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Differentiation 4 Marks Q.1If y=Aemx +Benx ,prove that d2y/dx2 –(m+n)dy/dx +mny =0Q.2If y= sin(logx) , prove that x2 d2y/dx2 +xdy/dx +y =0Q3If y=x+1x ,show that 2xdydx+y=2x .Q.4 If tan-11+x2-x w.r.t. x .Q.5 If y.x2+1=logx2+1-x ,show that x2+1dydx+xy+1=0 .Q.6 If x=asin2t (1+cos2t)and y=bcos2t(1-cos2t) ,show that (dy/dx)at x=π/4 =b/a .Q.7If y= x-3x23x2+4x+5 ,find dy/dx .Q.8If y=sin-1x21-x2+x1-x4 ,then prove that dydx=2x1-x4+11-x2 .q.9 If xy =yx ,find dy/dx .Q.10Differentiate, tan-11-x2x w.r.t. sin-12x1-x2 .Q.11If y= x(cosx)+(cosx) sinx , then find dy/dx .Q.12If y= xx ,show that d2ydx2-1ydydx2-yx=0 .Q.13 If x =2cosθ-cos2θ and y=2sinθ-sin2θ, find d2ydx2 at x= π2 .Q.14 Differntiate :xxsin-1x w.r.t. x .Q15If x =3sint – sin3t , y =3cost –cos3t find d2y /dx2 at t=π2 Q.16 If fx=3+x1+x2+3x ,find f'0 .Q.17If x=aθ+sinθ , y=a1-cosθ ,find d2ydx2 atπ2 .Q.18If x=a1+t21-t2 and y= 2t1-t2 ,find dydx . Q.19 If y=logx+x2+1 ,then prove that x2+1d2ydx2+xdydx=0. Q.20 Differentiate cos-11-x21+x2 w.r.t. tan-13x-x31-3x2 .Q.21If y= (logx)2 ,then prove that x2 y” +xy’ =2 .Q.22If tan-11+x2+1-x21+x2-1-x2 ,show that dydx=-x1-x4 .Q.23If y=1-x1+x ,prove that (1-x2 )dy/dx +y =0 .Q24 Differentiate :sin(xx) w.r.t. tan-11-x21+x2 .Q25 Find dy/dx , y=(logx)x +xlogx .Q.26If xy=ex-y ,prove that dy/dx =logx /(1+logx )2 .Q.27Find dy/dx , if y=log1+sinx1-sinx .Q28 Find dy/dx , if tan-1(sinx1-cosx) .Maxima and Minima(06MARKS)The sum of the perimeter of a circle and square is K , where K is constant. Prove that the sum of their areas is least when the side of the square is double the radius of the circle.A window is in the form of the rectangle surmounted by a semi circular opening .The total perimeter of the window is 30 m . Find the dimensions of the window so as the admit maximum light through the opening.Show that semi vertical angle of right circular curve of given surface area and maximum volume is sin-1 (1/3 ).Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is 8/27 of the volume of the sphere.A wire of length 28 m is to be cut into two pieces. One of the pieces is to be made a circle and other into a square. What should be the lengths of the two pieces so that the combined area of the square and the circle is minimum? If the sum of the length of the hypotenuse and a side of a right angled triangle is given , show that the area of the triangle is maximum when the angle between them is π/3.Show that the volume of the greatest cylinder which can be inscribed in a cone of height h and semi vertical angle α is 4/27 πh3tan2α .Prove that the least perimeter of an isosceles triangle which can be circumscribed to a circle of radius R is 6√3 R.Show that of all the rectangles inscribed in a given fixed circle the square has the maximum area . If the length of the three sides of a trapezium other than base are equal to 10 cm , then find the area of the trapezium when it is maximum .An apache helicopter of enemy is flying along the curve given by y = x2 + 7 . A solider placed at ( 3,7) wants to shoot down the helicopter when it is nearest to him . Find the nearest distance. TANGENT AND NORMAL (4-MARKS)Q.1Show that the curves 2x=y2 and 2xy = k cut at right angles if k2 =8 .Q.2 At what point on the curve x2+y2-2x-4y+1=0 is the tangent parallel to y- axis .Q.3Find the equation of the tangent to the curve x2+3y =which is parallel to the line y-4x+5=0 .Q.4Find the equation of the tangent and normal to the curve x=1-cosθ ,y=θ-sinθ atθ=π4. Q.5For the curve y=4x3-2x5 ,find all points at which the tangent passes through the origin.Q.6Find the equation of the tangent to the curve x=sin3t , y=cos2t at t= π4 .Q.7 Find the points on the curve y=x3 at which the slope of the tangent is equal to y- coordinate of the point .Q.8Find the point on the curve 9y2 =x3 where the normal to the curve makes equal intercepts with the axis.Q.9Show that the normal at any point to the curve x=a cosθ+θsinθ , y=a sinθ- θcosθ .Q.10 Find the equations of the tangent and normal to the curve y=x-7x-2x-3 at the point ,where it cuts x-axis . -------------------------------------------------------------------------------------------------- INCRASING AND DECRASING FUNCTION (4-MARKs) Q.1Find the intervals in which f(x) is increasing or decreasing :(1)If f(x) =x3+12x2+36x . (2) If F(x) =x3-12x2+36x+17(3)If f(x) =sinx –cosx ,0<x<2π .Q.2 Let I be any interval disjoint from (-1,1) .prove that the function f(x)=x+1/x is strictly increasing on I .Q.3 Show that the function f(x)=cos2x is a decreasing on(0,π/2) .Q.4 Find the intervals for which the function f(x) =(x+1)e-x is increasing or decreasing .Q.5Prove that the function f(x) =x3-6x2+12x-16 is always increasing on R .Q.6 Find the intervals in which the function f defined by f(x) =sinx +cosx , 0≤x≤2π ,strictly increasing or decreasing .Q.7Find the intervals on which the function f(x) =x/logx is increasing or decreasing .Q.8Show that the function tan-1(sinx+cosx) is increasing function in the interval (0,π/4) .Q.9Find the intervals on which the function f(x ) = (x-1)3 (x-2)2 is increasing or decreasing .Q.10Prove that y= 4sinx/2+cosx –x is increasing function of x in [0,π/2] .-------------------------------------------------------------------------------------------------------- CLASS – XII M. M. -40 Revision TestTIME -1.5 H SECTION-A (1X6 = 6) MARKSQ.1find dydx , when y= log (10x)+Iog (xx) +Iog(1010).Q.2find dydx , when , y= xsin-1x1-x2 .Q.3 find dydx , when , y=1-cosx1-cosxQ.4show that the function given by f(x) =e2x is strictly increasing on R .Q.5if x+y = 8 , find the maximum value of xy .Q.6 If y= e3logx , find dydx . SECTION-B (2X6 = 12) MARKS Q.7find dydx , when y=XX .Q.8find dydx , when y= log10X .Q.9find dydx , when x= at2 ,y=2at .Q.10using differentials find the approximate value of.0.0037Q11prove that the function given by f(x)=cosx is neither increasing nor decreasing in 0,2π .Q.12find an angle x , which increases twice as fast as its sine .--------------------------------------------------------------------------------------------------------------------------------------- SECTION-C (4X3 = 12) MARKSQ.13find the values of a,b,c so that the function f(x) defined below is continuous at x=0. sina+1x+sinxx , if x<0 f (x) = c, if x<0 x+bx2-xbx32 ,if x<0Q.14 find dydx , when y=xcosx+cosxsinx.Q.15 If x =(3sint- sin3t ) and y= (3cost-cos3t) , find d2ydx2 at π3 . Q16Find the intervals on which the following functions are (a)strictly increasing (b) strictly decreasing : (1) f(x) =5x3-15x2-120x+3 . -- ---------------------------------------------------------------------------------------------------------------------------- SECTION-D (5X2= 10) MARKSQ.17prove that the height of a right circular cylinder of maximum volume that can be inscribed in a sphere of radius r is 2r3 . Q.18determine the intervals in which the function f given by f (x) =sinx – cosx ,0≤x≤2π is increasing or decreasing . THE END PREPARED BY MR.A.K.BISEN (PGT MATHS) JNV,BORAI, DIST-DURG,C.G. ................
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