Jee 2014 Booklet1 Hwt Quadratic Equations & Inequations

August 28, 2017 | Author: varunkohliin | Category: Quadratic Equation, Zero Of A Function, Equations, Mathematical Concepts, Mathematical Analysis

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Jee 2014 Booklet1 Hwt Quadratic Equations & Inequations...

Description

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)

1b q     2 a p

(C)

1b q    2a 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

(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 :

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.

(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 =

abc 0

(B)

abc 0

abc0

(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

(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)

a2

1 5 a2 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

a2

(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

(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 :

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 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

(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 :

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.

If the equation (A)

2.

(C)

1

(D)

None of these

pq

(B)

qr

rq

(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 : xa xb

5, –4

(B)

5, 4

(C)

–5, 4

If the equation x 2  px  q  0 and x 2  px  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

(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) ab (B) ab 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

a0 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.

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 :

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.

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 abc

(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

(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

(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 :

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.

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 = ___________.