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Vidyamandir Classes

DATE :

MARKS : 10

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : QE [1]

ROLL NO.

START TIME :

END TIME :

STUDENT’S SIGNATURE :

TIME TAKEN:

PARENT’S SIGNATURE :

This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. 1.

(A) 2.

10.

0

(B)

1

None of these

9 4

(B)

9 4

(C)

1 4

(D)

1 4

0

(B)

1

(C)

2

(D)

3

a>5

(B)

07

k≤0

(B)

k≥0

(C)

k≥6

(D)

k≤6

If , are the roots of ax 2 bx c 0 ; h h are the roots of px 2 qx r 0 ; and D1, D2 the respective discriminants of these equations, then D1 : D2 = a2 p2

(B)

b2 q2

c2

(C)

r2

(D)

None of these

If , are the roots of ax 2 bx c 0 ; h h are the roots of px 2 qx r 0 , then h = (A)

9.

is greater than : (C) 2 (D)

2

If the equation x 2 2 k 1 x 9k 5 0 has only negative roots, then :

(A) 8.

None of these

(A) 7.

(D)

2

The real values of a for which the quadratic equation 2 x 2 a3 8a 1 x a 2 4a 0 possesses roots of opposite signs are given by : (A)

6.

–2

The value of a for which the sum of the squares of the roots of the equation x 2 a 2 x a 1 0 assumes the least value is : (A)

5.

(B)

If a and b ( 0) are the roots of the equation x 2 ax b 0 , then the least value of x 2 ax b 0 x R is : (A)

4.

2

If , are roots of the equation ax 2 3 x 2 0 a 0 , then (A)

3.

is less than : (C) 18

If , are roots of the equation 2 x 2 6 x b 0 b 0 , then

b q a p

(B)

1b q 2 a p

(C)

1b q 2a p

(D)

None of these

If , R, are the roots of the equation ax 2 bx c 0, k R lies between and , if : (A)

ak 2 bk c 0

(B)

a 2k 2 abk ac 0

(C)

a 2k 2 abk ac 0

(D)

None of these

If every pair from among the equations x 2 px qr 0, x 2 qx rp 0 and x 2 rx pq 0 has a common root, then the product of three common roots is : (A)

pqr

VMC/Quadratic Equation

(B)

2 pqr

p 2q 2r 2

(C)

1

(D)

None of these

HWT/Mathematics

Vidyamandir Classes

DATE :

MARKS : 10

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : QE [2]

ROLL NO.

START TIME :

END TIME :

STUDENT’S SIGNATURE :

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This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. However, question marked with ‘*’ may have More than One correct option. 1.

(A) 2.

3.

9.

(D)

2

None of these

2

2 a1 a2 a3 . . . . an

(C)

n a1 a2 . . . . an

(D)

None of these

If the quadratic equations ax 2 2cx b 0 and ax 2 2bx c 0 b c have a common root, then a + 4b + 4c is : –2

(B)

–1

(C)

0

(D)

1

The least integral value of a for which the equation x 2 2 a 1 x 2a 1 0 has both the roots positive, is : 3

(B)

4

(C)

1

(D)

5

The integer k for which the inequality x 2 2 4k 1 x 15 2 23k 7 0 is valid for any x, is : –2

(B)

3

(C)

–4

(D)

None of these

The values of a for which 2 x 2 2 2a 1 x a a 1 0 may have one root less than a and other root greater than a are given by : 1>a>0

(B)

–1 < a < 0

a≥ 0

(C)

(D)

a > 0 or a < –1

The condition that x3 px 2 qx r 0 may have two of its roots equal to each other but of opposite signs is : r = pq

(B)

r 2 p3 pq

r p 2q

(C)

(D)

None of these

The value of m for which one of the roots of x 2 3 x 2m 0 is double of one of the roots of x 2 x m 0 is : (A) 0 (B) –2 (C) 2 (D) None of these If the roots of ax 2 bx c 0 a 0 be each greater than unity, then : (A)

10.

–1/2

(B)

(A) *8.

(C)

a1 a2 . . . . an

(A) 7.

2

(A)

(A) 6.

(B)

2

(A) 5.

0

If a R and a1 , a2 , a3 . . . ., an R then x a1 x a2 . . . . x an assumes its least value at x =

(A) 4.

2 If the equation 3 x 27 31 p 15 x 4 0 has equal roots, then p =

abc 0

(B)

abc 0

abc0

(C)

(D)

None of these

If , be the roots of the equation x a x b c 0 c 0 , then the roots of the equation x c x c c are : (A)

a and b + c

VMC/Quadratic Equation

(B)

a + c and b

(C)

a + c and b + c

2

(D)

a + c and b

HWT/Mathematics

Vidyamandir Classes

DATE :

MARKS : 10

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : QE [3]

ROLL NO.

START TIME :

END TIME :

STUDENT’S SIGNATURE :

TIME TAKEN :

PARENT’S SIGNATURE :

This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. 1.

(A) 2.

(B)

(D)

None of these

0

(B)

1

(C)

2

(D)

None of these

The value of a for which the equation 1 a 2 x 2 2ax 1 0 has roots belonging to (0, 1) is : (A)

4.

0

9 The number of real solutions of the equation x 3 x x 2 is : 10

(A) 3.

2x 3 6 x2 x 6 1 is : x 1 x 1 1 (C) 2

The number of real roots of the equation

a

1 5 2

(B)

a2

1 5 a2 2

(C)

(D)

a 2

The values of a for which each one of the roots of x 2 4ax 2a 2 3a 5 0 is greater than 2, are : (A)

a 1,

(B)

a 1

a , 1

(C)

(D)

9 a , 2

5.

If the product of the roots of the equation x 2 2 2 kx 2e 2 log k 1 0 is 31, then the roots of the equation are real for k equal to : (A) 1 (B) 2 (C) 3 (D) 4

6.

If x R, the least value of the expression y (A)

7.

If x R then (A)

*8.

–1/2

then, (C)

–1/3

(D)

None of these

(C)

a 1, 1

(D)

None of these

(D)

None of these

(D)

a>4

2

x 2x a x 2 4 x 3a

a 0, 2

can take all values if : (B)

a 0 , 1

(B)

1

1

(C)

If the roots of the equation x 2 2ax a 2 a 3 0 are real and less than 3, then : (A)

10.

(B)

x2 2 x 1

If 0 < a < b < c, and the roots , of the equation ax 2 bx c 0 are imaginary, then : (A)

9.

–1

x2 6 x 5

a2

(B)

2≤ a ≤ 3

3≤ a ≤ 4

(C)

If f x ax 2 bx c, g x ax 2 bx c where ac 0, then f (x) g (x) = 0 has : (A) (C)

at least three real roots at least two real roots

VMC/Quadratic Equation

(B) (D)

no real solution two real roots and two imaginary roots

3

HWT/Mathematics

Vidyamandir Classes

DATE :

MARKS : 10

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : QE [4]

ROLL NO.

START TIME :

END TIME :

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This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. 1.

The number of real roots of the equation x 1 x 2 x 3 0 is : 2

(A) 2.

(C)

2, 3

(D)

3, 5

b 2 9ac

(B)

2b 2 9ac

2b 2 ac

(C)

(D)

b 2 ac

cx 2 bx a 0

(B)

bx 2 cx a 0

cx 2 ax b 0

(C)

(D)

bx 2 ax c 0

0

(B)

5

1 6

(C)

(D)

6

positive

(B)

negative

(C)

real

(D)

None of these

1

(B)

2

3

(C)

(D)

None of these

p 2 4q

(B)

p 2 4q 1

p 2 4q 1

(C)

(D)

None of these

If the equation a 5 x 2 2 a 10 x a 10 0 has real roots of the same sign, then : (A)

10.

4, 5

If the roots of the equation x 2 px q 0 differ by unity, then : (A)

9.

2, 3

None of these

If the equation x 2 3kx 2e 2 log k 1 0 has real roots such that the product of roots is 7, then the value of k is : (A)

8.

(B)

(D)

Both the roots of the equation x a x b x b x c x c x a 0 are always : (A)

7.

3

If one root of the equation 5 x 2 13 x k 0 is reciprocal of other, then the value of k is : (A)

6.

(C)

The quadratic equation whose roots are reciprocal of the roots of the equation ax 2 bx c 0 is : (A)

5.

2

The condition that one root of the equation ax 2 bx c 0 may be double of the other, is : (A)

4.

(B)

2

The roots of the equation log 2 x 2 4 x 5 x 2 are : (A)

3.

1

2

a > 10

5 a 5

(B)

a 10 and 5 a 6

(C)

2 2 1 is : x 1 x 1 2 (C)

(D)

None of these

The number of roots of the equation x (A)

1

VMC/Quadratic Equation

(B)

0

4

(D)

infinitely many

HWT/Mathematics

Vidyamandir Classes

DATE :

MARKS : 10

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : QE [5]

ROLL NO.

START TIME :

END TIME :

STUDENT’S SIGNATURE :

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This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. 1.

If the equation (A)

2.

(C)

1

(D)

None of these

pq

(B)

qr

rq

(C)

(D)

q+r

p 2m q 2l

(B)

pm 2 q 2l

p 2l q 2m

(C)

(D)

p 2m l 2q

–4

(B)

0

(C)

4

(D)

2

(D)

–5, –4

If x 2 3 x 2 is a factor of x 4 px 2 q , then the value of p and q are : (A)

6.

0

If x + 1 is a factor of x 4 p 3 x3 3 p 5 x 2 2 p 9 x 6 , then the value of p is : (A)

5.

(B)

If the ratio of the roots of the equation x 2 px q 0 be equal to the ratio of the roots of x 2 lx m 0 , then : (A)

4.

1

If , are roots of x 2 px q 0 and , are roots of x 2 px r 0 , then the value of is : (A)

3.

a b 1 has roots equal in magnitude but opposite to sign, then the value of a + b is : xa xb

5, –4

(B)

5, 4

(C)

–5, 4

If the equation x 2 px q 0 and x 2 px q 0 have common root, then it is equal to : (A)

p p q q

(B)

p p q q

q q p p

(C)

(D)

q q p p

7.

If the expressions x 2 11x a and x 2 14 x 2a have a common factor, then the values of ‘a’ are : (A) 0, 24 (B) 0, –24 (C) 1, –1 (D) –2, 1

8.

If the equations x 2 ax b 0 and x 2 bx a 0 have a common root, then the numerical value of a + b is : (A) 1 (B) 0 (C) –1 (D) None of these

9.

If the roots of the equation x 2 a 2 8 x 6a are real, then a belongs to the interval : (A) [2, 8] (B) [–2, 8] (C) [–8, 2]

10.

(D)

None of these

If the sum of the roots of the equation a 1 x 2 2a 3 x 3a 4 0 is –1, then the product of the roots is : (A)

0

VMC/Quadratic Equation

(B)

1

(C)

2

5

(D)

3

HWT/Mathematics

Vidyamandir Classes

DATE :

MARKS : 10

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : QE [6]

ROLL NO.

START TIME :

END TIME :

STUDENT’S SIGNATURE :

TIME TAKEN :

PARENT’S SIGNATURE :

This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. However, question marked with ‘*’ may have More than One correct option. 1.

If b1 b2 2 c1 c2 , then at least one of the equations

(A) (C)

x 2 b1x c1 0 and x 2 b1x c2 0 has : (A) (C) 2.

3.

*4.

(B) (D)

(A)

4 , 1 3

(B)

4, 1 3

(C)

4 , 1 3

(D)

4 , 1 3

*7.

b 2 4ac a and b are of opposite signs

The diagram shows the graph of y ax 2 bx c . Then,

If 2 i 3 is a root of x 2 px q 0 where p, q R, then : (A)

p 4 , q 7

(B)

p 4, q 7

(A)

a>0

(B)

b0

(D)

b 2 4ac 0

*8.

If the equations x 2 ax b 0 and x 2 bx a 0 have a common root, then : (A) ab (B) ab 0

a b 1

(D)

For real values of x, the expression assume all real values provided : (A) a≤ c ≤ b (B) (C) b≤ c ≤ a (D)

*6.

purely imaginary None of these

(B) (D)

The values of x satisfying x 2 4 x 3 2 x 5 0 are:

(C) 5.

real roots imaginary roots

a0 c>0

a b 1

x b x c x a

Let f x x 2 4 x 1 . Then, (A)

f x 0 for all x

(B)

f x 1 for all x ≥ 0

(C)

f (x) ≥ 1 when x ≤ –4

(D)

f (x) = f (–x) for all x

will *9.

b≥ a≥ c a≥ b≥ c

If S is the set of all real x such that

2 x 3x 2 x

is

positive, then S contains : (A)

3 , 2

(B)

3 1 , 2 4

(C)

1 1 , 4 2

(D)

1 , 3 2

The adjoining figure shows the graph of y ax 2 bx c . Then :

*10.

VMC/Quadratic Equation

2x 1 3

6

If x 2 2 x 3 0 and x 2 2 x 3 0 then : (A) x≥ 3 (B) x ≤ –1 ≤ (C) x –3 (D) x has no value in R

HWT/Mathematics

Vidyamandir Classes

DATE :

MARKS : 10

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : QE [7]

ROLL NO.

START TIME :

END TIME :

STUDENT’S SIGNATURE :

TIME TAKEN :

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This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct 1.

If

x2 2 x 7 6 , x R , then : 2x 3

(A) 2.

1

(A)

5.

6.

7.

8.

(B)

x

(C)

1

(C)

(C)

3 x 1 2

(D)

–1 < x < 11 or x

–1

(D)

0

3

(D)

4

3 2

2 x 1

1

(B)

(A)

a b c and a b c

(C)

a 2b c and

2

1 and a 2b c 2

(B)

1 abc

(D)

None of these

The number of positive real roots of x 4 4 x 1 0 is : (A) 3 (B) 2

(C)

1

(D)

0

The number of negative real roots of x 4 4 x 1 0 is : (A) 3 (B) 2

(C)

1

(D)

0

1

(D)

0

(D)

equal and imaginary

The number of complex roots of the equation x 4 4 x 1 0 is : (A) 3 (B) 2 (C)

If a < c < b, the roots of the equation a b x 2 2 a b 2c x 1 0 are : 2

imaginary

(B)

real

(C)

can’t say

If the ratio of the roots of x 2 bx c 0 is same as that of x 2 qx r 0 , then : (A)

10.

x > 11 or x < –1

The roots of the quadratic equation a b 2c x 2 2a b c x a 2b c 0 are :

(A) 9.

(B)

How many roots does the following equation process ?

3

4.

3 2

If x 2 x 1 0 then value of x 3n is : (A)

3.

x > 11 or x

r 2b qc 2

(B)

r 2 c qb2

2 The number of real roots of 32 x 7 x 7 9 is : (A) 0 (B) 2

VMC/Quadratic Equation

(C)

c 2r q 2b

(D)

b 2 r q 2c

(C)

1

(D)

4

7

HWT/Mathematics

Vidyamandir Classes

DATE :

MARKS : 10

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : QE [8]

ROLL NO.

START TIME :

END TIME :

STUDENT’S SIGNATURE :

TIME TAKEN :

PARENT’S SIGNATURE :

This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. 1.

The greatest value of (A)

4/9

4 2

4x 4x 9 (B)

is : 4

(C)

9/4

(D)

1/2

2.

If a and b are the non-zero distinct roots of x 2 ax b 0 , then the least value of x 2 ax b is : 2 9 9 (A) (B) (C) (D) 1 3 4 4

3.

The maximum and minimum values of (A)

4.

If x is real then (A)

5.

3, 1

(B)

x 2 14 x 9 x2 2 x 3 4, –5

are : (C)

–

(D)

, –

(C)

(3, 3)

(D)

1 , 3 3

x2 2 x 4

takes values in the interval : x2 2 x 4 1 1 (B) , 3 3 , 3 3

If , are the roots of x 2 bx c 0 , then the equation whose roots are b and c is : (A)

x2 x 0

(B)

x 2 x 0

(C)

x 2 x x 0

(D)

x 2 x 0

(C)

1

2 is :

log 5 log x 2 1 6.

The number of solutions of (A)

7.

8.

3

(D)

None of these

–2 < a < 0

1 1 Let , be the roots of x 2 bx 1 0 . Then the equation whose roots are and is : x2 0

(B)

x 2 2bx 4 0

x 2 2bx 4 0

(C)

(D)

x 2 bx 1 0

If x 2 x 1 is a factor of ax3 bx 2 cx d then the real root of ax3 bx 2 cx d 0 is : (A)

10.

log x 2 (B)

The values of ‘a’ for which the roots of the equation x 2 x a 0 are real and exceed ‘a’ are : (A) 0 < a < 1/4 (B) a < 1/4 (C) a < –2 (D)

(A) 9.

2

d a

(B)

d a

a d

(C)

(D)

None of these

(D)

b 0, c 0

If both the roots of the equation ax 2 bx c 0 are zero, then : (A)

b=c=0

VMC/Quadratic Equation

(B)

b = 0, c 0

b 0, c = 0

(C)

8

HWT/Mathematics

Vidyamandir Classes

DATE :

MARKS : 10

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : QE [9]

ROLL NO.

START TIME :

END TIME :

STUDENT’S SIGNATURE :

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This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. 1.

If

x 2 bx 1 has roots equal in magnitude and opposite in sign then the value of is : ax c 1

(A)

2.

7.

(B)

15

(B)

a1c2 a2c1

2

–5, –30

If x 7 4 3 , x

a≤ 1

(C)

9

(B)

a > –3

(C)

a1a2 c1c2 2

(B)

7

a1c1 a2c2 2

(C)

–5, 30

(D)

a < –3 or a > 1

(D)

8

(D)

a1c2 c1a2 2

(C)

5, 30

(D)

None of these

(C)

–3 < a < –3

(D)

a < –2

(C)

3

(D)

2

, 1

(D)

(1, 3)

(C)

2

(D)

None of these

(C)

2 5

(D)

5 2

1 x

4

(B)

6

The set of values of x which satisfy 5 x 2 3 x 8 and (2, 3)

(B)

x 2 4 x 1

, 1 2, 3

is :

(C)

9 The number of real solutions of the equation x 3 x x 2 is : 10

(A) 10.

a≥ 1

If the roots of x 2 x a 0 exceed a, then : (A) 2 –3

(B)

a 0 or a < –3

(D)

None of these

The value of ‘k’ for which one of the roots of x 2 x 3k 0 , is double of one of the roots of x 2 x k 0 is : (A) 1 (B) –2 (C) 2 (D) None of these 3c If the equation ax 2 2bx 3c 0 has no real roots and a b , then : 4 (A) c0 (C) c≥0

(D)

c=0

(D)

1, 3

The value of x satisfying log3 x 2 4 x 12 2 are : (A)

6.

(B)

The values of ‘a’ for which the equation 2 x 2 2 2a 1 x a a 1 0 has roots and where < a < are such that : (A)

3.

1

2, 4

(B)

1, 3

1, 3

(C)

x2 x 1 cannot be between ____________ and _____________for any x R. x 1

7.

Two equations x 2 cx d 0 and x 2 ax b 0 have one root common and second equation has equal roots if ___________.

8.

If x 1 i is a root of equation x 3 ix 1 i 0 then other root is _______.

9.

For real x, the least value of 2 x 2 5 x 3 is ____________.

10.

If the roots of the equations ax 2 2bx c 0 and bx 2 2 ac x b 0 are simultaneously real, then b2 = ___________.

VMC/Quadratic Equation

10

HWT/Mathematics

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DATE :

MARKS : 10

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : QE [1]

ROLL NO.

START TIME :

END TIME :

STUDENT’S SIGNATURE :

TIME TAKEN:

PARENT’S SIGNATURE :

This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. 1.

(A) 2.

10.

0

(B)

1

None of these

9 4

(B)

9 4

(C)

1 4

(D)

1 4

0

(B)

1

(C)

2

(D)

3

a>5

(B)

07

k≤0

(B)

k≥0

(C)

k≥6

(D)

k≤6

If , are the roots of ax 2 bx c 0 ; h h are the roots of px 2 qx r 0 ; and D1, D2 the respective discriminants of these equations, then D1 : D2 = a2 p2

(B)

b2 q2

c2

(C)

r2

(D)

None of these

If , are the roots of ax 2 bx c 0 ; h h are the roots of px 2 qx r 0 , then h = (A)

9.

is greater than : (C) 2 (D)

2

If the equation x 2 2 k 1 x 9k 5 0 has only negative roots, then :

(A) 8.

None of these

(A) 7.

(D)

2

The real values of a for which the quadratic equation 2 x 2 a3 8a 1 x a 2 4a 0 possesses roots of opposite signs are given by : (A)

6.

–2

The value of a for which the sum of the squares of the roots of the equation x 2 a 2 x a 1 0 assumes the least value is : (A)

5.

(B)

If a and b ( 0) are the roots of the equation x 2 ax b 0 , then the least value of x 2 ax b 0 x R is : (A)

4.

2

If , are roots of the equation ax 2 3 x 2 0 a 0 , then (A)

3.

is less than : (C) 18

If , are roots of the equation 2 x 2 6 x b 0 b 0 , then

b q a p

(B)

1b q 2 a p

(C)

1b q 2a p

(D)

None of these

If , R, are the roots of the equation ax 2 bx c 0, k R lies between and , if : (A)

ak 2 bk c 0

(B)

a 2k 2 abk ac 0

(C)

a 2k 2 abk ac 0

(D)

None of these

If every pair from among the equations x 2 px qr 0, x 2 qx rp 0 and x 2 rx pq 0 has a common root, then the product of three common roots is : (A)

pqr

VMC/Quadratic Equation

(B)

2 pqr

p 2q 2r 2

(C)

1

(D)

None of these

HWT/Mathematics

Vidyamandir Classes

DATE :

MARKS : 10

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : QE [2]

ROLL NO.

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This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. However, question marked with ‘*’ may have More than One correct option. 1.

(A) 2.

3.

9.

(D)

2

None of these

2

2 a1 a2 a3 . . . . an

(C)

n a1 a2 . . . . an

(D)

None of these

If the quadratic equations ax 2 2cx b 0 and ax 2 2bx c 0 b c have a common root, then a + 4b + 4c is : –2

(B)

–1

(C)

0

(D)

1

The least integral value of a for which the equation x 2 2 a 1 x 2a 1 0 has both the roots positive, is : 3

(B)

4

(C)

1

(D)

5

The integer k for which the inequality x 2 2 4k 1 x 15 2 23k 7 0 is valid for any x, is : –2

(B)

3

(C)

–4

(D)

None of these

The values of a for which 2 x 2 2 2a 1 x a a 1 0 may have one root less than a and other root greater than a are given by : 1>a>0

(B)

–1 < a < 0

a≥ 0

(C)

(D)

a > 0 or a < –1

The condition that x3 px 2 qx r 0 may have two of its roots equal to each other but of opposite signs is : r = pq

(B)

r 2 p3 pq

r p 2q

(C)

(D)

None of these

The value of m for which one of the roots of x 2 3 x 2m 0 is double of one of the roots of x 2 x m 0 is : (A) 0 (B) –2 (C) 2 (D) None of these If the roots of ax 2 bx c 0 a 0 be each greater than unity, then : (A)

10.

–1/2

(B)

(A) *8.

(C)

a1 a2 . . . . an

(A) 7.

2

(A)

(A) 6.

(B)

2

(A) 5.

0

If a R and a1 , a2 , a3 . . . ., an R then x a1 x a2 . . . . x an assumes its least value at x =

(A) 4.

2 If the equation 3 x 27 31 p 15 x 4 0 has equal roots, then p =

abc 0

(B)

abc 0

abc0

(C)

(D)

None of these

If , be the roots of the equation x a x b c 0 c 0 , then the roots of the equation x c x c c are : (A)

a and b + c

VMC/Quadratic Equation

(B)

a + c and b

(C)

a + c and b + c

2

(D)

a + c and b

HWT/Mathematics

Vidyamandir Classes

DATE :

MARKS : 10

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : QE [3]

ROLL NO.

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This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. 1.

(A) 2.

(B)

(D)

None of these

0

(B)

1

(C)

2

(D)

None of these

The value of a for which the equation 1 a 2 x 2 2ax 1 0 has roots belonging to (0, 1) is : (A)

4.

0

9 The number of real solutions of the equation x 3 x x 2 is : 10

(A) 3.

2x 3 6 x2 x 6 1 is : x 1 x 1 1 (C) 2

The number of real roots of the equation

a

1 5 2

(B)

a2

1 5 a2 2

(C)

(D)

a 2

The values of a for which each one of the roots of x 2 4ax 2a 2 3a 5 0 is greater than 2, are : (A)

a 1,

(B)

a 1

a , 1

(C)

(D)

9 a , 2

5.

If the product of the roots of the equation x 2 2 2 kx 2e 2 log k 1 0 is 31, then the roots of the equation are real for k equal to : (A) 1 (B) 2 (C) 3 (D) 4

6.

If x R, the least value of the expression y (A)

7.

If x R then (A)

*8.

–1/2

then, (C)

–1/3

(D)

None of these

(C)

a 1, 1

(D)

None of these

(D)

None of these

(D)

a>4

2

x 2x a x 2 4 x 3a

a 0, 2

can take all values if : (B)

a 0 , 1

(B)

1

1

(C)

If the roots of the equation x 2 2ax a 2 a 3 0 are real and less than 3, then : (A)

10.

(B)

x2 2 x 1

If 0 < a < b < c, and the roots , of the equation ax 2 bx c 0 are imaginary, then : (A)

9.

–1

x2 6 x 5

a2

(B)

2≤ a ≤ 3

3≤ a ≤ 4

(C)

If f x ax 2 bx c, g x ax 2 bx c where ac 0, then f (x) g (x) = 0 has : (A) (C)

at least three real roots at least two real roots

VMC/Quadratic Equation

(B) (D)

no real solution two real roots and two imaginary roots

3

HWT/Mathematics

Vidyamandir Classes

DATE :

MARKS : 10

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : QE [4]

ROLL NO.

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This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. 1.

The number of real roots of the equation x 1 x 2 x 3 0 is : 2

(A) 2.

(C)

2, 3

(D)

3, 5

b 2 9ac

(B)

2b 2 9ac

2b 2 ac

(C)

(D)

b 2 ac

cx 2 bx a 0

(B)

bx 2 cx a 0

cx 2 ax b 0

(C)

(D)

bx 2 ax c 0

0

(B)

5

1 6

(C)

(D)

6

positive

(B)

negative

(C)

real

(D)

None of these

1

(B)

2

3

(C)

(D)

None of these

p 2 4q

(B)

p 2 4q 1

p 2 4q 1

(C)

(D)

None of these

If the equation a 5 x 2 2 a 10 x a 10 0 has real roots of the same sign, then : (A)

10.

4, 5

If the roots of the equation x 2 px q 0 differ by unity, then : (A)

9.

2, 3

None of these

If the equation x 2 3kx 2e 2 log k 1 0 has real roots such that the product of roots is 7, then the value of k is : (A)

8.

(B)

(D)

Both the roots of the equation x a x b x b x c x c x a 0 are always : (A)

7.

3

If one root of the equation 5 x 2 13 x k 0 is reciprocal of other, then the value of k is : (A)

6.

(C)

The quadratic equation whose roots are reciprocal of the roots of the equation ax 2 bx c 0 is : (A)

5.

2

The condition that one root of the equation ax 2 bx c 0 may be double of the other, is : (A)

4.

(B)

2

The roots of the equation log 2 x 2 4 x 5 x 2 are : (A)

3.

1

2

a > 10

5 a 5

(B)

a 10 and 5 a 6

(C)

2 2 1 is : x 1 x 1 2 (C)

(D)

None of these

The number of roots of the equation x (A)

1

VMC/Quadratic Equation

(B)

0

4

(D)

infinitely many

HWT/Mathematics

Vidyamandir Classes

DATE :

MARKS : 10

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : QE [5]

ROLL NO.

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This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. 1.

If the equation (A)

2.

(C)

1

(D)

None of these

pq

(B)

qr

rq

(C)

(D)

q+r

p 2m q 2l

(B)

pm 2 q 2l

p 2l q 2m

(C)

(D)

p 2m l 2q

–4

(B)

0

(C)

4

(D)

2

(D)

–5, –4

If x 2 3 x 2 is a factor of x 4 px 2 q , then the value of p and q are : (A)

6.

0

If x + 1 is a factor of x 4 p 3 x3 3 p 5 x 2 2 p 9 x 6 , then the value of p is : (A)

5.

(B)

If the ratio of the roots of the equation x 2 px q 0 be equal to the ratio of the roots of x 2 lx m 0 , then : (A)

4.

1

If , are roots of x 2 px q 0 and , are roots of x 2 px r 0 , then the value of is : (A)

3.

a b 1 has roots equal in magnitude but opposite to sign, then the value of a + b is : xa xb

5, –4

(B)

5, 4

(C)

–5, 4

If the equation x 2 px q 0 and x 2 px q 0 have common root, then it is equal to : (A)

p p q q

(B)

p p q q

q q p p

(C)

(D)

q q p p

7.

If the expressions x 2 11x a and x 2 14 x 2a have a common factor, then the values of ‘a’ are : (A) 0, 24 (B) 0, –24 (C) 1, –1 (D) –2, 1

8.

If the equations x 2 ax b 0 and x 2 bx a 0 have a common root, then the numerical value of a + b is : (A) 1 (B) 0 (C) –1 (D) None of these

9.

If the roots of the equation x 2 a 2 8 x 6a are real, then a belongs to the interval : (A) [2, 8] (B) [–2, 8] (C) [–8, 2]

10.

(D)

None of these

If the sum of the roots of the equation a 1 x 2 2a 3 x 3a 4 0 is –1, then the product of the roots is : (A)

0

VMC/Quadratic Equation

(B)

1

(C)

2

5

(D)

3

HWT/Mathematics

Vidyamandir Classes

DATE :

MARKS : 10

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : QE [6]

ROLL NO.

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This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. However, question marked with ‘*’ may have More than One correct option. 1.

If b1 b2 2 c1 c2 , then at least one of the equations

(A) (C)

x 2 b1x c1 0 and x 2 b1x c2 0 has : (A) (C) 2.

3.

*4.

(B) (D)

(A)

4 , 1 3

(B)

4, 1 3

(C)

4 , 1 3

(D)

4 , 1 3

*7.

b 2 4ac a and b are of opposite signs

The diagram shows the graph of y ax 2 bx c . Then,

If 2 i 3 is a root of x 2 px q 0 where p, q R, then : (A)

p 4 , q 7

(B)

p 4, q 7

(A)

a>0

(B)

b0

(D)

b 2 4ac 0

*8.

If the equations x 2 ax b 0 and x 2 bx a 0 have a common root, then : (A) ab (B) ab 0

a b 1

(D)

For real values of x, the expression assume all real values provided : (A) a≤ c ≤ b (B) (C) b≤ c ≤ a (D)

*6.

purely imaginary None of these

(B) (D)

The values of x satisfying x 2 4 x 3 2 x 5 0 are:

(C) 5.

real roots imaginary roots

a0 c>0

a b 1

x b x c x a

Let f x x 2 4 x 1 . Then, (A)

f x 0 for all x

(B)

f x 1 for all x ≥ 0

(C)

f (x) ≥ 1 when x ≤ –4

(D)

f (x) = f (–x) for all x

will *9.

b≥ a≥ c a≥ b≥ c

If S is the set of all real x such that

2 x 3x 2 x

is

positive, then S contains : (A)

3 , 2

(B)

3 1 , 2 4

(C)

1 1 , 4 2

(D)

1 , 3 2

The adjoining figure shows the graph of y ax 2 bx c . Then :

*10.

VMC/Quadratic Equation

2x 1 3

6

If x 2 2 x 3 0 and x 2 2 x 3 0 then : (A) x≥ 3 (B) x ≤ –1 ≤ (C) x –3 (D) x has no value in R

HWT/Mathematics

Vidyamandir Classes

DATE :

MARKS : 10

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : QE [7]

ROLL NO.

START TIME :

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This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct 1.

If

x2 2 x 7 6 , x R , then : 2x 3

(A) 2.

1

(A)

5.

6.

7.

8.

(B)

x

(C)

1

(C)

(C)

3 x 1 2

(D)

–1 < x < 11 or x

–1

(D)

0

3

(D)

4

3 2

2 x 1

1

(B)

(A)

a b c and a b c

(C)

a 2b c and

2

1 and a 2b c 2

(B)

1 abc

(D)

None of these

The number of positive real roots of x 4 4 x 1 0 is : (A) 3 (B) 2

(C)

1

(D)

0

The number of negative real roots of x 4 4 x 1 0 is : (A) 3 (B) 2

(C)

1

(D)

0

1

(D)

0

(D)

equal and imaginary

The number of complex roots of the equation x 4 4 x 1 0 is : (A) 3 (B) 2 (C)

If a < c < b, the roots of the equation a b x 2 2 a b 2c x 1 0 are : 2

imaginary

(B)

real

(C)

can’t say

If the ratio of the roots of x 2 bx c 0 is same as that of x 2 qx r 0 , then : (A)

10.

x > 11 or x < –1

The roots of the quadratic equation a b 2c x 2 2a b c x a 2b c 0 are :

(A) 9.

(B)

How many roots does the following equation process ?

3

4.

3 2

If x 2 x 1 0 then value of x 3n is : (A)

3.

x > 11 or x

r 2b qc 2

(B)

r 2 c qb2

2 The number of real roots of 32 x 7 x 7 9 is : (A) 0 (B) 2

VMC/Quadratic Equation

(C)

c 2r q 2b

(D)

b 2 r q 2c

(C)

1

(D)

4

7

HWT/Mathematics

Vidyamandir Classes

DATE :

MARKS : 10

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : QE [8]

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This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. 1.

The greatest value of (A)

4/9

4 2

4x 4x 9 (B)

is : 4

(C)

9/4

(D)

1/2

2.

If a and b are the non-zero distinct roots of x 2 ax b 0 , then the least value of x 2 ax b is : 2 9 9 (A) (B) (C) (D) 1 3 4 4

3.

The maximum and minimum values of (A)

4.

If x is real then (A)

5.

3, 1

(B)

x 2 14 x 9 x2 2 x 3 4, –5

are : (C)

–

(D)

, –

(C)

(3, 3)

(D)

1 , 3 3

x2 2 x 4

takes values in the interval : x2 2 x 4 1 1 (B) , 3 3 , 3 3

If , are the roots of x 2 bx c 0 , then the equation whose roots are b and c is : (A)

x2 x 0

(B)

x 2 x 0

(C)

x 2 x x 0

(D)

x 2 x 0

(C)

1

2 is :

log 5 log x 2 1 6.

The number of solutions of (A)

7.

8.

3

(D)

None of these

–2 < a < 0

1 1 Let , be the roots of x 2 bx 1 0 . Then the equation whose roots are and is : x2 0

(B)

x 2 2bx 4 0

x 2 2bx 4 0

(C)

(D)

x 2 bx 1 0

If x 2 x 1 is a factor of ax3 bx 2 cx d then the real root of ax3 bx 2 cx d 0 is : (A)

10.

log x 2 (B)

The values of ‘a’ for which the roots of the equation x 2 x a 0 are real and exceed ‘a’ are : (A) 0 < a < 1/4 (B) a < 1/4 (C) a < –2 (D)

(A) 9.

2

d a

(B)

d a

a d

(C)

(D)

None of these

(D)

b 0, c 0

If both the roots of the equation ax 2 bx c 0 are zero, then : (A)

b=c=0

VMC/Quadratic Equation

(B)

b = 0, c 0

b 0, c = 0

(C)

8

HWT/Mathematics

Vidyamandir Classes

DATE :

MARKS : 10

IITJEE :

NAME :

TIME : 25 MINUTES

TEST CODE : QE [9]

ROLL NO.

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END TIME :

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This test contains a total of 10 Objective Type Questions. Each question carries 1 mark. There is NO NEGATIVE marking. Choose the correct alternative. Only one choice is correct. 1.

If

x 2 bx 1 has roots equal in magnitude and opposite in sign then the value of is : ax c 1

(A)

2.

7.

(B)

15

(B)

a1c2 a2c1

2

–5, –30

If x 7 4 3 , x

a≤ 1

(C)

9

(B)

a > –3

(C)

a1a2 c1c2 2

(B)

7

a1c1 a2c2 2

(C)

–5, 30

(D)

a < –3 or a > 1

(D)

8

(D)

a1c2 c1a2 2

(C)

5, 30

(D)

None of these

(C)

–3 < a < –3

(D)

a < –2

(C)

3

(D)

2

, 1

(D)

(1, 3)

(C)

2

(D)

None of these

(C)

2 5

(D)

5 2

1 x

4

(B)

6

The set of values of x which satisfy 5 x 2 3 x 8 and (2, 3)

(B)

x 2 4 x 1

, 1 2, 3

is :

(C)

9 The number of real solutions of the equation x 3 x x 2 is : 10

(A) 10.

a≥ 1

If the roots of x 2 x a 0 exceed a, then : (A) 2 –3

(B)

a 0 or a < –3

(D)

None of these

The value of ‘k’ for which one of the roots of x 2 x 3k 0 , is double of one of the roots of x 2 x k 0 is : (A) 1 (B) –2 (C) 2 (D) None of these 3c If the equation ax 2 2bx 3c 0 has no real roots and a b , then : 4 (A) c0 (C) c≥0

(D)

c=0

(D)

1, 3

The value of x satisfying log3 x 2 4 x 12 2 are : (A)

6.

(B)

The values of ‘a’ for which the equation 2 x 2 2 2a 1 x a a 1 0 has roots and where < a < are such that : (A)

3.

1

2, 4

(B)

1, 3

1, 3

(C)

x2 x 1 cannot be between ____________ and _____________for any x R. x 1

7.

Two equations x 2 cx d 0 and x 2 ax b 0 have one root common and second equation has equal roots if ___________.

8.

If x 1 i is a root of equation x 3 ix 1 i 0 then other root is _______.

9.

For real x, the least value of 2 x 2 5 x 3 is ____________.

10.

If the roots of the equations ax 2 2bx c 0 and bx 2 2 ac x b 0 are simultaneously real, then b2 = ___________.

VMC/Quadratic Equation

10

HWT/Mathematics

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