1. If \(2a = 3b = 4c\), then \(a : b : c\) will be.

(a) 2:3:4 (b) 6:4:3 (c) 4:3:2 (d) 3:4:6

2. If \[ 2a = 3b = 4c, \] then what will be the ratio \(a : b : c\)?

(a) 3:4:6 (b) 4:3:6 (c) 3:6:4 (d) 6:4:3

3. The roots of the equation \(ax^2+bx+c=0\) will be real and equal when –

(a) \(b^2>4ac \) (b) \(b^2=4ac \) (c) \(b^2≠ 4ac \) (d) \(b^2<4ac\)

4. If the product of the roots of the equation \(3x^2 - 5x + b = 0\) is \(4\), the value of \(b\) will be –

(a) \(\cfrac{5}{3}\) (b) \(\cfrac{3}{5}\) (c) 12 (d) -12

5. 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

6. The sum of the roots of the equation \(x^2 - 6x + 2 = 0\) will be –

(a) 2 (b) -2 (c) 6 (d) -6

7. If \(2a = 3b = 4c\), then \(a : b : c = 2 : 3 : 4\).

8. \(\triangle\)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.

9. If \(\tan\theta = \cos 30^\circ + \sin 60^\circ\), then \(\sin(90^\circ - \theta)\) will be –

(a) \(\cfrac{1}{3}\) (b) \(\cfrac{1}{4}\) (c) \(\cfrac{1}{2}\) (d) \(1\)

10. In triangle ABC, ∠ABC = 90° and BD ⊥ AC; if BD = 8 cm and AD = 5 cm, then the length of CD will be –

(a) \(\cfrac{16}{5}\) cm (b) \(\cfrac{32}{5}\) cm (c) \(\cfrac{64}{5}\) cm (d) \(\cfrac{128}{5}\) cm

11. If the product of the roots of the quadratic equation \(3x^2 – 4x + k = 0\) is 5, then what will be the value of \(k\)?

(a) 5 (b) -12 (c) 15 (d) -20

12. If \(2A=3B=4C\), then \(A:B:C=6:4:______\).

13. If \(2a=3b=4c\), then \(a:b:c=6:4:3\).

14. If \(\sqrt5+\sqrt6=a\), then the value of \((\sqrt5-\sqrt6)\) will be –?

(a) \(\cfrac{a}{2}\) (b) \(\cfrac{1}{a}\) (c) \(-\cfrac{1}{a}\) (d) \(-a\)

15. If \(3a = 4b = Sc\), then \(a:b:c\) will be _____.

16. If \( \tan \theta \cdot \tan \phi = 1 \), then the value of \( \sin\left(\frac{\theta + \phi}{3}\right) \) will be –

(a) \(1\) (b) \(\cfrac{1}{\sqrt2}\) (c) \(\cfrac{\sqrt3}{2}\) (d) \(\cfrac{1}{2}\)

17. In a partnership business, three friends invested capital amounts of ₹4,500, ₹6,000, and ₹9,000 respectively. After a certain period, their profit shares will be in the same ratio as their investments: \[ \text{Profit Ratio} = 4500 : 6000 : 9000 = 3 : 4 : 6 \] So, their respective shares of profit will be in the ratio 3 : 4 : 6.

18. The product of two rational numbers will always be a rational number. Want a quick example to see it in action? Let’s multiply ¾ and ⅔: ¾ × ⅔ = ½**, and ½ is still a rational number.