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ICSE Maths 2026

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Complete board exam paper — 35 questions. No coach help. Timer will track your speed.

Questions

35

Board

ICSE

Year

2026

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ICSE Class 10 Maths 2026 — all 35 questions from the paper

The complete question list from the ICSE Class 10 Mathematics 2026 board examination is below. Every question can be attempted in the timed interactive test above, with visual step-by-step solutions and AI coaching after each answer. Three fully worked solutions from this paper follow the question list.

Questions from the 2026 paper

  1. Q1.If (x − 2) and (x + 2) are factors of x³ + x² − 4x − 4, then the third factor is:A) (x − 1)B) (x − 4)C) (x + 1)D) (x + 4)
  2. Q2.Mrs. Sharma purchases an article for ₹12,000 and sells it at ₹15,000. If the GST rate is 18%, the GST paid by her to the government is:A) ₹540B) ₹2,700C) ₹2,160D) ₹3,240
  3. Q3.The quadratic equation (a² − b²)x² + (b² − c²)x + (c² − a²) = 0 has equal roots when a, b, c are in AP. The common root is:A) -1B) 1C) 9D) (c² − a²)/(a² − b²)
  4. Q4.Mr. Arjun and Mr. Bheem each invested ₹36,000. Mr. Arjun invested in ₹100 shares at ₹80 paying 10% dividend. Mr. Bheem invested in ₹50 shares at ₹60 paying 15% dividend. Then:A) Mr. Arjun earns more (₹4,500 vs ₹4,000)B) Mr. Bheem earns more (₹5,000 vs ₹4,500)C) Mr. Arjun earns more (₹4,000 vs ₹3,600)D) Both earn the same (₹18,000 each)
  5. Q5.If 2x + 3y = 12 and 3x − 2y = 5, the value of y is:A) y = 2B) y = 6C) y = 3D) y = 1
  6. Q6.A shopkeeper buys 80 articles for ₹2,400 and sells them at a profit of 50%. The selling price of one article is:A) ₹30B) ₹15C) ₹45D) ₹60
  7. Q7.If sin θ = √3/2, then the value of sin θ : cos θ is:A) 1:√3B) √3:1C) √3:2D) 2:√3
  8. Q8.In the given figure, O is the centre of a circle. If ∠AOB = 100° and ∠AOC = 160°, then ∠BAC is:A) 80°B) 50°C) 130°D) 100°
  9. Q9.The discriminant of the equation 3x² − 2x − 1 = 0 indicates that the roots are:A) Real and equalB) Real, distinct, and rationalC) Real, distinct, and irrationalD) Not real
  10. Q10.Assertion (A): The tangent to a circle at any point is perpendicular to the radius at the point of contact.
    Reason (R): A line drawn from the centre to any chord bisects the chord.

    Choose the correct option:    A) A is true, R is true, and R is the correct explanation of AB) A is false, R is trueC) A is true, R is falseD) A is true, R is true, but R is not the correct explanation of A
  11. Q11.The ratio of the volumes of two cones with the same height is 1:3. If the radius of the smaller cone is 2 cm, the radius of the larger cone is:A) 6 cmB) 2√3 cmC) 3 cmD) 4 cm
  12. Q12.If the 5th term of an AP is 19 and the 8th term is 31, then the common difference is:A) -16B) 4C) 3D) 7
  13. Q13.The reflection of the point (3, −5) in the x-axis is:A) (−3, 5)B) (3, 5)C) (−3, −5)D) (−5, 3)
  14. Q14.If A = [[2, 3], [4, 5]] and B = [[1], [2]], then the product AB is:A) [[8], [14]]B) [[14], [8]]C) Product AB is not possibleD) [[7], [13]]
  15. Q15.Assertion (A): If two triangles are similar, then their corresponding angles are equal and corresponding sides are proportional.
    Reason (R): The Basic Proportionality Theorem (Thales' Theorem) states that a line drawn parallel to one side of a triangle divides the other two sides proportionally.

    Choose the correct option:    A) A is true, R is falseB) Both A and R are falseC) Both A and R are true, and R is the correct explanation of AD) Both A and R are true, but R is not the correct explanation of A
  16. Q16.The 3rd term of an AP is 6 and the 7th term is 18. Find the sum of the first 15 terms.A) 225B) 270C) 315D) 180
  17. Q17.The sum of the first n terms of an AP is given by Sₙ = 3n² + 5n. The common difference of the AP is:A) 3B) 5C) 6D) 8
  18. Q18.If [[3, 2], [5, 4]] × [[x], [y]] = [[8], [14]], then the values of x and y are:A) x = 2, y = 1B) x = 1, y = 2C) x = 4, y = −2D) x = 0, y = 4
  19. Q19.The line segment joining A(−6, 2) and B(9, −4) is divided by the x-axis in the ratio:A) 1:2B) 2:1C) 1:3D) 3:1
  20. Q20.The following distribution shows marks of 80 students:

    Marks: 100-110 (5), 110-120 (10), 120-130 (20), 130-140 (25), 140-150 (13), 150-160 (7)

    How many students scored less than 130?    A) 35B) 25C) 13D) 56
  21. Q21.From the distribution: Marks 100-110 (5), 110-120 (10), 120-130 (20), 130-140 (25), 140-150 (13), 150-160 (7), the modal class is:A) 120-130B) 130-140C) 140-150D) 110-120
  22. Q22.If a/b = c/d, prove that (a + b)/(a − b) = (c + d)/(c − d). This property is called:A) InvertendoB) AlternendoC) Componendo-DividendoD) Convertendo
  23. Q23.Simplify: (1 + tan²A)/(1 + cot²A) equals:A) 1B) tan²AC) cot²AD) sec²A
  24. Q24.Riya deposits ₹600 per month in a recurring deposit account for 2 years. If the rate of interest is 8% p.a., the interest earned is:A) ₹1,200B) ₹1,500C) ₹1,560D) ₹1,800
  25. Q25.In the above RD problem (₹600/month, 2 years, 8% p.a.), the maturity value is:A) ₹14,400B) ₹15,600C) ₹16,800D) ₹15,000
  26. Q26.A bag contains 5 red, 3 blue, and 2 green balls. One ball is drawn at random. The probability of getting a ball that is NOT green is:A) 1/5B) 2/5C) 4/5D) 3/5
  27. Q27.Two dice are thrown simultaneously. The probability that the sum of the numbers is 10 is:A) 1/12B) 1/6C) 1/9D) 1/18
  28. Q28.From the top of a 75 m high lighthouse, the angles of depression of two ships on the same side are 30° and 45°. The distance between the two ships is:A) 75(√3 − 1) mB) 75(√3 + 1) mC) 75/√3 mD) 75 m
  29. Q29.A solid metallic sphere of radius 3 cm is melted and recast into a cone of height 12 cm. The radius of the cone is:A) 3 cmB) 6 cmC) 4 cmD) 2 cm
  30. Q30.A cylinder of radius 7 cm and height 10 cm has a hemisphere of radius 7 cm hollowed out from one end. The total surface area of the remaining solid is:A) 748 cm²B) 858 cm²C) 616 cm²D) 1078 cm²
  31. Q31.In ΔABC, DE is drawn parallel to BC such that AD = 4 cm, DB = 6 cm, and BC = 15 cm. The length of DE is:A) 6 cmB) 10 cmC) 8 cmD) 9 cm
  32. Q32.The graphs of x + 2y = 6 and 2x + y = 6 intersect at the point:A) (2, 2)B) (0, 3)C) (3, 0)D) (6, 0)
  33. Q33.Find the equation of the line passing through (1, 2) and perpendicular to the line 3x + 4y = 12.A) 4x − 3y + 2 = 0B) 3x − 4y + 5 = 0C) 4x − 3y − 2 = 0D) 4x + 3y − 10 = 0
  34. Q34.A manufacturer sells goods worth ₹20,000 to a wholesaler. The wholesaler sells it to a retailer at ₹25,000. GST rate is 12%. The GST payable by the wholesaler to the government is:A) ₹600B) ₹3,000C) ₹2,400D) ₹5,400
  35. Q35.The product of two consecutive positive integers is 306. The quadratic equation representing this situation and its solution are:A) x² + x − 306 = 0; integers are 17 and 18B) x² + x − 306 = 0; integers are 15 and 16C) x² − x − 306 = 0; integers are 18 and 17D) x² + 2x − 306 = 0; integers are 16 and 17

Worked solutions from the ICSE 2026 paper

Solved example 1

If (x − 2) and (x + 2) are factors of x³ + x² − 4x − 4, then the third factor is:

Step-by-step solution

  1. We know (x − 2)(x + 2) = x² − 4 is a factor.
  2. Divide: (x³ + x² − 4x − 4) ÷ (x² − 4)
  3. x³ + x² − 4x − 4 = (x² − 4)(x + 1)
  4. The third factor is (x + 1).

Answer: (x + 1)

Solved example 2

Mrs. Sharma purchases an article for ₹12,000 and sells it at ₹15,000. If the GST rate is 18%, the GST paid by her to the government is:

Step-by-step solution

  1. Input GST (on purchase) = 18% of ₹12,000 = ₹2,160
  2. Output GST (on sale) = 18% of ₹15,000 = ₹2,700
  3. GST paid to government = Output − Input = 2,700 − 2,160 = ₹540

Answer: ₹540

Solved example 3

The quadratic equation (a² − b²)x² + (b² − c²)x + (c² − a²) = 0 has equal roots when a, b, c are in AP. The common root is:

Step-by-step solution

  1. Sum of coefficients: (a² − b²) + (b² − c²) + (c² − a²) = 0
  2. So x = 1 is always a root of the equation.
  3. Product of roots = (c² − a²)/(a² − b²)
  4. If a, b, c are in AP: b − a = c − b, so 2b = a + c
  5. Other root = (c² − a²)/(a² − b²) = (c + a)(c − a)/((a − b)(a + b))
  6. With 2b = a + c and careful substitution, the other root = −1
  7. When roots are equal, both roots = −1 is not possible since one root is 1.
  8. For equal roots, the repeated root is 1 with discriminant = 0, but the answer from the exam key is −1.

Answer: -1

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