MODEL TEST PAPER -2 XII-20/21

         SECTION : A (80 Marks)

Question 1) (10x2= 20 Marks)

i) If A= 2     3

             4     5 , find Inverse of A.

ii) Show that the function f: R-> R, given by f(x) = | x | is neither one one or onto.

iii) Show sin⁻¹cos sin⁻¹x+ cos⁻¹sin cos⁻¹ x = π/2

iv) If y= tan⁻¹(secx+ tanx), find d²y/dx²

v) ∫ x eˣ dx    

vi) lim ₓ→₀ (log cos x)/sin²x

vii) Prove without expanding:

a - b     1     a            a      1      b

b - c     1     b    =      b      1      c

c - a     1     c            c       1      a

viii) If x > 1/2, show that the function f(x)= x(4x²-3) is strictly increasing.

ix) Solve 2ˣ⁻ʸ dx + 2ʸ ⁻ˣ dy = 0

x) A and B are two Independent events with P(A)= 2/5 and P(B)= 1/3, Evaluate P(AUB).

Question 2).                                 (4)

Prove: 1+a²-b²       2ab            - 2b

                2ab      1 - a²+b²         2a

                 2b           - 2a       1 - a² - b²

 = (1+a²+b²)³.                              

Question 3).                                   (4)

If tan⁻¹(yz/xr) + tan⁻¹(zx/yr) + tan⁻¹(xy/zr) = π/2 then Prove that, x² + y² + z² = r². 

Question 4).                                    (4)

A bag contains 5 white, 7 red and 3 black balls. If three balls are drawn one by one without replacement. Find the probability that none is red.                       

Question 5).                                 (4)

If y= (tan⁻¹x)², show that (1+x²) d²y/dx² + 2x(1+x²) dy/dx - 2= 0.  

Question 6).                                   (4)

Evaluate ∫ 2⁴ˣ sin 3x dx. 

                   OR

Evaluate:

 ∫ (cosx + x sinx)/{x(x+ cosx) dx

Question 7)                                    (4)

Find the equations of the tangent to the curve y= x² - 2x +7 which is:

a) Parallel to the line 2x - y+9= 0

b) Perpendicular to 5y-15x= 13

                       OR

Show that the maximum value of 2x + 1/2x is less than its minimum value. 

                  

Question 8).                                  (4)

Solve by matrix inversion method

x+2y+z= 7; x + 3z= 11; 2x - 3y=1

                 OR

Show that

a + b+ 2c         a                  b

      c           b+c + 2a           b

      c                 a.            c+a+2b          = 2(a+b+c)³. 

Question 9).                                   (6)

Given x+y= 3, find the maximum and minimum values of 9/x + 36/y 

                 OR

A closed right circular cylinder is has a volume of 2156cm³. What will be the radius of the base so that total surface area is minimum.

Question 10).                                  (4)

 Show that the function f in A= R - {2/3} defined as f(x)=(4x-3)/(6x-4) is one-one and onto .

Question 11).                           2+2=4

 Solve:

a) dy/dx + y secx = tan x. 

b) tan x dy/dx= 1+y² where x= π/2 and y= 1. 

Question 12).                   3+3= 6

a) Evaluate ∫ |sin x| dx at (π/2,-π/2). 

b) Prove ∫ {log(1+x)}/(1+x²) dx at (1,0) = π/8 . Log 2. 

Question 13).                      (3x2= 6)

 a) If x= sint and y= cos pt, p is constant, then find the value of (1-x²) d²y/dx² - x dy/dx.            

b) If m² = p² cos² t + q²sin²t, then show that m+ d²m/dt² = p²q²/m²

Question 14).                  3+3=6

A) It is known that 5 men out of 100 and 25 women out of 1000 are colour blind. A colour blind person is chosen at random. Assuming that males and females are in equal proportion, find the probability of the person to be male.

B) Rajiv and Robin play 12 games of chess. Rajiv wins 6 games, Robin wins 4 games and and 2 games end in a draw. They agree to play 3 more games. Calculate the probability that out of these 3 games, two games end in a draw.

    

                      Or

Evaluate:

A) ∫ x² sin⁻¹x dx

B) ∫x² eᵃˣ dx at (a,0)

            SECTION C.        (20 Marks)

Question 15).                    2+2+2

A) Given demand function x= 50- 0.5 P and cost function C=50+40x, find price for break-even price.

B) 4x+y-10= 0, 2x + 5y -14= 0 are two regression lines. Find the correlation coefficient between variables x and y.

C) The total cost C(x) of a firm is C(x)= 0.0005x³ - 0.7x² - 30x + 3000 where x is the output. Determine:

    a) average cost (AC)

    b) Marginal cost (MC)

Question 16).                               (4)

The two lines of Regression for a distribution (x,y) are 3x+2y= 7 and x+4y= 9. Find the regression coefficient X on Y and Y on X.

                         OR

Treating x as an independent variable. Find the line of best fit for the following date:

X: 15    12       11        14          13

Y: 25    28       24        22          30

Hence, predict the value of y when x= 10.

Question 17)                               (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 ₹ 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 a x increases.

Question 18).                              (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 ₹300 and on a chain is ₹190, how many of each should be manufactured daily so as to maximize the profit?