Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 3 - Theory of Equations>Exercise>Q 2

MEDIUM

12th Tamil Nadu Board

IMPORTANT

Earn 100

Prove that a straight line and parabola cannot intersect at more than two points.

Important Questions on Theory of Equations

MEDIUM

12th Tamil Nadu Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 3 - Theory of Equations>Exercise>Q 3

Solve the equation:

\sin^{2} x - 5 \sin x + 4 = 0

MEDIUM

12th Tamil Nadu Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 3 - Theory of Equations>Exercise>Q 4

Examine for the rational roots of:

2 x^{3} - x^{2} - 1 = 0

MEDIUM

12th Tamil Nadu Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 3 - Theory of Equations>Exercise>Q 5

Examine for the rational roots of

x^{8} - 3 x + 1 = 0

HARD

12th Tamil Nadu Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 3 - Theory of Equations>Exercise>Q 6

Solve the equation:

$x^{4} + 3 x^{3} - 3 x - 1 = 0$

HARD

12th Tamil Nadu Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 3 - Theory of Equations>Exercise>Q 7

Solve the equation

3 x^{3} - 26 x^{2} + 52 x - 24 = 0

if its roots form a geometric progression.

EASY

12th Tamil Nadu Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 3 - Theory of Equations>Exercise>Q 8

Find the quadratic equation whose sum of the roots is

\frac{4}{9}

and product of the roots is

9

.

EASY

12th Tamil Nadu Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 3 - Theory of Equations>Exercise>Q 9

Find the quadratic equation whose sum of the roots is

2 \sqrt{7}

and product of the roots is

8

.

HARD

12th Tamil Nadu Board

IMPORTANT

Mathematics Crash Course (Based on Revised Syllabus-2023)>Chapter 3 - Theory of Equations>Exercise>Q 10

Let

p_{1} (x) = x^{3} - 2020 x^{2} + b_{1} x + c_{1}

and

p_{2} (x) = x^{3} - 2021 x^{2} + b_{2} x + c_{2}

be polynomials having two common roots

α

and

β

. Suppose there exist polynomials

q_{1} (x)

and

q_{2} (x)

such that

p_{1} (x) q_{1} (x) + p_{2} (x) q_{2} (x) = x^{2} - 3 x + 2

. Then the correct identity is