1. If \( x \propto y^n\), then the correct relationship will be _____.

(a) \( x^n \propto y^n\) (b) \( x^{2n} \propto y^n\) (c) \( x^{2n} \propto y^{2n}\) (d) \( x^{n} \propto y^{2n}\)

2. If \(x \propto \cfrac{1}{y}\), then the value of \(x + y\) will be minimum when _____.

3. If the product of the roots of the equation \(x^2 - 3x + k = 10\) is -2, then the value of \(k\) will be _____.

4. If \(a \propto b\), \(b \propto \cfrac{1}{c}\), and \(c \propto d\), then what will be the proportional relationship between \(a\) and \(d\)?

5. If \(x \propto \sqrt{y}\) and \(z \propto \sqrt[3]{x}\), then what is the relationship between \(y\) and \(z\)?

(a) \(z^6 \propto y\) (b) \(y^6 \propto z\) (c) \(z^2 \propto y\) (d) \(y^2 \propto z\)

6. If \(x \propto \cfrac{1}{z}, z\propto \cfrac{1}{y}\), then the relation between \(x\) and \(y\) is _____.

7. If \(x \propto \cfrac{1}{y}\), then the value of \(x+y\) will be the smallest when \(x=y\).

8. If \(x \propto \cfrac{1}{y}\), then \(xy = 1\) will be true.

9. In the quadratic equation \(ax^2+bx+c=0 (a \ne 0)\), if \(b^2 = 4ac\), then the roots will be real and _____.

10. If the mean of \(x_1, x_2, x_3, ....., x_n\) is \(\bar{x}\), then the mean of \(x_1 + k, x_2 + k, x_3 + k, ....., x_n + k\) will be _____.

11. If \(x \propto y^{\cfrac{3}{2}}\), then \(y \propto\) ----- will be.

12. If \( x_1, x_2, x_3, ..., x_a \) have an average of \( \bar{x} \), then the average of \( ax_1, ax_2, ..., ax_a \), where \( a \ne 0 \), will be _____.

13. If \(x\) is in inverse proportion to \(y\), and \(y\) is in inverse proportion to \(z\), then \(x\) will The statement is true.

14. 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

15. If the average of the quantities \(x_1, x_2, x_3,.......,x_{10}\) is 20, then the average of \(x_1+4, x_2+4, x_3+4,...., x_{10}+4\) will be—?

(a) 20 (b) 24 (c) 40 (d) 10

16. If \(xy > 0\), then the point \((x, y)\) will be located in which quadrant?

(a) first quadrant (b) second quadrant (c) third quadrant (d) forth quadrant

17. If one root of the quadratic equation \(3x^2 + (k - 1)x + 9 = 0\) is 3, then what will be the value of \(k\)?

(a) -11 (b) 11 (c) 12 (d) 14

18. If \(x = y(\csc\theta + \cot\theta)\) and \(z = y(\csc\theta - \cot\theta)\), then which of the following relationships is correct?

(a) \(xy=z^2\) (b) \(xz=y^2\) (c) \(yz=x^2\) (d) \(xyz=1\)

19. 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) \(r_1-r_2=d\)

20. If \(x : y = 3 : 4\), then the value of \(\cfrac{x^2 - xy + y^2}{x^2 + xy + y^2}\) will be:

(a) 37:13 (b) 13:35 (c) 13:37 (d) 20:13

21. If \(x = 9 + 4\sqrt{5}\), then the value of \(\sqrt{x} - \frac{1}{\sqrt{x}}\) will be —

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

22. If \((x^2 + y^2) \propto (x^2 - y^2)\), then which of the following is correct?

(a) \(x\propto y\) (b) \(xy=\) constant (c) \(x^2\propto y\) (d) \(y^2\propto x\)

23. If \(u_i = \cfrac{x_i - 20}{10}\), \(\sum{f_iu_i} = 15\), and \(\sum{f_i} = 80\), then what will be the value of \(\bar{x}\)?

(a) 21.875 (b) 20.875 (c) 21.800 (d) 20.125

24. If class width = 20, assumed mean \(A = 25\), total frequency \(y = 50\), and \(\sum f_u = -5\), then the combined mean \(\bar{x}\) will be —

(a) 25 (b) 23 (c) 24 (d) 27

25. 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) \(n\bar{x}-k\)

26. If the mean of the numbers \(x_1, x_2, x_3, x_4, ..., x_n\) is \(\bar{x}\), then the value of \((x_1 - \bar{x}) + (x_2 - \bar{x}) + (x_3 - \bar{x}) + ... + (x_n - \bar{x})\) will be —

(a) 0 (b) 1 (c) 3 (d) 5

27. If \(\sum\limits_{i=1}^5 x_i = 5\) and \(\sum\limits_{i=1}^5 x_i^2 = 14\), then the value of \(\sum\limits_{i=1}^5 2x_i(x_i - 3)\) will be —

(a) 2 (b) -2 (c) 0 (d) 4

28. If a sum of money doubles in \(n\) years at a fixed annual compound interest rate, then in how many years will it become 4 times?

29. 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

30. If the combined mean of \(x_1, x_2, x_3, …, x_n\) is \(\bar{x}\), then the combined mean of \(\frac{x_1 - c}{h}, \frac{x_2 - c}{h}, …, \frac{x_n - c}{h}\) will be \(h\bar{x} + c\).