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 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

3. 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) \(3a\)

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

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

5. If the current price of a machine is ₹2p and its value decreases by 2r% each year, then after 2n years the price of the machine will be:

(a) ₹ p\((1-\cfrac{r}{100})^n\) (b) ₹ p\((1-\cfrac{r}{50})^n\) (c) ₹ p\((1-\cfrac{r}{50})^{2n}\) (d) ₹ 2p\((1-\cfrac{r}{50})^{2n}\)

6. If sinA+sinB=2, then what is the value of cosA-cosB

(a) 0 (b) 1 (c) 2 (d) 3

7. For the equation \(5x^2+9x+3=0\) , if the roots are \(α\) and \(β\), then what is the value of \(\cfrac{1}{α}+\cfrac{1}{β}\) ?

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

8. 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

9. 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

10. 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) \(\cfrac{16}{9}\)

11. 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

12. If the mode of the numbers 64, 60, 48, x, 43, 48, 43, 34 is 43. Then the value of \((x+3)\) is.

(a) 44 (b) 45 (c) 46 (d) 48

13. The current price of a machine is rupees 2P, and if its price decreases by 2r% each year, then after 2n years, the price of the machine will be:

(a) \(P\left(1-\cfrac{r}{100}\right)^n\) (b) \(2P\left(1-\cfrac{r}{50}\right)^n\) (c) \(P\left(1-\cfrac{r}{50}\right)^{2n}\) (d) \(2P\left(1-\cfrac{r}{50}\right)^{2n}\)

14. If the data arranged in ascending order, 8, 9, 12, 17, x+2, x+4, 30, 31, 34, 39 has a median of 24, then the value of x -

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

15. In a circle centered at O, the chords AB and CD have equal lengths. If \(\angle\)AOB = 60°, then the value of \(\angle\)COD is -

(a) 60° (b) 30° (c) 120° (d) 90°

16. If A + B = 90° and tanA = \(\cfrac{3}{4}\), then the value of cotB is -

(a) \(\cfrac{3}{4}\) (b) \(\cfrac{4}{3}\) (c) \(\cfrac{3}{5}\) (d) \(\cfrac{5}{3}\)

17. If 16, 15, 17, 16, 15, x, 19, 17, 14 have a mode of 15, then the value of x is-

(a) 15 (b) 16 (c) 17 (d) 19

18. 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

19. PQRS is a cyclic trapezium. PQ is a diameter of the circle, and PO || SR. If \(\angle\)QRS = 110°, then the value of \(\angle\)QSR is -

(a) 20° (b) 25° (c) 30° (d) 40°

20. If sinθ + cosθ = √2 (where 0° < θ < 90°), then the value of θ is

(a) 30° (b) 45° (c) 60° (d) 90°

21. If \(u_i = \cfrac{x_i - 35}{10}\), \(∑f_i u_i = 30\), and \(∑f_i = 60\), then the value of \(\bar{x}\) is –

(a) 40 (b) 20 (c) 80 (d) None of these

22. 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

23. If \(\cfrac{p}{q} = \cfrac{5}{7}\) and \(p - q = -2\), then the value of \(p + q\) is –

(a) 12 (b) 13 (c) 14 (d) 15

24. If \(\theta\) is a positive acute angle and \( \sin \theta - \cos \theta = 0 \), then the value of \(\cot 2\theta\) is –

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

25. If \(∑f_i(x_i - a) = 400\), \(∑f_i = 50\), and \(a\) (assumed mean) = 52, then the value of the combined mean \(\bar{x}\) is –

(a) 52 (b) 60 (c) 80 (d) 90

26. If the equation \(3x^2 - 6x + p = 0\) has real and equal roots, then the value of \(p\) is –

(a) \(\cfrac{5}{3}\) (b) -\(\cfrac{1}{3}\) (c) -3 (d) 3

27. If \(∑f_i d_i = 400\), \(∑f_i = 50\), and \(a =\) assumed mean \(= 52\), then the value of the combined mean is –

(a) 52 (b) 60 (c) 80 (d) 55

28. If the mode of the data set 16, 15, 17, 16, 15, x, 19, 17, 14 is 15, then the value of x is—

(a) 15 (b) 16 (c) 17 (d) 19

29. If \( r\cos\theta = 1 \) and \( r\sin\theta = \sqrt{3} \), then the value of \( \theta \) is—

(a) \(\cfrac{π}{2}\) (b) \(\cfrac{π}{3}\) (c) \(\cfrac{π}{4}\) (d) \(\cfrac{π}{6}\)

30. If \(sin θ = cos θ\), then the value of \(2θ\) is—?

(a) 30° (b) 60° (c) 45° (d) 90°