Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 4 - Theory of Equations>Exercise>Q 27

HARD

12th Telangana Board

IMPORTANT

Earn 100

If the equation $x^{5} - 10 a^{3} x^{2} + b^{4} x + c^{5} = 0$ has three equal roots, show that $a b^{4} - 9 a^{5} + c^{5} = 0$ .

Important Questions on Theory of Equations

MEDIUM

12th Telangana Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 4 - Theory of Equations>Exercise>Q 5

Two sportsmen

A

and

B

went out shooting and brought home

10

birds. The sum of the squares of the number of shots was

f (x) = 0

and the product of the number of shots fired by each was

48

times the product of the numbers of birds killed by each. If

A

had fired as often as

B

as often as

A,

then

B

would have killed

5

more birds than

A

. Find the number of birds killed by each.

EASY

12th Telangana Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 4 - Theory of Equations>Exercise>Q 9

Construct a cubic equation with roots

2, \frac{1}{2}

and

1

.

HARD

12th Telangana Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 4 - Theory of Equations>Exercise>Q 28

If the equation

x^{4} + a x^{3} + b x^{2} + c x + d = 0

has three equal roots, show that each of them is equal to

\frac{6 c - a b}{3 a^{2} - 8 b}

.

HARD

12th Telangana Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 4 - Theory of Equations>Exercise>Q 29

If

a, b, c

are the roots of

x^{4} + p x^{3} + q x^{2} + r x + s = 0,

find the value of

Σ a^{4}

.

MEDIUM

12th Telangana Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 4 - Theory of Equations>Exercise>Q 30

Solve the equation,

4 x^{3} + 20 x^{2} - 23 x + 6 = 0,

two of the roots being equal.

HARD

12th Telangana Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 4 - Theory of Equations>Exercise>Q 31

Show that the equation

x^{4} + p x^{3} + q x^{2} + r x + s = 0

may be solved as a quadratic if

r^{2} = p^{2} s

.

EASY

12th Telangana Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 4 - Theory of Equations>Exercise>Q 32

Solve the equation,

18 x^{3} + 81 x^{2} + 121 x + 60 = 0,

one root being half the sum of the other two.

EASY

12th Telangana Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 4 - Theory of Equations>Exercise>Q 33

If $a, b, c$ are the roots of the equation $x^{3} - p x^{2} + q x - r = 0$ ; find the value of $\frac{1}{a^{2}} + \frac{1}{b^{2}} + \frac{1}{c^{2}}$