# NCERT Class 10 Mathematics Higher Order Thinking Skills CBSE Board Sample Problems Part 1 (For CBSE, ICSE, IAS, NET, NRA 2023)

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**Number system**

1. Show that square of any positive odd integer is of the form for some integer q.

2. Find whether the decimal representation (expansion) of the numbers

(i)

(ii) terminating or non-terminating

3. The decimal expansion of following numbers, will terminate after how many places

(i)

(ii)

4. Prove that is an irrational number.

**Polynomials**

5. Find all zeroes of the polynomials if two of its zeroes are and

6. Find all zeroes of the polynomial if two of its zeroes are

7. Obtain all zeroes of the polynomial if two of its zeroes are and

**Linear Equations**

8. Solve the following pair of equations:

9. A chemist has one solution which is 50 % acid and a second solution which is 25 % acid. How much of each should be mixed to make 10 litres of a 40 % acid solution.

10. Check graphically whether the pair of equations is consistent. Also, find the points where these lines intersect x-axis.

11. A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Santa paid ₹ 27 for a book kept for seven days while Sunita paid ₹ 21 for the book she kept for five days. Find the fixed charge and the charge for each extra day.

**Quadratic Equations**

12. Find the value of k for which the roots of the quadratic equation are equal.

13. Find the value of p for which the roots of the quadratic equation are real where

14. Sum of the areas of two squares is 468 m2. If the difference of their perimeters is 24 m, find the sides of two squares.

15. A train travels 360 km at a uniform speed. If the speed had been 5 km/hr. more, it would have taken 1 hour less for the same journey. Find the speed of the train.

**Arithmetic Progressions**

16. The 19^{th} term of an AP is equal to 3 times its 6^{th} term. If the 9^{th} term of the AP is 19, find the AP.

17. Find the sum of first 19 terms of an AP whose 8^{th} term is 41 and 13^{th} term is 61.

18. Find the sum of all three-digit numbers, which are divisible by 9.

19. For what value of and are three consecutive terms of an AP.

20. If the sum of first n terms of an AP is given by find the common difference of the AP.

21. The sum of first three terms of an AP is 15. If the sum of their squares is 93, find the AP.

**Trigonometry**

22. Prove that

23. If prove that -

24. If and ,

25. Evaluate:

26. Prove that

**Coordinate Geometry**

27. Show that the points do not form a quadrilateral.

28. Find the Coordinates of Point P on AB such that where A (3,1) and B (-2,5)

29. Points P, Q, R and S in that order divides the line segment joining points A (2,5) and B (7, -5) in five equal parts. Find the coordinates of P, Q, R, and S.

30. The vertices of a triangle are If the area of triangle is 15 sq. units, then find the value (s) of K.