RD Sharma Chapter 4 : Quadratic Equation – Exercise 4.6 Solutions (Step-by-Step Guide)

1. Determine the nature of the roots of the following quadratic equations:

  (i) 2x² − 3x + 5 = 0 Watch Solution

  (ii) 2x² − 6x + 3 = 0 Watch Solution

  (iii) (3/5)x² − (2/3)x + 1 = 0 Watch Solution

  (iv) 3x² − 4√3x + 4 = 0 Watch Solution

  (v) 3x² − 2√6x + 2 = 0 Watch Solution

  (vi) 4x² + 4√3x + 3 = 0  Watch Solution

2. Find the values of k for which the roots are real and equal in each of the following equations:

  (i) kx² + 4x + 1 = 0 Watch Solution

  (ii) kx² − 2√5x + 4 = 0 Watch Solution

  (iii) 3x² − 5x + 2k = 0 Watch Solution

  (iv) 4x² + kx + 9 = 0 Watch Solution

  (v) 2kx² − 40x + 25 = 0 Watch Solution

  (vi) 9x² − 24x + k = 0 Watch Solution

  (vii) 4x² − 3kx + 1 = 0 Watch Solution

  (viii) x² − 2(5 + 2k)x + 3(7 + 10k) = 0 Watch Solution

  (ix) (3k + 1)x² + 2(k + 1)x + k = 0 Watch Solution

  (x) kx² + kx + 1 = −4x² − x Watch Solution

  (xi) (k + 1)x² + 2(k + 3)x + (k + 8) = 0 Watch Solution

  (xii) x² − 2kx + 7k − 12 = 0 Watch Solution

  (xiii) (k + 1)x² − 2(3k + 1)x + 8k + 1 = 0 Watch Solution

  (xiv) 5x² − 4x + 2 + k(4x² − 2x − 1) = 0 Watch Solution

  (xv) (4 − k)x² + (2k + 4)x + (8k + 1) = 0 Watch Solution

  (xvi) (2k + 1)x² + 2(k + 3)x + (k + 5) = 0 Watch Solution

  (xvii) 4x² − 2(k + 1)x + (k + 4) = 0 Watch Solution

  (xviii) 4x² − 2(k + 1)x + (k + 1) = 0 Watch Solution

3. In the following, determine the set of values of k for which the given quadratic equation has real roots:

  (i) 2x² + 3x + k = 0 Watch Solution

  (ii) 2x² + x + k = 0 Watch Solution

  (iii) 2x² − 5x − k = 0 Watch Solution

  (iv) kx² + 6x + 1 = 0 Watch Solution

  (v) 3x² + 2x + k = 0 Watch Solution

4. Find the values of k for which the following equations have real and equal roots:

  (i) x² − 2(k + 1)x + k² = 0 Watch Solution

  (ii) k²x² − 2(2k − 1)x + 4 = 0 Watch Solution

  (iii) (k + 1)x² − 2(k − 1)x + 1 = 0 Watch Solution

  (iv) x² + k(2x + k − 1) + 2 = 0 Watch Solution

5. Find the values of k for which the following equations have equal roots:

  (i) 2x² + kx + 3 = 0 Watch Solution

  (ii) kx(x − 2) + 6 = 0  Watch Solution

  (iii) x² − 4kx + k = 0 Watch Solution

  (iv) kx(x − 2√5) + 10 = 0 Watch Solution

  (v) kx(x − 3) + 9 = 0  Watch Solution

  (vi) 4x² + kx + 3 = 0 Watch Solution

6. Find the values of k for which the given quadratic equation has real and distinct roots:

  (i) kx² + 2x + 1 = 0 Watch Solution

  (ii) kx² + 6x + 1 = 0

7. For what value of k, (4 − k)x² + (2k + 4)x + (8k + 1) = 0, is a perfect square.
8. Find the least positive value of k for which the equation x² + kx + 4 = 0 has real roots.
9. (i) Find the values of k for which the quadratic equation (3k + 1)x² + 2(k + 1)x + 1 = 0 has equal roots. Also, find the roots.

            (ii) Write all the values of k for which the quadratic equation x² + kx + 16 = 0 has equal roots. Find the roots of the equation so obtained.

10. Find the values of p for which the equation (2p + 1)x² − (7p + 2)x + (7p − 3) = 0 has equal roots. Also, find these roots.
11. If −5 is a root of the quadratic equation 2x² + px − 15 = 0 and the quadratic equation p(x² + x) + k = 0 has equal roots, find the value of k.
12. If 2 is a root of the quadratic equation 3x² + px − 8 = 0 and the quadratic equation 4x² − 2px + k = 0 has equal roots, find the value of k.
13. If 1 is a root of the quadratic equation 3x² + ax − 2 = 0 and the quadratic equation a(x² + 6x) − b = 0 has equal roots, find the value of b.
14. Find the value of p for which the quadratic equation (p + 1)x² − 6(p + 1)x + 3(p + 9) = 0, p ≠ −1 has equal roots. Hence, find the roots of the equation.
15. Determine the nature of the roots of the following quadratic equations:

  (i) (x − 2a)(x − 2b) = 4ab

  (ii) 9a²b²x² − 24abcdx + 16c²d² = 0, a ≠ 0, b ≠ 0

  (iii) 2(a² + b²)x² + 2(a + b)x + 1 = 0

  (iv) (b + c)x² − (a + b + c)x + a = 0

16. Determine the set of values of k for which the following quadratic equations have real roots:

  (i) x² − kx + 9 = 0

  (ii) 2x² + kx + 2 = 0

  (iii) 4x² − 3kx + 1 = 0

  (iv) 2x² + kx − 4 = 0

17. If the roots of the equation (b − c)x² + (c − a)x + (a − b) = 0 are equal, then prove that 2b = a + c. [CBSE 2002 C]
18. If the roots of the equation (a² + b²)x² − 2(ac + bd)x + (c² + d²) = 0 are equal, prove that a/b = c/d. [CBSE 2017]
19. If the roots of the equations ax² + 2bx + c = 0 and bx² − 2√acx + b = 0 are simultaneously real, then prove that b² = ac.
20. If p, q are real and p ≠ q, then show that the roots of the equation (p − q)x² + 5(p + q)x − 2(p − q) = 0 are real and unequal.
21. If the roots of the equation (c² − ab)x² − 2(a² − bc)x + b² − ac = 0 are equal, prove that either a = 0 or a³ + b³ + c³ = 3abc.
22. Show that the equation 2(a² + b²)x² + 2(a + b)x + 1 = 0 has no real roots, when a ≠ b.
23. Prove that both the roots of the equation (x − a)(x − b) + (x − b)(x − c) + (x − c)(x − a) = 0 are real but they are equal only when a = b = c.
24. If a, b, c are real numbers such that ac ≠ 0, then show that at least one of the equations ax² + bx + c = 0 and −ax² + bx + c = 0 has real roots.
25. If the equation (1 + m²)x² + 2mcx + (c² − a²) = 0 has equal roots, prove that c² = a²(1 + m²).