Quadratic Equations

1 )

If 81 is the discriminant of 2x² + 5x – k = 0 then the value of k is_________

-7

View Solution

Answer :

2x² + 5x – k = 0

Δ = b² – 4ac = 81

∴ 25 + (4 × 2 × k) = 81

∴ 8k = 56

∴ k = 7

2 )

Discriminant of the quadratic equation √5x² – 3√3x – 2√5 = 0 is_______

-74

-67

View Solution

Answer :

Discriminate = Δ = b² – 4ac

Here b = – 3 √3, a = √5, c = – 2√5

∴ Δ = (– 3 √3)² – 4(√5) (– 2 √5)

= 27 + 40

Δ = 67

3 )

The value of k for which the quadratic equation k x² + 1 = k x + 3 x – 11 x² has real and equal roots are

{– 11, – 3}

{5, 7}

{5, – 7}

{– 5, – 7}

View Solution

Answer :

kx² + 1 = kx + 3x – 11x²

∴ (k + 11)x² – (k + 3)x + 1 = 0

∵ Δ = b² – 4ac

Δ = [– (k + 3)]² – 4(k + 11)(1)

= k² + 6k + 9 – 4k – 44

= k² + 2k – 35

Roots are real and equal

∴ Δ ≥ 0

∴ k² + 2k – 35 ≥ 0

∴ k² + 2k + 1 ≥ 36

|k + 1|² ≥ 36

|k + 1| ≥ 6

∴ – 6 ≥ (k + 1) ≥ 6

∴ – 7 ≥ k ≥ 5

i.e. k = – 7 and k = 5

∴ k ∈ {– 7, 5}

4 )

If the sum of the roots of ax² + bx + c = 0 is equal to the sum of the squares of their reciprocals then bc², ca², ab² are in

A.P

G,P

H.P

None of these

View Solution

Answer :

ax² + bx + c = 0

Let α, β are roots of equation

α + β = {(– b ) / a} αβ = (c/a)

α + β = (1 / α²) + (1 / β²)

∴ α + β = [{(α + β)² – 2αβ} / {α² β²}]

∵ {(– b) / a} = [({(– b) / a}² – 2 (c/a)) / (c / a)²]

∴ {(– b) / a} = [{(b² / a²) – (2c / a)} / {c² / a²⁰}]

∴ {(– b) / a} = [(b² – 2ac) / c²]

∴ – bc² = ab² – 2a²c

∴ 2a²c = ab² + bc²

a²c = [(ab² + bc²) / 2]

∴ ab², a²c and bc² are in arithmetic progression.

5 )

If the equation x² – m (2x – 8) – 15 = 0 has equal roots then m = ________

3, – 5

– 3, 5

3, 5

– 3, – 5

View Solution

Answer :

x² – m(2x – 8) – 15 = 0

∴ x² – 2mx + 8m – 15 = 0

roots are equal hence Δ = 0

∴ b² – 4 ac = 0

∴ (– 2m)² – 4(1) (8m – 15) = 0

∴ 4m² – 32m + 60 = 0

∴ m = 5, m = 3

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