JEE  >  Mathematics For JEE  >  Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) Download as PDF

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities)


Test Description

31 Questions MCQ Test Mathematics For JEE | Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities)

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) for JEE 2022 is part of Mathematics For JEE preparation. The Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) questions and answers have been prepared according to the JEE exam syllabus.The Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) MCQs are made for JEE 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) below.
Solutions of Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) questions in English are available as part of our Mathematics For JEE for JEE & Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) solutions in Hindi for Mathematics For JEE course. Download more important topics, notes, lectures and mock test series for JEE Exam by signing up for free. Attempt Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) | 31 questions in 60 minutes | Mock test for JEE preparation | Free important questions MCQ to study Mathematics For JEE for JEE Exam | Download free PDF with solutions
1 Crore+ students have signed up on EduRev. Have you?
Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 1

If α ≠ β but α2 = 5α – 3 and  β2 = 5β – 3 then the equation having α/β and β/α as its roots is [2002]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 1

  We have α2 = 5α – 3 and β2 = 5β – 3;
⇒ α & β are roots of equation,
x2 = 5x – 3 or x2 – 5x + 3 = 0
∴ α + β  = 5 and  αβ = 3

Thus, the equation havingas its roots is

= 0  or  3x2 – 19x +3 = 0

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 2

Difference between the corresponding roots of x2+ax+b=0 and x2+bx+a=0 is same and a ≠ b, then [2002]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 2

Let α, β and γ, δ be the roots of the equations
x2 + αx + β = 0 and
x2 + βx + α = 0 respectively.
∴ α + β = –α, αβ = β and γ + δ = –β, γ δ = α.
Given |α – β| = |γ – δ|
⇒ (α – β)2 = (γ – δ)2
⇒ (α + β)2 – 4αβ = (γ + δ)2 – 4γδ
⇒ α2  – 4β = β2 – 4α
⇒ (α2 – β2) + 4(α – β) = 0
⇒ α + β + 4 = 0 (∵ α ≠ β)

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 3

Product of real roots of the equation t2x2+|x|+9=0 [2002]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 3

Product of real roots = 

∴ Product of real roots is always positive.

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 4

If p and q are the roots of  the equation x2+px+q=0, then

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 4

p + q = – p and pq = q
⇒ q (p – 1) = 0 ⇒ q = 0 or p = 1.
If q = 0, then p = 0. i.e.p = q
∴ p = 1 and q = –2.

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 5

If a, b, c are distinct +ve real numbers and a2+b2+c2=1 then ab + bc + ca is [2002]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 5

∵ (a – b)2 +  (b – c)2 + (c – a)2 > 0
⇒ 2(a2 + b2 + c2 – ab – bc – ca) >0
⇒ 2 > 2(ab + bc + ca)
⇒ ab + bc + ca < 1

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 6

If the sum of the roots of the quadratic equation 2ax2 + bx +c= 0 is equal to the sum of the squares of  their reciprocals, then      are in [2003]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 6

 = 

As for given conditon, α + β =

On simplification 2a2c = ab2 + bc2

 are in A.P..

 are in H.P

 

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 7

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              [2003]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 7

Let the roots of given equation be α and 2α then

 

 = 

or  39a = 26 or 

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 8

The number of real solutions of the equation x2 - 3 x + 2 = 0 is

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 8

 No.of solution 4

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 9

The real number  x  when  added  to  its inverse gives  the minimum value of the sum at x equal to [2003]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 9

 or 

For max. or min., 

 = 2(+ve minima)            ∴x = 1

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 10

Let two numbers have arith metic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation[2004]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 10

Let two numbers be a and b then  and

∴ Equation with roots a  and b is

x2 - (a + b)x + ab=0 ⇒ x2 -18x + 16= 0

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 11

If (1- p) is a root of quadratic equation x2 + px + (1 -p)=0 th en its r oot are [2004]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 11

Let the second root be α.
Then α + (1 - p) =-p
⇒ α =-1 Also α .(1 - p) =1-p
⇒ (α - 1)(1- p) = 0
⇒ p = 1[∵α = -1]
∴ Roots are α = -1 and p - 1=0

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 12

If one r oot of th e equation x2 + px + 12=0 is 4, wh ile the equation x2 + px +q= 0 has equal r oots , then th e value of ‘q’ is[2004]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 12

4 is a root of x2 + px + 12=0
⇒ 16 + 4p +12 = 0
⇒ p =-7
Now, the equation x2 + px +q= 0 has equal roots.

∴ p2 - 4q = ⇒

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 13

In a triangle PQR, . If tan and – tan are the roots of ax2 + bx + c = 0, a ≠ 0 then [2005]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 13

are the roots of ax2 + bx +c= 0

⇒ – b = a – c   or   c = a + b.

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 14

If both the roots of the quadratic equation x2 - 2kx + k2 + k – 5 = 0 are less than 5, then k lies in the interval [2005]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 14

both roots are less than 5 then
(i) Discriminant ≥ 0
(ii) p(5) > 0
(iii) 

Hence (i ) 4k2– 4(k2 + k – 5) ≥ 0  
4k2 – 4k2 – 4k + 20 ≥ 04k ≤ 20
⇒ k ≤ 5

(ii) f(5) > 0 ; 25 – 10 k + k+ k – 5 > 0
or  k2 – 9k + 20 > 0
or  k (k – 4) –5(k – 4) > 0
or  (k – 5) (k – 4) > 0
⇒ k∈( –∞, 4 ) U ( –∞, 5)

(ii) 

The interection of (i), (ii) & (iii) gives k ∈ ( – ∞, 4 ).

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 15

If the roots of the quadratic equation x2 + px +q= 0 are tan30° and tan15°,  respectively, then the value of 2 + q – p is [2006]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 15

x2 + px +q= 0
Sum of roots = tan30° + tan15° = – p
Product of roots = tan30° . tan15° = q

 

⇒ – p =1-q ⇒ q-p=1
∴ 2+q-p=3

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 16

All the values of m for which both roots of the equation x2 - 2mx +m2 - 1=0 are greater than – 2 but less then 4, lie in the interval [2006]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 16

Equation x2 - 2mx +m2 - 1=0  
(x -m)2 -1=0 or (x -m+1)(x -m-1)= 0
x = m - 1,m+1
m – 1 > –2  and m + 1<4
⇒ m >- 1 and m < 3 or,,    -1 < m<3

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 17

If x is real, the maximum value of  is                        [2006]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 17

3x2(y -1) + 9x(y -1) + 7y -17=0

D ≥ 0 ∵ x is real

81( y - 1)2 - 4 x 3( y - 1)(7y - 17)≥ 0
⇒ ( y - 1)(y - 41) ≤ 0
⇒ 1 ≤ y ≤ 41
∴ Max value of y is 41

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 18

If the difference between the roots of the equation x2 + ax + 1 = 0 is less than , then the set of possible values of a is [2007]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 18

Let α and β are roots of the equation x2 + αx + 1 = 0
So, α + β = – α and αβ = 1
given

⇒ a2 – 9 < 0 ⇒ a2 < 9 ⇒ – 3 < a < 3
⇒ a ∈ (–3, 3)

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 19

Statement-1 : For every natural number n ≥ 2,

Statement-2 : For every natural number n ≥ 2,[2008]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 19

Statement 2 is 

 which is true

Now 

Also      ∴ Adding all, we get

Hence both the statements are correct and statement 2 is a correct explanation of statement-1.

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 20

The quadritic equations x2 –  6x  + a = 0 and x2 – cx + 6 = 0 have one root in common.  The other roots of the first and second equations are integers in the ratio 4 : 3. Then the common root is [2009]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 20

Let the roots of equation x2 – 6x + α = 0 be α and 4 β and that of the equation
x2 –cx + 6 = 0  be α and 3β .
Then α + 4β = 6 ; 4aβ = α and α + 3β = c ; 3αβ = 6
⇒ α = 8
∴ The equation becomes  x2 – 6x + 8 = 0
⇒ (x –2) (x – 4) = 0
⇒ roots are 2 and 4 ⇒ α = 2, β = 1
∴ Common root is 2.

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 21

If the roots of the equation bx2 + cx + a = 0 be imaginary, then for all real values of x, the expression 3b2x2 + 6bcx + 2c2 is : [2009]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 21

Given that roots of the equation bx2 + cx + a = 0 are imaginary
∴ c2 – 4ab < 0 ....(i)

Let y = 3b2x2 + 6 bc x + 2c2
⇒ 3b2x2 + 6 bc x + 2c2 – y = 0
As x is real, D ≥ 0
⇒ 36 b2c2 – 12 b2 (2c2 – y ) ≥ 0
⇒ 12 b2 (3 c2 – 2 c2+ y ) ≥ 0
⇒ c2 + y ≥ 0
⇒ y ≥ – c2
But from eqn.
(i), c2 < 4ab   or  – c2 > – 4ab
∴ we get y ≥ – c2 > – 4ab
⇒ y > – 4 ab

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 22

 If    then the maximum value of | Z | is equal to : [2009]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 22

Given that  

Now  |Z| = 

⇒ 

⇒ 

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 23

If α and β are the roots of the equation x2 – x + 1 = 0, then α2009 + β2009 = [2010]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 23

x2 -x + 1=0 

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 24

The equation esinx – e–sinx– 4 = 0 has : [2012]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 24

Given equation is esinx – e–sinx – 4 = 0
Put esin x = t in the given equation, we get t2 – 4t – 1 = 0

 ( ∵ t=esinx)

and 

So rejected                       So, rejected

Hence given equation has no solution.
∴ The equation has no real roots.

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 25

The real number k for which the equation, 2x3 + 3x + k = 0 has two distinct real roots in [0, 1] [JEE M 2013]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 25

f (x) = 2x3 + 3x + k
f'(x) = 6x2 + 3 > 0 ∀ x ∈R(∵ x2 > 0)
⇒ f(x) is strictly increasing function
⇒ f(x) = 0 has only one real root, so two roots are not possible.

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 26

The number of values of k , for which the system of equations : [JEE M 2013]
(k + 1) x + 8y = 4k
kx + (k + 3) y = 3k – 1
has no solution, is

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 26

From the given system, we have

(∵ System has no solution)

⇒ k2 + 4k + 3 = 8k
⇒ k = 1, 3
If  k = 1  then which is false

And if k = 3

then which is true, therefore k = 3

Hence for only one value of k. System has no solution.

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 27

If the equations x2 + 2x + 3 = 0 and ax2 + bx + c = 0, a,b,c ∈ R, have a common root, then a : b : c is [JEE M 2013]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 27

Given equations are x2 + 2x + 3 = 0 …(i)
ax2 + bx + c = 0 …(ii)
Roots of equation (i) are imaginary roots.
According to the question (ii) will also have both roots same as (i). Thus

  (say) 
Hence, required ratio is 1 : 2 : 3

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 28

If a ∈ R and the equation -3 ( x - [ x])2 + 2 ( x- [ x])+a 2 =0 (where [x] denotes the greatest integer ≤ x ) has no integral solution, then all possible values of a lie in the interval: [JEE M 2014]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 28

Consider –3(x – [x])2 + 2 [x – [x]) + a2 = 0
⇒ 3{x}2 – 2{x} –a2 = 0 (∵ x – [x] = {x})

Now, {x} ∈ (0,1) and (by graph)

Since , x is not an integer
∴ a ∈ (-1,1)- {0} ⇒ a ∈ (-1, 0)∪(0,1)

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 29

Let α and β be the roots of equation px2 + qx +r= 0, p ≠ 0. If p, q, r are in A.P. and  = 4  then the value of |α -β| is: [JEE M 2014]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 29

Let p, q, r are in AP ⇒ 2q = p + r ...(i)

Given  = 4 ⇒​

We have a + b = – q/p and ab = 

  4 ⇒​ q = -4r        ....(ii)

From (i), we have 2( – 4r) = p + r  
⇒  p = –9r
q = – 4r

Now 

 

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 30

Let α and β be the roots of equation x2 – 6x – 2 = 0. If an = αn – βn, for  n ≥ 1, then the value   is equal to :[JEE M 2015]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 30

Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 31

The sum of all real values of x satisfying the equation (x2 - 5 x+ 5) x2 +4x- 60 = 1 is : [JEE M 2016]

Detailed Solution for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) - Question 31

(x2 - 5 x + 5) x2 + 4x-60=1
Case I x2 – 5x + 5 = 1 and
x2 + 4x – 60 can be any real number
⇒ x = 1, 4
Case II x2 – 5x + 5 = –1 and
x2 + 4x – 60 has to be an even number
⇒ x = 2, 3 where 3 is rejected because for x = 3,
x2 + 4x – 60 is odd.
Case III x2 – 5x + 5 can be any real number and
x2 + 4x – 60 = 0
⇒ x = –10, 6
⇒ Sum of all values of x = –10 + 6 + 2 + 1 + 4 = 3

 

130 videos|359 docs|306 tests
Use Code STAYHOME200 and get INR 200 additional OFF
Use Coupon Code
Information about Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) Page
In this test you can find the Exam questions for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities) solved & explained in the simplest way possible. Besides giving Questions and answers for Test: 35 Year JEE Previous Year Questions: Quadratic Equation and Inequations (Inequalities), EduRev gives you an ample number of Online tests for practice