# KENDRIYA VIDYALAYA SANGATHAN-KOLKATA REGION 3 PRE-BOARD ... ?· 1 kendriya vidyalaya sangathan-kolkata…

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KENDRIYA VIDYALAYA SANGATHAN-KOLKATA REGION

3rd PRE-BOARD EXAMINATION: 2014-15

SUB: MATHEMATICS

TIME ALLOWED: 3 HOURS M.M :100

General Instructions:

1. All questions are compulsory.

2. The question paper consists of 26 questions divided into three sections-A, B and C. Section A comprises of 6

questions of one mark each, Section B comprises of 13 questions of four marks each and Section C comprises

of 7 questions of six marks each.

3. All questions in Section A are to be answered in one word, one sentence or as per the exact requirement of

the question.

4. There is no overall choice. However, internal choice has been provided in 4 questions of four marks each and

2 questions of six mark each. You have to attempt only one of the alternatives in all such questions.

5. Use of calculators is not permitted.

Section A

Q1. Evaluate: tan1

3 sec1

(2)

Q2 Find gof if f(x) =8 x3 , g(x)=

.

Q3. If [

] = [

] , find the value of y .

Q4. Evaluate: |

|

Q5. Find p such that p

zyx

321 and

142

zyx

are perpendicular to each other.

Q6. If determinant of a square matrix of order 3 is 4,find the determinant of its adjoint.

Section B

Q7.

( ) ( ) ( ) Show that is commutative and associative.

Find the identity element for on A, if any.

Q8. Prove (

) =

, x (

)

OR

Solve for x ( ) ( )

Q9. By using properties of determinants, show that:

• 2

|

| = ( )

Q10. If cos y = x cos(a + y) with cos a 1, prove that

( )

OR

Find

of the function ( ) ( )

Q11. If ( ) , show that ( ) ( )

Q12 Find the equation of plane passing through the line of intersection of the planes

x + 2y + 3z = 4 and 2 x + y z + 5 = 0 and perpendicular to the plane 5 x + 3y 6 z + 8 = 0

Q13.Find the intervals in which the function f given by f(x) = 2x3 3x

2 36x + 7 is

(a) strictly increasing (b) strictly decreasing

Q14.Show that [ + + + ] =2[ ]

OR

Find a unit vector perpendicular to each of the vectors ( + ) ( - ) where and

.

Q15.Evaluate:

( )( ) dx

Q16. Evaluate: (

) dx

Q17.If sum of two unit vectors is a unit vector ,then show that the magnitude of their difference is

Q18.. A man is known to speak truth 3 out of four times .He throws a die and report that it is a six. Find the

probability that it is actually six. Which value is discussed in this question?

Q19.Find the shortest distance between the lines

)k2j5-i(3k-ji2r

and )2(

kjijir

Section C

Q20.Two institutions decided to award their employees for the three values of resourcefulness, competence and

determination in the form of prizes at the rate of Rs. x ,Rs.y and Rs.z respectively per person. The first Institute

decided to award respectively 4,3 and 2 employees with a total prize money of Rs.37000 and the second

Institute decided to award respectively 5, 3 and 4 employees with a total prize money of Rs.47000.If all the

three prizes per person together amount to Rs.12000, using matrix method find the value of x, y and z. Write the

values described in the question.

Q21.Solve the differential equation

• 3

, given that y = 0 where x =

Q22..Using properties of definite integrals, evaluate:

OR

Using properties of definite integrals, evaluate:

( )

Q23. The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their

areas is least when the side of square is double the radius of the circle.

OR

Show that the volume of greatest cylinder that can be inscribed in a cone of height h and semi vertical angle

is, 23 tan27

4h .

Q.24 There are a group of 50 people who are patriotic, out of which 20 believe in non-violence. Two persons

are selected at random out of them, write the probability distribution for the selected persons who are non-

violent. Also find the mean of the distribution. Explain the importance of non- violence in patriotism.

Q25. Using integration Find the area lying above x-axis and included between the circle + = 8 x and

parabola = 4 x

OR

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

5x - 2y = 10, x + y 9 =0 , 2x 5y 4 =0

Q26. Reshma wishes to mix two types of food P and Q in such a way that the vitamin contents of the mixture

contain at least 8 units of vitamin A and 11 units of vitamin B. Food P costs Rs 60/kg and Food Q costs Rs

80/kg. Food P contains 3 units /kg of vitamin A and 5 units /kg of vitamin B while food Q contains 4 units /kg

of vitamin A and 2 units /kg of vitamin B. Determine the minimum cost of the mixture? What is the importance

of Vitamins in our body?