1. "Write the polynomial \(2x(x + 5) + 5\) in the form of a quadratic polynomial."

2. "Write the polynomial \(7x^5 - x(x + 2)\) in the form of a quadratic polynomial."

3. 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

4. In the polynomial \(2x^3 - 3x^2 + 4x + 5 = 0\), the coefficient of \(x^0\) is—?

(a) 2 (b) 3 (c) 4 (d) 5

5. Translate: \(\cfrac{x}{4 - x} = \cfrac{1}{3x}, \ (x \ne 0, \ x \ne 4)\) — If we express this equation in the form of a quadratic equation \(ax^2 + bx + c = 0\) where \(a \ne 0\), then let's determine the coefficient of \(x\).

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

6. Let’s translate that into English: Express \(3x^2 + 7x + 23 = (x + 4)(x + 3) + 2\) in the form of a quadratic equation \(ax^2 + bx + c = 0\), where \(a \ne 0\).

7. Translate: Express \((x + 2)^3 = x(x^2 - 1)\) in the form of a quadratic equation \(ax^2 + bx + c = 0\), where \(a \ne 0\), and write down the coefficients of \(x^2\), \(x\), and \(x^0\) (i.e., the constant term).

8. If the product of two consecutive positive odd numbers is 143, form the equation and find the two numbers using Sridhar Acharya's formula (the quadratic formula).

9. Five times a positive integer is 3 less than twice the square of that integer. Form the required quadratic equation to find the integer, and then solve the equation to determine its value.

10. If \(4\), \(5 + x\), \(2x + 1\), and \(14\) are in continued proportion and \(x\) is a positive number, then what is the value of \(x\)?

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

11. Which of the following is a quadratic polynomial sequence.

(a) \(\sqrt{x}-4\) (b) \(x^3+x\) (c) \(x^3+2x+6\) (d) \(x^2+5x+6\)

12. Divide 42 into two parts such that one part is equal to the square of the other part. — Form a quadratic equation with one variable from the given statement.

13. The distance between two stations is 300 km. A train travels from the first station to the second at a uniform speed. If the speed of the train had been 5 km/h more, it would have taken 2 hours less to reach the second station. — Form a quadratic equation with one variable from the given statement.

14. In a two-digit number, the unit digit is 6 more than the tens digit, and the product of the digits is 12 less than the number itself. — Form a quadratic equation with one variable from the given statement.

15. \(\cfrac{x}{4 - x} = \cfrac{1}{3x},\ (x ≠ 0,\ x ≠ 4)\) — Let us express this in the form of a quadratic equation \(ax^2 + bx + c = 0\), where \(a ≠ 0\), and determine the coefficient of \(x\).

16. Let us express \(3x^2 + 7x + 23 = (x + 4)(x + 3) + 2\) in the form of a quadratic equation \(ax^2 + bx + c = 0\), where \(a ≠ 0\).

17. The length of a rectangular field is 36 meters more than its breadth. The area of the field is 460 square meters. Form a quadratic equation in one variable from this statement and determine the coefficients of \(x^2\), \(x\), and \(x^0\).

18. The length of a rectangle is 2 meters more than its breadth, and the area of the rectangle is 24 square meters. Form a quadratic equation in one variable.

19. Express the equation \(x^3 - 4x^2 - x + 1 = (x + 2)^3\) in the standard form of a quadratic equation and write the coefficients of \(x^2\), \(x\), and \(x^0\).

20. Find the value of \(k\) such that one of the roots of the quadratic equation \(x^2 + kx + 3 = 0\) is \(1\). Show the calculation.

21. Form an equation with the product of two consecutive positive odd numbers equal to 143, and use Śrīdharāchārya’s formula (the quadratic formula) to find those two odd numbers.

22. A group had ₹195 in funds. After each of its members contributed an amount equal to the number of members, the total amount was distributed equally among all members such that each received ₹28. Using Śrīdharāchārya’s formula (the quadratic formula), find the number of members in the group.

23. If a right circular cylinder with radius \(2x\) cm and height \(x\) cm is melted to form a sphere, then the radius of the resulting sphere will be –

(a) \(\sqrt[3]3{x}\) cm (b) \(\sqrt[3]{3x}\) (c) \(x\) (d) \(2x\)

24. The coefficient of \(x\) in the quadratic equation \(x+\cfrac{1}{x}=6\).

(a) 6 (b) -6 (c) 0 (d) 1

25. The radii of two solid iron spheres are \(r_1\) and \(r_2\) respectively. The spheres are melted and recast into a single solid sphere. What will be the radius of the resulting sphere?

(a) \((r_1^3+r_2^3)\) (b) \((r_1^3+r_2^3)^3\) (c) \((r_1+r_2)^3\) (d) \((r_1^3+r_2^3)^{\cfrac{1}{3}}\)

26. If a ring-shaped iron plate has inner diameter \(d_1\) and outer diameter \(d_2\), then the area of the plate will be:

(a) \(\pi(d_1^2-d_2^2)\) (b) \(\cfrac{\pi}{4}(d_2^2-d_1^2)\) (c) \(\cfrac{\pi}{4}(d_1^2-d_2^2)\) (d) \(\cfrac{1}{4}(d_2^2-d_1^2)\)

27. If two circles with radii \(r_1\) and \(r_2\) touch each other externally, and the distance between their centers is \(d\), then which of the following is correct?

(a) \(r_1+d=r_2\) (b) \(r_2+d=r_1\) (c) \(r_1+r_2=d\) (d) \(r_1-r_2=d\)

28. 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

29. A cylindrical rod of height 64 cm is melted to form 12 solid spheres of equal radius. What is the radius of each sphere?

(a) 2 cm (b) 4 cm (c) 8 cm (d) 16 cm

30. A solid lead sphere with a diameter of 12 cm is melted to form three smaller spheres. If the diameters of the smaller spheres are in the ratio 3:4:5, what is the radius of the smallest sphere?

(a) 1.5 cm (b) 3 cm (c) 4 cm (d) 5 cm