﻿ Let α and β be the distinct roots of ax2 + bx + c = 0, then  equal to from Mathematics Complex Numbers and Quadratic Equations

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# Complex Numbers and Quadratic Equations

#### Multiple Choice Questions

11.

If the roots of the quadratic equation x2 + px + q = 0 are tan30° and tan15°, respectively then the value of 2 + q − p is

• 2

• 3

• 0

• 1

B.

3

x2 + px + q = 0
tan 30° + tan 15° = − p
tan 30° ⋅ tan 15° = q ⇒ − p = 1 − q
⇒ q − p = 1
∴ 2 + q − p = 3

145 Views

12.

If z2 + z + 1 = 0, where z is a complex number, then the value of • 18

• 54

• 6

E.

12

z2 + z + 1 = 0 ⇒ z = ω or ω2 ∴ The given sum = 1 + 1 + 4 + 1 + 1 + 4 = 12
109 Views

13.

If the coefficient of x7 in equals the coefficient of x-7 in then a and b satisfy the relation

• a – b = 1

• a + b = 1

• a/b =1

• ab =1

D.

ab =1 113 Views

14.

If both the roots of the quadratic equation x2 – 2kx + k2 + k – 5 = 0 are less than 5, then k lies in the interval

• (5, 6]

• (6, ∞)

• (-∞, 4)

• [4, 5]

C.

(-∞, 4) 418 Views

15.

If z1 and z2 are two non-zero complex numbers such that |z1 + z2| = |z1| + |z2| then argz1 – argz2 is equal to

• π/2

• 0

• -π/2

C.

0

z1 + z2| = |z1| + |z2| ⇒ z1 and z2 are collinear and are to the same side of origin; hence arg z1 – arg z2 = 0

128 Views

16.

If the equation anxn +an-1xn-1 +....... +a1x =0, a1 ≠ 0, n≥2, has a positive root x =  α, then the equation nanxn-1 + (n-1)an-1xn-2 +......+a1 = 0 has a positive root, which is

• greater than α

• smaller than α

• greater than or equal to α

• equal to α

B.

smaller than α

f(0) = 0, f(α) = 0
⇒ f′(k) = 0 for some k∈(0, α).

152 Views

17.

A body weighing 13 kg is suspended by two strings 5 m and 12 m long, their other ends being fastened to the extremities of a rod 13 m long. If the rod be so held that the body hangs immediately below the middle point. The tensions in the strings are

• 12 kg and 13 kg

• 5 kg and 5 kg

• 5 kg and 12 kg

• 5 kg and 13 kg

C.

5 kg and 12 kg  106 Views

# 18.Let α and β be the distinct roots of ax2 + bx + c = 0, then equal to 0  A.  100 Views

19.

All the values of m for which both roots of the equations x2 − 2mx + m2 − 1 = 0 are greater than −2 but less than 4, lie in the interval

• −2 < m < 0

• m > 3

• −1 < m < 3

• 1 < m < 4

C.

−1 < m < 3

Equation x2 − 2mx + m2 − 1 = 0
(x − m)2 − 1 = 0
(x − m + 1) (x − m − 1) = 0
x = m − 1, m + 1 − 2 < m − 1 and m + 1 < 4

m > − 1 and m < 3 − 1 < m < 3.

152 Views

20.

If x is so small that x3 and higher powers of x may be neglected, then may be approximated as

• • • • C.  227 Views