Theory of Equations
"Art of mathematics consists finding the special case which contains all germs of generality." -David Hilbert
In mathematics, the Theory of Equations comprises a major part of traditional algebra. Topics include polynomials, algebraic equations, separation of roots including Sturm's theorem, approximation of roots, and the application of matrices and determinants to the solving of equations.
From the point of view of abstract algebra, the material is divided between symmetric function theory, field theory, Galois theory, and computational considerations including numerical analysis.
The first chapter Systems of Equations is basically about Linear Algebra, that is the study of multivariable equation systems through matrices and vector spaces. It extends to complete study Matrices and related topics of Determinants, Eigenvalues,etc. along with their applications in equation systems.
The second chapter Polynomial Functions is about the study of polynomials and their properties such as Roots and Determinants, Coefficients and Symmetric Function of Roots, Galois Groups, Derivatives, Maxima and Minima, etc.
The third chapter Functional Equations
Chapters
[edit | edit source]=Textbooks
[edit | edit source]Practice Questions
[edit | edit source]1. Let a, b be the roots of the equation and be the roots of the equation ew. Then, the value of r is
2. If a, b, c, are the sides of a triangle ABC such that has real roots. then
3. If one root is square of the other root of the equation , then the relation between p and q is
4. For all ‘x’ , x2 + 2ax + (10–3a) > 0, then the interval in which ‘a’ lies is
(a) a < – 5
(b) –5 < a < 2
(c) a > 5
(d) 2 < a < 5
[IIT JEE 2004]
5. The set of all real numbers x for which is
6. For the equation , if one of the root is square of the other, the n p is equal to
(a) 1/3
(b) 1
(c) 3
(d) 2/3
[IIT JEE 2000]
7. If b > a, then the equation (x – a) (x – b) – 1 = 0 has
(a) Both roots in (a,b)
(b) Both roots in (– ¥ ,a)
(c) Both roots in (b, + ¥)
(d) One root in (– ¥, a) and the other in (b, ¥)
[IIT JEE 2000]
8. If a and b (a < b) are the roots of the equation , where c < 0 < b, then
9. If the roots of the equation are real and less than 3, then
(a) a < 2
(b) 2£a£3
(c) 3 < a < 4
(d) a > 4
[IIT JEE 1999]
10. The equation has
(a) No solution
(b) One solution
(c) Two solutions
(d) More then two solutions
[IIT JEE 1997]
11. Let a ,b be the roots of the equation Then the roots of the equation
(a) a,c
(b) b,c
(c) a,b
(d) a + c, b + c
[IIT JEE 1992]
12. Let f(x) be quadratic expression which is positive for all real values of x. If g(x) = f(x)+ f'(x)+ f” (x), then for any real x
(a) g(x) < 0
(b) g(x) > 0
(c) g(x) = 0
(d) g(x) ³ 0
[IIT JEE 1990]
13. Let a, b, c be real number, a ¹ 0. If a is a root of a2x2 + bx + c = 0, b is the root of a2x2 – bx – c = 0 and 0 < a < b, then the equation a2x2 + 2bx + 2c = 0 has a root g that always satisfies
14. If a and b are the roots of and are roots of then the equation has always
(a) Two real roots
(b) Two positive roots
(c) Two negative roots
(d) One positive and one negative root
[IIT JEE 1989]
15. If a, b and c are distinct positive numbers, then the expression is
(a) Positive
(b) Negative
(c) Non-positives
(d) Non-negative
[IIT JEE 1986]
16. The equation has
(a) No root
(b) One root
(c) Two equal roots
(d) Infinitely many roots
[IIT JEE 1984]
17. a + b + c = 0, then the quadratic equation has
(a) At least one root in (0, 1)
(b) One root in (2, 3) and the other in (–2, –1)
(c) Imaginary roots
(d) None of the above
[IIT JEE 1983]
18. The largest interval for which is
(a) – 4 < x £ 0
(b) 0 < x < 1
(c) – 100 < x < 100
(d) – ¥ < x < ¥
[IIT JEE 1982]
19. If x1, x2, ……, xn are any real numbers and n is any positive integer, the
20. The number of real solutions of the equation |x|2 –3 |x| + 2 = 0 is
(a) 4
(b) 1
(c) 3
(d) 2
[IIT JEE 1982]
Answer keys:
[edit | edit source]1. d 2. a 3. a 4. b 5. b
6. c 7. d 8. b 9. a 10. a
11. c 12. b 13. d 14. d 15. b
16. a 17. a 18. d 19. d 20. a