The integral  is equal to 

Two sides of a rhombus are along the lines, x−y+1=0 and 7x−y−5=0. If its diagonals intersect at (−1, −2), then which one of the following is a vertex of this rhombus?

• (−3, −9)

• (−3, −8)

• (1/3, -8/3)

• (1/3, -8/3)

The centres of those circles which touch the circle, x²+y²−8x−8y−4=0, externally and also touch the x-axis, lie on:

• a circle

• an ellipse which is not a circle

• a hyperbola.

• a hyperbola.

The integral 

The integral   is equal to 

• 2

• 4

• 1

• 1

The number of points having both coordinates as integers that lie in the interior of the triangle with vertices (0,0), (0,41) and (41,0) is

• 901

• 861

• 820

• 820

The integral  dx is equal to 

The Integral  is equal to

• π-4

• 
• 
• 

9.

At present, a firm is manufacturing 2000 items. It is estimated that the rate of change of production P with respect to the additional number of workers x is given by . If the firm employs 25 more workers, then
the new level of production of items is

• 2500

• 3000

• 3500

• 3500

If ∫f x dx =Ψ (x),then ∫x⁵ f(x³)dx is equal to