Aptitude - Quadratic Equations
1.
Which of the following equations has real roots ?
Answer
Given equation is (x − 1)(2x − 5) = 0
⇒ 2x2 − 7x + 5 = 0
∴ D = (−7)2 − 4 x 2 x 5 = (49 − 40) = 9 > 0.
∴ Given equation has real roots.
Answer :
Option AExplanation :
Given equation is (x − 1)(2x − 5) = 0
⇒ 2x2 − 7x + 5 = 0
∴ D = (−7)2 − 4 x 2 x 5 = (49 − 40) = 9 > 0.
∴ Given equation has real roots.
2.
The roots of the equation x2 − 8x + 15 = 0 are :
Answer
x2 − 8x + 15 = 0 ⇒ x2 − 5x − 3x + 15 = 0
⇒ x(x − 5) − 3(x − 5) = 0
⇒ (x − 5)(x − 3) = 0 ⇒ x = 5 or x = 3
Answer :
Option BExplanation :
x2 − 8x + 15 = 0 ⇒ x2 − 5x − 3x + 15 = 0
⇒ x(x − 5) − 3(x − 5) = 0
⇒ (x − 5)(x − 3) = 0 ⇒ x = 5 or x = 3
3.
If a and b are the roots of the equation x2 − 6x + 6 = 0, then the value of (a2 + b2) are :
Answer
(a + b) = 6 and ab = 6
∴ (a2 + b2) = (a + b)2 − 2ab = (6)2 − 2 x 6 = (36 − 12) = 24
Answer :
Option CExplanation :
(a + b) = 6 and ab = 6
∴ (a2 + b2) = (a + b)2 − 2ab = (6)2 − 2 x 6 = (36 − 12) = 24
4.
A quadratic equation whose roots are 2 and −15 is given by :
Answer
α + β = 2 + (−15) = −13, αβ = 2 x (−15) = −30.
Required equation is x2 − (α + β)x + αβ = 0.
∴ x2 + 13x − 30 = 0
Answer :
Option BExplanation :
α + β = 2 + (−15) = −13, αβ = 2 x (−15) = −30.
Required equation is x2 − (α + β)x + αβ = 0.
∴ x2 + 13x − 30 = 0
5.
For what values of k, the equation x2 + 2(k − 4)x + 2k = 0 has equal roots ?
Answer
Since the roots are equal, we have D = 0.
∴ 4(k − 4)2 − 8k = 0 ⇒ (k − 4)2 − 2k = 0
⇒ k2 + 16 − 10k = 0 ⇒ k2 − 10k + 16 = 0
⇒ (k − 8)(k − 2) = 0 ⇒ k = 8 or k = 2
Answer :
Option BExplanation :
Since the roots are equal, we have D = 0.
∴ 4(k − 4)2 − 8k = 0 ⇒ (k − 4)2 − 2k = 0
⇒ k2 + 16 − 10k = 0 ⇒ k2 − 10k + 16 = 0
⇒ (k − 8)(k − 2) = 0 ⇒ k = 8 or k = 2
Jump to page number :