Mock Test paper (1)

MOCK TEST PAPER(1) For ISC 2019
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SECTION A              (80 Marks)
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Question1)                   ( 10×2=20)

(a) if the matrix A=      6    x   2
                                       2   -1   2        
                                     -10   5   2
     is a singular matrix. Find the
     value of x.

b) solve cos⁻¹(sin (cos⁻¹x))=π/3. 

c) Let R be a relation defined on
     the set of natural numbers N as
     follows ::
     R=((x,y)⇒x∈ℕ,y∈ℕand
     2x+y=24)
     Find the domain and range of
     the relation R.
     Also, find if R is an equivalence
     relation or not.  
   
d) lim x -≻π/2 (x tanx - π/2 sec x).

e) differentiate tan⁻¹√((a-x)/(a+x)).

f) ∫³₋₃  | x+2 | dx.

g) A fair die is thrown once. What
    is the probability that either an
   even number or a number greater
   than three will turn up.

h) A and B appear for an interview
    for two vacancies.Theprobability
    of A´s selection is 1/4and B′s
    selection is 2/3.
    Find the probability that only one
    of them will be selected.

i) Find the value(s) of which
   y=(x²-2x)² is an increasing
   function.

j) solve the differential equation.
   cosec³xdy - cosec y dx=0.

Question 2)
By using properties of determinants prove       
                 a         sinx     cosx
              -sinx       -a          1
              cosx        1          a 
is independend of x.                (4)   
        
Question 3)                                (4)
If R →R defined as f(x) = (2x-7)/4
  is an ab invertible function,
   find f⁻¹(x).
                                    
Question 4)                                    (4)
Prove
    Sin(2tan⁻¹3/5-sin⁻¹7/25)=304/425

Question 5)                                   (4)
Using Rolle′s theorem find a point
  on the curve y= sin x+ cosx -1,
   x∈(0,π/2)where the tangent
   is parallel to the  x-axis.             
                     or
   If the function f(x) given by
    f(x)=        3ax+b.     If x>1
                        11.         If x=1
                   5ax-2b.   If x <1 is
   continuous at x=1, find the value
   of a and b.    
                                  
Question 6                                     (4)
    Evaluate ∫     ( 2 sin 2x -cosx) dx
                          ( 6-cos²x-4 sinx)  
                         Or
   ∫ ²₀ (x+4) dx as limit of sum.  

Question 7)                                    (4)
   If e^y(x+1)=1,
     show that d²y/dx²=(dy/dx)².

Question 8)                                  (4)   
   The equation of the tangent at
   (2,3) on the curve y²=ax³+b is
    y=4x-5. Find the values of
    a and b.
                           or
   A 5m long ladder is leaning
   against a wall. The bottom of the
   ladder is pulled along the ground,
   away from the wall, at the rate of
   2 cm/s. How fast is its height on
   the wall decreasing when the
   foot of the ladder is 4m away
   from the wall ?

Question9)                                  (4)
   Solve the following differential
   equation for a particular solution:
     y - x dy/dx = x + y dy/dx.  
         
Question10)                               (4)
   In a class of 75 students,15 are
   above average, 45 are average
   and the rest below-average
   achievers.The probability that an
  above average achieving students
  fails is 0.005, that an average
  achieving students fails is 0.05
and the probability of a
  below-average achieving student
  failing is 0.15. If a student is
  known to have passed, what is
  the probability that he is a
  below-average achiever ? 
                            or
   The probability that a bulb
   produced by a factory will fuse in
   100 days of use is 0.05.
   Find the probability that out of 5
   such bulbs, after 100 days of
   use.

Question11)                               (6)

Find A-¹, where A =  4    2    3
                                    1    1    1
                                    3     1  -2
                       or
Using elementary transformations,
  Find the inverse of the matrix
    A=  8   4     3
           2   1     1
           1   2     2   and use it to solve
    the following system of linear
    equations. 8x+4y+3z=11,
     2x+y+z=5, x+2y+2z=7.

Question 12)                            (6)
   A wire of 50m length is cut into
   two pieces. One piece of the wire
   is being in the shape of a square
   and the other in the shape of a
   circle. What should be the length
   of each piece so that the
   combination area of the two is
   minimum ?
                           or
The length of the sides of a triangle are 9+x², 9+x² and 18-2x² units. Calculate the area of the triangle in terms of x and the value of x which makes this area is
maximum.

Question 13)                              (6)
            ∫ (secx)/(1+cosecx) dx

Question 14.a)                         (3)

   Box 1 contains 2 white and 3 
   black balls, Box2 contains 4
   white and 1 black ball and Box3
   contains 3 white and 4 black
   balls. A dice having 3 red, 2
   yellow and 1 green face, is
  thrown to select the box. If the red
  face turns up, we pick up Box1, if
  a yellow face turns up we pick up
  Box 2; otherwise, we pick up Box
  3. Then, we draw a ball from the
  selected box. If the ball drawn is
  white, what is the probability that
  the dice had turned up with a red
  face ?

b)                                               (3)

    Five dice are thrown
    simultaneously. If the
   occurrenceof an odd number in a
   single dice is considered a
   success, find the probability of
   maximum three successes. 

              SECTION-B        (20 Marks)
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Question15a)                             
 
    Find the area of the
    parallelogram having diagonals
   (3i+j-2k) and (i-3j+k).      (2)

   b) Find the angle between the
     following pairs of lines:
     (-x+2)/-2= (y-1)/7=(z+3)/-3 and
      (x+2)/-1=(2y-8)/4=(z-5)/4 and
      check whether the lines are
      parallel or perpendicular.       (2)

Question16)                                  (4)

If a,b,c are mutually perpendicular
   vectors of equal magnitude,
    prove that (a+b+c) is equally
   inclined with vectors a,b,c.
                          or
   Find the value of M for which the
   four points with position vectors
   6i-7j, 16i-19j-4k, Mj-6k
   and 2i-5j +10k are coplanar.

Qustion 17)                               (4)

   Find the area of the  
   parallelogram having the
   diagonals represented
   by the vectors 3i+j-2k and i-3j+4k.
                            or
   Find the equations of the two
   lines through the origin which
   intersect the line x-3/2=
  (y-3)/1=((z/1) at angles of π/3  
                                                       (4)

Question18)                             (6)

   Find the area enclosed between
   the curves y²=4ax and x²= 4ay.

        SECTION--C            (20 Marks)
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Question19a)
  It is known that cost of producing
  100 units of a commodity is 
   Rs250 and cost of producing 200
   units is Rs300. Assuming
   average variable cost is constant,
   find the cost function      (4)          

b) Out of the following two
    regression lines, find the line of
    regression of x on y: x+4y=5,
     3x+y=11                                   (2)

c) The total cost function C is
    given by C(x)=x²/25.   + 2x. Find
    1) the average cost function.
    b) the marginal cost function  (2)

Question 20)
   compute regresion coefficient of
       y on x for the following data:
       {(x,y) : (5,2) , (7,4) , (8,3) ,
       (4,2),(6,4)                                 (4)
                            or
       Two random variables have
    regression lines 3x+2y-26=0 and
    6x+y=31. Calculate: 1) the mean
    value of x and y 2) the coefficient
     of correlation 3) the standard
     deviation of y, given variance of
     x is 25.                                          

Question 21)                                   (4)

The marginal cost function of manufacturing x units of a commodity is 6+10x-6x². The total cost of producing one unit of the commodity is Rs 12. Find the total and average cost functions.
                        or
If C=2x{(x+4)/(x+1)}+6 is the total cost of production of x units of a commodity, show that marginal cost falls continuously as x increases.

Questiin 22)                                 (6)

A manufacturer produces two products And B. Both the products are processed on two different machines. The available capacity of the first machine is 12 hours and that of the second machine is 9 hours per day. Each unit of product A requires 3 hours on both machine and each unit of product B requires 2 hours on the first machine and 1 hour on the second machine. Each unit of product A is sold at profit of Rs7 and that of B at a profit of Rs4. Find graphically the production level per day for maximum profit.