ISC 2019 Mathematics Question Paper for Class 12

About this paper

This is the ISC Class 12 Mathematics paper, code 860, from the 2019 CISCE board examination. It carries 100 marks and runs for three hours, with fifteen minutes extra for reading only. All of Section A is compulsory and is worth 80 marks: Question 1 has ten two-mark parts, and internal choices are given in three four-mark questions and two six-mark questions. Candidates then answer either Section B or Section C, each worth 20 marks, with an internal choice in two four-mark questions. Section B is vectors, planes, lines and area. Section C is break-even and cost, regression, and a carpenter's linear programming problem in teak, plywood and rosewood. Tables and graph paper are provided. This copy has the questions only, without solutions.

Exam

ISC Class 12

Subject

Mathematics

Year

2019

Paper code

860

Maximum marks

100

Time allowed

3 hours + 15 min reading

Questions

22

Pages

6

Answers included

No, questions only

Paper pattern

• Section A

  80 marks

  Questions 1–14, all compulsory. Question 1 has ten parts of two marks each; internal choice in three four-mark questions and two six-mark questions

• Section B or Section C

  20 marks

  Attempt every question in one section only. Section B (Questions 15–18) is vectors, planes, lines and area; Section C (Questions 19–22) is cost, regression and linear programming. Each has an internal choice in two four-mark questions

Topics covered

• Functions and inverse trigonometric functions
• Determinants and matrices
• Continuity, differentiability and mean value theorem
• Increasing functions and derivatives
• Related rates and maxima
• Indefinite and definite integrals
• Differential equations
• Probability and binomial distribution
• Vectors and three-dimensional geometry
• Area under curves
• Cost, break-even and regression
• Linear programming

Questions from this paper

1. Prove that f(x) = x³ − 6x² + 12x + 5 is increasing on the set of real numbers.
2. A 13 m ladder leans against a wall and its foot is pulled away at 2 m/s. How fast is the height on the wall decreasing when the foot is 5 m from the wall?
3. Two balls are drawn one by one, without replacement, from an urn containing 3 white, 5 red and 2 black balls. What is the probability that at least one ball is red?
4. Bag A has 4 white and 3 black balls, and Bag B has 3 white and 5 black balls. Two balls are transferred from A to B. What is the probability of then drawing a white ball from B?
5. A closed rectangular metal box with a square base has volume 4096 cm³. Polishing the outer surface costs ₹4 per cm². Find the dimensions that minimise the cost.
6. Find the point on the line 2x + 3y = 6 which is closest to the origin.