Important Questions Class 10 Maths for CBSE 2026-27

Class 10 Maths important questions help students practise concepts, formulas, proofs and applications from the full NCERT syllabus.
Important Questions Class 10 Maths cover MCQs, short answers, long answers and case studies based on the CBSE 2026-27 paper pattern.

Class 10 Maths 2026-27 needs steady practice across formulas, proofs, calculations and application-based questions. These Important Questions Class 10 Maths help students revise the full NCERT syllabus through the same question flow used in the CBSE board paper. The theory paper carries 80 marks and includes 38 questions across MCQs, very short answers, short answers, long answers and case studies. Students should practise Algebra, Geometry, Trigonometry, Mensuration, Statistics and Probability with clear steps. These CBSE Class 10 Maths Important Questions are arranged section-wise so students can revise chapter-wise concepts, improve step marking and prepare for board-style problem solving.

Key Takeaways

• Theory Paper: CBSE Class 10 Maths theory paper carries 80 marks across 38 questions.
• Algebra: Algebra carries 20 marks through Polynomials, Linear Equations, Quadratic Equations and AP.
• Geometry: Triangles and Circles together carry 15 marks in the standard paper.
• Case Studies: Section E includes 3 case-based questions of 4 marks each.

Important Questions for Class 10 Maths Structure 2026-27

Section A: Class 10 Maths MCQs

Section A has 20 objective-type questions. Class 10 Maths MCQs usually test formulas, definitions, quick calculations and concept identification from Real Numbers, Algebra, Geometry, Trigonometry and Probability.

Q1. The HCF of 96 and 404 is:

1. 2
   b. 4
   c. 8
   d. 12

Answer: b. 4

Using Euclid’s division algorithm:

404 = 96 × 4 + 20

96 = 20 × 4 + 16

20 = 16 × 1 + 4

16 = 4 × 4 + 0

So, HCF = 4.

Q2. If p(x) = x² - 5x + 6, then its zeroes are:

1. 1 and 6
   b. 2 and 3
   c. -2 and -3
   d. 0 and 6

Answer: b. 2 and 3

x² - 5x + 6 = (x - 2)(x - 3)

So, the zeroes are 2 and 3.

Q3. The pair of linear equations x + y = 5 and 2x + 2y = 10 has:

1. No solution
   b. Unique solution
   c. Infinitely many solutions
   d. Two solutions only

Answer: c. Infinitely many solutions

The second equation is exactly twice the first equation. Both equations represent the same line.

Q4. The discriminant of the quadratic equation x² - 4x + 4 = 0 is:

1. 0
   b. 4
   c. 8
   d. 16

Answer: a. 0

Discriminant = b² - 4ac

= (-4)² - 4(1)(4)

= 16 - 16

= 0

Q5. The 10th term of the AP 3, 7, 11, 15, ... is:

1. 35
   b. 37
   c. 39
   d. 41

Answer: c. 39

Here, first term a = 3 and common difference d = 4.

10th term = a + 9d

= 3 + 9 × 4

= 39

Q6. The distance between points (0,0) and (3,4) is:

1. 3 units
   b. 4 units
   c. 5 units
   d. 7 units

Answer: c. 5 units

Distance = √[(3 - 0)² + (4 - 0)²]

= √(9 + 16)

= √25

= 5 units

Q7. If two triangles are similar, then their corresponding sides are:

1. Equal only
   b. Proportional
   c. Perpendicular
   d. Unrelated

Answer: b. Proportional

In similar triangles, corresponding angles are equal and corresponding sides are proportional.

Q8. The tangent at any point of a circle is perpendicular to the:

1. Diameter through any point
   b. Radius through the point of contact
   c. Chord through the centre
   d. Secant through the point

Answer: b. Radius through the point of contact

The radius drawn to the point of contact is perpendicular to the tangent.

Q9. sin 30° equals:

1. 1/2
   b. √3/2
   c. 1
   d. 0

Answer: a. 1/2

sin 30° = 1/2.

Q10. tan 45° equals:

1. 0
   b. 1/2
   c. 1
   d. √3

Answer: c. 1

tan 45° = 1.

Q11. The area of a sector of angle 90° in a circle of radius r is:

1. πr²
   b. πr²/2
   c. πr²/4
   d. 2πr

Answer: c. πr²/4

Area of sector = angle/360 × πr²

= 90/360 × πr²

= πr²/4

Q12. The volume of a cylinder with radius r and height h is:

1. πr²h
   b. 2πrh
   c. πrh²
   d. 4πr²

Answer: a. πr²h

Volume of cylinder = base area × height.

So, volume = πr²h.

Q13. The mean of 5, 7, 9, 11 and 13 is:

1. 8
   b. 9
   c. 10
   d. 11

Answer: b. 9

Mean = (5 + 7 + 9 + 11 + 13)/5

= 45/5

= 9

Q14. If an event is impossible, its probability is:

1. 0
   b. 1
   c. 1/2
   d. 2

Answer: a. 0

The probability of an impossible event is always 0.

Q15. If one card is drawn from a standard deck, the probability of getting a king is:

1. 1/13
   b. 1/26
   c. 4/13
   d. 1/4

Answer: a. 1/13

There are 4 kings in 52 cards.

Probability = 4/52 = 1/13.

Q16. If a polynomial has degree 2, its graph can have at most:

1. 1 zero
   b. 2 zeroes
   c. 3 zeroes
   d. 4 zeroes

Answer: b. 2 zeroes

A quadratic polynomial can have at most two zeroes.

Q17. Assertion: The number 7 × 11 × 13 + 13 is a composite number.

Reason: A number with more than two factors is composite.

1. Both Assertion and Reason are true, and Reason explains Assertion
   b. Both are true, but Reason does not explain Assertion
   c. Assertion is true, Reason is false
   d. Assertion is false, Reason is true

Answer: a. Both Assertion and Reason are true, and Reason explains Assertion

7 × 11 × 13 + 13 = 13(7 × 11 + 1)

So, it has 13 as a factor.

Q18. Assertion: If two tangents are drawn from an external point to a circle, they are equal.

Reason: Tangent lengths from an external point to a circle are equal.

1. Both Assertion and Reason are true, and Reason explains Assertion
   b. Both are true, but Reason does not explain Assertion
   c. Assertion is true, Reason is false
   d. Assertion is false, Reason is true

Answer: a. Both Assertion and Reason are true, and Reason explains Assertion

This is the standard tangent theorem.

Q19. Assertion: The quadratic equation x² + 4x + 5 = 0 has no real roots.

Reason: Its discriminant is negative.

1. Both Assertion and Reason are true, and Reason explains Assertion
   b. Both are true, but Reason does not explain Assertion
   c. Assertion is true, Reason is false
   d. Assertion is false, Reason is true

Answer: a. Both Assertion and Reason are true, and Reason explains Assertion

Discriminant = 4² - 4(1)(5)

= 16 - 20

= -4

So, roots are not real.

Q20. Assertion: Probability of an event can never be greater than 1.

Reason: Probability lies between 0 and 1.

1. Both Assertion and Reason are true, and Reason explains Assertion
   b. Both are true, but Reason does not explain Assertion
   c. Assertion is true, Reason is false
   d. Assertion is false, Reason is true

Answer: a. Both Assertion and Reason are true, and Reason explains Assertion

For any event E, 0 ≤ P(E) ≤ 1.

Section B: Very Short Answer Questions from Class 10 Maths Important Questions

Section B has 5 questions of 2 marks each. Class 10 Maths Important Questions in this section need short step-by-step solutions with correct formulas.

Q21. Find the HCF of 135 and 225 by Euclid’s division algorithm.

225 = 135 × 1 + 90

135 = 90 × 1 + 45

90 = 45 × 2 + 0

So, HCF = 45.

Q22. Find the zeroes of x² - 7x + 12.

x² - 7x + 12 = (x - 3)(x - 4)

So, x - 3 = 0 or x - 4 = 0.

Therefore, the zeroes are 3 and 4.

Q23. Find the value of k if the quadratic equation x² - 2kx + 9 = 0 has equal roots.

For equal roots, discriminant = 0.

Here, a = 1, b = -2k and c = 9.

b² - 4ac = 0

(-2k)² - 4(1)(9) = 0

4k² - 36 = 0

k² = 9

So, k = ±3.

Q24. Find the distance between A(2,3) and B(6,6).

Distance = √[(6 - 2)² + (6 - 3)²]

= √[4² + 3²]

= √(16 + 9)

= √25

= 5 units.

Q25. A die is thrown once. Find the probability of getting an even number.

Possible outcomes = 1, 2, 3, 4, 5, 6.

Even outcomes = 2, 4, 6.

Number of favourable outcomes = 3.

Probability = 3/6 = 1/2.

Section C: Short Answer Questions from CBSE Class 10 Maths Important Questions

Section C has 6 questions of 3 marks each. CBSE Class 10 Maths Important Questions here usually test multi-step calculations, proofs and formula application.

Q26. Prove that √5 is irrational.

Assume that √5 is rational.

Then √5 = a/b, where a and b are coprime integers and b ≠ 0.

Squaring both sides:

5 = a²/b²

So, a² = 5b².

This means 5 divides a², so 5 divides a.

Let a = 5c.

Then a² = 25c².

So, 25c² = 5b².

This gives b² = 5c².

So, 5 divides b², and 5 divides b.

This means a and b have 5 as a common factor.

This contradicts that a and b are coprime.

Therefore, √5 is irrational.

Q27. Solve the pair of equations 2x + 3y = 13 and x + y = 5.

From x + y = 5:

x = 5 - y.

Substitute in 2x + 3y = 13:

2(5 - y) + 3y = 13

10 - 2y + 3y = 13

y = 3

Now, x + y = 5.

x + 3 = 5

x = 2

Therefore, x = 2 and y = 3.

Q28. Find the sum of the first 20 terms of the AP 5, 8, 11, 14, ...

Here, first term a = 5.

Common difference d = 3.

Number of terms n = 20.

Sum formula:

Sn = n/2 [2a + (n - 1)d]

S20 = 20/2 [2(5) + (20 - 1)3]

= 10 [10 + 57]

= 10 × 67

= 670.

Q29. In triangles ABC and DEF, ∠A = ∠D, ∠B = ∠E and AB/DE = BC/EF. Prove that ΔABC ~ ΔDEF.

Given, ∠A = ∠D and ∠B = ∠E.

By angle sum property, ∠C = ∠F.

So, all corresponding angles are equal.

Therefore, ΔABC ~ ΔDEF by AAA similarity criterion.

The given side ratio also supports corresponding side proportionality.

Q30. If sin A = 3/5, find cos A and tan A.

Given sin A = 3/5.

So, opposite side = 3 and hypotenuse = 5.

By Pythagoras theorem:

Adjacent side² = 5² - 3²

= 25 - 9

= 16

Adjacent side = 4.

So, cos A = adjacent/hypotenuse = 4/5.

tan A = opposite/adjacent = 3/4.

Q31. Find the mean of the following data: 10, 20, 30, 40, 50.

Mean = sum of observations/number of observations.

Sum = 10 + 20 + 30 + 40 + 50 = 150.

Number of observations = 5.

Mean = 150/5 = 30.

Section D: Long Answer Questions from Important Questions for Class 10 Maths

Section D has 4 long-answer questions of 5 marks each. Important Questions for Class 10 Maths in this section need full working, theorem use and clear final answers.

Q32. Solve the quadratic equation 2x² - 7x + 3 = 0 by factorisation.

Given equation:

2x² - 7x + 3 = 0

Split the middle term:

2x² - 6x - x + 3 = 0

Group the terms:

2x(x - 3) - 1(x - 3) = 0

(2x - 1)(x - 3) = 0

So, 2x - 1 = 0 or x - 3 = 0.

From 2x - 1 = 0:

2x = 1

x = 1/2

From x - 3 = 0:

x = 3

Therefore, the roots are 1/2 and 3.

Q33. Find the coordinates of the point which divides the line segment joining A(2, -3) and B(8, 9) in the ratio 1:2 internally.

Let the point P divide A(2, -3) and B(8, 9) in the ratio 1:2.

Using section formula:

x = (m x₂ + n x₁)/(m + n)

y = (m y₂ + n y₁)/(m + n)

Here, m = 1 and n = 2.

x = [1(8) + 2(2)]/(1 + 2)

= (8 + 4)/3

= 12/3

= 4

y = [1(9) + 2(-3)]/(1 + 2)

= (9 - 6)/3

= 3/3

= 1

So, P = (4, 1).

Q34. A cone has radius 7 cm and height 24 cm. Find its curved surface area and total surface area.

Given, radius r = 7 cm and height h = 24 cm.

Slant height l is found by:

l² = r² + h²

l² = 7² + 24²

l² = 49 + 576

l² = 625

l = 25 cm

Curved surface area of cone = πrl

= (22/7) × 7 × 25

= 550 cm²

Total surface area of cone = πr(l + r)

= (22/7) × 7 × (25 + 7)

= 22 × 32

= 704 cm²

Therefore, curved surface area = 550 cm² and total surface area = 704 cm².

Q35. Prove that tangents drawn from an external point to a circle are equal.

Let PA and PB be two tangents drawn from an external point P to a circle with centre O.

Join OA, OB and OP.

Since radius is perpendicular to tangent at the point of contact:

OA ⟂ PA and OB ⟂ PB.

So, ∠OAP = 90° and ∠OBP = 90°.

In triangles OAP and OBP:

OA = OB because both are radii.

OP = OP because it is common.

∠OAP = ∠OBP = 90°.

Therefore, ΔOAP ≅ ΔOBP by RHS congruence.

So, PA = PB by corresponding parts of congruent triangles.

Hence, tangents drawn from an external point to a circle are equal.

Section E: Class 10 Maths Case Study Questions

Section E has 3 case-based questions of 4 marks each. Class 10 Maths Case Study Questions usually test real-life application from AP, Trigonometry, Mensuration, Statistics, Probability and Coordinate Geometry.

Q36. Case Study: Arithmetic Progression in Seating Arrangement

A school auditorium has seats arranged in rows. The first row has 20 seats, the second row has 24 seats, the third row has 28 seats, and so on. The number of seats increases by 4 in each next row.

Q36(a). What is the common difference?

The common difference is 4.

Q36(b). Find the number of seats in the 10th row.

Here, a = 20, d = 4 and n = 10.

nth term = a + (n - 1)d

10th term = 20 + 9 × 4

= 20 + 36

= 56

So, the 10th row has 56 seats.

Q36(c). Find the total number of seats in the first 10 rows.

Sum of n terms = n/2 [2a + (n - 1)d]

S10 = 10/2 [2(20) + 9(4)]

= 5 [40 + 36]

= 5 × 76

= 380

So, there are 380 seats in the first 10 rows.

Q37. Case Study: Height of a Building

A student observes the top of a building from a point on the ground. The angle of elevation is 30°. The distance of the student from the foot of the building is 30√3 m.

Q37(a). Which trigonometric ratio is used here?

tan θ is used because height and base are involved.

Q37(b). Write the relation between height and distance.

tan 30° = height/distance.

Q37(c). Find the height of the building.

Let the height be h.

tan 30° = h/(30√3)

1/√3 = h/(30√3)

h = 30 m

So, the height of the building is 30 m.

Q38. Case Study: Probability in a Game

A box contains 5 red balls, 3 blue balls and 2 green balls. One ball is drawn at random.

Q38(a). Find the total number of balls.

Total balls = 5 + 3 + 2 = 10.

Q38(b). Find the probability of getting a blue ball.

Number of blue balls = 3.

Probability = 3/10.

Q38(c). Find the probability of getting a ball that is not green.

Number of green balls = 2.

Number of balls not green = 10 - 2 = 8.

Probability = 8/10 = 4/5.

Chapter-Wise Practice from Class 10 Maths Chapter Wise Important Questions

Class 10 Maths Chapter Wise Important Questions should cover all 14 NCERT chapters for the 2026-27 exam. Students should begin with Real Numbers because Euclid’s division algorithm and irrationality proofs build number-system reasoning.

Algebra needs the most attention because it carries the highest unit weightage. Students should practise Polynomials, Pair of Linear Equations in Two Variables, Quadratic Equations and Arithmetic Progressions together.

Geometry should be revised through theorem-based questions from Triangles and Circles. Coordinate Geometry needs repeated practice of distance formula and section formula.

Trigonometry should include ratios, identities and heights and distances. Mensuration needs formulas from Areas Related to Circles and Surface Areas and Volumes.

Statistics and Probability should be practised at the end of revision. These chapters can help students score well when formulas and calculation steps are clear.

Class 10 Maths Standard Important Questions for 2026-27

Class 10 Maths Standard Important Questions need stronger step-by-step practice because the paper tests reasoning, calculation accuracy and application. The standard paper includes questions that require formula selection, proof writing and multi-step solving.

Students should focus on Algebra, Geometry, Trigonometry and Mensuration because these units carry higher combined weightage. These topics also appear in 3-mark, 5-mark and case-based questions.

For Class 10 Maths 2026-27 preparation, students should solve mixed questions after completing chapter-wise practice. This helps them switch between formulas, theorems and application questions faster.

Class 10 Maths Sample Questions for Final Revision

Class 10 Maths Sample Questions are useful after completing NCERT examples and exercises. Students should attempt MCQs, 2-mark steps, 3-mark problems, 5-mark long answers and case-based questions under timed practice.

A good revision set should include Real Numbers proofs, quadratic equations, AP sums, coordinate geometry, triangle similarity, tangents, trigonometry values, mensuration and probability. These areas cover most common Class 10 Maths Board Questions.

Students should check every solution for formula, substitution, calculation and final unit. This improves step marking in the board exam.

Useful Important Questions Class 10 Maths Links