Mock Test paper (2)

MOCK TEST PAPER(2) for ISC 
                        2019
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      SECTION A.        (80 Marks)
                     ********
Question 1.                   (10×2=20)

a)         1        1       1
             1   1+sina  1
             1       1   1+sin a   
using determinants find maximum
value of. 

c) Solve for x, cos(tan⁻¹x) =
                                  sin(cot⁻¹3/4).

d) Evaluate. lim ₓ→₀ (sinx-x)/x³.

e) Given that the events Aand B are
   such that P(A)=1/2 and P(B)=p
   and P(A∪B)=3/5. find p if A and B
   are 1) mutually exclusive
          2) indepedent.

f) ∫ (x sinx)/(1+sin x) dx at (π,0).

g) for what value of x is given
     matrix
        2x+4     4       a singular matrix
         x+5      3

h) if y=xʸprove xdy/dx= y²/(1-ylogx)

i) The probabilities of A, B and C
   solving aproblem are 1/2,1/3 and
   1/4 respectively. Find the
   probability that the problem will
   be solved.

Question 2)                                   (4)

   Let f: [0, undefined) -> R be a
   function defined by
   f(x) = 9x²+6x-5. Prove that f is not
   invertible. Modify only the
   codomain  of f to make f
   invertible and then find its
   inverse.

Question 3)                           (4)

Using properties of determinants, show that
a    a+b          a + b +c
2a  3a+2b    4a+ +3b+2c
3a  6a+3b    10a+ 6b+3c      = a³

Question 4)                             (4)

prove cos⁻¹(63/65)+2tan⁻¹(1/5)
                             =   sin⁻¹(3/5).
 
Question 5)                         (4) 

show that the function f(x)= | x-2 |, x ∈ R, is continuous

Question 6)                              (4)

if y= xsin⁻¹x/√(1-x²) prove
          (1-x²) dy/dx = x+  y/x.

Question7)                                 (4)

Evaluate  ∫ 2y²/(y²+4)  dy.  
                         or

evaluate ∫¹₋₁  eˣ dx as the limit of a sum.

Question 8)                               (4)

Find the equation of tangents to the curve y=cos(x+y), - 2π ≤ x ≤ 2π that are parallel to the line x+2y=0
                          or
Find the approximate change in the volume V of a cube of side x metres caused by increasing the side by 3%.

Question 9)                          (4)

Solve the differential equation.
x² - dy + (xy+y²)dx = 0
when x=1, y =1.
 
Question10)                             (4)

In a class of 60 students, 30 opted for Mathematics. 32 opted for Biology and 24 opted for both Mathematics and Biology. If one of these students is selected at random. find the probability
       1) The student opted for
            Mathematics
       2) The student has opted for
Mathematics but not for Biology. 

Question 11)                          (6)

Solve the equation by Martins′ Rule
x-3y+z=0 , y-z =2, 2x-3z=10 .
  
Question12)                        (6)

Prove that the area of a right angled triangle of given hypotenuse is maximum when the triangle is isosceles.
                        Or
Show that the rectangle of maximum area that can be inscribed in a circle of radius r units is a square of side √2 r units.

Questuon13)                            (6)

∫((x² +1)(x²+2)) dx/((x²+3)(x²+4)).

Question14)                             (3)
a) An insurance company insured 2000 drivers, 4000 car drivers and 6000 truck drivers. The probability of an accident involving a scooter drivers, car drivers and a truck drivers is 0.01, 0.03,and 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver ?  

       SECTION C.    (20Marks)
                       * *******

Question19)a)

The fixed cost of a product is Rs18000 and the variable cost per unit is Rs550. If the demand function is p(x)= 4000 - 15 x
find the break- even values.       (2)
                              
b)                                                   (2)

Given x+4y=4 and 3x+y=16/3 are regression lines. Find the line of regression of x on y. 

c)                                                 (2)

The cost function for a commodity is C(x) = Rs(200+29x-x²/2).
      1) Find the marginal cost
      2) Calculate the marginal cost when x=4 and intercept it.         (2)

Question20)                                 (4)

Two regression lines are represented by 2x+3y-10=0 and 4x+y-5=0. Find the line of regression of y on x.  
                            or
Fit a straight line to the following data, treating y as the dependent variable.
X: 1  2  3  4  5
Y: 7  6  5  4  3 Hence, estimate the value of y when x=3.5.

Question 21)                            (4)

The marginal cost function of a firm is 33 log x. Find the total cost function when the cost of producing one unit is Rs11.  
                       Or
If the marginal cost of a commodity is equal to half its average cost, show that fixed cost is zero.
If the cost of producing 9 units of the commodity is Rs 60, find the cost function.

Question 22)                             (6)

A small firm manufacturers gold rings and chains. The combined number of rings and chains manufactured per day is almost 24. It takes one hour to make a ring and half an hour for a chain. The maximum number of hours available per day is 16. If the profit on a ring is Rs. 300 and on a chain is Rs. 190 how many of each should be manufactured daily so as to maximize the profit ?