CBSE SAMPLE QUESTION PAPER Mathematics Class XII

 

Time Allowed   3 hrs                                                                                                  Max .Marks 100

 

 General Instructions:

1.         All questions are compulsory.

2.         The question paper consists of 29 questions divided in to three sections A, B and C. Sections A comprises of 10 questions of one             mark each, Section B comprises of 12 questions of 4 marks each and section C comprises of 7 questions of 6 marks each . 

3.         There is no overall choice.However, internal choice has been provided in 4 questions of 4marks each and 2 questions of 6 mark             each.

4.         Use of calculator is not permitted. You may ask for logarithmic tables, if required.

 

                        Section - A

1.         Give examples of matrices A and B such that AB¹0 but BA ? 0.

2.         Let A be a square matrix. Then check whether A + AT is a symmetric matrix.

3.         Show that the elements on the main diagonal of a skew -symmetric matrix are all zero.

4.         Find the differential co-efficient of f( log x)  w.r.t. x , where f(x) = log x.

5.         Show that the f(x) = 2x + 3 is strictly increasing function on R.

6.         Evaluate: ò [x] dx.\

7.         Find the general solution of the differential equation (dy/dx) = (y/x).

8.         If a. i = a. (i +j) = a. (i + j+k) =1, then find vector a.

9.         A and B draw two cards each one after another, from a pack of well shuffled pack of 52 cards. Find the probability that all the              four cards drawn are of same suit.

10.       If P(A) =0.7, P(B) =0.7 and P(B/A)= 0.5, then find P(A  U  B).

 

                        Section - B

11.       Evaluate :

            tan-1((3sin2a)/ (5+ 3 cos2a) + tan-1 (¼ tana),

                                    where - P /2<a < P/2.

12.       Differentiate:

            Cot-1[Ö1+x2 -1/ x] w.r.t. tan-1 [x + (1+Ö1-x2)], x= 0.

13.       Prove using properties of determinants:

                                                x      x2      1+px3

                                                y      y2      1+py3    = (1+pxyz) (x-y) (y-z) (z-x)

                                                z      z2      1+pz3

                        OR

            If A= 3  -4 , using principle of mathematical induction, show       

                     1  -1

            that       An=

                                    1 + 2n                   -4n         V    n€ N.

                                   

                                    n                           1-2n

 

14.       Find the equations of tangent and the normal to the curve x=1-cosq,    y= q-sin q at q = P/4.

                                                            Or

          Find the intervals in which the function f(x) =[(4x2+1)/x] is  (i) increasing (ii) decreasing.

15.       Discuss the continuity of the function f(x) at the point x=1/2.

                                    x      ;   0<= x< ½

                        F(x)=   ½   ;   x=  ½

                                   1-x ;   ½ < x < = 1

                      

Q16.    Evaluate: òdx/Ösin3 x  sin(x+a)

Q17.    Evaluate the following integral as limit of sums 0ò2 (x2+x) dx

Q18.    Sand is pouring from a pipe at the rate of 12 cm3/sec. The falling sand forms a cone on the ground in such a way that the height of             the cone is always one sixth  of the radius of the cone, how fast is the height of the sand cone increasing when the height is 4 cm?

Q19.    Evaluate: p/2ò0 xsinx/(1+sinx) dx

Q20.    Solve the differential equation (x2 -1)dy/dx + 2(x+2)y = 2(x+1)

                                    OR

            Show that y = e3x (A+Bx) is the solution of the differential equation d2y/dx2 - 6 dy/dx + 9y = 0

Q21.    Bag A contains 4 red and 5 black balls and bag B contains 3 red and 7 black balls. One ball is drawn from bag A and two from             bag B. Find the probability that out of 3 balls drawn two are black and one is red.

Q22.    If the sum of the mean and variance of a binomial distribution for 5 trials is 18, find the distribution.

 

                        SECTION - C

Q23.    Let * be the binary operation on N given by a*b = LCM of a and b. Find

            i)  5*7, 20*16

            ii) Is * commutative?

            iii) Is * associative?

            iv) Find the identity of * in N.

            v) Which elements of N are invertible for the operation *?

Q24.    Show that the altitude of a right circular cone of maximum volume which can be inscribed in a sphere of radius R is 4/3 R. 

                                    OR

            A wire of length 28m is to be cut into two pieces .One of the two pieces is to be made into a square and the other into a circle             what should be the lengths of the two pieces so that the combined area of the square and the circle is minimum.

Q 25.   Find the area of the region bounded by the parabola y = x2 +1 and the lines  

            y =x, x=0 and x=2.

                                    OR

            Find the area of the region

            [(x,y): 0<= y<= x2+1; 0<=y<x+1,0<=x<=2]

Q 26.   Solve the following system of equation by matrix method:

            (2/x) +(3/y)+(10/z) =4 ;  (4/x)-(6/y)+(5/z)=1 and  (6/x)+(9/y)- (20/z)=2.

Q 27.   Find the equation of the line passing through the point (-1, 3,-2) and perpendicular to each of the lines each of the lines

            (x/1)=(y/2) = (z/3) and (x+2)/ (-3) = (y-1)/2 = (z+1)/5

Q 28.   A random variable X has following probability distribution.

                        X         0          1          2          3          4          5          6          7

                        P(X)     0          K         2k        2k        3k        K2       2k2      7k2+k

 

            (i) Find k  

            (ii) Evaluate P(X < 3), P(X<=6) and P (0<X<3).

Q 29.   To maintain one's health a person must fulfill minimum daily requirements for the following three nutrients calcium , protein and             calories. His diet consists of only food item I and II whose price and nutrient contents are shown below.

             Price               I is 0.60 per unit         II is 1 per unit                         Min Requirement

                                                                                   

            Calcium                        10                                  4                                              20

            Protein                           5                                  5                                               20

            Calories                         2                                  6                                               12

 

Find the combination of food items so that the cost may be minimum? Solve it graphically.

Tips: 1. Students should give preference to those questions in Answering for which they are fully confident. 2. The sequence of answering the questions can be according to there choice but the question number should be indicated properly. 3. They can use short cut methods to check their answers simultaneously. 4. The figures in linear programming, Areas of bounded reason, Max. Min. etc. will help their solutions if they are neat and accurate. 5. To get bonus marks, they must attempt all the questions (to get step marks). ÙæðÅUÑ- ØãU °·¤ âñ{ÂÜ ÂðÂÚU ãñU ÁM¤ÚUè ÙãUè´ ç·¤ §âè ×ð´ âð ÂÚUèÿææ ×ð´ ÂýàÙ ¥æ°Ð

Mr. Adesh Sharma P.G.T (Maths) K.V    I.M.A (Dehradun)