1. If \(a, b, c, d\) are in continued proportion, then what is the value of \(\frac{abc(a + b + c)}{ab + bc + ca}\)?

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

2. If \(a, b, c, d\) are in continued proportion (i.e., in geometric progression), then the value of \(\cfrac{adc(a + b + c)}{ab + bc + ca}\) is —

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

3. If \(a, b, c, d\) are in continued proportion, then prove that \[ (a^2 + b^2 + c^2)(b^2 + c^2 + d^2) = (ab + bc + cd)^2 \]

4. If \(a, b, c,\) and \(d\) are in continuous proportion, then prove that \((b-c)^2+(c-a)^2+(b-d)^2=(a-d)^2\).

5. If \(x, y, z\) are in continued proportion, then what is the value of \[ x^2y^2z^2\left(\frac{1}{x^3}+\frac{1}{y^3}+\frac{1}{z^3}\right)? \]

(a) \(x+y+z\) (b) \(x^2+y^2+z^2\) (c) \(x^3+y^3+z^3\) (d) None of the above

6. If 8, 14, \(7x + 2\), and 28 are in continued proportion, then what is the value of \(x\)?

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

7. If the numbers \((2 + x)\), \((8 + x)\), and \((26 + x)\) are in continued proportion, then what is the value of \(x\)?

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

8. If \(a, b, c,\) and \(d\) are in continued geometric proportion, then prove that \[ (b - c)^2 + (c - a)^2 + (b - d)^2 = (a - d)^2 \]

9. If the roots of the equation \(ax^2 + b + c = 0\) are \(\sin α\) and \(\cos α\), then what is the value of \(b^2\)?

(a) \(a^2-2ac\) (b) \(a^2+2ac\) (c) \(a^2-ac\) (d) \(a^2+ac\)

10. If \(a, b, c,\) and \(d\) are in continued proportion, then prove that \((b - c)^2 + (c - a)^2 + (b - d)^2 = (a - d)^2\).

11. If sin 51° = \(\cfrac{a}{\sqrt{a^2 + b^2}}\), then what is the value of tan 51° + tan 39°?

(a) \(\cfrac{a^2-b^2}{ab}\) (b) \(\cfrac{a^2+b^2}{2ab}\) (c) \(\cfrac{a^2+b^2}{ab}\) (d) \(\cfrac{a^2-b^2}{2ab}\)

12. If \( \sin\theta + \csc\theta = 2 \), then what is the value of \( \sin^2\theta + \csc^2\theta \)?

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

13. If \( \sin 51^\circ = \cfrac{a}{\sqrt{a^2 + b^2}} \), then what is the value of \( \tan 51^\circ + \tan 39^\circ \)?

(a) \(\cfrac{a^2+b^2}{ab}\) (b) \(\cfrac{a^2-b^2}{ab}\) (c) \(\cfrac{a-b}{ab}\) (d) \(\cfrac{a+b}{ab}\)

14. If \(a, b, c, d\) are in continued geometric progression, prove that \[(a^2 + b^2 + c^2)(b^2 + c^2 + d^2) = (ab + bc + cd)^2\]

15. In triangle ABC, let X and Y be the midpoints of sides AB and AC respectively. If \(BC + XY = 12\) units, then what is the value of \(BC - XY\)?

16. If \(a, b, c\) are in continued geometric progression (i.e., in successive proportion), then prove that \[ \cfrac{1}{b^2} = \cfrac{1}{b^2 - a^2} + \cfrac{1}{b^2 - c^2} \]

17. If \(4\), \(5 + x\), \(2x + 1\), and \(14\) are in continued proportion and \(x\) is a positive number, then what is the value of \(x\)?

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

18. If \(x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ\), then what is the value of \(x\)?

19. If \(a \propto (b + c)\), \(b \propto (c + a)\), and \(c \propto (a + b)\), and \(k, l, m\) are non-zero distinct constants respectively, then what is the value of \[ \frac{k}{k+1} + \frac{l}{l+1} + \frac{m}{m+1} \, ? \]

(a) 1 (b) 3 (c) 5 (d) 7

20. If \(a, b, c, d\) are in continued geometric progression, then prove that \[ (b - c)^2 + (c - a)^2 + (b - d)^2 = (a - d)^2 \]

21. If \(a, b, c, d\) are in continuous proportion, then show that \((a^2 - b^2)(c^2 - d^2) = (b^2 - c^2)^2\).

22. If \(a, b, c\), and \(d\) are in continued proportion, then prove that \((b - c)^2 + (c - a)^2 + (b - d)^2 = (a - d)^2\)

23. If \(a, b, c,\) and \(d\) are in continued geometric progression, then prove that \((b - c)^2 + (c - a)^2 + (b - d)^2 = (a - d)^2\)

24. If \(\sin 51^\circ = \cfrac{a}{\sqrt{a^2 + b^2}}\), then what is the value of \(\tan 39^\circ\)?

25. If \(0^\circ < \theta \leq 90^\circ\), then what is the minimum value of \((4\csc^2\theta + 9\sin^2\theta)\)?

26. If \( \sin 51^\circ = \frac{a}{\sqrt{a^2 + b^2}} \), then what is the value of \( \tan 51^\circ + \tan 39^\circ \)?

27. If the mean of the frequency distribution is 62.8 and the total frequency is 50, then what are the values of \(x\) and \(y\) given the following table: | Class | 0–20 | 20–40 | 40–60 | 60–80 | 80–100 | 100–120 | |-------------|------|--------|--------|--------|---------|----------| | Frequency | 5 | \(x+y\) | 10 | \(y−x\) | 7 | 8 |

28. If \(a, b, c\) are in continued geometric progression, then prove that \[ a^2 b^2 c^2 \left(\frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3 \]