1. 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.

2. If in triangle \( \triangle ABC \), the sum of the medians AD, BE, and CF is \(x\), and the sum of the sides is \(y\), then the relationship between \(x\) and \(y\) is —

(a) \(x\gt y\) (b) \(x\lt y\) (c) \(x= y\) (d) None of the above

3. If a cube has a side length of \(a\) units and a diagonal length of \(d\) units, then the relationship between \(a\) and \(d\) is—?

(a) \(\sqrt2a=d\) (b) \(\sqrt3a=d\) (c) \(a=\sqrt3d\) (d) \(a=\sqrt2d\)

4. If the average of the numbers 6, 7, x, 8, y, and 16 is 9, then—?

(a) x + y = 21 (b) x + y = 17 (c) x - y = 21 (d) x - y = 19

5. y is equal to the sum of two variables—one that varies directly with x and another that varies inversely with x. When x = y, then y = –1, and when x = 3, then y = 5. Determine the relationship between x and y.

6. If the product of three consecutive positive geometric numbers is 64, then the mean proportional between the first and the third is ______.

7. If the combined average of the numbers 7, \(x - 3\), 10, \(x + 3\), and \(x - 5\) is 15, then the median is _____.

(a) 16 (b) 10 (c) 18 (d) 24

8. If the average of the numbers 6, 7, x, 8, y, and 14 is 9, then

(a) x+y=21 (b) x+y=19 (c) x-y=21 (d) x-y=19

9. If the combined mean of \(n\) quantities is \(\bar{x}\), and the sum of the first \((n-1)\) quantities is \(k\), then the \(n\)th quantity will be —

(a) \(k+n\) (b) \(\bar{x}+k\) (c) \(\bar{x}-k\) (d) \(n\bar{x}-k\)

10. 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

11. If the combined average of the expressions ???? − 3 , ???? − 1 , 7 , ???? , 2 ???? − 1 , and 3 ???? − 5 is 3.5, then what will be the median of these expressions?

(a) 3.5 (b) 4.5 (c) 5 (d) 5.5

12. If the mode and the combined mean of a statistical distribution are ₹12.30 and ₹18.48 respectively, then the median of the distribution will be —

(a) ₹ 16 (b) ₹ 17 (c) ₹ 16.42 (d) ₹ 17.42

13. If the curved surface area of a sphere is \(S\) square units and its volume is \(V\) cubic units, then determine the relationship between \(S\) and \(V\).

14. If AD is the median of triangle ABC and E is any point on AD, then—

(a) \(\triangle\)AEB=\(\triangle\)AEC (b) \(\triangle\)AEB<\(\triangle\)AEC (c) \(\triangle\)AEB=\(\triangle\)BED (d) \(\triangle\)AEC=\(\triangle\)DEC

15. From a ghat on one side of a 600-meter wide river, two boats set off toward the opposite bank in two different directions. If the first boat moves at an angle of 30° with the riverbank, and the second boat moves at a 90° angle to the path of the first boat, then what will be the distance between the two boats after they reach the opposite bank?

16. If \(n\) is an even number, then the median will be the average of the \(\left(\cfrac{n}{2}\right)\)-th and \(\left(\cfrac{n}{2}−1\right)\)-th observations.

17. y is equal to the sum of two variables—one is directly proportional to x, and the other is inversely proportional to x. When x = 1, y = -1; and when x = 3, y = 5. Determine the relationship between x and y.

18. If the total surface area of a cube is \(s\) square units and the length of its diagonal is \(d\) units, then the relationship between \(s\) and \(d\) will be –

(a) \(s=6d^2\) (b) \(3s=7d\) (c) \(s^3=d^2\) (d) \(d^2 = \cfrac{s}{2}\)

19. If the total surface area of a cube is \(s\) square units and the length of its diagonal is \(d\) units, then the relationship between \(s\) and \(d\) is \[ s = 6d^2 \]

20. If the total surface area of a cube is \(s\) square units and the length of its diagonal is \(d\) units, then the relationship between \(s\) and \(d\) is \(s^3 = d^2\).

21. Rambabu divided his total money between two banks in such a way that he received equal interest from both banks after one year. The annual interest rates of the two banks are 4% and 5%, respectively. The amount of money deposited in the second bank is — of the amount deposited in the first bank.

(a) \(\frac{1}{2}\) part (b) \(\frac{1}{3}\) part (c) \(\frac{1}{6}\) part (d) twice

22. If the numerical values of the volume and lateral surface area of a cone are equal, then the relationship between its height and base radius is?

(a) \(r^2-h^2=2^2\) (b) \(2r^2+h^2=3^2\) (c) \(r^2+h^2=3^2\) (d) none of the above

23. Standing in the middle of a field, Mohit saw a flying bird—first to the north at an elevation angle of 30°, and then 2 minutes later to the south at an elevation angle of 60°. If the bird is flying along the same straight path at a height of 50√3 meters, calculate its speed in kilometers per hour.

24. If the fourth of 5 consecutive geometric numbers is 54 and the fifth is 162, then find the first number.

25. A right-angled triangle where the hypotenuse is 12 cm and one of the other sides is 5 cm. — Draw the triangle and then draw its circumcircle. Mark the position of the circumcenter and measure and write the radius of the circumcircle. [Only drawing symbols required]

26. The lengths of two sides are 7.6 cm and 6 cm, and the included angle between them is 75°. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).

27. A right-angled triangle in which the hypotenuse is 9 cm and one of the other sides is 5.5 cm. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).

28. Draw an equilateral triangle with side length 7 cm. Then draw both the circumcircle and the incircle of the triangle. Using a scale, determine the lengths of the circumradius and the inradius. Also, write whether there is any relationship between them.

29. If the data arranged in ascending order is 8, 9, 12, 17, x+2, x+4, 30, 31, 34, 39 and the median is 24, then the value of x is—

(a) 22 (b) 21 (c) 20 (d) 24

30. y is equal to the sum of two variables—one is directly proportional to x, and the other is inversely proportional to x. When \(x = 1\), \(y = -1\); and when \(x = 3\), \(y = 5\). Determine the relationship between x and y.