XII_Maths-Code B_December 2, 2013

8/12/2019 XII_Maths-Code B_December 2, 2013

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Code 041 (S) BTime: 3 hrs M. M. 100

GENERAL INSTRUCTIONS(i) All questions are compulsory.(ii) The question paper consists of 29 questions divided into three sections, A, B and C. Section

A comprises of 10 questions of one mark each, Section B comprises of 12 questions of fourmarks each and Section C comprises of 7 questions of six marks each.

(iii) All questions in Section A are to be answered in one word, one sentence or as per the exactrequirement of the question.

(iv) There is no overall choice. However, an internal choice has provided in 4 questions of fourmarks each and 2 questions of six mark each. You have to attempt only one of thealternatives in all such questions.

(v) Use of calculators is not permitted. You ask for logarithmic tables, if required.

SECTION AQuestion numbers 1 to 10 carry 1 mark each.

1. For what value of k, the matrix2 k 3

5 1

is not invertible?

2. If A is matrix of order 2 X 3 and B is matrix of order 3 x 5, what is the order of matrix

(AB)?

3. Find f(x) satisfying the following: x 2 xe (sec x tan x)dx e f (x) c

4. In a triangle ABC, the sides AB and BC are represented by vectors

2i j 2k, i 3j 5k

respectively. Find the vector representing side CA.

5. Find the value such that the line x 2 y 1 z 39 6

is perpendicular to the plane

3x y 2z =7.

6. Evaluate X5 dx

7. What is the value of 33I , where I 3 is the identity matrix of order 3?

8. Find x if 1 1tan 4 cot x2

9. If

a 2i j 3k and

b 6i j 9k and a / /b then find


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10. Write the number of all one-one functions from the set A ={a, b, c} to itself.

SECTION B

Question numbers 11 to 22 carry 4 marks each.11. Three balls are drawn one by one without replacement from a bag containing 5 white

and 4 green balls. Find the probability distribution of number of green balls drawn.

12. Find whether the lines 1

r (i j k) (2i j) and 2

r (2i j) (i j k) intersect or not.If intersecting, find their point of intersection.

13. If the sum of two unit vectors is a unit vector, show that the magnitude of theirdifference is 3 .

14. Find the intervals in which the following function is strictly increasing or strictlydecreasing f(x) = 20 9x + 6x 2 x3.

ORFor the curve y = 4x 3 2x5, find all points at which the tangent passes through origin.

15. Form the differential equation of the family of circles having radii 3.

16. Solve the following differential equation: 2 2 2 2 dy1 x y x y xy 0dx

.

17. Evaluate: dxsin x(5 4cos x)

OR

Evaluate:1 x

dx1 x

18. If y x b dyx y a , find dx .OR

If x=a (cos t + t sin t ) and y = b (sin t cos t ), find2

2

d ydx

19. (a) Show that f x x 3 , R is continuous but not differentiable at x= 3

(b) If tan2 2

2 2

x ya

x y

, then prove thatdy ydx x

.

20. If3 5

A4 2

, show that A 2 5A 14I = O. Hence find A -1

21. Show that the function f : R R define f (x) = 2x 3 7, for R is bijective.


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OR

If 2 2 4 4 2 21 1

x y t and x y tt t

then prove that 3dy 1dx x y

.

22. Prove that tan -1 1 + tan -12 + tan -13=

Section-CQuestion numbers 23to 29 carry 6 marks each.

23. An aeroplane can carry a maximum of 200 passengers. A profit of Rs. 1000 is made oneach executive class ticket and a profit of Rs. 600 is made on each economy classticket. The airline reserves at least 20 seats for the executive class. However, at least 4times as many passengers prefer to travel by economy class, than by the executiveclass. Determine how many tickets of each type must be sold, in order to maximizeprofit for the airline. What is the maximum profit? Make a L.P.P and solve itgraphically. Which class should be chosen by the passengers and why?

24. Find the co-ordinates of the foot of the perpendicular and the perpendicular distanceof the point (1, 3, 4) from the plane 2x y+ z + 3 = 0. Find also, the image of the pointin the plane.

25. Evaluate2

2

1

(x x 2)dx as a limit of sums.

OR

Evaluate 0

1 2

1

sin x 1 x x 1 x dx , 0 1.

26. Draw a rough sketch of the region enclosed between the circles x 2 + y2 = 4 and(x-2) 2+y2 = 1. Using integration, find the area of the enclosed region.

27. If the length of three sides of a trapezium, other than the base are equal to 10cm each,then find the area of trapezium when it is maximum.

28. If 12 1 3

A 4 1 0 , Find A

7 2 1

and hence solve the following system of equations:

2x + y + 3z = 34x y = 3

-7x + 2y + z = 2OR

Using elementary transformation, find the inverse of the matrix:1 1 2

0 2 3

3 2 4


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29. A diet for a sick person must contain at least 4000 units of vitamins, 50 units ofminerals and 1400 calories. Two foods X and Y are available at a cost of

`

4 and`

3 perunit respectively. 1 unit of food X contains 200 units of vitamins, 1 unit of mineralsand 40 calories whereas 1 unit of food Y contains 100 units of vitamins, 2 units ofminerals and 40 calories. Find what combination of foods X and Y should be used tohave least cost, satisfying the requirements. Make it a LPP and solve it graphically.What is use of vitamins and minerals in our human bodies?

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XII_Maths-Code B_December 2, 2013