1. In a right-angled quadrilateral, if the number of vertices is denoted by \(x\), the number of sides by \(y\), and the number of diagonals by \(z\), then what is the value of \(x + 3y - 5z\)?

(a) 14 (b) 44 (c) 20 (d) 24

2. Let the length of the space diagonal of the cube be \(x\), the length of a face diagonal be \(y\), and the edge of the cube be \(a\). Then, \[ x = a\sqrt{3} \Rightarrow a = \cfrac{x}{\sqrt{3}} \quad \text{— (i)} \] And, \[ y = a\sqrt{2} \Rightarrow a = \cfrac{y}{\sqrt{2}} \quad \text{— (ii)} \] Comparing equations (i) and (ii): \[ \cfrac{x}{\sqrt{3}} = \cfrac{y}{\sqrt{2}} \Rightarrow x = \cfrac{\sqrt{3}}{\sqrt{2}} \cdot y \Rightarrow x = \cfrac{\sqrt{6}}{2} \cdot y \] \(\therefore\) The length of the space diagonal of a cube is \(\cfrac{\sqrt{6}}{2}\) times the length of its face diagonal.

(a) undefined (b) positive (c) negative (d) zero

3. Three friends—A, B, and C—started a business by investing \(x\), \(2x\), and \(y\) taka respectively. If the profit at the end of the term is \(z\) taka, then what will be A’s share of the profit?

(a) ₹ \(\cfrac{xz}{3x+y}\) (b) ₹ \(\cfrac{2xz}{3x+y}\) (c) ₹ \(\cfrac{z}{2x+y}\) (d) ₹\(\cfrac{xyz}{3x+y}\)

4. y is equal to the sum of two variables—one is directly proportional to x, and the other is inversely proportional to x. When \(x = 1\), \(y = -1\); and when \(x = 3\), \(y = 5\). Determine the relationship between x and y.

5. In triangle \( \triangle ABC \), a line parallel to side BC intersects sides AB and AC at points P and Q respectively. If \( AB = 3 \times PB \) and \( BC = 18 \) cm, then what is the length of \( PQ \)?

(a) 10 cm (b) 9 cm (c) 12 cm (d) 8 cm

6. If the measure of an angle in degrees is \(x\) and in radians is \(y\), then what is the value of \( \frac{x}{y} \)?

(a) \(\cfrac{π}{180}\) (b) \(\cfrac{π}{90}\) (c) \(\cfrac{90}{π}\) (d) \(\cfrac{180}{π}\)

7. In triangle ABC, angle B is a right angle. The hypotenuse is \(\sqrt{15}\), and the sum of the other two sides is 4. What is the value of \((\cos A + \cos C)\)?

(a) \(\cfrac{8}{\sqrt{13}}\) (b) \(\cfrac{-8}{\sqrt{15}}\) (c) \(\cfrac{-8}{\sqrt{13}}\) (d) \(\cfrac{8}{\sqrt{15}}\)

8. In triangle \(\triangle ABC\), AB = AC. Points E and F are the midpoints of sides AB and AC respectively. AD is perpendicular to BC, and AD = 4 cm. If EF = 3 cm, then what is the length of BD?

(a) 4 cm (b) 3 cm (c) 6 cm (d) 7 cm

9. In triangle \(\triangle ABC\), D and E are the midpoints of sides AB and AC respectively. If the area of triangle ABC is 36 square centimeters, then what is the area of triangle ADE?

(a) 6 square centimeters (b) 9 square centimeters (c) 12 square centimeters (d) 16 square centimeters

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

11. y is equal to the sum of two variables—one that varies directly with x and another that varies inversely with x. When x = y, then y = –1, and when x = 3, then y = 5. Determine the relationship between x and y.

12. In \( \triangle ABC \), points \( D \) and \( E \) lie on sides \( AB \) and \( AC \) respectively, such that \( DE \parallel BC \) and \( AD:DB = 3:1 \). If \( EA = 33 \) cm, then the length of \( AC \) is _____.

13. In \(\triangle ABC\), the center of the incircle is \(O\), and the incircle touches the sides \(AB\), \(BC\), and \(CA\) at points \(P\), \(Q\), and \(R\) respectively. Given that \(AP = 4\) cm, \(BP = 6\) cm, \(AC = 12\) cm, and \(BC = x\) cm, determine the value of \(x\).

14. Points \(X\) and \(Y\) are taken on sides \(PQ\) and \(PR\) of triangle \(∆PQR\), respectively. Given: \(PQ = 8\) units, \(YR = 12\) units, \(PY = 4\) units, and the length of \(PY\) is 2 units less than the length of \(XQ\). Determine, with reasoning, whether segment \(XY\) is parallel to side \(QR\).

15. If the volume of a cone is \(x\), the base area is \(y\), and the height is \(z\), then the value of \(\cfrac{x}{yz}\) will be 3.

16. 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

17. 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

18. If \(sinθ−cosθ=0,\) \( (0°<θ<90°)\) and \(secθ+cosecθ=x\), then the value of \(x\) is—?

(a) \(1\) (b) \(2\) (c) \(\sqrt2\) (d) \(2\sqrt2\)

19. Here’s the English translation of your math problem: In triangle \( \triangle ABC \), \( \angle ABC = 90^\circ \), \( BC = 24 \) cm, and \( E \) is the midpoint of \( AC \). If \( ED \perp BC \), then what is the length of \( BD \)?

(a) 6 cm (b) 8 cm (c) 9 cm (d) None of the above

20. In triangle ABC, the circumcenter is O; points A and B, C lie on opposite sides of the center. If \(\angle BOC = 120^\circ\), then what is the measure of \(\angle BAC\)?

(a) 50° (b) 60° (c) 70° (d) 80°

21. In triangle \( \triangle ABC \), if \( \angle B \) is a right angle and \( BC = \sqrt{3} \times AB \), then what is the value of \( \sin C \)?

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

22. If \(\sum f_iu_i = 10\), class width = 20, \(\sum f_i = 40 + k\), the combined mean is 54, and the assumed mean is 50, then what is the value of \(k\)?

23. If \(\sum{f_ix_i} = 216\), \(\sum{f_i} = 16\), and the combined mean is \(13.5 + p\), then what is the value of \(p\)?

(a) 0 (b) 1 (c) 0.1 (d) 0.01

24. y is equal to the sum of two variables—one is directly proportional to x, and the other is inversely proportional to x. When x = 1, y = -1; and when x = 3, y = 5. Determine the relationship between x and y.

25. \(y\) varies directly with the square of \(x\), and \(y = 9\) when \(x = 9\); if \(y = 4\), then what is the value of \(x\)?

26. If \(x\) and \(y\) are the factors of \(30\), then which of the following is correct –

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

27. If the equation \(ax^2 - 5x + c = 0\) has both the sum and product of its roots equal to \(10\), then which of the following is correct?

28. \(y\) is directly proportional to the square root of \(x\), and \(y = 9\) when \(x = 9\). Determine the value of \(x\) when \(y = 6\).

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

29. If \(O\) is the circumcenter of \(\triangle ABC\) and \(\angle BAC = 50^\circ\), then find the measure of \(\angle OBC\).

(a) 100° (b) 30° (c) 40° (d) 50°

30. The lengths of two sides are 7.6 cm and 6 cm, and the included angle between them is 75°. — Draw the triangle and then draw its incircle. Measure and write the length of the inradius (i.e., the radius of the incircle).