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Syllabus

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

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.

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..

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

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.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.Prove that the least perimeter of an isosceles triangle in which circle or radius r can be inscribed is 6root3 r

1) Water is running into a conical vessel, 15cm deep and 5cm in radius at the rate of 0.1cm

^{3}/s. When the water is 6cm deep find at what rate(i) water level is raising (ii) the water surface area is is increasing (iii) the wetted surface of the vessel is increasing (iv) slant height is increasing

2) A man 1.5m tall moves away from a pole 9.5m tall at a rate of 3m/s. Find the rate at which elevation of top of the pole is changing when the man is 6m away from the pole?

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

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..??

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 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'Show that the semi-vertical angle of the cone fo masimum volume and of given slant height is tan inverse root 2

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

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

A 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.

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.

Find the angle between the parabolas y

^{2}= 4ax & x^{2}= 4by at their point of intersection other than the origin ?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)

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?

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)..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.

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)

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.

Find the condition for the curves x

^{2}/a^{2}-y^{2}/b^{2}=1 and xy=c^{2 }to intersect orthogonally.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

x

^{x}y^{y}z^{z}=c,show that x=y=z,d^{2}z/dx*dyFind the volume of the largest right circular cylinder that can inscribed in a sphere of radius r cm.

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.

^{x}is e^{1/e}Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius

ris.Show that the height of the cone of maximum volume that can be inscribed in a sphere of radius 12 cm is 16 cm.

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 3/2bthe 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 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.]An 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 ..

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.

. 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)

Show that a cylinder of given volume, open at the top, has minimum total surface area if its height is equal to radius of the base. (2009c)

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

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

at what points of the ellipse 16x^2 + 9y^2 = 400 does the ordinate decrease at the same rate at which the abcissa increases ??

Divide 15 into two parts such that the square of one number multiplied with the cube of the other number is maximum.

A cylinder is such that sum of its height and circumference of its base is 10 m . Find the greatest volume of the cylinder.

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

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

find the interval in which the function is increasing or decreasing.f(x) = (4sinx - 2x - xcosx ) / (2+ cosx)

ind the coordinate of a point on the parabola y=x^{2 }+7x+2 which is closest to the straight line y=3x-3?Show that the rectangle of max. perimeter which can be inscribed in a circle of radius 'a' is a square of side a√2.

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 volume of greatest cylinder that can be inscribed in a cone of height h, and semivertical angle is 30 degree is 4*pi*hcube/81

Find the equations of the tangents to the curve 3x^{2}- y^{2}= 8, which pass through the point (4/3.0).FIND THE dy/dx, x=at

^{2}, y= 2atA 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.

Prove that y=( 4 sin theta/ 2+cos theta )- theta is an increasing function of theta in[0, pie/2]. NCERT Q 9,6.2, in this ques dy/dx is calculated 2 times .explain how it was calculated second time and why?

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?

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.if y=ax+b/(cx+d), prove that 2 y1 y3 =3 (y2)^2 where y1,y2,y3 are first second and third derivative respec.

^{3}(2x-1)^{-x/a.}At a point where it crosses the y axis.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?prove that the curve 4x=y

^{2}and 4xy=k cut at right angles if k^{2}=512. 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 ?_{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}