Mathematics Test 19

Result

Mathematics Test 19

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Weekly Quiz Competition

Question 1
4 / -1

If the lines 3x – 4y – 7 = 0 and 2x – 3y – 5 = 0 are two diameters of a circle of area 49π square units, the equation of the circle is

A
x² + y² – 2x + 2y – 62 = 0

B
x² + y² – 2x + 2y – 47 = 0

C
x² + y² + 2x – 2y – 47 = 0

D
x² + y² + 2x – 2y – 62 = 0

Solution

3x – 4y – 7 = 0
2x – 3y – 5 = 0
x = 1
y = –1
radius = 7
(x – 1)² + (x + 1)² = 49
⇒ x² + y² – 2x + 2y – 47 = 0
Question 2
4 / -1

Let α, β be real and z be a complex number. If z² + αz + β = 0 has two distinct roots on the line Re z = 1, then it is necessary that:

A
Β ∈ (0, 1)

B
Β ∈ (-1, 0)

C
|β| = 1

D
Β ∈ (1, ∞)

Solution

Let roots be 1 + ia and 1 - ia
So (1 + ia) + (1 - ia) = -α
and (1 + ia) (1 - ia) = β
⇒ β = 1 + a²
⇒ β ∈ (1, ∞)