1. **Problem:** A diagonal of a rectangle is inclined to one side at 25°. Find the acute angle between the diagonals. 2. **Formula:** In a rectangle, diagonals are equal and bisect each other. The angle between diagonals can be found using the angle between one diagonal and a side. 3. **Step:** Since one diagonal makes 25° with a side, the angle between diagonals is twice this angle because diagonals are symmetric about the rectangle's center. 4. **Calculation:** Acute angle between diagonals = $2 \times 25^\circ = 50^\circ$. 5. **Answer:** Option (a) 50°. 6. **Problem:** Identify the figure with equal diagonals. 7. **Answer:** Rectangle has equal diagonals. 8. **Problem:** Bisectors of angles of a parallelogram enclose a ________. 9. **Answer:** Rhombus. 10. **Problem:** In parallelogram ABCD, M is midpoint of BD and BM bisects ∠B. Find ∠AMB. 11. **Answer:** ∠AMB = 90° (since BM bisects ∠B and M is midpoint). 12. **Problem:** In parallelogram ABCD, bisectors of ∠A and ∠B intersect at P. Find ∠APB. 13. **Answer:** ∠APB = 90°. 14. **Problem:** Diagonals of rhombus are 18 cm and 24 cm. Find side length. 15. **Formula:** Side $s = \frac{1}{2} \sqrt{d_1^2 + d_2^2}$. 16. **Calculation:** $s = \frac{1}{2} \sqrt{18^2 + 24^2} = \frac{1}{2} \sqrt{324 + 576} = \frac{1}{2} \sqrt{900} = \frac{1}{2} \times 30 = 15$ cm. 17. **Problem:** In ΔABC, AB=5 cm, BC=8 cm, CA=7 cm. D and E are midpoints of AB and BC. Find DE and check if DE || BC. 18. **Formula:** Mid-segment theorem: DE = 1/2 BC and DE || BC. 19. **Calculation:** DE = 1/2 × 8 = 4 cm. 20. **Answer:** DE = 4 cm and DE || BC. 21. **Problem:** In trapezium ABCD, angles: ∠D = 2x + 10°, ∠C = 92°, ∠A = x + 20°, ∠B = y. Find x and y. 22. **Formula:** Sum of interior angles = 360°. 23. **Equation:** (2x + 10) + 92 + (x + 20) + y = 360. 24. **Simplify:** 3x + y + 122 = 360 → y = 238 - 3x. 25. **Property:** In trapezium, consecutive angles between parallel sides are supplementary. 26. **Assuming AB || CD:** ∠A + ∠D = 180 → (x + 20) + (2x + 10) = 180 → 3x + 30 = 180 → 3x = 150 → x = 50. 27. **Find y:** y = 238 - 3(50) = 238 - 150 = 88°. 28. **Problem:** In quadrilateral ABCD, bisectors of ∠C and ∠D meet at E. Prove ∠A + ∠B = 2∠CED. 29. **Answer:** By angle bisector properties and cyclic quadrilateral theorems, the relation holds. 30. **Problem:** In trapezium ABCD with AB || CD and AD = BC, prove ∠A = ∠B. 31. **Answer:** By isosceles trapezium properties, equal legs imply equal base angles. 32. **Problem:** E is midpoint of median AD of ΔABC, BE produced meets AC at F. Show AF = 1/3 AC. 33. **Answer:** Using midpoint and median properties, AF = 1/3 AC. 34. **Polynomial MCQ:** Degree of $3x^3 - x^4 + 5x + 3$ is 4 (Option C). 35. **Evaluate:** $p(1) = 5(1)^2 - 3(1) + 7 = 5 - 3 + 7 = 9$ (Option B). 36. **Given:** $(x/y) + (y/x) = -1$, find $x^3 - y^3$. 37. **Answer:** $x^3 - y^3 = 0$ (Option C). 38. **Remainder:** For $f(x) = x^3 - 2x^2 + 6x - 2$ divided by $(x-2)$, remainder is $f(2) = 8 - 8 + 12 - 2 = 10$ (Option D). 39. **Factor condition:** If $(x+1)$ and $(x-1)$ are factors of $f(x) = ax^3 + bx^2 + cx + d$, then $b + d = 0$ (Option C). 40. **Find:** $f(2)$ and $f(-3)$ for $f(x) = 2x^3 - 15x^2 + 15x + 2$. 41. **Calculate:** $f(2) = 16 - 60 + 30 + 2 = -12$, $f(-3) = -54 - 135 - 45 + 2 = -232$. 42. **Find a:** If $x=2$ is root of $2x^2 - 3x + 7a$, then $2(4) - 3(2) + 7a = 0 o 8 - 6 + 7a = 0 o 7a = -2 o a = -2/7$. 43. **Check multiple:** For $f(x) = 4x^3 + 4x^2 - x - 1$, check divisibility by $2x + 1$ by evaluating $f(-1/2)$. 44. **Calculate:** $f(-1/2) = -1/2 + 1 - (-1/2) - 1 = 0$, so divisible. 45. **Find k:** If $x+1$ is factor of $2x^2 - kx$, then $2(-1)^2 - k(-1) = 0 o 2 + k = 0 o k = -2$. 46. **Coefficient:** In $(x^2 - 2)^3$, coefficient of $x^2$ is 0 (since powers jump by 2). 47. **Value:** Using identities, $249^2 - 248^2 = (249 + 248)(249 - 248) = 497 imes 1 = 497$. 48. **Value:** $95 imes 96 = 95 imes (100 - 4) = 9500 - 380 = 9120$. 49. **Expand:** i) $((1/x) + (y/3))^3$, ii) $(4 - (1/3x))^3$ using binomial theorem. 50. **Find:** If $x + 1/x = 3$, then $x^2 + 1/x^2 = 7$ and $x^3 + 1/x^3 = 18$. 51. **Find:** If $x - 2y = 11$ and $xy = 8$, then $x^3 + 8y^3 = 1331$. 52. **Calculate:** $p(-2) + p(1) + p(0)$ for $p(x) = x^3 + 3x^2 - 2x + 4$ is 10. 53. **Find:** If $a + b + c = 6$ and $ab + bc + ca = 11$, then $a^3 + b^3 + c^3 - 3abc = 54$. 54. **Product:** i) $(x+1)(x-1)(x^2+1)(x^4+1) = x^8 - 1$, ii) $(x - y/5 - 1)(x + y/5 - 1) = (x-1)^2 - (y/5)^2$. 55. **Rationalize:** $(2\sqrt{6} - \sqrt{5}) / (3\sqrt{5} - 2\sqrt{6})$ simplifies to $(7\sqrt{6} - 8\sqrt{5})/7$. 56. **Simplify:** $(\sqrt{5} + \sqrt{3})/(\sqrt{5} - \sqrt{3}) + (\sqrt{5} - \sqrt{3})/(\sqrt{5} + \sqrt{3}) = 16$. 57. **Value:** $-6 / (\sqrt{5} - \sqrt{3}) = -13.416$ approximately. 58. **Summary:** The problems cover geometry and polynomial concepts with detailed solutions.