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Class 12th CBSE Mathematics 29 Questions 100 Marks

Class 12th CBSE Mathematics 29 Questions 100 Marks

**Sample Paper **

**Class – XII**

**Subject – ****MATHEMATICS**

**Time : 3 hrs Max. Marks : 100**

** General Instructions:**

(i)

(ii) The question paper consists of 29 questions divided into three sections – A, B and C. Section A comprises of 10 questions of 1 mark each; Section B comprises of 12 questions of 4 marks each and Section C comprises of 7 questions of 6 marks each.

** **

** **

__ __

__SECTION – A__

1**. **Show that the binary operation defined by a*b = ab + 1 on Q is commutative.

2. Solve : tan-12x + tan-13x = π/4.

3. Find a matrix X such that B – 2A + X = O, where A =.

4. If A is a square matrix of order 3 such that = 64, find.

5. Construct a 2 2 matrix whose elements are given by: =.

6. Evaluate.

7. Evaluate

8. If

9. The Cartesian equation of a line AB is. Find the direction cosines of a line parallel to AB.

10. If, find a unit vector parallel to the vector __ __

12.

**OR**

13.

14.

15.

16.

17.

**OR **

18.

**OR **

19.

20.

21.

**OR **

22.

__SECTION-C__

** **

23. Given that A = and B = find AB. Hence using this product solve the system of equations : x – y + z = 4, x – 2y – 2z = 9, 2x + y + 3z = 1

**OR**

Using elementary row transformation, find the inverse of the matrix.

24. Show that a right circular cylinder, which is open at the top and has a given surface area, will have the greatest volume if its height is equal to the radius of its base.

**OR**

Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is 8/27 of the volume of the sphere.

25. Evaluate: ò dx.

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

2x + y = 4, 3x – 2y = 6 and x – 3y + 5 = 0.

**OR**

Make a rough sketch of the region given below and find the area using the method of integration :

27. Find the image of the point (1, 2, 3) in the plane x + 2y + 4z = 38. Also find the perpendicular distance from the point to the plane.

**OR**

A line makes angles α, β, γ and δ with the diagonals of a cube, prove that cos2α + cos2β + cos2γ + cos2δ = 4/3

28. An aeroplane can carry a maximum of 200 passengers. A profit of Rs. 400 is made on each first class ticket and a profit of Rs. 300 is made on each economy class ticket. The airline reserves at least 20 seats for first class. However, at least 4 times as many passengers prefer to travel by economy class to by the first class. Determine how many of each type ticket must be sold in order to maximize the profit for the airline. What is the maximum profit? Frame an L.P.P and solve it graphically.

29. In a bolt factory, machines A, B and C, manufacture respectively 25%, 35%, 40% of the total bolts. Of their output 5%, 4% & 2% respectively are defective bolts. A bolt is drawn at random and is found to be defective. Find the probability that it is manufactured by machine B.

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