Practice the questions given in the Worksheet on nature of the roots of a quadratic equation.
We know the nature of the roots of a quadratic equation depends completely on the value of its discriminant.
1. Without solving, comment upon the nature of roots of each of the following equations:
(a) 7x\(^{2}\)  9x + 2 = 0
(b) 6x\(^{2}\)  13x + 4 = 0
(c) 25x\(^{2}\)  10x + 1 = 0
(d) x\(^{2}\) + 2√3 x  9 = 0
(e) x\(^{2}\)  ax + b\(^{2}\) = 0
(f) 2x\(^{2}\) + 8x + 9 = 0
2. Find the discriminant of the following equations.
(a) x(x  2) + 1 = 0
(b) \(\frac{1}{x + 2}\) + \(\frac{1}{x  2}\) = 2
3. Prove that none of the following equations has any real solution.
(a) x\(^{2}\) + x + 1 = 0
(b) x(x  1) + 1 = 0
(c) x + \(\frac{4}{x}\)  1 = 0, x ≠ 0
(d) x(x + 1) + 3(x + 3) = 0
(e) \(\frac{x}{x + 1}\) + \(\frac{3}{x  1}\) = 0; x ≠ 1, 1
4. Find the value of ‘p’, if the following quadratic equation has equal roots: 4x\(^{2}\)  (p  2)x + 1 = 0
5. Prove that each of the following equation has only one solution. Find the solution.
(a) 4y\(^{2}\)  28y + 49 = 0
(b) \(\frac{1}{4}\)x\(^{2}\) + \(\frac{1}{3}\)x + \(\frac{1}{9}\) = 0
(c) 8x(2x  5) + 25 = 0
6. Find the value of λ for which the equation λx\(^{2}\) + 2x + 1 = 0 has real and distinct roots.
7. For what value of k will each of the following equations give equal roots? Also, find the solution for that value of k.
(a) 3x\(^{2}\) + kx + 2 = 0
(b) kx\(^{2}\)  4x + 1 = 0
(c) 5x\(^{2}\) + 20x + k = 0
(d) (k  12)x\(^{2}\) + 2(k  12)x + 2 = 0
8. The equation 3x\(^{2}\)  12x + z  5 = 0 has equal roots. Find the value of z.
9. Find k for which the equation 4x\(^{2}\) + kx + 9 = 0 will be satisfied by only one real value of x. Also find the solution.
10. Find the value of ‘z’, if the following equation has equal roots:
(z  2)x\(^{2}\)  (5 + z)x + 16 = 0
11. Find the nature of roots of the following equation. If they are real, find them.
(a) 3x\(^{2}\)  2x + \(\frac{1}{3}\) = 0
(b) 3x\(^{2}\)  6x + 2 = 0
Answers for the Worksheet on nature of the roots of a quadratic equation are given below.
Answers:
1. (a) Rational and unequal
(b) Irrational and unequal
(c) Rational (real) and equal
(d) Irrational and unequal (since, b = 2√3 is irrational)
(e) Irrational and unequal
(f) Imaginary roots
2. (a) 0
(b) 17
4. p = 2 or 6
5. (a) \(\frac{7}{2}\)
(b) \(\frac{2}{3}\)
(c) \(\frac{5}{4}\)
6. All real values of λ < 1.
7. (a) ±2√6; when k = 2√6, solution = \(\frac{2}{√6}\) and when k = 2√6, solution = \(\frac{2}{√6}\)
(b) 4; solution = \(\frac{1}{2}\)
(c) 20; solution = 2
(d) 14; solution = 1
8. z = 17
9. ± 12; when k = 12, solution = \(\frac{3}{2}\) and when k = 12, solution = \(\frac{3}{2}\)
10. z = 3 or 51
11. (a) Real, Roots = \(\frac{1}{3}\), \(\frac{1}{3}\)
(b) Real, Roots = \(\frac{√3  1}{√3}\), \(\frac{√3 + 1}{√3}\)
9 Grade Math
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