Test: Quadratic Equations - JEE MCQ

For Real roots
D = b² -4ac ≥ 0
or b² ≥ 4ac
For eqn(1); k² ≥ 4.1.64 ⇒ k ≥ 16 ……….(I)
For eqn (2) (-8)² ≥ 4.1.k ⇒ k ≤ 16 ……….(II)
Clearly k = 16 satisfies (I) & (II)
∴ (b) is correct.

Let 1st root = 1
Then 2nd root = 2
The Eqn. is x² – (1 + 2) + 1 × 2 = 0
or x² – 3x + 2 = 0
Comparing it with ax² + bx + c = 0 we get;
a = 1; b = -3 ; c = 2
Go by choices (GBC)
(a) 2b² = 3ac
2.(-3)² =3.1.2 (False)
(c) 2b² =9.ac
2. (-3)² = 9.1.2
⇒ 18 = 18 (True)
Hence, Option (c) is (true)

∵ Roots are in A.R
Let roots are a – d; a; a + d
So, (a – d)+a + (a + d) = 15
or; 3a = 15
or; a = 5
And Product of roots
(a – d ). a . (a + d ) = 45
or (5 – d);5. (5 + d) = 45
or 25 – d² = 9
or; d² = 25 – 9 = 16
or; d = √16 = 4
Hence; roots are
a – d, a, a + d = 5 – 4; 5; 5 + 4
= 1; 5 ; 9.
The value of K
= Sum of product of two roots in a order
= (1 × 5) + (5 × 9) + (9 × 1)
= 5 + 45 + 9 = 59
(b) is correct.

∵ x² + 3x – 4 = 0
or; x² – 4x + x – 4 = 0
or; x(x – 4) + 1(x – 4) = 0
or; (x – 4)(x + 1) = 0
x = 4; -1
Eqn. having roots 1/2 & 1/−1 = 1/4 & – 1 is.
or x² – (1/4 – 1) + 1/4(-1) = 0
or x² + 3/4x – 1/4 = 0
Multiplying by 4 ; we get
4x² + 3x -1 = 0
Comparing it with kx² + 3x -1 = 0
We get K = 4
Tricks : Eqn. having roots the reciprocal of the roots of ax² + bx + c = 0 is cx² + bx +a = 0 i.e. 1st and last term interchanges.

Sum of the roots = p + q = −p and product of the roots pq = q
∴ p = 1 [q ≠ 0]
∴1 + q = −1 ⇒ q = −2
∴ p = 1,q = −2

let α, β are roots of x² – kx + 8 = 0
∴ α + β = -b/a = −(−k)/1 = k & α. β = c/a = 8/1 = 8
(α – β)² = (α + β)² – 4αβ = 4²
⇒ k² – 4 × 8 = 16
or k² = 48 ⇒ k = ±√16×3 ⇒ k = ±4√3
(d) is correct.

The given equation can be written as x² − 2(a + b)x + 3ab = 0
Since roots are equal in magnitude and opposite in sign.
∴ Sum of roots =0
∴a + b = 0

x² + ax + c = (x − γ)(x − δ)
Thus,

We have
(x − a₁)² + (x − a₂)² +….. +(x − a_n)² = nx² − 2x(a₁ + a₂ + …. + a_n) + (a²₁ + a²₂ + ….. + a²_n)
Clearly, y = nx² − 2x (a₁ + a₂ + …… + a_n) + (a²₁ + a₂² + …….. + a²_n) represents a parabola which opens upward. So, it attains its minimum value at the vertex i.e. at

As −b/2a = 3/2 ∉ (0,1), so both roots can not lie between 0 and 1 . So no values of k possible.