1. \(y\) is directly proportional to the square root of \(x\), and \(y = 9\) when \(x = 9\). Determine the value of \(x\) when \(y = 6\).

(a) 4 (b) 3 (c) 2 (d) 1

2. The time period of a simple pendulum is directly proportional to the square root of its length. If the time period of a pendulum with length \(L\) is \(T\), then show that the time period of a pendulum with length \(4L\) is \(2T\).

3. What is the value of \(k\) such that the sum of the squares of the roots of the equation \(6x^2 + x + k = 0\) is \(\frac{25}{36}\)

4. I will solve the quadratic equation \(5x^2 + 23x + 12 = 0\) using an alternative method—that is, by multiplying both sides of the equation by 5 and then finding the roots through the method of completing the square.

5. \(y\) is in direct proportion with the square root of \(x\), and \(y = 9\) when \(x = 9\). Find the value of \(x\) when \(y = 6\).

6. Prove that:The squares of the roots of the equation \(x^2 + x + 1 = 0\) are also roots of the same equation \(x^2 + x + 1 = 0\). Would you like me to walk through the proof again in English, or are you asking for a translated version of your original Bengali explanation? I’ve got both options ready.

7. If the sum of the squares of the roots of the equation \(6x^2 + x + k = 0\) is \(\frac{25}{36}\), then the value of \(k\) will be \(12\).

8. For the equation \(5x^2+9x+3=0\) , if the roots are \(α\) and \(β\), then what is the value of \(\cfrac{1}{α}+\cfrac{1}{β}\) ?

(a) 3 (b) -3 (c) \(\cfrac{1}{3}\) (d) -\(\cfrac{1}{3}\)

9. For the equation \( 3x^2 + 8x + 2 = 0 \), if the roots are \( \alpha \) and \( \beta \), then what is the value of \( \frac{1}{\alpha} + \frac{1}{\beta} \)?"

(a) -\(\cfrac{3}{8}\) (b) \(\cfrac{2}{3}\) (c) -4 (d) 4

10. The total surface area of a cube is 216 square centimeters. Find the volume of the cube.

11. For the quadratic equation \(x^2 - bkx + 5 = 0\), if one of the roots is 5, then the value of \(k\) will be.

(a) \(-\cfrac{1}{2}\) (b) -1 (c) 1 (d) 0

12. 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

13. 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

14. The product of the roots of the equation \(x^2−7x+3=0\) is—?

(a) 7 (b) -7 (c) 3 (d) -3

15. If one root of the quadratic equation \(3x^2 + (k - 1)x + 9 = 0\) is 3, then what will be the value of \(k\)?

(a) -11 (b) 11 (c) 12 (d) 14

16. In the parallelogram ABCD, point P is the midpoint of side CD. If the area of triangle \( \triangle APD \) is 25 square centimeters, then what is the area of the parallelogram?

(a) 100 square cm (b) 75 square cm (c) 150 square cm (d) 50 square cm

17. 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

18. If \(25x^2 - 20ax + b\) is a perfect square, which of the following relations is true?

(a) \(b=2a^2\) (b) \(b=4a^2\) (c) \(b=a^2\) (d) \(b=3a^2\)

19. In triangle \(\triangle ABC\), D and E are the midpoints of sides AB and AC respectively. If the area of triangle ABC is 36 square centimeters, then what is the area of triangle ADE?

(a) 6 square centimeters (b) 9 square centimeters (c) 12 square centimeters (d) 16 square centimeters

20. The radius of the base of a right circular cone is equal to the radius of a sphere. The volume of the cone (in cubic centimeters) is equal to the curved surface area of the sphere (in square centimeters). What is the height of the cone?

(a) 4 cm (b) 3 cm (c) 8 cm (d) 12 cm

21. If E and F are the midpoints of sides AB and AD respectively in parallelogram ABCD, and the area of parallelogram ABCD is 1600 square centimeters, then what is the area of triangle AEF?

(a) 400 sq cm (b) 200 sq cm (c) 300 sq cm (d) None of the above

22. If the quadrilateral formed by joining the midpoints of the sides of parallelogram ABCD has an area of 100 square cm, then what is the area of the parallelogram ABCD?

(a) 400 sq cm (b) 200 sq cm (c) 600 sq cm (d) 800 sq cm

23. In triangle ABC, E and F are the midpoints of sides AB and AC respectively. If the area of triangle AEF is 50 square cm, then what is the area of triangle ABC?

(a) 100 sq cm (b) 200 sq cm (c) 150 sq cm (d) 300 sq cm

24. Find the square root of: \(33 - 4\sqrt{35}\)

(a) \(\pm(\sqrt{7}-\sqrt{5})\) (b) \(\pm(2\sqrt{7}-\sqrt{5})\) (c) (3\sqrt{7}-\sqrt{5})\) (d) \(\pm(4\sqrt{7}-\sqrt{5})\)

25. If one root of the equation \(x^2 - ax - 15 = 0\) is 3, then what is the value of \(a\)?

(a) -2 (b) 2 (c) 3 (d) 0

26. The dimensions of a right rectangular prism are in the ratio 6:5:4. If its total surface area is 3700 square centimeters, what is its volume in cubic centimeters?

(a) 1500 cubic cm (b) 51000 cubic cm (c) 50100 cubic cm (d) 15000 cubic cm

27. If the total surface area of a solid sphere is 462 square centimeters, what is its diameter?

(a) 7 cm (b) 14 cm (c) 21 cm (d) 3.5 cm

28. If the radius of a sphere is increased by 2 cm, its curved surface area increases by 352 square centimeters. What was the original radius of the sphere?

(a) 5 cm (b) 6 cm (c) 7 cm (d) 5.6 cm

29. If the total surface area of a cube is 216 square centimeters, what is its volume?

(a) 216 tcubic cm (b) 212 cubic cm (c) 316 cubic cm (d) 256

30. If the length, breadth, and height of a right rectangular prism are in the ratio 3:2:1 and its total surface area is 88 square centimeters, then what is its volume?

(a) 120 cubic cm (b) 64 cubic cm (c) 48 cubic cm (d) 24 cubic cm