1. 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
2. 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
3. If the roots of the equation \(x^2 + (p - 3)x + p = 0\) are real and equal, then prove—without solving—that the value of \(p\) will be either \(1\) or \(9\).
4. If the roots of the quadratic equation \( ax^2 + 2bx - c = 0 \) (where \( a \ne 0 \)) are real and equal, then \( b^2 \) will be ——.
5. The roots of the equation \(ax^2+bx+c=0\) will be equal in magnitude but opposite in sign if-
(a) \(c=0, a≠0\) (b) \(b=0, a≠0\) (c) \(c=0, a=0\) (d) \(b=0, a=0\)
6. If the equation \(3x^2 - 6x + p = 0\) has real and equal roots, then the value of \(p\) is –
(a) \(\cfrac{5}{3}\) (b) -\(\cfrac{1}{3}\) (c) -3 (d) 3
7. For what value of \(k\) will the equation \(2x^2 + 3x + k = 0\) have real and equal roots?
8. If the sum of the roots of a quadratic equation is 7 and their difference is 3, then the equation will be –
(a) \(x^2–7x+3=0\) (b) \(x^2–7x-3=0\) (c) \(x^2–7x+10=0\) (d) \(x^2-7x+7=0\)
9. For which value or values of \(k\) will the following equation have real and equal roots: \[ (3k + 1)x^2 + 2(k + 1)x + k = 0 \]
10. What should be the value of \(k\) so that the roots of the quadratic equation \(9x^2 + 3kx + 4 = 0\) are real and equal?
11. Calculate the value(s) of \(k\) for which the quadratic equation \(49x^2 + kx + 1 = 0\) will have real and equal roots.
12. For which value(s) of \(k\) will the quadratic equation \(3x^2 - 5x + 2k = 0\) have real and equal roots?
13. For which value(s) of \(k\) will the quadratic equation \(9x^2 - 24x + k = 0\) have real and equal roots?
14. Calculate the value(s) of \(k\) for which the quadratic equation \(2x^2 + 3x + k = 0\) will have real and equal roots.
15. Calculate the value(s) of \(k\) for which the quadratic equation \(x^2 - 2(5 + 2k)x + 3(7 + 10k) = 0\) will have real and equal roots.
16. For which value(s) of \(k\) will the quadratic equation \((3k + 1)x^2 + 2(k + 1)x + k = 0\) have real and equal roots?
17. Determine the value of \(p\) for which the equation \(x^2 + (p - 3)x + p = 0\) will have equal and real roots.
18. A quadratic equation in one variable \(x\), with real coefficients, has equal coefficients for \(x^2\) and the constant term. Show with reasoning that any other quadratic equation whose roots are the reciprocals of the original equation's roots must be identical to the original equation.
19. If the roots of the equation \( ax^2+bx+c=0 \,(a\ne 0) \) are real and equal, then
(a) \(c=\cfrac{-b}{2a}\) (b) \(c=\cfrac{b}{2a}\) (c) \(c= \cfrac{-b^2}{4a}\) (d) \(c = \cfrac{b^2}{4a}\)
20. If the product of the roots of the equation \(3x^2 - 5x + b = 0\) is \(4\), the value of \(b\) will be –
(a) \(\cfrac{5}{3}\) (b) \(\cfrac{3}{5}\) (c) 12 (d) -12
21. If the volumes of two vertical circular cylinders are equal, and their heights are in the ratio 4:9, then the ratio of their radii will be –
(a) 3:2 (b) 2:3 (c) 4:9 (d) 8:9
22. If the volumes of two vertical solid cylinders are equal, and their heights are in the ratio 1:2, then the ratio of the lengths of their radii will be –
(a) 1: √2 (b) √2:1 (c) 1:2 (d) 2:1
23. The sum of the roots of the equation \(x^2 - 6x + 2 = 0\) will be –
(a) 2 (b) -2 (c) 6 (d) -6
24. y is equal to the sum of two variables—one that varies directly with x and another that varies inversely with x. When x = y, then y = –1, and when x = 3, then y = 5. Determine the relationship between x and y.
25. Without solving, determine all values of \(p\) for which the equation \(x^2 + (p - 3)x + p = 0\) has real and equal roots.
26. If the roots of the quadratic equation \(ax^2 + 2bx + c = 0\) \((a ≠ 0)\) are real and equal, then \(b^2 =\) _____ .
27. If the roots of the quadratic equation \((1 + m^2)x^2 + 2mcx + (c^2 − a^2) = 0\) are real and equal, show that \(c^2 = a^2(1 + m^2)\).
28. The graph of the equation \(cx + d = 0\) (where \(c\) and \(d\) are constants, and \(c \ne 0\)) will be the equation of the y-axis when:
(a) d =-c (b) d =c (c) d =0 (d) d =1
29. The graph of the equation \(ay + b = 0\) (where \(a\) and \(b\) are constants, and \(a \ne 0\)) will be the equation of the x-axis when:
(a) b = a (b) b = -a (c) b = 2 (d) b = 0
30. I will write a two-digit number whose digits add up to 14, and when 29 is subtracted from the number, the digits become equal. Let’s form simultaneous equations and find out what the two-digit number is.
