(A)   (B)  (C)  (D)  solution:  Equation of the curve is ……….(i)      Slope of the tangent to curve (i) at any point  =   Slope of the normal = negative reciprocal =  [ Slopes of lines making equal...

Equation of the curve is ………..(i)       Slope of the normal at  = ……….(ii) Again slope of normal at given point (1, 2) = ……….(iii) From eq. (ii) and (iii), we have          From eq. (i),        ...

Equation of the given curve is  ……….(i)        Slope of the tangent to the given curve at point (1, 1) =  (say)  Slope of the normal =   Equation of the normal at (1, 1) is      Therefore, The...

Equation of the curve is ……….(i)        Slope of the tangent to the given curve at point  =       ……….(ii) Now         …..(iii) Putting the values of  and  in eq. (i),      Therefore, The correct...

Let  be the depth of the wheat in the cylindrical tank of radius 10 m at time    V = Volume of wheat in cylindrical tank at time  cu. m It is given that  = 314 cu. m/hr         1 m/h Therefore,The...

 Let  be the radius of the right circular cone of height  Let the radius of the inscribed cylinder be  and height  In similar triangles APQ and ARC,          Volume of cylinder (V) = ……….(ii)  V = ...

Let  be the radius and  be the height of the cylinder inscribed in a sphere having centre O and radius R.  In right triangle OAM,  AM2 + OM2 = OA2     ……….(i) Volume of cylinder (V) =  ……….(ii)  V...

Let I be the interval  Given:  for all  in an interval I. Let  I with  By Lagrange’s Mean Value Theorem, we have,  where   where  Now    ……….(i) Also,  for all  in an interval I   From eq. (i),   ...

 Let  be the radius of base of cone and  be the height of the cone inscribed in a sphere of radius  OD = AD – AO =  In right angled triangle OBD, OD2 + BD2 = OB2     ……….(i) Volume of cone (V)...

 ……….(i) =  =  Now   = 0   or    or      [Turning points] Now  = 0 + 1 = 1 1 + 0 = 1  = 1 Therefore, absolute maximum is  and absolute minimum is...

Given: ……….(i) =  =  =  Now  = 0   or  or    or  or  Now, for values of  close to  and to the left of . Also for values of  close to  and to the right of . Therefore,  is the point of local maxima....

Let P be a point on the hypotenuse AC of a right triangle ABC such that PL AB =  and PM BC =  and let BAC = MPC = , then in right angled   AP = PL =  And in right angled PMC,  PM = PM Let AC =  then...

 Let  m be the radius of the semi-circular opening of the window. Then one side of rectangle part of window is  and  m be the other side of rectangle. Perimeter of window = Semi-circular arc AB +...

Let  be the radius of the circle and  be the side of square. According to question,  Perimeter of circle + Perimeter of square =     ……….(i) Let  be the sum of areas of circle and square.     [From...

Given: Depth of tank = 2 m Let  m be the length and  m be the breadth of the base of the tank. Volume of tank = 8 cubic meters       Cost of building the base of the tank at the rate of ` 70 per sq....

 Equation of the ellipse is  ……….(i) Comparing eq. (i) with  we have  and   and  Any point on ellipse is P Draw PM perpendicular to axis and produce it to meet the ellipse in the point Q.  OM =  and...

  =  =    = ……….(i) Now   = 0  = 0 Here,  is positive for all real    or  [Turning points] Therefore,  or  divide the real line into three sub intervals  and  For ,  from eq. (i) at  (say),...

=  =  =  =    =  =    = ……….(i) Now  for all real  as . Also  > 0 (i)  is increasing if , i.e., from eq. (i),   lies in I and IV quadrants, i.e.,  is increasing for  and  and (ii)  is decreasing...

The parametric equations of the curve are     And  Slope of tangent at point  =  Slope of normal at any point  =  And Equation of normal at any point  i.e., at  =  is          Distance of normal...

Equation of the curve is ……….(i)    Slope of the tangent to the curve at the point (1, 2) to curve (i) is  =1  Slope of the normal to the curve at (1, 2) is   Equation of the normal to the curve (i)...

Let BC =  be the fixed base and AB = AC =  be the two equal sides of given isosceles triangle. Since  cm/s  ……….(i) Area of  x BC x AM =  =  =  [By chain rule] =  cm2/s Now when    cm2/s Therefore,...

Here  ……….(i)    ……..(ii) And  =    ……….(iii) Now            From eq. (iii),  =  =  < 0   is a point of local maxima and maximum value of  is...

Assume ……….(i)   =  ……….(ii) Changing  to  and  to  in eq. (i), we have  ……….(iii) Here  and    From eq. (iii),       0.677 (b)  Let  ……….(i)   =  ……….(ii) Changing  to  and  to  in eq. (i), we have...