MODEL TEST PAPER-1 12(20-21)

          MODEL TEST PAPER-1 

          SECTION A.     (80 Marks)

Question 1) Choose the Correct alternative: (MCQ).        1x10= 10

i) If sec⁻¹x= cosec⁻¹y state which of the following is the value of:  (cos⁻¹1/x+cos⁻¹1/y)
A) π   B) 2π/3    C) 3π/6   D) π/2

ii) If A= [aᵢⱼ ] is a 2x2 matrix such that aᵢⱼ= i+ 2j, then will be
A)1   3    B) 2   4   C) 3   5  D) none
    2   4         3   5        4   6 

iii) The value of ∫ e⁵ˡᵒᵍ ˣ dx is
A) (e⁵ˡᵒᵍ ˣ)/5+c             

B) (e⁵ˡᵒᵍ ˣ)/(5 log x)+ c

C) x⁵/5 + c.               D) x⁶/6 + c

iv) The slope of the tangent to the rectangular hyperbola xy= c²  at the point (ct, c/t) is:
A) -1/t   B) -1/t²   C) 1/t  D) 1/t²

v) If the odds in favour of an event are 9:4, then its probability of occurrence is:
A) 9)13  B) 4/13  C) 4/9   D) 5/13

vi) the standard deviation of a binomial binomial a binomial of a binomial binomial a binomial binomial distribution with  parameters n and p is--
A) np                              B)√(np)
C) √{np(1-p)}.                D)2√np

vii) If f(x)= [x] and g(x) then the value of f{g(8/5)} - g{f(-8/5)} is:
A) 2       B) 1       C) -1       D)-2

viii) The value of dx/{1+√tan x} is
A) 0.    B) 1.    C) π/6.    D) π/12

ix) If A= 0     2   and KA= 0    3a
               3   -4                   2b  24
Then K, a, b are respectively
A) 6,12,-18                 B) -4,6, 9
C) -6, -4,-9                  D) 6, -4, 9

x) If A' is the transpose of a square matrix A, then,
A) |A| ≠ |A'|             B) |A| + |A'|
C) |A| + |A'| =0.      D) |A| = |A'| only when A is symmetric matrix.

Question 2).     (10x2= 20 Marks)

i) A relation R is defined on the set of natural numbers N as follows: (x,y) ∈R => x+y= 12, for all x,y ∈ N. Prove that R is not transitive on N.

ii) ∫ sin x/cos 2x  dx

iii) Prove{cos(sin⁻¹x)}²={sin(cos⁻¹)}²

iv) If A= cos t       sin t
             - sin t       cos t    prove that AA'= I

v) Find the Integrating factor of the differential equation (x+y+1) dy/dx = 1

vi) Five cards are drawn successively with replacement from a well-shuffled deck of cards. What is the probability that all the five cards are spades?

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) lim ₓ→₀{log(1+ax)}/sin bx

ix) If y= logₓtan x, find dy/dx

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

Question 3).                               (4)

Show with determinants:
a² + 1      ab     ac
   ab      b²+1    bc
   ca         bc    c²+1 = 1+a²+b²+c².

Question 4).                                (4)

If sin⁻¹x + sin⁻¹y+ sin⁻¹z= π, then show that x√(1-x²) + y√(1-y²)+ z√(1-z²) = 2xyz.                OR

Prove: tan{π/4 + 1/2 cos⁻¹(a/b)} + tan{π/4 - 1/2 cos⁻¹x(a/b)}=2b/a

Question 5)                                 (4)

If y= (sin⁻¹x)/√(1-x²), then prove (1-x²) dy/dx² - 3x dy/dx - y = 0

Question 6)                                 (4)

Evaluate: ∫ (x⁴+1)/(x⁶+1) dx     OR

Evaluate: ∫dx/(Cox + √3 sinx)

Question 7)                              (4)
Find the Equation of the tangent and normal to the curve: x= a sec³ t and y= a cos³ t at t= π/4                                       
                       OR

Find the intervals in which the function f(x)= 20 - 9y+ 6x²- x³ is.   
A) strictly increasing
B) strictly decreasing

Question 8).                            (6)

A window is in the shape of a rectangle with a semi-circular covering at the top. If the perimeter of the window is p(constant), find its maximum area.
                      OR
Of all the closed right circular cylindrical cans of volume 128π cm³, find the dimensions of the can which has minimum surface area.

Question 9)                             (4)
A dice is thrown 3 times. If getting a six is considered a success, find the probability of
A) 3 success
B) atleast two success.

          SECTION - C.   (20 Marks)

Question 10)
A)  Given that the cost function and revenue function respectively as C(x)= x+40 & R(x)= 10x - 0.2x²,  find the break even point         (2)

B) Equations of two lines of Regression are 4x+ 3y+7=0 and 3x+4y+8= 0. find the regression coefficient of y on x.               (2)

C) The demand function of a monopolist is given by p= 100 - x - x². find
i) the revenue function.          (2)
ii) marginal revenue function.  

Question 11).                          (4)
Find the equations of two lines of regression from the following observations:
(3,6),(4,5),(5,4),(6,3),(7,2).
Find the estimate of y for x=2.5

                    OR
The lines of regression of a set of data are 8x-10y+66= 0 and 40x- 17y= 224. The variance of X is 9. Find : i) the mean value of x and y

ii)  Coefficient of x on y and y on x, hence find correlation coefficient.
iii) Standard deviation of y.

Question 12)
A profit making company wants to launch a new product. It observes that the fixed cost of the new product is Rs 35000 and the variable cost per unit is Rs 500. The revenue received on the sale of X units is given by 500x - 100x².  Find
(i) The profit function,
ii)  break even point.                   (4)

                           OR

A Farm has the following total cost and demand functions:
  C(x)= x³/3 - 7x² + 111x + 50 and p= 100- x
Find the profit maximizing output.

Question 13).                            (6)

Solve the following LPP by graphical method
Minimise Z= 18c + 10y subject to 4x + y ≥ 20, 2x+ 3y≥30, x, y ≥ 0.