1. 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

2. If \(\log_{\sqrt{2}} x = a\), then the value of \(\log_{2\sqrt{2}} x\) will be:

(a) \(\cfrac{a}{3}\) (b) \(a\) (c) \(2a\) (d) \(3a\)

3. If \(\log_x \frac{1}{3} = -\frac{1}{3}\), then the value of \(x\) will be:

(a) \(27\) (b) (9\) (c) \(3\) (d) \(\cfrac{1}{27}\)

4. 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

5. If \(a:b = 2:3\), then the value of \(5a:6b\) will be \(1:1\).

6. 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.

7. 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

8. If, tanθ=cot3θ, then the value of sin2θ be:

(a) \(\cfrac{1}{√2}\) (b) \(\cfrac{√3}{2}\) (c) \(\cfrac{1}{2}\) (d) 0

9. if a: \(\cfrac{27}{64}=\cfrac{3}{4}\):a then the value of a be-

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

10. 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

11. 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

12. If \( \tan A \tan B = 1\), then the value of \( \tan \cfrac{(A+B)}{2} \) will be –

(a) 1 (b) √3 (c) \(\cfrac{1}{√3}\) (d) None of these

13. 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

14. If \(a + b : \sqrt{ab} = 1 : 1\), then what is the value of \(\sqrt{\cfrac{a}{b}} + \sqrt{\cfrac{b}{a}}\)?

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

15. If \(a+\cfrac{1}{a}=\sqrt{3}\), then the value of \(a^3+\cfrac{1}{a^3}\) will be —

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

16. 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

17. If 7, x, y, 189 are in continued proportion, then the values of x and y respectively will be:

(a) 63,21 (b) 21,23 (c) 21,63 (d) 23,21

18. If \(a = \frac{\sqrt{3}}{2}\), then the value of \(\sqrt{1+a} + \sqrt{1-a}\) will be —

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

19. 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

20. 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

21. 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

22. If \( \tan\theta + \cot\theta = 2 \), then the value of \( \theta \) will be —

(a) \(\cfrac{\pi}{2}\) (b) \(\cfrac{\pi}{4}\) (c) \(\pi\) (d) \(\cfrac{\pi}{6}\)

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

24. If \(x = 2 + \sqrt{3}\), then the value of \(x + \frac{1}{x}\) will be \(2\sqrt{3}\).

25. If \(tan 35° \cdot tan 55° = \sin θ\), then the smallest positive value of \(θ\) will be ——.

26. \(\triangle\)ABC ~ \(\triangle\)DEF; BC and EF are corresponding sides. If BE : EF = 1 : 3, then the ratio of the areas of \(\triangle\)ABC and \(\triangle\)DEF will be 1 : 27.

27. If \(x(2 + \sqrt{3}) = y(2 - \sqrt{3}) = 1\), then the value of \(\frac{1}{x + 1} + \frac{1}{y + 1}\) will be—

(a) \(1\) (b) \(\sqrt{3}\) (c) \(2\sqrt{3}\) (d) \(2\)

28. If one root of the quadratic equation \(x^2 + ax + 12 = 0\) is 1, then the value of \(a\) will be —.

29. If \(\sin\theta + \cos\theta = \sqrt{2}\), then the value of \(\theta\) will be—

(a) \(\cfrac{\pi^c}{2}\) (b) \(\cfrac{\pi^c}{3}\) (c) \(\pi^c\) (d) \(\cfrac{\pi^c}{4}\)

30. If the ratio of the principal amount to its increased value after one year is 25:27, then what will be the annual interest rate?

(a) 2% (b) 4% (c) 6% (d) 8%