ISC 2025 Mathematics Improvement Examination Question Paper for Class 12

About this paper

This is the ISC Class 12 Mathematics paper, code 860, from the 2025 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, with fourteen questions and the same internal-choice rule as the main paper: two questions each of two, four and six marks. Candidates then attempt every question in either Section B or Section C, each worth 15 marks. The paper includes a forest population-density model, a distribution of teachers who never make an error, Naman arriving late for a wedding, and a farmers' cooperative dividing 50 hectares between two crops under a limit on herbicide. Tables and graph paper are provided. This copy has the questions only.

Exam

ISC Class 12

Subject

Mathematics

Year

2025

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, lines and area; Section C is regression, cost and linear programming. Internal choice in one two-mark and one four-mark question

Topics covered

• Determinants and matrices
• Continuity, derivatives and normals
• Inverse trigonometric functions
• Definite and indefinite integrals
• Differential equations
• Probability distributions and Bayes' theorem
• Vectors and three-dimensional geometry
• Area under curves
• Regression
• Cost, revenue and break-even
• Linear programming

Questions from this paper

1. Find the equation of the normal to the curve y = x² − 3x + 1 at the point (3, 1).
2. The population density of wild animals, in thousands per km², at a distance r km from the centre of a forest is P = 10 + 20r − 10r². Find the density at the centre and the rate of change of density.
3. Of 30 teachers on an examination committee, 20 never make an error. Two are chosen at random. If X is the number chosen who never make an error, find the possible values of X, its probability distribution and its mean.
4. Find the shortest distance between the lines r = (i + 2j + 3k) + λ(2i + 3j + 4k) and r = (2i + 4j + 5k) + μ(4i + 6j + 8k).
5. The cost of producing x units is C(x) = 500x² + 2500x + 5000, and each unit sells for ₹6000. Find the break-even point, and show that marginal cost increases with x.
6. A cooperative has 50 hectares for crops X and Y, with profits of ₹10,500 and ₹9,000 per hectare. Herbicide is used at 20 litres and 10 litres per hectare, and not more than 800 litres in all. How much land should each crop get to maximise profit? Formulate the linear programming problem and solve it graphically.