MODEL TEST PAPER-1 For (XII)

Question 1).                   2x6= 12

a) Find the maximum value of 

1        1           1

1    1+ sinx     1 

1        1     1+ cosx

b) Solve the equation for x:

cos(tan⁻¹x) = sin(cot⁻¹3/4)

c) Evaluate lim ₓ→₀ (sinx-x)/x³

d) ∫ x sinx/(1+ sinx)dx at (π, 0)

e) For what value of x is the given matrix 

2x+4        4

 x+5         3 a singular matrix

f) if Y= x ʸ , prove that 

x dy/dx = y²/(1 - y logx)

2) Using property of                     (4) determinants, prove that 

1+ a            1          1 

1                1+b       1 

1                  1       1+c

= abc(1 + 1/a +1/b+ 1/c)         

3) If cos⁻¹(x/a)+ cos⁻¹(y/b) = k

prove that x²/a² - (2xy cos k)/ab + y²/b² = sin² k.                                (4)

4) If y= {x+√(1+x²)ⁿ,                      (4)

then show that 

(1+x²) d²y/dx² + x dy/dx = n²y

5) ∫(3x+1)/√(5- 2x - x²) dx.            (4)

6) Find the equations of the tangent to the curve y²=x²-2x+7 which is 

i) parallel to the line 2x- y+9=0

ii) perpendicular to the line 5y -15x = 13.                                                (4)

                         OR

Find the intervals in which the function f given by f(x)= sinx - cosx, 0≤ x ≤2π is strictly increasing or strictly decreasing.

7) Solve:

A) dy/dx =(xy+y)/(xy+x).               (2)

B) dy/dx + 1= eˣʸ.                           (2)

8) Using metrics, solve the following equations:

5x+3y+z= 16, 2x+y+3z= 19, x+2y+4z= 25. (6)

                      OR

 If A = 1     -1       0 

           2      5       3

           0      2       1 , 

find inverse of A, 

9) Prove that the area of right angle triangle triangle triangle of right angle triangle triangle triangle of given hypotenuse is a maximum, when the triangle is isosceles.    (6)     

                          OR 

show that of all the rectangles inscribed in a given fixed circle, the square has the maximum the maximum has the maximum area.

10) Evaluate:

A) ∫ x² sin⁻¹x dx.                          (3)

B) ∫ x/(x² + 4x +3) dx.                 (3)

11) The fixed cost of the product is Rs18000 and the variable cost per unit is Rs550. If the demand function is p(x)= 4000 - 150x, find the break even even values.       (2)

12) Given x+ 4y = 4 and 3x+y=16/3

are regression lines. find the line of regression of x on y.                     (2)

13) the cost function for a commodity is commodity is 

C(x)=₹(200+20x - x²/2) 

A) Find the marginal cost(MC)

B) calculate the marginal cost when x= 4 and interpret it.                    (2)

14) Two regression lines are represented by 2 X + 3 Y - 10= 0 and 4X + Y - 5= 0. Find the line of regression of Y on X.                    (4)

                          OR

15) Fit a straight line 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

16) The marginal cost function of a firm is MC= 33 log x. Find the total cost function when the cost of producing one producing one cost of producing one producing one unit is ₹11.                                    (4)

                          OR

If the marginal cost of a commodity is equals to to half its average average cost, show that fixed cost is zero. If the cost of producing 9 units of the commodity is ₹60. find the cost function.

17) A manufacturer produces two products A and B. Both the products are are processed on two different machines. The available capacity of the first first machine is 12 hours and that of the second second machine the second machine is 9 hours per day. Each unit of a product A requires 3 hours on both machines and each unit of product B requires 2 hours on the first machine and 1 hour on the second second machine. Each unit of product A is sold at profit of ₹7 and that of B at a profit of ₹4. Find graphically the production level per day for maximum profit.               (6)