CBSE Board Class 12 Math Previous Year Question Papers 2010

1. All question are compulsory.
2. The question paper consists of 29 questions divided into three sections A,B and C. Section
– A comprises of 10 question 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 .
3. Question numbers 1 to 10 in Section – A are multiple choice questions where you are to
select one correct option out of the given four.
4. There is no overall choice. However, internal choice has been provided in 4 question of
four marks and 2 questions of six marks each. You have to attempt only one lf the
alternatives in all such questions.
5. Use of calculator is not permitted.
6. Please check that this question paper contains 4 printed pages.
7. Code number given on the right hand side of the question paper should be written on the
title page of the answer-book by the candidate.
 
Time : 3  Hours      
Maximum Marks : 100
CLASS – XII                CBSE         MATHEMATICS
                                                       
Q.1  Check whether the relation R in R defined by R = {( ) }
a,b : a ≤ b is transitive. Ans =
not ( ,4 −3)∈ R & (− 1,3 )∈ R ⇒ ( 1,4 )∉ R

Q.2.Evaluate : ∫ cos cos cos 2 cos 2 dx x x  α α

Q.3 Find the value of k for which the matrix
k 2 3 4 has no inverse. 

Q.4  Write the principal branch of  x 1 sec

Q.5  Find the value of x  if the area of  ∆ is 35 square cms with vertices (x,4 ),(2, -6 )and
(5,4 ). 

Q.6  Evaluate : [ ] ∫ 1 + 2 tan (tan + sec ) . /1 2 x x x dx 

Q.7  Write down a unit vector in XY-plane, making an angle of 30° with the positive
direction of x-axis. 

Q.8 If → → a and b are non-collinear vectors, find the value of x for which the vectors→ → → → → →
I = 2( x + )1 a− b and m = (x − )2 a+ b are collinear. 

Q.9 If  a b c r r r = + , then is it true that  a b c  r r r
= + ? Justify your answer. 

Q.10  Find the perpendicular distance from ( 2,5,6) on XY plane . Ans : 6 unit
                                           
Q.11 .Solve the following equation :

Q.12 .If f(x ) and g (x)  be two  invertible function defined as

Q.13  Using the properties of determinants, prove the following:

Q.14  An air force plane is ascending vertically at the rate of 100 km/ h. If the radius of
the earth is r km, how fast is the area of the earth , visible from the plane ,
increasing  at 3 minutes after it started ascending ? Given that the visible area A at
height h is given by  .

Q.15 If  y m x = − sin( sin ) 1

Q.16 .Find the distance of the point (2,3,4) from the line
6 2 2 3 x 3 y z = − =  + measured parallel to the plane 3x + 2y + 2z - 5 = 0.      

Q.17  Find all the local maximum values and local minimum values of the function

Q.18 Evaluate ∫ − − . 1 cos 4 sin 4 2 2 e dx x x x
 
Q.19  Solve  the differential equation :  3( ) ( ) .0
2 2 xy + y dx + x + xy dy 

Q.20  A girl walks 4 km towards west, then she walks 3 km in a direction 30° east of
north and stops. Determine the girl’s displacement from her initial point of
departure. 

Q.21 Form the differential equation of the family of curve  ; x 2 x 3x y = ae + be + ce where
a, b, c are some arbitrary constants. 

Q.22Evaluate : ∫− dxx x 1 3

Q.23 .Let A be a square symmetric matrix, Show that : (i) ( )' 2 1
A + A is a symmetric
matrix. (ii)  ( )' 2 1
A − A is a skew symmetric matrix. Also prove that any square
matrix can be uniquely expressed as the sum of a symmetric matrix and a skewsymmetric matrix.
                                                                    OR
                                                        
Q.24  Reduce in symmetrical form , the equation of the line x – y + 2z = 5 , 3x+ y + z =6 

Q.25  Assume that the chances of a patient having a heart attack is 40%. It is also
assumed that a meditation and yoga course reduce the risk of heart attack by 30%
and prescription of certain drug reduces its chances by 25%. At a time a patient
can choose any one of the two options with equal probabilities. It is given that
after going through one of the two options the patient selected at random suffers a
heart attack. Find the probability that the patient followed a course of meditation
and yoga? 

Q.26  Draw the rough sketch of  the region enclosed between the circles  4
2 2 x + y = and ( )2 1 2 2 x − + y = . Using integration, find the area of the enclosed  region 
       OR
Find the area of the lying  circle x y 2ax
2 2 + = lying above x-axis and interior of the parabola  y = ax

     
Q.27  Prove that all normals  to the curve  x = a cos t + at sin t, y = a sin t − at cos t are at a
distance a from the origin. Ans;dy/dx= slope of tangent = tan t ;  slope of normal =
- cot t  then equation of normal : x cos t + y sin t = a & distance from origin is  = a

Q.28Evaluate: x x dx ∫ π0 log sin . 2 1 log 2 2 π
 
Q.29  A fruit grower can use two types of fertilizer in his garden, brand P and brand Q.
The amounts (in kg) of nitrogen, phosphoric acid, potash, and chlorine in a bag of
each brand are given in the table. Tests indicate that the garden needs at least 240
kg of phosphoric acid, at least 270 kg of potash and at most 310 kg of chlorine. If
the grower wants to minimize the amount of nitrogen added to the garden, how
many bags of each brand should be used? What is the minimum amount of
nitrogen added in the garden?