Equations and Inequalities Test

Result

Equations and Inequalities Test - 21

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Weekly Quiz Competition

Question 1
4 / -1

E₁ : a + b + c = 0 , if 1 is root of aχ² + bχ + c = 0, E₂ : b² - a² = 2ac, if sin θ and cos θ are the roots of aχ² + bχ + c = 0, which of the following is correct ?

A
E₁ is correct E₂ is correct

B
E₁ is correct E₂ is incorrect

C
E₁ is incorrect E₂ is correct

D
E₁ is incorrect E₂ is incorrect

Solution
Question 2
4 / -1

If a, b and c are distinct positive real numbers in AP, then the roots of the equation aχ² + 2bχ + c = 0 are

A
imaginary

B
rational and equal

C
rational and distinct

D
irrational

Solution
Question 3
4 / -1

If a = cos(2π/7) + i sin(2π/7), then the quadratic equation whose roots are α = a + a² + a⁴ and β = a³ + a⁵ + a⁶, is

A
χ² - χ + 2 = 0

B
χ² + 2χ + 2 = 0

C
χ² + χ + 2 = 0

D
χ² + χ - 2 = 0

Solution
Question 4
4 / -1

If the roots of the equation ⋋² +8⋋ + μ² + 6μ = 0 are real, then μ lies between

A
- 2 and 8

B
- 3 and 6

C
- 8 and 2

D
- 6 and 3

Solution
Question 5
4 / -1

Let p and q be real numbers. If α is the root of χ² + 3p² χ + 5q² = 0, β is the root of χ² + 9p²χ + 15q² = 0 and 0 < α< β, then the equation χ² + 6p²χ + 10q² = 0 has root γ that always satisfies

A

B
β < γ

C

D
α < β < γ

Solution
Question 6
4 / -1

If the equations x² + 2χ + 3 = 0 and aχ² + bχ + c = 0; a, b, c ∈ R, have a common root, then a : b : c is equal to

A
1: 2 : 3

B
3 : 2 : 1

C
1 : 3 : 2

D
3 : 1 : 2

Solution
Question 7
4 / -1

Let a, b and c be real numbers, a ≠ 0, if α ≠0. If α is a root of a²x² + bx + c = 0, β is a root of a²χ² - bχ - c = 0 and 0 < α< β. Then, the equation a²χ² + 2bχ - c = 0 has a root γ that always satisfies

A
γ = a

B
α < β < γ

C
α < γ < β

D

Solution
Question 8
4 / -1

A value of b for which the equations χ² + bχ - 1 = 0, χ² + χ + b = 0 have one root in common, is

A
- √2

B
- i√3

C
i√5

D
√2

Solution
Question 9
4 / -1

If α and β are the roots of the equation aχ² + bχ + c = 0, then the equation whose roots are k/α and k/β is

A
cχ² + kbχ + k²a = 0

B
cχ² + k²bχ + ka = 0

C
kcχ² + bχ + k²a = 0

D
k²cχ² + bχ + ka = 0

Solution
Question 10
4 / -1

If both the roots of the equation χ² - 6aχ + 2 – 2a + 9a² = 0 exceed 3, then

A
a < ½

B
a > ½

C
a < 1

D
a > 11/9

Solution