Top 60 Important Questions for Class 10 Maths Board Exam 2026 (Without Solutions) | Full Syllabus Practice Questions

Get Top 60 Important Questions for Class 10 Maths Board Exam 2026 from full syllabus. Chapter-wise most repeated & high-weightage practice questions (without solutions). Best for CBSE, ICSE & State Board students for quick revision.

TOP 60 Questions Class 10 Maths

1. Prove that √3 is irrational number.
2. Prove that 2-5√7 is irrational.
3. Prove that 7x6x5x4x3+7 is composite.
4. Find LCM and HCF of 24,36 and 48 by prime factorization method.
5. Check whether 6ⁿ can end with the digit 0 for any natural number n.
6. On a morning walk, 3 persons step out together and their steps measure 30 cm, 36 cm and 40 cm respectively.What is the minimum distance each should walk so that each can cover the same distance in complete steps ?
7. Find zeroes of quadratic polynomial 4x² – 4x-3 and verify the relation between the zeroes and its coefficients.
8. Find the polynomial whose zeroes are 5 + √19 and 5 -√19.
9. If α and β are the zeroes of the quadratic polynomial f(x) = x² + x – 2, find the value of 1/α +1/β .
10. Find the value of k such that the polynomial x²– (k+6)x + 2(2k-1) has sum of its zeroes equal to half of their product.
11. Five years hence, the age of Jacob will be three times that of his son. Five years ago, Jacob’s age was seven times that of his son. What are their present ages?
12. The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.
13. The monthly income of A and B are in the ratio of 9: 7. If their monthly expenditures are in the ratio 4:3. If each saves Rs 1600 per month. Find the monthly incomes of each?
14. Solve the following pair of linear equations by the elimination method and the substitution method 3x + 4y = 10 and 2x–2y =2.
15. How many solutions following lines have ? Tell about consistency/inconsistency and also tell that lines are parallel or intersecting or coincident. 3x – 5y – 4 = 0 and 9x = 2y + 7
16. Find the discriminant of the quadratic equation 2x²-4x+3 , and hence find the nature of its roots.
17. Find the roots of the quadratic equation by quadratic formula. 3x²-6x+2
18. Find the value of k for which the quadratic equation x² +kx +16=0 has equal roots.
19. Find 2 consecutive odd positive integers, sum of whose squares is 290.
20. A train travels a distance of 480 km at a uniform speed .If the speed had been 8 km per hour less, then it would have taken 3 hours more to cover the same distance. Find the actual speed of the train.
21. The altitude of a right angled triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.
22. Find the 20th term from the last of the A.P. : 3,8,13,………253.
23. How many three- digit numbers are divisible by 7 ?
24. If sum of first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of n terms.
25. Find sum of 15 terms of an A.P. whose nth term is 9-5n.
26. 200 logs are stacked in such a way, 20 logs in the bottom row, 19 in the next row, 18 in the row next to it and so on. In how many rows are the 200 logs placed and how many logs are there in the top row ?
27. Find the 31st term of an AP whose 11th term is 38 and the 16th term is 73.
28. Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18 respectively.
29. ( BPT) Prove that if a line is drawn parallel to one side of a triangle, then it divides the other two sides in the same ratio.
30. Sides AB and BC and median AD of a triangle ABC are respectively proportional to sides PQ and QR and median PM of triangle PQR. Show that ∆ABC ~ ∆PQR.
31. If ∆ABC ~ ∆RPQ, AB = 3 cm, BC = 5 cm, AC = 6 cm, RP = 6 cm and PQ = 10, then find QR
32. ABCD is a trapezium in which AB || DC and its diagonals intersect each other at the point O. Show that AO /BO = CO/DO
33. Prove that the angle between the two tangents drawn from external point to a circle is supplementary to the angle subtended by the line segment joining the points of contact at the centre.
34. PQ is a chord of length 8 cm of a circle of radius 5 cm. The tangents at P and Q intersect at a point T. Find the length TP.
35. Find a point on the y-axis which is equidistant from the points A(6, 5) and B(–4, 3).
36. Find the ratio in which the y-axis divides the line segment joining the points (5, – 6) and (–1, – 4). Also find the point of intersection.
37. If the points A(6, 1), B(8, 2), C(9, 4) and D(p, 3) are the vertices of a parallelogram, taken in order, find the value of p.
38. Find the coordinates of the point which divides the join of (–1, 7) and (4, –3) in the ratio 2 : 3
39. If sin A = 3/4 , calculate cos A and tan A.
40. If tan(A+B)=√3 and tan(A-B) = 1/√3 ; 0o < A+B≤900 ;A > B, find A and B.
41. Find value of sin 45⁰ geometrically.
42. A circus artist is climbing a rope 12m long which is tightly stretched and tied from the top of a vertical pole to the ground. Find the height of the pole, if the angle made by the rope with the ground is 30 degree.
43. A 1.5 m tall boy is standing at some distance from a 30 m tall building. The angle of elevation from his eyes to the top of the building increases from 30° to 60° as he walks towards the building. Find the distance he walked towards the building.
44. A triangle ABC is drawn to circumscribe a circle of radius 4 cm such that the segments BD and DC into which BC is divided by the point of contact D are of lengths 8 cm and 6 cm respectively (see figure). Find the sides AB and AC.

45. Prove that the lengths of tangents drawn from an external point to circle are equal.

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46. Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that ∟PTQ =2∟OPQ
47. A quadrilateral ABCD is drawn to circumscribe a circle. Prove that AB + CD = AD + BC
48. Prove that the parallelogram circumscribing a circle is a rhombus.
49. In fig. XY and X’Y’ are two parallel tangents to a circle with center O and another tangent AB with point of contact C intersecting XY at A and X’Y’ at B. Prove that angle AOB= 90⁰
50. Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.
51. chord of a circle of radius 10 cm subtends a right angle at the centre. Find the area of the corresponding : (i) minor segment (ii) major sector. (Use Pi = 3.14)
52. A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel.
53. A solid iron pole consists of a cylinder of height 220 cm and base diameter 24 cm, which is surmounted by another cylinder of height 60 cm and radius 8 cm. Find the mass of the pole, given that 1 cm3 of iron has approximately 8g mass. (Use pi= 3.14).
54. From a solid cylinder of height 7 cm and base diameter 12 cm, a conical cavity of same height and same base radius is hollowed out. Find the total surface area and the volume of the remaining solid
55. The literacy rate of females in 50 cities is given in the following frequency distribution:

Literacy rate (in % )	20-30	30-40	40-50	50-60	60-70	70-80	80-90	90-100
No. of cities	3	2	6	15	8	7	5	4

Find the mean, median and mode of above data.

56. Two dice are thrown simultaneously. Find the probability of getting:
    (i) an even number as the sum (ii) a total of at least 10 (iii) a doublet of even number
    (iv) a multiple of 3 as the sum (v) the sum as a prime number
57. Red queens and black jacks are removed from a pack of 52 playing cards. A card is drawn at random from the remaining cards, after reshuffling them. Find probability that card draw is
    (i) a king (ii) of red colour (iii) a face card (iv) a queen
58. What is the probability that a non-leap year selected at random will contain 53 Sundays ?
59. If the median of the distribution given below is 28.5 , find the values of x and y.
    

Class intervals	0-10	10-20	20-30	30-40	40-50	50-60	Total
Frequency	5	x	20	15	y	5	60

60. What is the relationship between mean, median and mode ?