## Class 10 Math Important Questions

The CBSE class 10th mathematics focuses on the basic geometry, trigonometry and the concept of numbers. The subject helps to build a basic aptitude and concepts that would be beneficial for the students who wish to appear for aptitude based tests in future. This article is on CBSE class 10 Math Important questions.

- The class 10 Math formulae are otherwise
**very general**, but sometimes it is seen in the previous question papers that the questions are asked from the middle of such numericals which you have to identify. - The
**concepts/formulae**sheet should be kept**handy.** - You should know the
**usage of all the Math formulae.** - If you really want to score an A1 in math exam, then it is really important to completely be in sync with your NCERT book. Almost the whole questions paper comprises of the concepts and formulae that are given in the NCERT book.
- If you are solving a problem based question, it is
**advisable to read the problem again and again**to get the**exact idea**of what you’re being asked to solve. - Write down on a rough paper, exactly what is given in the question paper and what you are asked to find. Then in a systematic way, try and find what is asked.
- Once revision is done, start solving the
**sample papers, unsolved papers**and practice papers within the given amount of time.

## Some CBSE class 10 Math Important questions have been compiled below:

1. Prove that 1/(2+√3) is an irrational number.

2. Prove that √5 is an irrational number.

3. Show that any positive odd integer is of the form (6p+1), (6p+3) or (6p+5), where p is some integer.

4. Find the HCF and LCM of 306 and 54. Verify that HCF × LCM = Product of the two numbers.

5. A merchant has 120 litres of oil of one kind, 180 litres of another kind and 240 litres of third kind. He wants to sell the oil by filling the three kinds of oil in tins of equal capacity. What should be the greatest capacity of such a tin?

6. Use Euclid’s Lemma to show that square of any positive integer is of form 4 m or 4m+1 for some integer m.

7. If the ratio of the corresponding sides of two similar triangles is 2:3, then what is the ratio of their corresponding height?

8. The areas of two similar triangles and are 25cm^{2} and 49 cm^{2} respectively. If QR = 9.8 cm, find BC.

*9. DE is parallel to BC. If AD = 12.4 cm, DB = 6.2cm, AE = 2x and EC = 6x – 2. Find the value of x.*

10. Prove that if a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points then the other two sides are divided in the same ratio.

11. In the given figure, in ABC, DE || BC so that AD = 2.4cm, AE = 32cm and EC=4.8cm. Find AB.

12. State and prove Basic Proportionality Theorem. Using the above theorem, if ABCD is a trapezium whose diagonals intersect each other at O show that AO/OC = BO/OD.

13. It the given figure, if Δ ABE ≅ Δ ACD, prove that ΔADE ~ ΔABC.

14. Prove that the square of the hypotenuse is equal to the sum of the squares of the other two sides.Using the above result show that sum of the squares of the sides of a rhombus is equal to the sum of the squares of its diagonals.

15. Prove that the ratio of areas of 2 similar triangles is equal to the ratio of squares of their corresponding sides. Using the above result prove that area of an equilateral triangle described on one side of a square is half the area of triangle described on one of its diagonals.

16. Find the missing frequencies f1 and f2 in the following frequency distribution table, it is given that the mean of the distribution is 56.

C.I | 0 – 20 | 20 – 40 | 40 – 60 | 60 – 80 | 80 –100 | 100 – 120 | Total |

F | 16 | f1 | 25 | f2 | 12 | 10 | 90 |

17. Find the value of k, for which given value is a zero of the given quadratic polynomial

(a) (x^{2}+2kx-3); x = -1/2 (b) x^{2}+4ax-k; x= -a

18. Verify that -1, 1, 2 are zeros of a cubic polynomial x^{3} – 2x^{2} – x+2 & verify the relationship between the zeros & its coefficients.

19. Form a quadratic polynomial whose (i) zeros are 2 & -3 (ii) zeros are -4/5 & 1/3.

20. Solve the equations 15x -6y = 30 ; 17x + 10y =118.

21. Solve the equations ax + by = c; bx – ay = 0.

22. A fraction becomes 9/11, if 2 is added to both the numerator & denominator. If 3 is added to both the numerator & denominator it becomes 5/6. Find the fraction.

23. Solve (By cross multiplication) 2/u + 3/v = 13 ; 5/u – 4/v = -2.

24. Find the values of p & q for which the following system has infinite solutions.

2x + 3y = 7 ;

(p + q)x + (2p – q)y = 21.

25. I am three times as old as my son. Five years later, I shall be two and a half times as old as my son. How old I am and how old is my son?

26. A and B are friends and their ages differ by two years. A’s father D is twice as old as A, & B is twice as old as his sister C. The ages of D and C differ by 40 years. Find the ages of A and B?

27. Five years hence father’s age will be three times age of his son. Five years ago father was seven times as old as his son. Find their present ages.

28. Five years ago, Neeta was thrice as old as Gita. Ten years later, Neeta will be twice as old Gita. How old are Gita & Neeta now?

29. If two zeroes of the polynomials are x^{4 }– 6x^{3 }– 26x^{2} + 138x – 35 are 2±√3, find the other zeroes.

30. On dividing x^{3 }– 3x^{2} + x + 2 by polynomials g(x), the quotient & remainder were x – 2 & – 2x+4 respectively. Find g(x).

31. If the polynomials x^{4} + 2x^{3} + 8x^{2} + 12x + 18 is divided by another polynomial x^{2}+ 5, the remainder comes out to be p x +q. Find the values of p and q .

32. Find the values of m and n for which the following system of equations has infinitely many solutions:

3x+4y = 12;

(m + n)x +2(m-n)y=5m-1.

33. Solve for x & y: x/a +y/b =1 ; a(x-a) – b( a + b)=2a^{2}+b^{2}.

34. Solve for x & y: b x/a +ay/b = a^{2}+b^{2}; x+ y =2ab.

35. Solve for x & y: x/a – y/b = a-b; ax +by = a^{3} + b^{3}

36. Solve for x & y: 3(2x + y ) = 7xy ; 3(x +3y) = 11xy.

37. Solve for x & y: 3/(x + y) + 2/(x – y) =2 ; 9/(x +y) + 4/(x- y) = 1; (x + y) ≠0 (x – y) ≠0.

38. A two digit number is obtained by either multiplying the sum of the digits by 8 & adding 1, or by multiplying the difference of the digits by 13 & adding 2. Find the number. How many such numbers are there?

39. The difference between two numbers is 15 &the difference between their squares is 465. Find the numbers.

40. In a rectangle if length is increased by 7 units & breadth is decreased by 3 units or if length is decreased by 7 units & breadth is increased by 5 units, in both the cases the area remains same. Find the dimensions of the rectangle. Also find the area of the rectangle.

41. A fraction is such that if the numerator is multiplied by 3 & denominator is reduced by 3, we get 18/11, but if the numerator is increased by 8 & denominator is doubled, we get 2/5. Find the fraction.

42. Solve(By Cross-Multiplication)

(a – b) x + (a+ b) y = a^{2} – 2ab – b^{2 };

(a + b)( x + y) = a^{2} + b

## CBSE class 10 math important questions (Sample Paper)

To get a detailed overview of question paper pattern, students can go through the sample question paper provided below:

**[Time: 3hrs.] [M. M.: 80]**

**General Instructions**

**(1) All questions are compulsory.**

**2) The questions paper consists of thirty questions divided into 4 sections A, B, C, D. Section ‘A’ comprises of ten questions of 1 marks each, Section ‘B’ comprises of five questions of 2 marks each, Section ‘C’ comprises of ten questions of 3 marks each and Section ‘D’ comprises of five questions of 6 marks each.**

**SECTION – A (10 marks)**

1. Centroid of triangle whose vertices are A(−4,6), B(2, −2) and C(2, 5) is . a) (0, 2) b) (0, 3) c) (1, 3) d) (1, 2)

2- The ratio between the volumes of two spheres is 8 : 27. What is the ratio between their surface areas?

a). 2 : 3 b). 4 : 5 c). 5 : 6 d) . 4 : 9

3. If first term of an AP is a and nth term is b, then its common difference is a). (b-a)/n+1 b). (b-a)/n-1 c). (b-a)/n d). none of these

4. In a lottery there are 7 prizes and 21 blanks. The probability of getting a prize is. a). 1/2 b). 1/3 c). 1/4 d). 1/5

5. A funnel is the combination of a). Cone and a cylinder b). Frustum of a cone and cylinder c). Hemisphere and cylinder d). Hemisphere and cone

6. Which was the first book on Probability?

7. If the product of roots of the equation ax2 + bx + c = 0 is unity, then a). a = c b). c = b c). b = a d). b 2 = 4ac

8. The length of the tangent drawn from a point 8cm away from the centre of a circle of radius 6cm is: a).7cm b). 27cm c). 10cm d). 5cm

9. A letter is chosen at random from the letters of the word ‘ASSASSINATION’. Find the probability that the letter chosen is a consonant. a). 1/13 b). 7/13 c). 6/13 d). 2/13

10. If k , 2k-1 and 2k+1 are three consecutive terms of an A.P. , Find the value of k.

**SECTION B (10marks):**

11. Use factorization method to solve: 3x 2 − 2√6x + 2 = 0.

12. Divide a line segment of length 8 cm internally in the ratio 4:5. Also, give justification of the construction.

13. Without drawing the graphs , state whether the following pair of linear equations will represent intersecting lines , coincident lines or parallel lines. 6x – 3y + 10 =0 2x – y + 9 = 0 Justify your answer.

14. Prove that the perpendicular at the point of contact to the tangent to a circle passes through the centre.

15. Solve for x, 4√6 x2 -13 x -2√6 = 0 by using a completing the square.

**SECTION C (30marks)**

16. Prove that the diagonals of a rectangle with vertices (0, 0), (a, 0), (a, b) and (0,b) bisect each other and are equal.

17. What is the probability that a leap year, selected at random will contain 53 Sundays?

18. Find the roots of 4x 2 + x – 5 = 0 by the method of completing the square.

19. The sum of the 4th and 8th terms of an AP is 24 and the sum of the 6th and 10th terms is 44. Find the first three terms of the AP.

20. Two concentric circles are of radii 5cm and 3 cm. Find the length of the chord of the larger circle which touches the smaller circle.

21. The cost of fencing a circular field at the rate of Rs 24 per meter is Rs 5280. The field is to be ploughed at the rate of Rs 0.50 per m2 . Find the cost of ploughing the field. [π = 22 7 ]

22. Prove that opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

23. A tree 12m high is broken by the wind in such a way that its top touches the ground and makes an angle of 60o with the ground. At what height from bottom the tree is broken by the wind. Give the answer to the second place of decimal.

24. For what value of n, the nth term of the following two A.P.’s are equal?

25. Construct an isosceles triangle whose base is 8 cm and altitude 4 cm and then another triangle whose side are 1 1 2 times the corresponding sides of the isosceles triangle.

**SECTION D (30marks)**

26. Water is flowing at 7m/sec through a circular pipe of internal diameter 2 cm into a cylindrical tank the radius of whose base is 40cm. Find the increased in water level in 30 minutes. OR A toy is in the form of a cone mounted on a hemisphere of radius 3.5cm. If the total height of the toy is 15.5cm, find its total surface area.(use =22/7)

27. Show that the points (1, 7), (4, 2), (–1, –1) and (– 4, 4) are the vertices of a square.

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

29. Construct a triangle of sides 4 cm, 5cm and 6cm and then a triangle similar to it whose side’s are2/3 of the corresponding sides of the first triangle Give the justification of the construction.

30 .Sarah Purchases every year National Saving Certificates of value exceeding the last year purchases by Rs.25. After 20 years, she find the total value of certificates purchased by her is Rs.7250. Find the value of the certificates purchased.

- Students can check out sample papers released by CBSE by visiting from their official site.
- It is also important to know the marking scheme as CBSE has provision of awarding marks for correct steps. Students can check out the official marking scheme on the official website.
- For more CBSE Board Exam study material, visit the official CBSE website.

Hope you liked our article on CBSE class 10 Math Important questions.

Also check out our article on How To Tackle CBSE Class 10th Maths here.