Math test(1) CLASS 12

1) a) Find the inverse of.  2     -2
                                              4      3
                                                       (2)
b) If f: N -> R be a function defined
    as f(x) = 4x²+12x+15, show that
    f: N -> S, where S is the range of f
    is invertible.
   Find the inverse of f.              (2)

c) eˣ+eʸ=eˣ⁺ʸ prove y₁= - eʸ⁻ˣ.  (2)

d) prove.                                      (2)
tan⁻¹ 1/2+ tan⁻¹1/5 + tan⁻¹1/8=π/4

e)Evaluate limₓ-₀ tan8x/sin 2x.  (2)

2) using the properties of Determinant find
1    a      a²-bc
1     b     b²-ca
1     c     c²-ab                              (4)

3)  show sin⁻¹12/13+cos⁻¹4/5 +
      tan⁻¹63/16 =π                     (4)              

4) If eʸ(x +1) =1 show y₂ =( y₁)²   (4)

5) i) ∫ x²eᵃˣ dx.    ii) ∫ xlogx dx.    OR

        ∫ dx/(x³+x²+x+1)              (4)

6) Find the point on the curve
     y= x³-11x+5 at which the
     equation of the tangent is
     y= x -11                               ( 4)
                         OR
If f(x) = (4x+3)/(6x -4) , x≠2/3. what is the inverse of f.

7) If A= 2  -3   5
             3    2  -4
             1    1  -2 , find A⁻¹ . Using A⁻¹, solve the following system of equations. 2x-3y+5z=16, 3x+2y-4z=-4, x+y-2z=-3            (6)  

                       OR

If A = 1       2      3
          2       3      1
         -1       1      1    using elementary transfirmation, find A⁻¹ and verify A⁻¹A = I = AA⁻¹.

8) Show that the semi-vertical angle of the cone of maximum volume and of given slant height is tan⁻¹√2
                     OR                          (6)
Find the area of the great rectangle that can be inscribed in the ellipse
x²/a² + y²/b² = 1.

9) ∫ (tanθ +tan³θ)/(1+tan³θ)dθ. (6)

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

11) A furniture dealer deals in only two items: tables and chairs. He has Rs20000 to invest and a space to store at most 80 pieces. A table costs him Rs500 and a chair costs him Rs200. He can sell a table for Rs950 and a chair for Rs280. Assume that he can sell all the items that he buys. Formulate this problem as an LPP so that he can maximize his profit.                   (6)