CBSE Question Paper 2012 class 12 Mathematics

General Instruction:

• All question are compulsory.
• The question paper consists of 29 questions divided into three section A, B and C. Section A comprises of 10 questions of one mark each, Section B comprises of 12 questions of four marks each and Section C comprises of 7 questions of six marks each.
• All question in Section A are to be answered in one word, one sentence or as per the exact requirement of the questions.
• There is no overall choice. However, internal choice has been provided in 4 questions of four marks each and 2 questions of six marks each. You have to attempt only one of the alternatives in all such questions.
• Use of calculators is not permitted. 

Section A

1. If a line has direction ratios 2, -1, -2, then what are its direction cosines ?

2. Find  when the projection of  on  is 4 units.

3. Find the sum of vectors  and .

4. Evaluate:

5. Evaluate

6. If  write the minor of the element a₂₃.

7. If write the value of x.

8. Simplify:

9. Write the principal value of .

10.  Let  be a ‘binary’ operation on N given a b = LCM (a, b) for all  Find 5 7.

Section B

11.  If  find .

OR

  If sin y = x sin(a + y), prove that

12.  How many times must a man toss a fair coin, so that the probability of having at least one head is more than 80%?

13.  Find the vector and cartesian equations of the line passing through the point (1, 2, -4) and perpendicular to the two lines  and

14.  If  are three vectors such that  and and  find the value of

15.  Solve the following differential equation:

16.  Find the particular solution of the following differential equations;

 given that y=1 when x = 0.

17.  Evaluate:

OR

Evaluate:

18.  Find the point on the curve y = x³ – 11x + 5 at which the equation of tangent is y = x – 11.

OR

Using differentials, find the approximate value of

19.  If  show that

20.  Using properties of determinants, prove that

21.  prove that

OR

Prove that

22.  Let A = R – {3} and B = R – {1}. Consider the function  defined by   Show that f is one-one and onto and hence find f^-1.

Section -C

23.  Find the equation of the plane determined by the points A(3, -1, 2), B(5, 2, 4) and C(-1, -1, 6) and hence find the distance between the plane and the point P(6, 5, 9).

24.  Of the students in a college, it known that 60% reside in hostel and 40% are day scholars (not residing in hostel). Previous year results report that 30% of all students who reside in hostel attain ‘A’ grade and 20% of day scholars attain ‘A’ grade in their annual examination. At the end of the year, one student is chosen at random from the college and he has an ‘A’ grade, what is the probability that the students is a hostlier?

25.  A manufacturer produces nuts and bolts. It takes 1 hour of work on machine A and 3 hours on machine B to produce a package of nuts. It takes 3 hours on machine A and 1 hour on machine B to produce a package of bolts. He earns a profit of Rs.17.50 per package onnuts and Rs.7 per package of bots. How many packages of each should be produced each day so as to maximize his profits if he operates his machines for at the most 12 hours a day? Form the above as a linear programming problem and solve it graphically.

26.  prove that

OR

Evaluate  as a limit of a sum.

27.  Using the method of integration, find the area of the region bounded by the lines

  and

28.  Show that the height of a closed right circular cylinder of given surface and maximum volume, is equal to the diameter of its base.

29.  using matrices, solve the following system of linear equations:

X – y + 2z = 7

3x + 4y – 5z = -5

2x – y + 3z = 12

OR

Using elementary operations, find the inverse of the following matrix: