12th Centum Question Paper

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Padasalai’s Centum Coaching team – Special Question Paper

            STD: XII                          Mathematics                    Marks  : 150

                                                                                  Time   :  3 hrs

                                                          PART - I                                               (40x1=40)

        (i) Answer all the questions. (ii) Choose and write the correct answer.

1.     If (A)  (A,B), then the system is …..

a)     consistent and has infinitely many solutions

b)    consistent and has a unique solution

c)     consistent

d)    inconsistent

2.     Inverse of   is ………....

(a

3.     The system of equations ax+y+z = 0; x+by+z = 0; x+y+cz = 0; has a non-trivial solution then    = ………….

       (a) 1                          (b) 2                               (c)  -1                          (d) 0

4.     The angle between the asymptotes to the hyperbola    -  = 1 is

 (a)  -2tan^-1(  )           (b) -2tan^-1(  )        (c)  2tan^-1(  )      (d)  2tan^-1(  )

5.     If   = 64 then    is ……

            (a)  32           (b)   8            (c)  128                 (d)  0

6.     If the projection of   on  and projection of   on  are equal then the angle   between   +  and  -  is  …..

(a)             (b)              (c)             (d) 

7.     If  is any vector, then value of the ( ( x )² =?

(a)  a²         (b)  2a²         (c)  3a²        (d)  4a²

8.     The non-parametric vector equation of a plane passing through three points , whose position vector is  and parallel to  and  is……….

(a)  [-,  ,] = 0    (b)   [    ] = 0  (c) [,,  x] =   (d)  [    ] = 0

9.     If  =3, =4 and   = 1,   = ?

    (a)  5             (b)  6            (c)  7           (d)  8

10.                         The least position integral value of n for which  = 1 is …..

             (a)  2         (b)  3           (c)  4           (d)  5

11.                         The number of values of (cos + isin  where p and q are non-zero   

  integers prime to each other …..

(a)  p       (b)  q        (c)  p + q        (d)  p - q

12.                         Simplify   = …..

(a)  cos62 + isin62 (b) cos19 + isin19  (c) cos + isin   (d) cos62 - isin  

13.                         What values of  (1-+)⁵ +(1+-)⁵  ……….

( a) 30          (b)  32            (c) 35              (d) 36

14.                         The line 2x+3y +9 = 0  touches the parabola y² = 8x at the point …

(a)  (0,-3)           (b)   (2,4)        (c)  (-6, )             (d)  (  , -6)  

15.                         If y= mx+c is a tangent to the ellipse 9x²+16y²=144 then c =  …..

(a)  2               (b)  4                 (c)  5             (d)  6

16.                         The vertex of the parabola x²+8x+12y+4= 0 is …..

(a)  (-4, 1)      (b)  (4, -1)        (c)  (-4, -1)       (d)  (4, 1)

17.                         The foci of the ellipse  +  = 1 are …..

(a)  (5, 0)           (b) (0, )        (c) ( 0, 5)           (d)  (0, )

18.                           is

(a)   2         (b)  0                (c)               (d)  1

     19. The parametric equations of the curve  +  =  are

            (a)  x = a sin³  ; y = a cos³          (b)    x = a cos³  ; y = a sin³  

(c)  x = a³ sin  ; y = a³ cos          (d)  x = a³ sin  ; y = a³ cos  

20. What is the surface area of the of a sphere when the volume is increasing at 

  the same rate of the radius?

(a)   1         (b)              (c)  4           (d) 

21. Which of the following function is increasing in (0, )

(a)            (b)               (c)  -x²           (d)  x^-2

22. The curve y =     is

             (a) Concave upward for x  0    (b) concave downward for x  0 

(c) everywhere concave upward  (d) everywhere concave downward  

23. The value of  dx is

(a)  0        (b)  2                (c)  log 2             (d)  log 4

24. The curve 9y² = x²(4-x²) is symmetrical axis

(a)   x - axis         (b)  y - axis                (c)  y = 0     (d) both the axes 

25. The percentage error in the root 11^th root of the number 28 is approximately

.……. times the percentage  error in 28.

(a)             (b)              (c)  11           (d)  28

26. In which region the curve  y² (a+x)= y²(3a-x) does not lie?

(a)   x  a     (b)  0  x  3a    (c) x  -a and x  3a   (d)  -a  x  3a

27. The volume generated by rotating the triangle with vertices at (0,3) , ( 3,0)

  and (3,3) about x- axis is

(a)   18        (b)  2          (c)  36        (d)  9     

28. The area between the ellipse   +  = 1 and its auxiliary circle is

(a)   b(a-b)   (b)  2a(a-b)     (c)  a(a-b)     (d)  2b(a-b)      

29. The length of the  arc of the curve    +  = 4 is

(a)   48           (b)  24        (c)  12          (d)  96

30. The differential equation of the family of lines y = mx is

(a)    = m     (b)  ydx – xdy = 0   (c)   = 0    (d) ydx – xdy = 0  

31. The differential equation formed by eliminating A and B from the relation

 y = e^x (A cos x + B sin x) is

(a)  y₂ +y₁ = 0  (b) y₂ -y₁ = 0 (c) y₂ -2y₁ +2y= 0   (d) y₂ -2y₁ -2y= 0

32. The integrating factor of the differential equation   -y tan x = cos x is

(a)   sec x    (b)  cos x          (c)  e ^tanx     (d)  cot x

33. Which of the following is not a group?

(a)   (Z_{n ,}+_n)     (b)  (Z _,+)         (c)  (Z_, .)           (d)  (R _,+)

34.  Which of the following is a contradiction?

(a)   p ∨ p     (b)  p ∧ q        (c) ∼ p ∨ q        (d)  p∧∼ p

35. In the multiplicative group of  n^th the roots of unity, the inverse of ^k is

 (k  n)    (a)      (b)  ^-1     (c)  ^n-k     (d)  

36.  ‘  ’ is a binary operation on

(a)   N       (b)  R        (c)  Z           (d)   - {0}

37. Given E(X + c) = 8 and E(X - c) = 12 then the value of c is

(a)   -2         (b)  4       (c)  -4          (d)  2

38. If  f(x) is a p.d.f of a normal variate X and X∼ N(, ) then   dx

(a)   Undefined    (b)  1          (c)  .5       (d)  - .5

39. The distribution function F(X) of a random variable X is

           (a) a decreasing function   (b) a non decreasing function

           (c) a constant function      (d) increasing first and then decreasing

40. If  f(x) is a p.d.f. of a normal distribution with mean µ then  is

            (a)  1              (b) 0.5                   (c)  0                 (d)  0.25

                                              SECTION-B                                        10X6=60

      (i)Answer any ten questions. (ii) Question no.55 is compulsory and choose any  

         Nine questions from the remaining. (iii) Each question carries six marks.

     41.  If A= and B= then verify that (AB)^-1 = B^-1A^-1.

     42.  Find the Rank of a matrix   .

     43. Obtain the vector Cartesian equations of the sphere whose centre is (1,-1, 1)

           and the radius is the same as that of the sphere  = 5.

     44. If P represents the variable complex number Z, satisfying the condition

         arg  =  find the locus of P.

    45.  Prove that for any two complex numbers z₁ and z₂  (i)  =  .

          (ii) arg (z₁ . z₂) = arg z₁ + arg z₂.

    46. Find the angle between the asymptotes to the hyperbola.

          3x²-5xy-2y²+17x +y+14 = 0.

  47.  Obtain Maclaurin’s series for   log_e (1 + x).

  48.  Find the point of inflection and determine the intervals of convexity and   

         Concavity of the Gaussian curve y =  ^.

  49. Verify Euler’s theorem for  f(x, y) =  .

  50. Evaluate d x.

  51.  Solve   + y = x.

  52. Show that p ↔ q ≡ ((∼ p) ∨ q) ∧ ((∼ q) ∨ p).

  53. Prove that the cancellation laws of a group.

  54. Marks in an aptitude test given to 800 students of a school was found to be

        Normally distributed. 10% of the students scored below 40 marks and 10% of

       the students scored above 90 marks. Find the number of students scored  

       between 40 and 90.

 55. a) Find the mean and variance of the distribution

          f(x) =   .         (OR)

   (b) Prove that the area of a quadrilateral ABCD is   where AC and  

         BD are  its diagonals.        

                                     SECTION-C                                        10 x 10=100

(i)Answer any ten questions. (ii) Question no.70 is compulsory and

choose any nine questions from the remaining. (iii) Each question carries ten marks.

  56.  Using rank method, for what values of k, has the system of equations    

          kx+y+z = 1, x+ky+z = 1,  x+y+kz = 1   (i) unique solution  (ii) more then

          one solution  (iii) no solution.

  57.  Prove that by vector method  sin (A-B) = sin A cos B - cos A sin B.

  58.  Find the vector and Cartesian equations of the plane containing the line

           =   =   and passing through the point (-1,1,-1).

 59.  If a=cos2 + isin2, b = cos2 +i sin2, c=cos2 +isin2 prove that

a)      + = 2cos ()

b)      =  cos2().

 60. Find the eccentricity, centre, vertices and foci of the ellipse

       9x² + 25y² -18x -100y-116 = 0 and draw its curve.

 61. Find the equation of the rectangular hyperbola which has for one of its

        Asymptotes the line x + 2y − 5 = 0 and passes through the points (6, 0)

        and (− 3, 0).

 62. If the curve y² = x and xy = k are orthogonal then prove that 8k² = 1.

 63. Show that the volume of the largest right circular cone that can

        be inscribed in a sphere of radius a is    (volume of the sphere).

 64. Trace the cure  y² = 2x³. 

 65. Find the common area enclosed by the parabolas 4y² = 9x and 3x² = 16y .

 66. Show that the surface area of the solid obtained by revolving the arc of the     

      curve y = sin x from x = 0 to x =  about x-axis is 2  [ 2 + log (1 + 2)] .  

67.  Solve :   – 3  + 2y = 2e^3x  when  x = log2, y = 0 and when x = 0, y = 0.

68.  Show that    

       where ω³ = 1, ω ≠ 1 form a group with respect to matrix multiplication. 

69.  The mean score of 1000 students for an examination is 34 and S.D. is 16.

       (i) How many candidates can be expected to obtain marks between 30 and 60

       assuming the normality of the distribution and (ii) determine the limit of the 

      marks of the central 70% of the candidates.                                         

      [P(0<Z<0.25)=0.0987,   P(0<Z<1.63)=0.4484]   

70. (a)  A comet is moving in a parabolic orbit around the sun which, is at the       

     focus of the parabola, when the comet is 80 million km from the sun, the line    

    segment from the sun to the comet makes an angle of    radians with the axis of

    orbit. Find (i) the equation of comet’s orbit  (ii) how close the comet comes  

    nearer to the sun.(Takes the orbit as open rightward)         (OR)

          (b) The rate at which the population of a city increases at any time is  

   Proportional to the population at that time. If there were 1, 30,000 people in the  

   City in 1960 and 1, 60,000 in 1990 what population may be anticipated in 2020.

    [ log_e  = . 2070 , e ^{. 42} = 1.52]

                                                         ALL THE BEST

                                 

Prepared by

     C. Anbarasu M.sc., B.Ed., M.Phil.

          Genius Tuition Centre, 

          72, Sanganoor road,

          Andhra bank (opp.)

Coimbatore - 641 006.

Cell No. : 95667 72455, 97897 01947.