Theory of Equations - Previous Year Questions
Previous Year Questions on Theory of Equations
Previous year questions on Theory of Equations provide students with a detailed understanding of equations and their roots. The Theory of Equations, rooted in Classical Algebra, has shaped mathematics since ancient times. Organized Mathematics Questions help students improve their problem-solving skills. Explore related topics like Mathematics Notes for comprehensive study materials.
Vidyasagar University
2023-24 (CBCS)
- Find the sum of \(99^{th}\) powers of the roots of the equations \(x^{7}-1=0\). [2]
- If \(\alpha\) be multiple root of the polynomial equation \(f(x)=0\) of order \(r\), then prove that \(\alpha\) is a multiple root of the equation \(f^{\prime}(x)=0\) of order \(r-1\). [2]
- Prove that the roots of the equations \(\frac{1}{x-1}+\frac{2}{x-2}+\frac{3}{x-3}=x \) are all real. [5]
- Solve the equation \(2x^{3}-9x^{2}+7x+6=0 \), given that two roots \(\alpha, \beta\) are connected by the relation \(2\alpha+\beta=1 \). [6]
- If \(\alpha, \beta, \gamma, \delta\) are the roots of the equation \(t^{4}+t^{2}+1=0\) and \(n\) is a positive integer, prove that \(\alpha^{2n+1}+\beta^{2n+1}+\gamma^{2n+1}+\delta^{2n+1}=0\). [5]
- Find the relation among the coefficients of the of the equation \(ax^{3}+3bx^{2}+3cx+d=0\) if its roots be in arithmetic progression. [3]
- Find the values of \(a\) for which the equation \(ax^{3}-6x^{2}+9x-4=0\) may have multiple roots. [2]
- If \(\alpha+ \beta+ \gamma=0\), then prove that \(\frac{\alpha^{5}+\beta^{5}+\gamma^{5}}{5}=\frac{\alpha^{3}+\beta^{3}+\gamma^{3}}{3}.\frac{\alpha^{2}+\beta^{2}+\gamma^{2}}{2} \). [4]
- If \(\alpha, \beta, \gamma\) be the roots of the equation \(x^{3}-2x^{2}+3x-1=0\), find the equation whose roots are \( \frac{\beta\gamma-\alpha^{2}}{\beta+\gamma-2\alpha}\), \( \frac{\gamma\alpha-\beta^{2}}{\gamma+\alpha-2\beta}\), \( \frac{\alpha\beta-\gamma^{2}}{\alpha+\beta-2\gamma}\). [4]
- Solve \((1+x)^{2n}+(1-x)^{2n}=0\). [5]
- If the roots of the equation \(x^{3}+px^{2}+qx+r=0\) are in A.P. where \(p,q,r\) are real numbers, prove that \(p^{2}\geq 3q\). [4]
- If the equation \(x^{4}+px^{2}+qx+r=0\) has three equal roots then show that \(8p^{3}+27q^{2}=0\). [2]
- Solve the equations \(x+py+p^{2}z=p^{3}\), \(x+qy+q^{2}z=q^{3}\), \(x+ry+r^{2}z=r^{3}\). [2]
- Find the equations whose roots are cubes of the roots of the cubic \(x^{3}+3x^{2}+2=0\). [2]
- If \(\alpha\) be a root of the equation \(x^{3}-3x-1=0\), prove that the other roots are \(2-\alpha, \alpha^{2}-\alpha-2\). [2]
- If the roots of the equation \(a_{0}x^{n}+na_{1}x^{n-1}+\frac{n(n-1)}{2!}a_{2}x^{n-2}+…+a_{n}=0\) be in A.P., Show that they can be determined from the expression \(-\frac{a_{1}}{a_{0}}\pm \frac{r}{a_{0}}\sqrt{\frac{3\left(a_{1}^{2}-a_{0}a_{1} \right)}{n+1}} \) by giving \(r=1,3,5,…,n-1\) when \(n\) is even and \(r=0,2,4,…,n-1\) \(n\) is odd. [8]
- For what integral values of \(m\), \(x^{2}+x+1\) is a factor of \(x^{2m}+x^{m}+1\)? [6]
- Solve the equation \(x^{4}-5x^{3}+11x^{2}-13x+6=0\) using the fact that two of its roots \(\alpha\) and \(\beta\) are connected by the relation \(2\alpha+2\beta=7\). [8]
- If \(\alpha, \beta, \gamma\) be the roots of \(x^{3}-qx+r=0\), find the equation whose roots are \( \frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}-\frac{1}{\gamma^{2}}, ~-\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}+\frac{1}{\gamma^{2}}, ~\frac{1}{\alpha^{2}}-\frac{1}{\beta^{2}}+\frac{1}{\gamma^{2}} \). Hence calculate the value of \( \left(\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}-\frac{1}{\gamma^{2}} \right)\left( ~-\frac{1}{\alpha^{2}}+\frac{1}{\beta^{2}}+\frac{1}{\gamma^{2}}\right)\left(\frac{1}{\alpha^{2}}-\frac{1}{\beta^{2}}+\frac{1}{\gamma^{2}}\right) \). [8]
- Solve the equation \(3x^{3}-26x^{2}+52x-24=0\) given that the roots are in geometric progression. [3]
- Show that the equation \((x-a)^{3}+(x-b)^{3} +(x-c)^{3}+(x-d)^{3}=0\), where \(a,b,c,d\) are not all equal, has only one real root. [5]
- If \(\alpha,\beta, \gamma\) be the roots of the equation \(x^{3}+px^{2}+qx+r=0\), then form the equation whose roots are \(\alpha+\frac{1}{\alpha}\), \(\beta+\frac{1}{\beta}\), \(\gamma+\frac{1}{\gamma}\). [5]
- Solve the equation \(x^{4}+12x-5=0\) by Ferrari’s method. [5]
- Apply Descartes’ rule of signs to ascertain the minimum number of complex roots of the equation \(x^{6}-3x^{2}-2x-3=0 \). [2]
- Solve the equation \(x^{3}-15x^{2}-33x+847=0 \) by Cardan’s method. [5]
- Solve the equation \(x^{4}-12x^{3}+47x^{2}-72x+36=0\) given that the product of the two of the roots is equal to the product of the other two.[4]
- State Descartes’ rule of signs. Obtain the equation whose roots exceed the roots of the equation \(x^{4}+3x^{2}+8x+3=0\) by \(1\). Use Descartes’ rule of signs to both he equations to find the exact number of real and complex roots of the given equation.[1+2+3]
- Solve the equation \(x^{5}+x^{4}+x^{3}+x^{2}+x+1=0 \). [2]
- Show that the condition that the sum of two roots of the equation \(x^{4}+mx^{2}+nx+p=0 \) be equal to the product of the other roots is \(\left(2p-n \right)^{2}=\left(p-n \right)\left(p+m-n \right)^{2} \). [5]
- If the equation whose roots are squares of the roots of the cubic \(x^{3}-ax^{2}+bx-1=0 \) is identical with this cubic, prove that either \(a=b=0\) or \(a=b=3\) or \(a,~b\) are the roots of the equation \(t^{2}+t+2=0\). [5]
- If the equation \(x^{3}+px^{2}+qx+r =0\) has a root \(\alpha+i~\alpha\) where \(p,q,r \) and \(\alpha \) are real, prove that \(\left(p^{2}-2q \right)\left(q^{2}-2pr \right)=r^{2} \). Hence solve the equation \(x^{3}-x^{2}-4z+24=0\). [5]
FAQs
-
What is the Theory of Equations?
It is the branch of algebra that deals with finding roots of polynomial equations and their properties. -
Why study previous year questions on Theory of Equations?
These questions help students understand exam patterns and strengthen their conceptual knowledge. -
What are the relations between roots and coefficients?
The sum and product of roots of a polynomial equation are directly related to its coefficients. -
Where can I find Theory of Equations Notes?
You can find them here. -
How does this topic apply in real life?
It is used in engineering, physics, and computational mathematics for modeling and analysis. -
What is Descartes’ rule of signs?
It determines the number of positive and negative real roots of a polynomial equation. -
Are there numerical methods for solving polynomial equations?
Yes, methods like Newton-Raphson and synthetic division are commonly used. -
Can Theory of Equations connect to other topics?
It is closely related to Complex Numbers and Inequalities. -
What is a polynomial root?
A root is a solution to a polynomial equation where the polynomial equals zero. -
Are solutions to previous year questions available?
Yes, solved solutions are often provided in books and academic resources.
Related Questions
Trending Today
Semeter-1 Mathematics Honours (Vidyasagar University)
[manual_filtered_papers subject="Math" university="VU" semester="1"]
Previous Year's Mathematics Honours (Vidyasagar University) Questions papers
[manual_filtered_papers subject="Math" university="VU"]
Categories
[categories]
Related Notes
Related Questions
Related Quizzes
No posts found!
