1. If the roots of the quadratic equation \((b - c)x^2 + (c - a)x + (a - b) = 0\) are equal, prove that \(2b = a + c\).

2. If the roots of the quadratic equation \((b-c)x^2+(c-a)x+(a-b)=0\) are equal, prove that \(2b = a + c\).

3. If the roots of the quadratic equation \((b - c)x^2 + (c - a)x + (a - b) = 0\) are equal, prove that \(2b = a + c\).

4. If the roots of the equation \( a(b - c)x^2 + b(c - a)x + c(a - b) = 0 \) are equal, prove that \(\cfrac{1}{a} + \cfrac{1}{c} = \cfrac{2}{b}\).

5. If the roots of \((a^2-bc)x^2+2(b^2-ca)x+(c^2-ab)=0\) are equal, show that \(b=0\) or \(a^3+b^3+c^3=3abc\).

6. If the roots of the quadratic equation \(2(a^2 + b^2)x + 2(a + b)x + 1 = 0\) are equal, prove that \(a = b\).

7. If the roots of the quadratic equation \((a^2 + b^2)x^2 + 2(ac + bd)x + (c^2 + d^2) = 0\) are equal, then prove that \(\cfrac{a}{b} = \cfrac{c}{d}\).

8. If the roots of the quadratic equation \(ax^2+bx+c=0\) are in the ratio \(1:2\), then prove that \(2b^2=9ac\).

9. **"If the roots of the quadratic equation \((a^2 + b^2)x^2 - 2(ac + bd)x + (c^2 + d^2) = 0\) are equal, then prove that \(\cfrac{a}{b} = \cfrac{c}{d}\)."**

10. If the roots of the quadratic equation \(ax^2 + bx + c = 0\) are in the ratio \(1 : r\), then prove that \(b^2r = ac(r + 1)^2\).

11. If the roots of the quadratic equation \(ax^2 + 2bx + c = 0\) \((a ≠ 0)\) are real and equal, then \(b^2 =\) _____ .

12. If the roots of the equation \(ax^2 + bx + c = 0\) \((a \ne 0)\) are reciprocals of each other, then \(c =\) _____________.

13. If the roots of the quadratic equation \(ax^2 + bx + c = 0\) are in the ratio \(1 : r\), then show that \[ \frac{(r + 1)^2}{r} = \frac{b^2}{ac} \]

14. If the roots of the quadratic equation \(ax^2 + bx + c = 0\) are in the ratio \(1 : s\), then show that \[ \frac{(s + 1)^2}{s} = \frac{b^2}{ac} \]

15. If the roots of the equation \(ax^2 + b + c = 0\) are \(\sin α\) and \(\cos α\), then what is the value of \(b^2\)?

(a) \(a^2-2ac\) (b) \(a^2+2ac\) (c) \(a^2-ac\) (d) \(a^2+ac\)

16. If the roots of the quadratic equation \(ax^2+bx+c=0\) are in the ratio \(1:p\), prove that \(\cfrac{(p+1)^2}{p} = \cfrac{b^2}{ac}\).

17. If \(\alpha\) and \(\beta\) are the two roots of the equation \(ax^2 + bx + c = 0\), then \(\cfrac{\alpha}{\beta}\) and \(\cfrac{\beta}{\alpha}\) are also roots of a quadratic equation. Determine that quadratic equation.

18. If the roots of the quadratic equation \((a^2 + b^2)x^2 - 2(ac + bd)x + (c^2 + a^2) = 0\) are equal, prove that \(\cfrac{a}{b} = \cfrac{c}{d}\).

19. If the roots of the equation \(ax^2 + bx + c = 0\) \((a \ne 0)\) are real and unequal, then \(b^2 - 4ac\) will be —

(a) >0 (b) =0 (c) <0 (d) none of the above

20. If the roots of the equation \(ax^2 + bx + c = 0\) \((a \ne 0)\) are equal, then —

(a) \(c=\cfrac{b}{2a}\) (b) \(c=-\cfrac{b}{2a}\) (c) \(c=- \cfrac{b^2}{4a}\) (d) \(c= \cfrac{b^2}{4a}\)

21. If the roots of the equation \(ax^2 + bx + c = 0\) \((a \ne 0)\) are reciprocal to each other, then \(c =\) _____________

22. If the roots of the quadratic equation \(ax^2 + bx + c = 0\), where \(a \ne 0\), are in the ratio 1:2, then show that \(2b^2 = 9ac\).

23. If the roots of the quadratic equation \(ax^2+bx+c=0\) are real and unequal, the value of \(b^2-4ac\) will be:

(a) >0 (b) <0 (c) 0 (d) None of these

24. If α and β are the roots of the equation \(ax^2 + bx + c = 0\), then what is the value of \[ \left(1 + \frac{α}{β}\right)\left(1 + \frac{β}{α}\right)? \]

25. If the roots of the equation \(ax^2 + bx + 35 = 0\) are -5 and -7, then find the values of \(a\) and \(b\).

26. If the equation \(ax^2 - 5x + c = 0\) has both the sum and product of its roots equal to \(10\), then which of the following is correct?

27. If the equation \(ax^2 + bx + c = 0\) has equal roots, then what is the value of \(c\)?

(a) \(\cfrac{-b}{2a}\) (b) \(\cfrac{b}{2a}\) (c) \(\cfrac{-b^2}{4a}\) (d) \(\cfrac{b}{4a}\)

28. If the equation \(ax^2 + 2bx + c = 0\) has equal roots, then what is the value of \(c\)?

(a) \(\cfrac{b^2}{a}\) (b) \(\cfrac{b^2}{4a}\) (c) \(\cfrac{a^2}{b}\) (d) \(\cfrac{a^2}{4b}\)

29. If the roots of the quadratic equation \((1+m^2)x^2 + 2mcx + (c^2 - a^2) = 0\) are real and equal, prove that \(c^2 = a^2(1+m^2)\).

30. If the quadratic equation \(ax^2 + 7x + b = 0\) has roots \(\cfrac{2}{3}\) and \(-3\), then find the values of \(a\) and \(b\).