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Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

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Q.1. Solve for x : Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE          (1978)

Ans. 

Sol.  Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEESubjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEESubjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEESubjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE x = 3/2

 

Q.2. If (m , n) = Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE (1978)
 where m and n are positive integers (n ≤ m), show that (m, n + 1) = (m – 1, n + 1) + xm – n – 1 (m – 1, n).

Ans. 

Sol. RHS = (m – 1, n + 1) + xm – n –1 ( m - 1,n)

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

=  (m, n +1) = L.H.S. Hence Proved

 

Q.3. Solve for x :Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE (1978)

Ans. 

Sol. Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Squaring both sides, we get

x +1 = 1+ x -1+ Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ 1 = 4 (x – 1)

⇒ x  = 5/4

 

Q.4. Solve the following equation for x : (1978)

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. 

Sol.  Given a > 0, so we have to consider two cases :

a ≠ 1and a = 1.  Also it is clear that x > 0 and  x ≠ 1, ax ≠ 1, a2x ≠ 1.

Case I : If a > 0, ≠ 1

then given equation can be simplified as

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Putting loga x = y, we get

2 (1 +  y) (2 + y) + y (2 + y) + 3y (1 + y) = 0

⇒ 6y2 +11y + 4 = 0 ⇒y = -4/3 and-1/2

⇒ loga x =-4 / 3 and logax =-1/2

⇒ x = a -4/3 and x=a-1/2

Case II : If a = 1 then equation becomes

2 logx1 + logx1 + 3 logx1 = 6 logx1 = 0

which is true ∀ x > 0,≠1

Hence solution is  if  a = 1, x > 0, ≠ 1

if a > 0, ≠ 1 ; x= a-1/2,a-4/3

Q.5. Show that the square of Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  is a rational number.                      (1978)

Ans. Sol.  

Let Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEwhich is a rational number..

 

Q.6. Sketch the solution set of the followin g system of inequalities: x2 + y2 – 2x ≥ 0; 3x – y – 12 ≤ 0; y – x ≤ 0; y ≥ 0. (1978)

Ans. 

Sol. Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ (x – 1)2 + y2 ≥ 1 which represents the boundary and exterior region of the circle with centre at (1,0) and radius as 1.
For 3x – y ≤ 12, the corresponding equation is 3x – y = 12; any two points on it can be taken as (4, 0), (2, – 6). Also putting (0, 0) in given  inequation, we get 0 ≤ 12 which is true.

∴  given inequation represents that half plane region of line 3x – y = 12 which contains origin.
For y ≤ x, the corresponding equation y = x has any two points on it as (0, 0) and (1, 1). Also putting (2, 1) in the given i nequation, we get 1 ≤ 2 wh ich is true, so y ≤ x represents that half plane which contains the points (2, 1). y ≥ 0 represents upper half  cartesian plane.
Combining all we find the solution set as the shaded region in the graph.

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

 

Q.7. Find all integers x for which (1978)
 (5x – 1) < (x + 1)2 < (7x – 3).

Ans. 

Sol. There are two parts of this question (5x – 1) < (x + 1)2  and  (x + 1)<  (7x – 3)

Taking first part (5x -1) < (x +1)2 ⇒ 5x -1 < x2 + 2x+1

⇒ x2 - 3x + 2 > 0 ⇒ (x-1)(x - 2)>0

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE       (using wavy method)

⇒ x < 1  or x > 2 ....(1)

Taking second part

(x +1)2 < (7x -3) ⇒ x2 - 5x +4< 0

⇒ ( x - 1)(x - 4)<0

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE         (using wavy method)

⇒ 1 <  x  < 4 ....(2)

Combining (1) and (2) [taking common solution], we get 2 < x < 4 but x is an integer therefore x = 3.

 

Q.8. If α, β are the roots of x+ px + q = 0 and γ, δ are the roots of x2 + rx + s = 0, evaluate (α - γ) (α - δ) (β - γ) (β - δ) in terms of p, q, r and s.
 Deduce the condition that the equations have a common root.
 (1979)

Ans.

 Sol. ∵ α, β are the roots of x2 + px + q = 0
∴  α + β = – p,     αβ = q
∵  γ,δ are  the roots of x+ rx + s = 0
∴ γ + δ=-r, γδ=s

Now, (α - γ)(α -δ)(β- γ)(β-δ

= [α2 - (γ + δ)α + γδ][β2 - (γ + δ)β + γδ]

= [α2 + rα+ s][β2 + rβ+s]

[∵ α , β are roots of x2 + px + q = 0

∴ α2 + pα + q = 0 and β2 + pβ +q= 0]

= [(r - p)α + (s- q)][(r - p)β + ( s-q)]

= (r - p)2 αβ + (r- p)( s -q)(α + β) + ( s-q)2

= q(r - p)2 - p(r - p)(s -q) + (s-q)2

Now if the equations x2  +  px + q = 0 and x2 + rx + s = 0 have a common root say α, then α2 + pα + q = 0 and α2 + rα + s = 0

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ (q – s )2 = (r – p )(ps – qr) which is the required condition.

 

Q.9. Given n4 < 10n for a fixed positive integer n ≥ 2,  prove that (n + 1)4 < 10n + 1. (1980)

Ans. 

Sol. Given that n4 < 10n for a fixed  + ve integer n ≥ 2.

To prove that (n + 1)4 < 10n + 1

Proof : Since  n4 < 10n ⇒ 10n4 < 10n +1 ....(1)

So it is sufficient to prove that (n + 1)4 < 10n4

Now Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEESubjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ (n + 1)4 < 10n4 .... (2)

From  (1) and (2), (n +1)< 10n + 1

 

Q.10. Let  y = Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE         (1980)
 Find  all the real values of x for which y takes real values.

Ans. Sol. 

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

y will take all real values if   Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

By wavy method

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

[2 is not included as it makes denominator zero, and hence y an undefined number.]

 

Q.11. For what values of m, does the system of equations
 3x + my = m
 2x – 5y = 20
 has solution satisfying the conditions x > 0, y > 0. (1980)

Ans. Sol.  The given equations are 3x + my – m = 0 and 2x – 5y – 20 = 0 Solving these equations by cross product method, we get

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE NOTE THIS STEP

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

For Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  ....(1)

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE                   

For  Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE....(2)

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Combining (1) and (2), we get the common values of m as follows :

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEESubjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

 

Q.12. Find the solution set of the system (1980)
 x + 2y + z = 1;
 2x – 3y – w = 2;

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. Sol. The given system is
x + 2y + z = 1 ....(1)
2x - 3y - ω =2 ....(2)
where  x, y, z, ω ≥ 0
Multiplying eqn. (1) by 2 and subtracting from (2), we get

7y + 2z + ω = 0 ⇒ ω = - (7 y+ 2z)

Now if y, z > 0, ω < 0 (not possible)

If  y = 0, z = 0 then x = 1 and ω = 0 .

∴ The only solution is x = 1, y = 0, z = 0, ω = 0 .

 

Q.13. Show that the equation esin x –e – sinx – 4 =0 has no real solution. (1982 - 2 Marks)

Ans.

Sol.  esin x - e- sinx - 4=0

Let esin x = y then e - sinx = 1/y

∴ Equation becomes,Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ y2 – 4y – 1 = 0  ⇒Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

But y is real +ve number,

∴   Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ esin x = 2+ Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  ⇒ sin x = loge (2+Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE)

ButSubjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEHence, sin x > 1
Which is not possible.

∴ Given equation has no real solution.

 

Q.14. mn squares of euqal size are arranged to from a rectangle of dimension m by n,  where m and n are natural numbers.Two squares will be called ‘neighbours’ if they have exactly one common side. A natural number is written in each square such that the number written in any square is the arithmetic mean of the numbers written in its neighbouring squares.
 Show that this is  possible only if all the numbers used are equal. (1982 - 5 Marks)

Ans. Sol. For any square there can be at most 4, neighbouring squares.

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Let for a square having largest number d, p, q, r, s be written then

According to the question, p + q + r + s = 4d

⇒ (d – p) + (d – q) + (d – r) + (d – s) = 0

Sum of four +ve numbers can be zero only if these are zero individually

∴ d – p = 0 = d – q = d – r = d – s

⇒ p = q = r = s = d

⇒ all the numbers written are same.
Hence Proved.

 

Q.15. If one root of the quadratic equation ax2 + bx + c = 0 is equal to the n-th power of the other, then show that

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE (1983 - 2 Marks)

Ans. Sol. 

Let α, β be the roots of eq. ax2 + bx + c = 0

According to the question, β = an

Also α + β = – b/a  ; αβ = c/a

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEESubjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

then α + β = – b/a ⇒ Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

or Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Hence Proved.

 

Q.16. Find all real values of x which satisfy x2 - 3x + 2 > 0 and x2 - 2 x - 4 ≤ 0 (1983 - 2 Marks)

Ans. Sol.  x2 -3x+ 2 >0, x2 -3x- 4 ≤ 0

⇒ (x – 1) (x – 2)  > 0 and (x – 4) (x + 1) < 0

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  andx ∈ [-1, 4]

∴ Common solution is Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

 

Q.17. Solve for x;   Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE = 10  (1985 - 5 Marks)

Ans. Sol. The given equation is

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE....(1)

Let      Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE....(2)

then  Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEESubjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE(Using (2))

∴ The given equation (1) becomes  Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE = 10

⇒ y2 - 10y + 1=0 ⇒ Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ y = 5 + Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Consider, y = 5+  Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ x2 – 3 = 1 ⇒ x2 = 4  ⇒ x = ± 2

Again consider

y = 5 - Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒  Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE⇒ x2 – 3 = – 1

⇒ x2 = 2  ⇒ x  = ±Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Hence the solutions are 2, – 2, Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE, -Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

 

Q.18. For a ≤ 0, determine all r eal roots of the equation x2 - 2a | x - a | - 3a= 0 (1986 - 5 Marks)

Ans. 

Sol. The given equation is, x2 – 2a | x – a | – 3a= 0

Here two cases are possible.
Case I : x – a > 0 then | x – a | = x – a

∴ Eq. becomes x2 – 2a (x – a) – 3a= 0

or x2 – 2ax – a2 = 0 ⇒  Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ x = a±aSubjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Case II : x – a < 0 then | x – a | = – (x – a)
∴ Eq. becomes
x2 + 2a (x – a) – 3a2 = 0

or x2 + 2ax – 5a2 = 0  ⇒  Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

  ⇒Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  ⇒Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Thus the solution set is Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

 

Q.19. Find the set of all x for which Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE (1987 - 3 Marks)

Ans. Sol. We are given  Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒   Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒  Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒  Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEESubjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒  Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ (3x + 2) (x + 1) (x + 2) (2x + 1) < 0 ....(1)
NOTE THIS STEP 

: Critical pts are x = – 2/3,  –1, – 2, – 1/2
On number line

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Clearly Inequality (1) holds for,

x ∈(– 2, – 1) ∪ (– 2/3, – 1/2)
[asx ≠ -2, -1, -2 / 3,-1 / 2]

 

Q.20. Solve | x2 + 4 x + 3 | +2x + 5=0 (1988 - 5 Marks)

Ans. Sol.  The Given equation is, | x2 + 4x + 3  | + 2x + 5 = 0 Now there can be two cases.

Case I : x+ 4x + 3≥0 ⇒ (x +1)(x + 3)≥0 ⇒ x ∈ (-∞, - 3] ∪ [-1,∞) ....(i)

Then given equation becomes,

⇒ x2 + 6x + 8=0

⇒ ( x + 4)(x + 2)= 0 ⇒ x  = – 4, – 2

But x = – 2 does not satisfy (i), hence rejected

∴ x = – 4 is the sol.

Case II :   x2 + 4x + 3 < 0

⇒ (x + 1) (x + 3) < 0

⇒ x ∈ (– 3, – 1) ....(ii)

Then given equation becomes, – (x2 + 4x + 3) + 2x + 5 = 0

⇒ – x– 2x + 2 = 0 ⇒ x2 + 2x – 2 = 0

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE ⇒  Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Out of which    x = – 1 – Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE  is sol.
Combining the two cases we get the solutions of given equation as      x = – 4, – 1–Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

 

Q.21. Let a, b, c be real. If ax2 + bx + c = 0 has two real roots α and β, where α < -1 and β > 1 , then show that

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE (1995 -  5 Marks)

Ans. Sol. Given that for a, b, c ∈ R, ax2 + bx + c = 0 has two real roots a and b, where a < – 1 and b > 1. There may be two cases depending upon value of a, as shown below.
In each of cases (i) and (ii) af (–1) < 0 and af (1) < 0

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ a (a – b + c) < 0 and a (a + b + c) < 0 Dividing by a2 (> 0), we get

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE       ....(1)

and  Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE....(2)

Combining (1) and (2) we get

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEEHence Proved.

 

Q.22. Let S be a square of unit area. Consider any quadrilateral which has one vertex on each side of S. If a, b, c, and d denote the lengths of the sides of the quadrilateral, prove that 2 ≤ a2+b2+c2+d2 ≤ 4. (1997 - 5 Marks)

 Ans. 

Sol. a2 = p2 + s2, b2 = (1– p)2 + q2
c2 = (1– q)2 + (1 – r)2,  
d2 = r2 + (1– s)
∴ a2 + b2 + c2 + d2 = {p2 + (1– p)2}+{q2 – (1– q)2}
+ { r2 + (1– r)2 } +{s2 + (1– s)2 }

where p, q, r, s all vary in the interval [0, 1].
Now consider the function

y2 = x2 + (1– x)2, 0 ≤ x≤ 1,

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Hence y is minimum at x = Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE and its minimum

value is Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Clearly value is maximum at the end pts which is 1.

∴ Minimum value of a2 + b2 + c2 + d2 =Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE and maximum value is 1 + 1+ 1 + 1= 4. Hence proved.

 

Q.23. If α, β are the roots of ax2 + bx + c = 0, ( a ≠ 0 ) and α + δ, β + δ are the roots of Ax2 + Bx + C = 0,  ( A ≠ 0) for some constant δ, then prove that Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE (2000 - 4 Marks)

Ans. Sol. We know that, (α-β)2 = [(α+δ) - (β +δ)]2

⇒ (a+β)2 - 4αβ = (α +δ+β+δ)2 -4(α+δ)(β +δ)

⇒ Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEESubjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

[Here α + β=Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

(α+δ) (β +δ)

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

Hence proved.

 

Q.24. Let a, b, c be real numbers with a ≠ 0 and let α, β be the roots of the equation ax2 + bx + c = 0. Express the roots of a3x2 + abcx + c3 = 0 in terms of α, β. (2001 - 4 Marks)

Ans. Sol. Divide the equation by α3, we get

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE

⇒ x2 – (α + β). (αβ) x + (αβ)3 = 0
⇒ x2 – α2βx – αβ2 x  + (αβ)3 = 0
⇒ x (x – α2β) – αβ2 (x – α2β) = 0
⇒ (x – α2β) (x – αβ2) = 0
⇒ x = α2 β, αβ2
which is the required answer.

 

Q.25. If x2 + (a – b) x + (1 – a – b) = 0 where a, b ∈ R then find the values of a for which equation has unequal real roots for all values of b. (2003 - 4 Marks)

Ans. Sol. The given equation is, x2 + (a – b) x + (1– a – b) = 0, a, b ∈ R
For this eqn to have unequal real roots ∀ b D > 0
⇒ (a – b)2 – 4 (1– a – b) > 0
⇒ a2 + b– 2ab – 4 + 4a + 4b > 0
⇒ b2  + b (4 – 2a) + a2 + 4a – 4 > 0
Which is a quadratic expression in b, and it will be true ∀ b∈R if discriminant of above eqn less than zero.

i.e., ( 4 –2a)2  – 4 (a+ 4a – 4) < 0
⇒ (2 – a)2 – (a+ 4a – 4) < 0
⇒  4 – 4a + a2 – a2 – 4a + a < 0
⇒  – 8a + 8 <  0
⇒ a > 1

 

Q.26. If a, b, c are positive real numbers. Then prove that (a + 1)7 (b + 1)7 (c + 1)7>77 a4b4c4 (2004 - 4 Marks)

Ans. Sol. Given that a, b, c are positive real numbers. To prove that (a + 1)7 (b + 1)7 (c + 1)7  > 7 7 a4b4c
Consider L.H.S. = (1+ a)7. (1+ b)7. (1+ c)7                        
=  [(1 + a) (1+ b) (1+ c)]7[1 + a + b + c + ab + bc + ca + abc]
> [a + b + c + ab + bc + ca + abc]7 ....(1)
Now we know that AM ≥ GM using it for +ve no’s a, b, c, ab, bc, ca and abc, we get

Subjective Type Questions: Quadratic Equation and Inequations (Inequalities) | JEE Advanced Notes | Study Maths 35 Years JEE Main & Advanced Past year Papers - JEE
 ⇒ (a + b + c + ab + bc + ca+ abc)7 ≥ 77 (a4b4c4)a
From (1) and (2),
we get [(1 + a) (1 + b) ( 1 + c)]7  > 77a4 b4 c4
Hence Proved.

 

Q.27. Let a and b be the roots of the equation x2 – 10cx – 11d = 0 and those of x2 – 10ax – 11b = 0 are c, d then the value of a + b + c + d, when a ≠ b ≠ c ≠ d, is. (2006 - 6M)

Ans. Sol. Roots of   x2 – 10cx – 11d = 0 are a and b ⇒ a + b = 10c and ab = – 11d
Similarly c and d are the roots of x2 – 10ax – 11b = 0
⇒ c + d = 10a and cd = – 11b
⇒ a + b + c + d = 10 (a + c) and abcd = 121 bd
⇒ b + d = 9(a + c) and ac = 121
Also we have a2 – 10 ac – 11d = 0 and c2 – 10ac – 11b = 0
⇒ a2 + c2 – 20ac – 11 (b + d) = 0
⇒ (a + c)2 – 22 × 121 – 99 (a + c) = 0
⇒ a + c = 121 or – 22 For a + c = – 22,
we get a = c
∴ rejecting this value we have a + c = 121
∴ a + b + c + d =10 (a + c) = 1210

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