1. If the data arranged in ascending order is 6, 9, 11, 12, x+2, x+3, 17, 20, 21, 24 and the median is 21, then the value of x will be —
(a) 13.5 (b) 15.5 (c) 18.5 (d) 21.5
2. If for a set of data, \[ \sum_{i=1}^n (x_i - 7) = -8 \quad \text{and} \quad \sum_{i=1}^n (x_i + 3) = 72, \] then find the values of \(\bar{x}\) (the mean) and \(n\) (the number of data points).
3. 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: \[ m_1 t_1^2 d_2^3 = m_2 t_2^2 d_1^3 \]
4. If the mode of the data set 16, 15, 17, 16, 15, x, 19, 17, 14 is 15, then the value of x will be 15.
5. If the mode of the data 16, 15, 16, 16, 15, x, 19, 17, 14 is 16, then the value of x must be 16.
(a) 15 (b) 16 (c) 17 (d) 19
6. Draw an isosceles triangle with a base length of 10 cm and an equal angle of 45°. Draw the incircle of the triangle and write the value of the inradius. (Only marking for construction should be included.)
7. The weight data of 35 students in Nibedita’s class is as follows: ```html
8. When the angle of elevation of the sun is 45°, the length of the shadow of a vertical pillar is a certain value. When the angle of elevation decreases to 30°, the length of the shadow becomes 60 meters longer than before. Determine the height of the pillar.
9. The median of a set of data values will always be one of the values from that data set.
10. 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
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 mode of the data set 16, 15, 17, 16, 15, x, 19, 17, 14 is 15, then the value of x is—
13. If the assumed mean is 22, class width is 10, total frequency is 80, and the value of \(\sum{f_iu_i}\) is 16, then the actual (or true) mean will be —
(a) 23 (b) 24 (c) 25 (d) 26
14. From a statistical frequency distribution table, the given values are: \(n = 84,\) \(L = 30,\) \(f = 30,\) \(h = 30,\) \(cf = 40\) Therefore, the median of the distribution will be —
(a) 30 (b) 32 (c) 34 (d) 36
15. The numbers 11, 12, 14, \(x - 2\), \(x + 4\), \(x + 9\), 32, 38, 47 are arranged in ascending order, and their median is 24. Find the value of \(x\).
16. If the median of the following data is 32 and the total frequency is 100, find the values of \(x\) and \(y\): | Class Interval | 0–10 | 10–20 | 20–30 | |----------------|------|--------|--------| | Frequency | 10 | \(x\) | 25 | | Class Interval | 30–40 | 40–50 | 50–60 | |----------------|--------|--------|--------| | Frequency | 30 | \(y\) | 10 |
17. 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**.
18. In your uncle's factory, the value of a machine depreciates at a rate of 10% per year. If the current value of the machine is ₹6000, what will its value be after 3 years?
19. When the sun's angle of elevation is 45°, the length of the shadow of a vertical pillar on a horizontal plane is a certain value. When the angle of elevation becomes 30°, the shadow length increases by 60 meters compared to the previous case. Find the height of the pillar.
20. What is the value of the angle \(x^\circ\) formed between the diagonals of a rhombus?
21. A man left ₹28,000 for his 13-year-old son and 15-year-old daughter with the instruction that, at the age of 18, the amount each receives — including simple interest at an annual rate of 10% — should be equal. Determine the amount allocated to each child.
22. If a book is sold for ₹880 and the person incurs a 12% loss, at what price should it be sold to gain a 10% profit?
(a) 1050 (b) 1100 (c) 1000 (d) 1160
23. A classroom needs to be built for 40 students in such a way that each student gets 5 square meters of floor area and 25 cubic meters of space. If the length of the room is 20 meters, what should be its height?
(a) 5 meters (b) 7 meters (c) 9 meters (d) 11 meters
24. 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.
25. The daily water requirements of three families living in three flats are 1200 liters, 1050 liters, and 950 liters respectively. A tank needs to be installed such that, even after meeting these demands, 25% of the total water remains in reserve. A space of 2.5 meters in length and 1.6 meters in width is available for the tank. Determine how deep the tank should be.
26. Let \(x, y, z\) be three variables such that the value of \(y + z - x\) is constant and \((x + y - z)(z + x - y) \propto yz\). Prove that \((x + y + z) \propto yz\).
27. The price of a machine in a factory is ₹1,80,000. The value of the machine depreciates by 10% every year. What will be the value of the machine after 3 years?
28. In the figure, triangle ABC is inscribed in a circle and touches the circle at points P, Q, and R. If AP = 4 cm, BP = 6 cm, AC = 12 cm, and BC = x cm, then what is the value of x?
29. If a certain principal amount becomes 5 times its original value in 20 years due to interest, then the rate of interest is –
(a) 20% (b) 25% (c) 15% (d) 10%
30. The numerical value of the volume of a cube is equal to the numerical value of the sum of its edges. The total surface area of the cube is - -
(a) 12 square unit (b) 36 square unit (c) 72 square unit (d) 144 square unit
