R, then
(C) – 5 < a < 2
(D) 2 < a < 3
If one root of the equation x2 + px + q = 0 is square of other, then the relation between p, q is : (A) p3 – q (3p – 1) + q2 = 0 (B) p3 + q (3p + 1) + q2 = 0 (C) p3 + q (3p – 1) + q2 = 0 (D) p3 – q (3p + 1) + q2 = 0
IIT-JEE - 2003 8.
If f(x) = x 2 + 2bx + 2c2 and g(x) = – x 2 – 2cx + b 2 are such that min f(x) > max g(x), then the relation between b and c, is (A) no relation (B) 0 < c < b/2 (C) |c| < 2 |b| (D) |c| > 2 |b|
9.
x 2 + (a b) x + (1 a b) = 0, a, b are real and unequal b R
R. Find the condition on ' a ' for which both roots of the equation
IIT-JEE - 2002 10.
The set of all real num bers x for which x2 – |x + 2| + x > 0 is (A) (– , – 2 ) (2, ) (B) (– , – 2 ) (C) (– , – 1) (1, ) (D) ( 2 , )
(
2 , )
IIT-JEE - 2001 11.
12.
The number of solution(s) of log 4(x – 1) = log 2(x – 3) is/are (A) 3 (B) 1 (C) 2
(D) 0
Let a, b, c be real numbers with a 0 and let , be the roots of the equation ax2 + bx + c = 0. Expres s the roots of a 3x 2 + abcx + c 3 = 0 in terms of ,
33
IIT-JEE - 2000 13. (a)
For the equation 3x2 + px + 3 = 0, p > 0 if one of the roots is square of the other, then p is equal to: (A) 1/3 (B) 1 (C) 3 (D) 2/3
(b)
If & ( < ) are the roots of the equation x 2 + bx + c = 0, where c < 0 < b, then (A) 0 < < (B) < 0 < < (C) < < 0 (D) < 0 < <
(c)
If b > a, then the equat ion (x a) (x b) 1 = 0, has: (A) both roots in [a, b] (B) both roots in ( a) (C) both roots in [b ) (D) one root in ( a) & other in (b,
14.
If
, are the roots of ax 2 + bx + c
Ax 2 +
= 0 (a 0) and
)
+ , + are the roots of,
Bx + C = 0 (A 0) for some constant , then prove that
b 2 4ac a2
=
B 2 4 AC A 2
.
IIT-JEE - 1999 15.
If the roots of the equation x2 2ax + a2 + a 3 = 0 are real & less than 3, then (A) a < 2 (B) 2 a 3 (C) 3 < a 4 (D) a > 4
IIT-JEE - 1997
22
2 20
16.
The sum of all the real roots of the equation x
17.
Let S be a square of unit area. Consid er any quadrilateral which has one vertex on each side of S. If a, b, c & d denote the lengths of the sides of the quadrilateral, prove that 2 a 2 + b2 + c2 + d2 4.
18.
Find the set of all solutions of the equation 2|y| – | 2 y–1 – 1| = 2 y–1 + 1
19.
The equation x 1 – x 1 = 4 x 1 has (A) no solution (B) one solution
x
(C) two solutions
is ______.
(D) more than two solutions
IIT-JEE - 1995 20.
Let a, b, c be real. If ax2 + bx + c = 0 has t wo real roots & 1+ c/a + b/a .
where < 1 & > 1, then show that
4-B (PREVIOUS AIEEE/DCE QUESTIONS) 21.
If the difference between the roots of the equation x2 + ax + 1 = 0 is less than values of a is (A) (–3, 3)
(B) (–3, )
(C) (3, )
5 , then the set of possible
(D) (–, -3)
22.
If the roots of the quadratic equation x2 + px + q = 0 are tan 30° and tan 15° respectively, then the value of 2 + q – p is : (A) 3 (B) 0 (C) 1 (D) 2
23.
All the values of m for which both roots of the equation x2 – 2mx + m2 – 1 = 0 are greater than – 2 but less than 4 lie in the interval : (A) m > 3 (B) – 1< m < 3 (C) 1 < m < 4 (D) – 2 < m < 0 34
24.
In a triangle PQR, R = (A) b = a + c.
2
P 2
. If tan
Q are the 2
and tan
(B) b = c.
roots of ax2 + bx + c = 0, a 0, then :
(C) c = a + b.
(D) a = b + c.
25.
The value of a for which the sum of the squares of the roots of the equation x2 – (a – 2)x – a – 1 = 0 assume the least value is : (A) 2. (B) 3. (C) 0 (D) 1.
26.
If the roots of the equation x2 – bx + c = 0 be two consecutive integers, then b2 – 4c equals : (A) 1 (B) 2 (C) 3 (D) – 2
27.
If both the roots of the quardratic equation x2 – 2kx + k2 + k – 5 = 0 are less than 5, then k lies i n the interval: (A) [4, 5] (B) (–, 4) (C) (6, ) (D) (5, 6]
28.
If the equation an x n + an – 1 xn–1 +...............+ a1x = 0, a1 0, n 2, has a positive root x = , then the equation nanxn – 1 + (n – 1)an – 1xn – 2 + .........+ a1 = 0 has a positive root, which is : (A) equal to (B) greater than or equal to (C) smaller than (D) greater than
29.
Let two numbers have arithemetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation : (A) x2 + 18x + 16 = 0 (B) x2 – 18x + 16 = 0 (C) x2 + 18x – 16 = 0 (D) x2 – 18x – 16 = 0
30.
If (1 – p) is a root of quadratic equation x2 + px + (1 – p) = 0, then is roots are : (A) 0,1 (B) –1,1 (C) 0, –1 (D) –1,2
31.
If one root of the equation x2 + px + 12 = 0 is 4, while the equation x2 + px + q = 0 has equal roots, then the value of ‘q’ is : (A) 49/4 (B) 12 (C) 3 (D) 4
32.
If 2a + 3b + 6c = 0, then at least one root of the equation ax2 + bx + c = 0 lies in the interval : (A) (0,1) (B) (1,2) (C) (2,3) (D) (1,3)
33.
If the sum of the roots of t he quadratic equation ax2 + bx + c = 0 is equal to the sum of the squares of their reciprocals, then
a c
,
b a
and
c b
are in :
(A) arithmetic progresion. (C) harmonic progression. 34.
35.
The number of the real solutions of the equation x2 – 3|x| + 2 = 0 is : (A) 2 (B) 4 (C) 1
2 3
(B) –
2 3
(C)
1 3
(D) –
1 . 3
Let a, b, c be real, if ax2 + bx + c = 0 has two real roots and , where < –2 and > 2, then : (A) 4 +
37.
(D) 3.
The value of ‘a’ for which one root of the quadratic equation (a2 – 5a + 3)x2 + (3a – 1)x + 2 = 0 is twice as large as the other, is : (A)
36.
(B) geometric progression. (D) arithmetico-geometirc progression.
2b a
+
c a
= 0
(B) 4 –
2b a
+
c a
= 0
(C) 4 +
2b a
–
c a
< 0
The condition for a2x4 + bx3 + cx2 + dx + f 2 may be perfect square is : (A) 2a2c = a3f (B) 4a2c – b2 = 8a3f (C) 4a2c = 8a3f
(D) 4 –
2b a
+
c a
< 0
(D) none of these 35
38.
If a 0 then roots of x2 – 2a |x – a| – 3a2 = 0 is : (A) a
(B) (– 1 +
6)a
(C) ( 6
1) a
(D) none of these
39.
The values of x and y besides y can satisfy the equation (x, y real numbers) x2 – xy + y2 – 4x – 4y + 16 = 0 (A) 2, 2 (B) 4, 4 (C) 3, 3 (D) none of these
40.
Let and be the roots of the equation x2 + x + 1 = 0, then the equation whose roots are 19, 7 is : (A) x2 + x + 1 = 0 (B) x2 + x – 1 = 0 (C) x2 – x + 1 = 0 (D) x2 – x – 1 = 0
41.
If two equations x2 + a 2 = 1 – 2ax and x2 + b 2 = 1 – 2bx have only one common root, then (A) a – b = 2 (B) a – b = 1 (C) a – b = – 1 (D) |a – b| = 1
42.
The equation x 2 – 2 2 kx + 2 × e 2 log k – 1 = 0 has the product of roots equal to 31, then, for what value of k it has real roots? (A) 3 (B) 2 (C) 1 (D) 4
43.
If (x2 – 3x + 2) is a factor of x 4 – px 2 + q = 0, then the values of p and q are (A) 5, –4 (B) 5, 4 (C) –5, 4 (D) –5, –4
44.
For what value of a and b the equation x 4 – 4x 3 + ax 2 + bx + 1 = 0 has four real roots ? (A) (– 6, –4) (B) (–6, 5) (C) (–6, 4) (D) (6, –4)
45.
If
, are roots of the equation ax 2 + bx + c
3abc b 3 (A) a 46.
47.
a3 b 3 (B) 3abc
= 0, then the value of 3 + (C)
3abc b 3 a3
3 is (D)
(3abc b 3 ) a3
If b > a, then the equation (x – a) (x – b) – 1 = 0 has (A) both roots in (a, b) (B) both roots in (– , a) (C) both roots in (b, ) (D) one root in (– , a) and other in (b, If a + b + c = 0, then the quadratic equation 3ax2 + 2bx + c = 0 has (A) at least one root in [0, 1] (B) both roots imaginary (C) one root in (1, 2), either in (– 1, 0) (D) None of these
36
.
)
21. (i) (– , 4]
EXERCISE # 1-A 1.
B
2.
B
3.
C
4. A
5. A
6. A
7.
C
1, 3 2 2
22. (i) 8. A
9. A
10. B
11. B
12. B
13. A
14. C
15. C
16. C
17. D
18. D
19. B
20. B
21. B
22. A
23. B
24. D
25. A
26. A
27. A
28. B
29. B
30. D
31. C
32. C
33. B
34. ABD
35. BD 36. BCD
37. ABD
2.
x x
(1, 3) (2, 1) ( 2/3, 1/2)
(iv)
x
1 (– , – 2) , 1 (4, ) 4
(ii) –
8
4.
11 m and 9 m
7.
11. 3 ± 2 2
5.
(– 4, 7)
[–1, 1) (2, 4]
(ii) (iii) (iv)
x
2 1 , 3 2 ,
(v)
x
2 3 , 4
27. (i)
x
3 2, 2
x
3x 2 – 19x + 3 = 0.
9. – 1 ±
15. 4
2,–1±
1
(iii)
(0, 10 –1] [10 2, )
(iv)
1 < x <
(ii)
1
5
2
1
or
2
5
< x < 2
(– , –5) (–5, –1) (3, )
28. (i)
(iii)
(log2 5, )
(ii)
16. a = 0, 24
17. 3
20. (i)
(– , –1] {1} [4, ) (–4, –1) (–1, 0) (0, 2) (3, ) x (–2, –1) (2, 3) x (– , –2] {1} x x
(–1, 0) (0, 3) x (– , –4] [–1, 1] [4, ) x (–5, –2) (–1, )
26. (i)
7
(i) ac x 2 + b(a + c) x + (a + c) 2 = 0 (ii) a2 x2 + (2ac – 4a 2 – b 2) x + 2b2 + (c – 2a) 2 = 0
8, 3
(ii) (iii)
(iii) (iv)
3.
6.
(– , 2) (5, )
25. (i) (ii)
EXERCISE # 1-B
7 (i) – 4
1 , 2 2
x
23. (i)
24. x
a = 2; No real value of x.
(ii)
(iii) [3, 6]
38. ABCD 39. AD
40. AD
1.
(ii) [2, 3]
(ii) (iii) (iv)
[2, ) ( , 1) (1, ) x (– 2, –1) (– 1, 0)
30. K
( 2, 3)
31. a
32. a
(1, 5) – {3}
34. –
(0, 1) (2, )
(–2, 2) (r 1)3 r 2
37
35. – 3
EXERCISE # 2-A
EXERCISE # 3
1.
B
2.
B
3. A
4.
8.
C
9.
C
10. D
11. A
16. B
17. B
18. D
15. A
21. ABD
B
B
7. A
1.
(A) (p),
(B) (s),
(C) (q),
(D) (r)
12. A
13. B
14. A
2.
(A) (r),
(B) (p),
(C) (s),
(D) (p, q)
19. A
20. B
5.
D
6.
22. AB 23. BD 24. CD 25. AD
(ii) {2, 5}
(i) 2
7.
( , 0) (9/2, )
4.
B
5.
D
7.2 B
7.3 C
8.
False
11. True
EXERCISE # 2-B 1.
3. A
12. False
6.1 A 9.
13. 1
6.2 A
6.3 D
7.1 C
True
10. True
14. 2
15. 3
16. 0
11 3 , = – 4 4
17. =
EXERCISE # 4 8.
(i) x = ± 2, ±
2
(ii) x = a (1 –
2 ), x = a ( 6 – 1)
1.
B
2.
D
3.
(A) (p), (r), (s) ; (C) (q), (s) ;
9.
k
11. x
(B) (q), (s) ;
(D) (p), (r), (s)
(0, 4) 4. A
5.
1210
6.
C
7. A
8.
D
9.
10. B
11. B
12. =
2 and = 2 or = 2 and = 2
a> 1
R if y = 1, x = 2 if y = 3
12. a =
7, b = 8 ;
13. 6 < K < 6.75
roots (3, 4), (3, 5), (3, 12) 14. a
[9, )
13. (a) C
16. x 3 + qx – r = 0
17. (i) a
(– , – 4)
(iii) a
65 4, 4
65 , ƒ (ii) a 4 (iv) a
18. {–1}
(b) B
[1, )
(c) D
15. A
16. 4
19. A
21. A
22. A
23. B
24. C
25. D
26. A
27. B
28. C
29. B
30. C
31. A
32. A
33. C
34. B
35. A
36. D
37. B
38. D
39. A
40. D
41. A
42. D
43. B
44. D
45. C
46. D
47. A
38
