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Syllabus

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

SHOW THAT THE SEMI-VERTICAL ANGLE OF RIGHT CIRCULAR CONE OF GIVEN SURFACE AREA AND MAX VOLUME IS SIN INVERSE(1/3).

_{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}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..

^{3}+x^{2}+x+1 has neither a maximum value nor a minimum valueshow that the triangle of maximum area that can be inscribed in a given circle is an equilateral triangle

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

6.Find the point on the curve y^{2}= 4x which is nearest to the point (2, 1).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.the sum of their perimeters 600 m. Prove that the sum their areas is the least, when the side of the

square garden is double the radius of the circular garden

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

If y = 1/2 log (1-cos 2x / 1 + cos 2x), prove that dy/dx = 2 cosec 2x

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 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

Determine the intervals in which the function f(x)= x

^{4}-8x^{3}+22x^{2}-24x +21 is decreasing or increasing.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.

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.

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?

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)

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.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.

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)..^{n}, where k is any constant. If v be the velocity and f be the acceleration at any time, prove that v^{2}= nfx/(n-1).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 open tank is to be constructed with a square based and vertical sides so as to contain 500 cube metres of water. What should be the dimension of the tank, if the area of metal sheet used in its construction is to be minimum

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.

Q.3. Cost of producing 75 units of a commodity is Rs. 275 and cost of producing 150 units is Rs. 300. Assuming that TC is linear, find the

(i) cost function.

(ii) average total cost of producing 75, 150, 225 units respectively,

(iii) average fixed cost of producing 75, 150, 225 units respectively.

(iv) average variable cost of producing 75, 150, 225 units respectively.

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.

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

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!

^{3}(2x-1)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 3/2bdifferentiation of log e with base x?

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

Q. Verify Lagrange's mean value theorem

f (x) = ${\mathrm{tan}}^{-1}xon\left[0,1\right]$

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

Q.18. A sheet of paper is to contain 18 $c{m}^{2}$ of printed matter. The margins at the top and bottom are 2 cm each, and at the sides 1 cm each. Find the dimensions of the sheet which require the least amount of paper.

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 ..

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

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

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

f(x)=x-sinxandg(x)=x-tanxwherex belongs to (0,pi/2). Thenfor these values of x:-a)f(x)g(x)>0

b)f(x)g(x)<0

c)f(x)/g(x)>0

d)None of these

The

answer given is B part. Theconcept involved is of increasing and decreasing functions. Please explain how to do it properly.find the equation of tangent and normal to the curve x=1-cos theta, y= theta-sin theta at theta=pi/4.

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

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.]Find the condition for the curves x

^{2}/a^{2}-y^{2}/b^{2}=1 and xy=c^{2 }to intersect orthogonally.ind the coordinate of a point on the parabola y=x^{2 }+7x+2 which is closest to the straight line y=3x-3?^{4}-4x^{3}-45x^{2}+51 isa)strictly increasing

b)strictly decreasing

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.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'^{2}+2y = 8 which is perpendicular to the line x-2y+1 =0A 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.

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 right circular cone of least curved surface area and given volume has an altitude equal to root 2 times the radius of the base.

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

Find the equation of the normal to the curve y=1+sinx/cosx at x= pi/4.

FIND THE dy/dx, x=at

^{2}, y= 2atIf 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?

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?

^{-x/a.}At a point where it crosses the y axis.If x=cost(3 - 2 cos^2t) and y= sint(3 - 2 sin^2t) find dy/dx at t = pi / 4