1. Let the length of the space diagonal of the cube be \(x\), the length of a face diagonal be \(y\), and the edge of the cube be \(a\). Then, \[ x = a\sqrt{3} \Rightarrow a = \cfrac{x}{\sqrt{3}} \quad \text{— (i)} \] And, \[ y = a\sqrt{2} \Rightarrow a = \cfrac{y}{\sqrt{2}} \quad \text{— (ii)} \] Comparing equations (i) and (ii): \[ \cfrac{x}{\sqrt{3}} = \cfrac{y}{\sqrt{2}} \Rightarrow x = \cfrac{\sqrt{3}}{\sqrt{2}} \cdot y \Rightarrow x = \cfrac{\sqrt{6}}{2} \cdot y \] \(\therefore\) The length of the space diagonal of a cube is \(\cfrac{\sqrt{6}}{2}\) times the length of its face diagonal.

(a) undefined (b) positive (c) negative (d) zero

2. Let \(a\) be a positive number, and given: \[ a : \frac{27}{64} = \frac{3}{4} : a \] Find the value of \(a\).

3. If A : B = 2 : 3, B : C = 4 : 5, and C : D = 6 : 7, then what will be the value of D : A?

(a) 16 : 35 (b) 35 : 16 (c) 2 : 3 (d) 4 : 5

4. Translate and write: If by applying Sridharacharya's formula to the equation \(5x^2 + 2x - 7 = 0\), we get \(x = \cfrac{k \pm 12}{10}\), then what will be the value of \(k\)? Calculate and write it.

5. If \(a\) is a positive number and \[ a : \cfrac{27}{64} = \cfrac{3}{4} : a, \] then find the value of \(a\).

(a) \(\cfrac{81}{256}\) (b) 9 (c) \(\cfrac{9}{16}\) (d) \(\cfrac{16}{9}\)

6. If the current population of a village is **p**, and the annual growth rate is **2r%**, then after **n years**, the population will be: \[ \text{Population after } n \text{ years} = p \left(1 + \frac{2r}{100} \right)^n \]

(a) \(p\left(1+\cfrac{r}{100}\right)^n\) (b) \(p\left(1+\cfrac{r}{50}\right)^n\) (c) \(p\left(1+\cfrac{r}{100}\right)^{2n}\) (d) \(p\left(1-\cfrac{r}{100}\right)^n\)

7. In a partnership business, three friends invested capital amounts of ₹4,500, ₹6,000, and ₹9,000 respectively. After a certain period, their profit shares will be in the same ratio as their investments: \[ \text{Profit Ratio} = 4500 : 6000 : 9000 = 3 : 4 : 6 \] So, their respective shares of profit will be in the ratio 3 : 4 : 6.

8. If the radii of two cones are in the ratio 2:3 and their heights are in the ratio 5:3, then what will be the ratio of their volumes?

(a) 4:9 (b) 9:4 (c) 27:20 (d) 20:27

9. In a partnership business, the capital ratio between A and B is 2:3, and the capital ratio between B and C is 6:5. If A's profit share is ₹400, what will be C's profit share?

(a) ₹ 500 (b) ₹ 600 (c) ₹ 700 (d) ₹ 800

10. The modal class of the above frequency distribution is 15–20. So, the mode is calculated as: \[ \text{Mode} = l + \left(\frac{f_1 - f_0}{2f_1 - f_0 - f_2}\right) \times h \] Where: - \(l = 15\) (lower boundary of modal class) - \(f_1 = 28\) (frequency of modal class) - \(f_0 = 18\) (frequency of class before modal class) - \(f_2 = 17\) (frequency of class after modal class) - \(h = 5\) (class width) Substituting the values: \[ = 15 + \left(\frac{28 - 18}{2 \times 28 - 18 - 17}\right) \times 5 = 15 + \frac{10}{21} \times 5 = 15 + \frac{50}{21} = 15 + 2.38 = 17.38 \quad \text{(approx)} \] ✅ Therefore, the mode is approximately **17.38**.

11. The length, width, and height of a cuboidal room are respectively \(a\), \(b\), and \(c\) units, and given that \(a + b + c = 25\) and \(ab + bc + ca = 240.5\), what is the length of the longest rod that can be placed inside the room?

12. Given: AC is the diameter of a circle centered at O, ABC is an inscribed triangle, and OP ⊥ AB (where P lies on the circle). **Prove that:** \[ OP : BC = 1 : 2 \]

13. \(a\) is a positive number, and in the proportion \(a : \cfrac{27}{64} = \cfrac{3}{4} : a\), what is the value of \(a\)?

(a) \(\cfrac{9}{11}\) (b) \(\cfrac{3}{4}\) (c) \(\cfrac{9}{20}\) (d) \(\cfrac{9}{16}\)

14. 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

15. 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

16. 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

17. In a business, the capital ratio of A and C is 2:1, and the capital ratio of A and B is 3:2. If the annual profit is ₹1,57,300, what will be B's share of the profit?

(a) ₹ 48,000 (b) ₹ 58,400 (c) ₹ 48,400 (d) ₹68,000

18. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2+7x+3=0\), prove that: \[ \alpha^3+\beta^3+7(\alpha^2+\beta^2)+3(\alpha+\beta)=0 \]

19. The ratio of the capital investments of A, B, and C is 2:3:5, and the ratio of their investment durations is 3:4:5. Then, the profit will be distributed in the ratio _____.

20. If the ratio of the radii of two right circular cylinders is 2:3 and the ratio of their heights is 5:6, what will be the ratio of their volumes?

21. Here’s the English translation: *If the mean of a statistical distribution is 4.1, \(∑f_i.x_i = 132 + 5k\), and \(∑f_i = 20\), then what is the value of \(k\)?* Would you like help solving it too? I’d be glad to walk through it with you.

22. In compound interest, if the interest rates for the first, second, and third years are \(r_1\%\), \(r_2\%\), and \(2r_3\%\) respectively, then the amount after 3 years on a principal of ₹\(P\) will be: \[ P\left(1 + \cfrac{r_1}{100}\right)\left(1 + \cfrac{r_2}{100}\right)\left(1 + \cfrac{r_3}{100}\right) \]

23. If \[ 2a = 3b = 4c, \] then what will be the ratio \(a : b : c\)?

(a) 3:4:6 (b) 4:3:6 (c) 3:6:4 (d) 6:4:3

24. For the following data of handicraft scores awarded to 45 girl students in our village, calculate the **median** of the frequency distribution:

25. If a planet's gravitational force on its satellite is directly proportional to the planet's mass (M) and inversely proportional to the square of their distance (D), and the square of the satellite's orbital period (T) is directly proportional to the cube of the distance and inversely proportional to the gravitational force, then for two sets of values \(m_1, d_1, t_1\) and \(m_2, d_2, t_2\) corresponding to M, D, and T respectively, prove that: \[ m_1 t_1^2 d_2^3 = m_2 t_2^2 d_1^3 \]

26. Here’s the English translation of your math problem: If the ascending data set is \[ 27,\ 31,\ 46,\ 52,\ x,\ y + 2,\ 71,\ 79,\ 85,\ 90 \] and the median of the set is \(64\), then what is the value of \(x + y\)?

(a) 125 (b) 126 (c) 127 (d) 128

27. In a circle with center O, a tangent PT is drawn from an external point P to the circle, with T being the point of tangency. If PT = 12 cm and OP = 13 cm, the diameter of the circle will be:

(a) 5 cm (b) 8 cm (c) 6 cm (d) 10 cm

28. Two concentric circles have radii of 13 cm and 15 cm, respectively. A chord AB of the larger circle intersects the smaller circle at points P and Q. If PQ = 10 cm, then AB will be:

(a) 28 cm (b) 20 cm (c) 18 cm (d) 16 cm

29. 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

30. If the ratio of the capital investments of A, B, and C is \(\cfrac{1}{2} : \cfrac{1}{3} : \cfrac{1}{4}\) and the total profit is ₹520, then C's share of the profit will be

(a) ₹120 (b) ₹160 (c) ₹180 (d) ₹140