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Syllabus

Determine the interval, where f(x)=sin x - cos x, 0<x<2pie is strictly increasing or decreasing?

If straight line

x cos(alpha) + y sin(alpha) = ptouches the curvex, then prove that^{2}/a^{2}+ y^{2}/b^{2}= 1a.^{2}cos^{2}(alpha) + b^{2}sin^{2}(alpha) = p^{2}SHOW THAT THE SEMI-VERTICAL ANGLE OF RIGHT CIRCULAR CONE OF GIVEN SURFACE AREA AND MAX VOLUME IS SIN INVERSE(1/3).

Show that semi-vertical angle of right circular cone of given surface area and maximum volume is

sin.^{-1}(1/3)Sir please solve this as soon as possible..

show that the triangle of maximum area that can be inscribed in a given circle is an equilateral triangle

Prove that curve(x/a)

^{n}+(y/b)^{n}=2 touches the straight line x/a+y/b=2 at (a,b) for all values of n belongs to N at the point (a,b)..Prove that the least perimeter of an isosceles triangle in which circle or radius r can be inscribed is 6root3 r

If the sum of the lengths of the hypotenuse and a side of a right triangle is given, show that the area of the triangle is maximum when the angle b/w them is pie/3.

An open box with a square base is to be made out of a given quantity of card board of area c^{2}square units. Show that the maximum volume of the box is c^{3}/ 6√3 cubic units.water is dripping out from a conical funnel of semi vertical angle 45 at uniform rate of 2 cm^2/s (is its sure areea) through a tiny hole at vertical of the bottom wht is the rate of decrese of slant height when the slant height of water is 4cm

Examine for maximum and minimum values of the function

sinx siny sin(x+y)

Water is leaking from a conical funnel at the rate of 5 cm3/sec.if the radius of the base of the funnel is 10cm and altitude is 20cm,Find the rate at which water level is dropping when it is 5cm from top..??

show that the condition that the curves ax

^{2}+by^{2}=1 and a'x^{2}+b'y^{2}=1 should intersect orthogonally (at90^{0}) such that 1/a-1/b=1/a'-1/b'11a) Find maximum area of an isosceles triangle inscribed in the ellipse

xwith its vertex at one end of the major axis.^{2}/ a^{2}+ Y^{2 }/ b^{2}= 1Show that the semi-vertical angle of the cone fo masimum volume and of given slant height is tan inverse root 2

Show that the volume of greatest culinder that can be inscribed in a cone of height h and semi vertical angle alpha is 4/27 pi h cube tan square alpha

x

^{x}y^{y}z^{z}=c,show that x=y=z,d^{2}z/dx*dyA man of height 2 metres walks at a uniform speed of 5 km/hr away from a lamp post which is 6 metres high.Please Find the rate at which the length of his shadow increases.

Find the angle between the parabolas y

^{2}= 4ax & x^{2}= 4by at their point of intersection other than the origin ?A large window has the shape of a rectangle surmounted by an equilateral triangle.if the perimeter of the window is 12m, find the dimensions of the rectangle that will produce the largest area of the window?

A given quanitity of metal is to be cast into a half cylinder with a rectangular base and semi circular ends. Finfd ratio of length of the cylinder to the diameter of its semi-circular ends.

The points on the curve 9

y^{2}=x^{3}, where the normal to the curve makes equal intercepts with the axes are(A) (B)

(C) (D)

^{x}is e^{1/e}A rectangle is inscribed in a semi circle of radius 'r' with one of its sides on the diameter of the semi circle. Find the dimensions of the rectangle so that its area is maximum. Also find this area.

find the percentage error in the area of an ellipse if 1% error is made in measuring the major and minor axes?

A helicopter is flying along the curve y= x2+2.A soldier is placed at the point (3,2). Find the nearest distance between the soldier and the helicopter.(2010Sp)

the cost of fuel for running a train is proportional to square of speed generated in km/h.it cost rs 48/h when the train is moving at speed of 16km/h.what is its economical speed if the fixed charges are rs 300/hour over and above the running cost!

Prove that the radius of the base of right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half that of the cone

An isosceles triangle of vertical angle 2a is inscribed in a circle of radius r.Show that the area of triangle is maximum when a=(pi)/6.

Prove that the area of a right angled triangle of given hypotenuse is maximum when the

show that the line x/p+ y/q =1, touches the curve y=e^(-x/p) at the point where it crosses the y axis

^{2} = 4x is what ?Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius

ris.A point on the hypotenuse of a right angled triangle is at distances 'a' and 'b' from the sides .

raised to 2/3 +araised to 2/3}the whole raised to 3/2bAn inverted cone has a depth of 40 cm and a base of radius 5cm .water is poured into it at a rate of 3/2 cubic centimetres per minute.find the rate at which the level of water in the cone is rising when the depth is 4 cm ..

find the value of latus rectum of parabola which focus is ( 3 3 ) and directrix of streight line is 3x-4y-2=0

A window is in the form of rectangle surmounted by a semicircular opening. The total perimeter of the window is 10 m. Find the dimensions of the window to admit maximum light through the whole opening.

find the equation of tangent and normal to the curve x=1-cos theta, y= theta-sin theta at theta=pi/4.

. An open tank with a square base and vertical sides is to be constructed from a metalsheet so as to hold a given quantity of water. Show that the total surface area is least when depth of the tank is half its width.(2010c)

prove : semi verticle angle of right circular cone of given volume and least curved surface area is cot^-1 (root2) .

Show that of all the rectangles inscribed in a given fixed circle, the square has the maximum area.

^{2}) snd determine area enclosed between curve and x axisat what points of the ellipse 16x^2 + 9y^2 = 400 does the ordinate decrease at the same rate at which the abcissa increases ??

2 raise to the power x cot x upon root x

differentiate with respect to x

Find the volume of the largest right circular cylinder that can inscribed in a sphere of radius r cm.

find the point on the parabola y=x^2 + 7x+2 which is closest to the straight line y=3x-3.

Solve this:Water is poured into inverted conical vessel of which the radius of the base is 2m and height 4m, at the rate of 77 liter/minute. The rate at which the water level is rising at the instant when the depth is 70 cm is:(use $\mathrm{\pi}$ = 22/7).

(1) 10 cm/min (2) 20 cm/min (3) 40 cm/min (4) none

_{show by using mean value theorem the B-A/1+B^2 < tan^-1B - tan^-1A < B-A/1+A^2,where B>A>0}ind the coordinate of a point on the parabola y=x^{2 }+7x+2 which is closest to the straight line y=3x-3?y=x^{4}– 10 andxchanges from 2 to 1.99, what is the approximate change iny? Also, find the changed value ofy.Prove that the curves

x=y^{2}andxy = kcut at right angles if 8k^{2}= 1. [Hint: Two curves intersect at right angle if the tangents to the curves at the point of intersection are perpendicular to each other.]^{3}(2x-1)Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height

hand semi vertical angleαis one-third that of the cone and the greatest volume of cylinder istan^{2}α.A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 cm/s. How fast is its height on the wall decreasing when the foot of the ladder is 4 m away from the wall?

Show that the rectangle of max. perimeter which can be inscribed in a circle of radius 'a' is a square of side a√2.

FIND THE dy/dx, x=at

^{2}, y= 2atThe normal to the curve

x^{2}= 4ypassing (1, 2) is(A)

x+y= 3 (B)x−y= 3(C)

x+y= 1 (D)x−y= 1A wire of length 36cm is cut into two pieces,one of the pieces is turned in d form of a square and d other in d form of equilatral tringle.find the length of each piece so that d sum of areas of d two be minimum.? reply fast.

Show that the area of the triangle formed by the tangent and the normal at tha point(a,a) on tha curve y

^{2}(2a-x)=x^{3}and the line x=2a,is 5a^{2/}4.show that the right circular cone of least curved surface area and given volume has an altitude equal to root 2 times the radius of the base.

If the length of three sides of a trapezium ,other than the base are equal to 10 cm each, then find the area of trapezium when it is maximum?

Find the condition for the curves x

^{2}/a^{2}-y^{2}/b^{2}=1 and xy=c^{2 }to intersect orthogonally.^{-x/a.}At a point where it crosses the y axis.Show that the height of the cone of maximum volume that can be inscribed in a sphere of radius 12 cm is 16 cm.

^{2}= x A jet plane of an enemy is flying along the curve

y= x2 + 2. A soldier is placed at the point (3,2) . What is the nearest distance b/w the soldier and the jet plane?. For the curves y = 4x

^{3}– 2x^{5}, find all points at which tangent passes through the origin ?. For the curves y = 4x^{3}– 2x^{5}, find all points at which tangent passes through the origin ?. For the curves y = 4x

^{3}– 2x^{5}, find all points at which tangent passes through the origin ?the sum of the surface areas of a sphere and a cube is given. show that when the sum of their volume is least, the diameter of the sphere is equal to the edge of the cube.