If f and g are two functions over real numbers defined as :
f(x) = 3x+1
g(x) = x²+2
then find f+g , f-g , fg and f/g

Find the inverse of the function f(x) = 3x-1

Let the mapping f: A  -> B is defined by
f(x) = log (1+x)
and the mapping G : B -> C is defined by g(x) = e^x
Find ( gof)x.

Evaluate lim   3x³+2x²-x
                x->0      x2-x

Show that f(x) = e^x has no maxima or minima.

         
If y = x2 tanx , find dy/dx

Find the intervals in which the function f(x) = Sinx - Cosx is decreasing and the interval in which it is increasing, 0<x <2p

Find the equation of the tangent to the curve
  y = Cot²x - 2Cotx + 2   at   x = p/4                                           (4)     

                                                                                                            
If y = Sinx², find dy/dx from definition.      (6)

Evaluate : òx³/2x+1 dx

Evaluate : òSin²x Cos²x dx

Evaluate : òÖx² + 81 dx

Calculate the area under the graph of the function f(x) = 7x + 5 between the ordinates x =1, x =3.                 (2marks)

Evaluate : ò₀¹ tan^-1x / 1+x²  dx

Evaluate : ò^p/4_{-p/4 x¹¹Sin⁶x dx}

Prove that ò₀¹ log(1/x -1) dx = 0                                                                                                (4 marks)

Evaluate : òx²/Öx⁶+4 dx                                                                                                               (4 marks)

Evaluate : ò₀¹ tan^-1x / 1+x² dx                                                                                                   (4 marks)

Evaluate : ò₀⁹ f(x) dx where f(x) = sinx, 0<  x < p/2
                                                            = 1, if p/2  <  x < 3
                                                           = ex^-3, if 3 <  x < 9                                                             (4 marks)

                                  
Evaluate : ò1/x² - a² dx                                                                                                            (4 marks)

                              
Evaluate : ò₀^p/2 xdx/Sinx + Cosx                                                                                             (4 marks)

                                                
Find the order and degree of each of the following differential equations. State also if they are linear or non-linear.

 (i)        t² d²s   -    st ds     = s
                  dt²              dt

 (ii)        d²y/dx²  +  x(dy/dx)²  = 0                                                                                        (2 marks

       Solve: dy = 3x²-5x+1                                                                
                   dx                                                                                                           (2 marks)

Give that : dy = 2x³ - x^{2 + x - 5}                                                                
                    dx                                                                                                           (2 marks)

Solve the differential equation

       (x+y+1) dy  = 1
                        dx

(4 marks)

              ->     ^       ^                                                                                                  ->
A force F = 4 i + 4 k acrs through the point A(0,20). Find the moment m of F about the point B(4,04).

Find the area of the parallelogram whose adjacent sides are represented by the vectors
->  ^       ^      ^            ->      ^   ^
a = i + 2 j + 3 k   and   b = 3 i + k

Find the locus of the point which is at a distance of 7 from the point ( 1 , -2 , 6 ).

Find the angle between the planes 2x - y + z += 5 and x + 2y - z = 3

Find the equation of the plane containing the line
  x    =    y-1    =    z-1    and the point ( -1,0,2 )
-2             3              -1

                                                                                      ->       ->                                                 -> -> -> 
What will be the angle between the two vectors a and b if there is a set of 3 vectors a , b , c such
        -> ->   ->               ->           ->                  ->
that a +b + c = 0 and |a| = 3 , |b| = 5 and |c| = 7.

     Find the distance of the point p(+1 , -3 , 2) from the line
x  =  y-2  =  z-3
2         1           3

 Let A,B,C,D be any four points in space.
                        ->  ->     ->   ->      ->  ->  
Prove that | AB*CD + BC*AD + CA*BD | = 4 (Area of ABC)

Suppose a coin is tossed 5 times and the random variable x is the number of heads observed. Find the probability distribution of x, regarding the appearance of heads in any toss as a success.

Find theprobability distribution of x, the number of aces when 2 cards are drawn from a well shuffled deck of 52 cards with replacement.

A coin is tossed six times. What is the probability of obtaining 4 or more heads ?

Five dice are thrown 729 times. How many times do you expect atleast four dice to show five of six ?

Two cards are drawn simultaneously. Find the probability distribution of number of queens.

A man takes a step forward with probability 0.4 and backward with probability 0.6 . Find the probability that at the end of the eleven step away from the starting point.