ISC 2026 Mathematics Improvement Examination Question Paper for Class 12

About this paper

This is the ISC Class 12 Mathematics paper, code 860, from the 2026 Improvement Examination. It carries 80 marks and runs for three hours, with fifteen minutes extra for reading only, and it has 22 questions. Section A, worth 65 marks, is compulsory and has fourteen questions, with internal choices in two questions each of two, four and six marks. Candidates then attempt either Section B or Section C, four questions worth 15 marks. Among the questions are a cylindrical tank of diameter 20 m, footballs made in Chennai, Mumbai and Delhi, a cylinder fitted inside a sphere, and the scores of 15 students in a quiz. Tables and graph paper are provided. This copy has the questions only.

Exam

ISC Class 12

Subject

Mathematics

Year

2026

Paper code

860

Maximum marks

80

Time allowed

3 hours + 15 min reading

Questions

22

Pages

12

Answers included

No, questions only

Paper pattern

• Section A

  65 marks

  Questions 1–14, compulsory, including multiple-choice items. Internal choice in two questions of two marks, two of four marks and two of six marks

• Section B or Section C

  15 marks

  Four questions in the section attempted. Section B is vectors, planes and area; Section C is cost, regression and linear programming. Internal choice in one two-mark and one four-mark question

Topics covered

• Matrices and determinants
• Inverse trigonometric functions
• Relations and bijective functions
• Rates of change
• Increasing functions
• Probability
• Differential equations
• Applications of derivatives
• Vectors and three-dimensional geometry
• Area between curves
• Cost, revenue and regression
• Linear programming

Questions from this paper

1. Prove that f: Z → Z defined by f(x) = x − 5 is bijective.
2. Using determinants, find x if the points (x, −2), (5, 2) and (8, 8) are collinear.
3. A cylindrical tank has diameter 20 m and fills at 314 cubic metres per hour. Find the rate of change of the depth of the water.
4. Fifteen students scored 14, 17, 15, 14, 21, 17, 19, 20, 16, 18, 20, 17, 16, 19 and 20 in a quiz. If X is the score, form the frequency table and the probability distribution of X.
5. Find the equation of the plane through (2, −3, 1) and (−1, 1, −7) which is perpendicular to the plane x − 2y + 5z + 1 = 0.
6. The cost of x units is 5x² + 30x + 5. Find the range of output for which average cost is increasing.