1. e is a ______ number.
2. The product of two conjugate surds is a _____ number.
3. \(5\sqrt{11}\) is an __________ number. (Rational/Irrational)
4. If the average of the first four numbers out of five is 26, and the average of the last four numbers is 25, then find the difference between the first and the last number.
5. In a partnership business, the total profit is ₹1200. One partner invested ₹6000 and earned a profit of ₹900. Then, the capital invested by the other partner is ______.
6. If the three angles of a triangle are in the ratio 2:3:4, then the measure of the largest angle in degrees is ________.
7. If the product of the roots of the equation \(x^2 - 3x + k = 10\) is -2, then the value of \(k\) will be _____.
8. The angle subtended by a segment of a circle greater than a semicircle is _____.
9. If the simple interest for \(n\) years at an annual rate of \(r\%\) is \(\frac{pnr}{25}\) rupees, then the principal amount will be _____ rupees.
10. If the bases of two triangles lie on the same straight line and the other vertex of both triangles is common, then the ratio of their areas is ______to the ratio of the lengths of their bases.
11. The simplest value of \(\cfrac{\cos 53^\circ}{\sin 37^\circ}\) is _____.
12. The number of surfaces of a solid right circular cylinder is _____.
13. If the variables \(x_1, x_2, ..., x_{100}\) are arranged in ascending order, then their median is ______.
14. Anisur invested ₹500 for 9 months and David invested ₹600 for 5 months in a joint business. Their profit-sharing ratio will be _____ .
15. Two angles are called complementary if their sum is _____ degrees.
16. The maximum value of sin 3θ is _____ .
17. If a solid sphere is melted to form a solid right circular cylinder, then the _____ of the sphere and the cylinder will be equal.
18. The ages (in years) of some students are: 10, 11, 9, 7, 13, 8, 14; the median of their ages is _____ years.
19. If 9 is added to three times a positive number, the sum equals twice the square of that number. Find the number.
20. The perpendicular bisector of any chord of a circle is the ________ of that circle.
21. If a two-digit positive number is multiplied by its unit digit, the product is 189, and the tens digit is twice the unit digit, then find the number.
22. If \(5 + \sqrt{x} = \sqrt{2} - y\), then the value of \((x + y)\) is _________.
23. The measure of all angles in a circular segment is ______.
24. In a partnership business, when partners invest capital for different durations without any other conditions, it is called a _______.
25. The number of real roots of the equation \((x - 1)^2 + (x - 2)^2 + (x - 3)^2 = 0\) is ______.
26. If the ratio of the heights of two right circular cones is 1:4 and the ratio of their radii is 2:1, then the ratio of their volumes will be ______.
27. A two-digit number has its unit digit 5 more than its tens digit. Also, the product of the two digits is 14 less than the number itself. Find the number.
28. If the product of three consecutive positive geometric numbers is 64, then the mean proportional between the first and the third is ______.
29. If the current population of a village is \(P\) and the annual population growth rate is 20%, then the population 2 years ago was ________.
30. In the cyclic quadrilateral ABCD, AB is the diameter and O is the center of the circle. If \(\angle\)ADC = 120°, then \(\angle\)BAC = _____
