Class 10 Half-Yearly Maths – Top 50 Important Questions

Class 10 Half-Yearly Maths – Top 50 Important Questions

Chapter 1: Real Numbers

1. Use Euclid’s Division Lemma to find the HCF of 210 and 55.
2. Find the LCM and HCF of 12, 15, and 21 by the prime factorization method.
3. Prove that √3 is irrational.
4. Show that 5𝑛 cannot end with digit 0 for any natural number 𝑛.
5. Find the largest number which divides 130, 169, and 312 leaving the same remainder.

Chapter 2: Polynomials

6. If α and β are the zeroes of the polynomial 𝑥² – 5𝑥 + 6, find the value of α + β and αβ.
7. Find the quadratic polynomial whose zeroes are 2 and –3.
8. If the zeroes of the polynomial 𝑓(𝑥) = 4𝑥² – 2𝑥 + 3 are α and β, write the value of (α + β) and (αβ).
9. Find the zeroes of 𝑝(𝑥) = 2𝑥² – 7𝑥 + 3 and verify the relationship.
10. Divide 2𝑥⁴ – 3𝑥³ + 3𝑥² + 2𝑥 – 2 by (𝑥² – 2).

Chapter 3: Pair of Linear Equations in Two Variables

11. Solve by substitution: 7𝑥 – 15𝑦 = 2, 21𝑥 – 45𝑦 = 6.
12. Solve: 𝑥 + 𝑦 = 5, 2𝑥 – 𝑦 = 4.
13. If 3 chairs and 1 table cost ₹1,500 and 6 chairs and 2 tables cost ₹3,000, find the cost of one chair and one table.
14. Solve by cross-multiplication: 2𝑥 + 3𝑦 – 17 = 0, 𝑥 – 2𝑦 – 5 = 0.
15. A boat goes 16 km downstream in 2 hours and returns the same distance upstream in 4 hours. Find the speed of the boat in still water and the speed of the stream.

Chapter 4: Quadratic Equations

16. Solve: 2𝑥² – 7𝑥 + 3 = 0.
17. The sum of the reciprocals of Rehman’s ages (in years) 3 years ago and 5 years from now is 1/3. Find his present age.
18. If 7 is added to twice a number, it becomes 55 more than the number. Find the number.
19. Find the roots of the equation 𝑥² – 5𝑥 + 6 = 0 by factorization.
20. Solve for 𝑥: √2𝑥² + 7𝑥 + 5√2 = 0.

Chapter 5: Arithmetic Progressions (AP)

21. Find the 15th term of the AP: 7, 10, 13, …
22. Find the sum of the first 40 terms of the AP: 5, 8, 11, …
23. The 9th term of an AP is –32 and its 14th term is –47. Find the AP.
24. How many terms of the AP 18, 16, 14, … be taken so that their sum is 0?
25. A sum of ₹700 is to be divided among 35 children so that each child gets ₹20 less than the child before. Find the value of the first and last child’s share.

Chapter 6: Triangles

26. State and prove the Pythagoras Theorem.
27. In the given figure, DE ∥ BC. Find the value of 𝑥. (Diagram-based Q).
28. Prove that the ratio of areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
29. A girl of height 90 cm is standing 3 m away from a lamp post. If the length of the shadow is 2 m, find the height of the lamp post.
30. The diagonals of a trapezium divide each other proportionally. Prove it.

Chapter 7: Coordinate Geometry

31. Find the distance between A(–2, 3) and B(2, –1).
32. Find the coordinates of the point dividing the line joining (7, –2) and (3, –6) in the ratio 3:1.
33. Find the coordinates of the centroid of the triangle whose vertices are (1, 2), (3, –2), and (5, 4).
34. Find the area of the triangle whose vertices are (–2, –3), (3, 2), and (–1, –8).
35. Find the value of 𝑘 if the points A(1, 1), B(–2, 7), and C(3, 𝑘) are collinear.

Class 10 Half-Yearly Maths – Top 50 Important Questions

Chapter 8: Introduction to Trigonometry

36. If cos A = 3/5, find the values of sin A and tan A.
37. Evaluate: (1 + cot² θ) sin² θ.
38. Prove: (1 + tan² A) / (1 + cot² A) = tan² A.
39. If 3 cot θ = 4, find the value of (sin θ + cos θ).
40. Evaluate without using a calculator: sin² 30° + cos² 60° + tan² 45°.

Chapter 9: Some Applications of Trigonometry

41. A man is standing on the deck of a ship, which is 10 m above water level. He observes the angle of elevation of the top of a lighthouse as 60° and the angle of depression of the base of the lighthouse as 30°. Find the height of the lighthouse.
42. A 7 m long ladder rests against a vertical wall. The foot of the ladder is 4 m away from the wall. Find the angle of elevation of the ladder with the ground.
43. From a point on the ground, the angle of elevation of the top of a tower is 60°. From another point 40 m vertically above the first, the angle of elevation of the top is 30°. Find the height of the tower.
44. From the top of a 60 m building, the angles of depression of the top and bottom of a tower are 30° and 60°. Find the height of the tower.
45. A balloon is observed at an angle of elevation of 60° at a height of 20√3 m. Find the horizontal distance between the observer and the balloon.

Chapter 10: Statistics

46. The marks obtained by 50 students are given. Draw a histogram. (Question with data table).
47. The mean of 20 observations is 25. On checking, it is found that an observation 20 was wrongly taken as 30. Find the correct mean.
48. Find the mode of the marks: 10, 15, 10, 20, 25, 20, 20, 25, 10.
49. The mean of five numbers is 27. If one number is excluded, their mean becomes 25. Find the excluded number.
50. The median of the following data is 28. Find the missing frequency. (Data-based Q).

Class 10 Half-Yearly Maths – Top 50 Important Questions

Read Also: Maths formula class 10