1. If the roots of the equation ( ???? + 2 ) ???? 2 − ( ???? − 3 ) ???? + 3 ???? − 1 = 0 are equal in magnitude but opposite in sign, what is the value of ???? ?

(a) -3 (b) 1 (c) 3 (d) None of the above

2. If the quadratic equation \(x^2 - x = k(2x - 1)\) has roots that are equal in magnitude but opposite in sign, determine the value of \(k\).

3. Find the value of \(k\) if the roots of the equation \(x^2 - x = k(2x - 1)\) are equal and opposite in sign.

4. 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 \(9\).

5. 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

6. If the roots of the equation \(ax^2+bx+c=0\) are \(\alpha\) and \(\beta\), find the value of \(\left(1+\cfrac{\alpha}{\beta}\right)\left(1+\cfrac{\beta}{\alpha}\right)\).

7. If three real numbers a, b, and c are in continued geometric progression, then the value of the mean proportional b will be undefined when a and c have opposite signs.

8. If the equation \((x+2)^3 = x(x-1)^2\) is expressed in the form of the quadratic equation \(ax^2 + bx + c = 0\) \((a ≠ 0)\), the coefficient of \(x^0\) (the constant term) will be.

(a) -8 (b) -1 (c) 3 (d) 8

9. The roots of the equation \(ax^2+bx+c=0\) will be real and equal when –

(a) \(b^2>4ac \) (b) \(b^2=4ac \) (c) \(b^2≠ 4ac \) (d) \(b^2<4ac\)

10. 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

11. 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

12. If the volumes of two solid right circular cylinders are equal and their heights are in the ratio 1:2, what will be the ratio of the lengths of their radii?

(a) 1:√2 (b) √2:1 (c) 1:2 (d) 2:1

13. If the time period and the annual simple interest rate are equal, then at what annual interest rate will the interest amount be \(\cfrac{1}{25}\) of the principal?

(a) 25% (b) 10% (c) 2% (d) 2.5%

14. If the mode and the combined mean of a statistical distribution are ₹12.30 and ₹18.48 respectively, then the median of the distribution will be —

(a) ₹ 16 (b) ₹ 17 (c) ₹ 16.42 (d) ₹ 17.42

15. Translate of your statement in English: If the roots of the equation \((b - c)x^2 + (c - a)x + (a - b) = 0\) are equal, then prove that: \(a + c = 2b\).

16. Two friends start a partnership business by investing ₹40,000 and ₹50,000 respectively. They agree that 50% of the profit will be shared equally, and the remaining 50% will be divided in the ratio of their investments. If the first friend’s share of the profit is ₹800 less than the second friend’s share, then what is the first friend’s profit?

17. From a ghat on one side of a 600-meter wide river, two boats set off toward the opposite bank in two different directions. If the first boat moves at an angle of 30° with the riverbank, and the second boat moves at a 90° angle to the path of the first boat, then what will be the distance between the two boats after they reach the opposite bank?

18. If a solid sphere is melted to form a solid right circular cylinder, then the _____ of the sphere and the cylinder will be equal.

19. Three friends purchase a bus by investing ₹1,20,000, ₹1,50,000, and ₹1,10,000 respectively. The first friend works as the driver, while the other two work as conductors. They decide that \(\frac{2}{5}\) of the income will be distributed based on work in the ratio 3 : 2 : 2, and the remaining amount will be divided according to their capital investment. If the total income in a certain month is ₹29,260, determine how much each person will receive.

20. At the beginning of the year, Srikanta and Saifuddin purchased a minibus and started operating it by investing ₹2,40,000 and ₹3,00,000 respectively. Four months later, their friend Peter joined them with ₹81,000. Srikanta and Saifuddin withdrew amounts proportional to their original investments from Peter’s contribution. At the end of the year, the total profit was ₹39,150. How much profit will each of them receive?

21. If two cones have equal base radii and their heights are in the ratio 2:3, then the ratio of their volumes will be ______.

22. For which value or values of \(k\) will the following equation have real and equal roots: \[ (3k + 1)x^2 + 2(k + 1)x + k = 0 \]

23. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2 - 3x + 5 = 0\), then find the value of \((\alpha + \beta)\left(\frac{1}{\alpha^2} + \frac{1}{\beta^2}\right)\).

24. In a business, if the capital ratio of A, B, and C is 2:3:5 and their time of investment ratio is 3:4:5, then in what ratio will the profit be distributed?

(a) 3:4:5 (b) 5:8:9 (c) 8:9:11 (d) 6:12:15

25. If \( \alpha \) and \( \beta \) are the roots of the equation \( x^2 - 22x + 105 = 0 \), find the value of \( \alpha - \beta \).

26. If the radius of a right circular cylinder is 2 units, then for any height, the numerical values of the cylinder’s volume and curved surface area will be equal.

27. If \( a \), \( b \), and \( c \) are three consecutive numbers in a geometric progression, and \( a \) and \( c \) have opposite signs, then \( b \) will be undefined.

28. If one root of the equation \(x^2 + px + 12 = 0\) is \(2\), and both roots of the equation \(x^2 + px + q = 0\) are equal, then find the value of \(q\).

29. 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?

30. In a business, three friends have their capital investment in the ratio 2:3:4. If in the following year they increase their capital by 10%, 15%, and 20%, respectively, in what ratio will the profit be distributed among them at the end of the second year?

(a) 44 : 69 : 96 (b) 69 : 44 : 96 (c) 96 : 69 : 44 (d) 45 : 96 : 63