Recent questions tagged ch6

Find the dimentions of the rectangle of perimeter 36cm which will sweep out a volume as large as possible,when revolved about one of its sides.Also find the maximum volume.

asked Jan 11, 2013 by sreemathi.v

1 answer

An open box with square base is to be made of a given quantity of card board of area $ c^2$.Show that the maximum volume of the box is $\Large \frac{c^3}{6\sqrt 3}$ cubic units.

asked Jan 11, 2013 by sreemathi.v

1 answer

If the straight line $x\cos\alpha+y\sin\alpha=p$ touches the curve $\Large\frac{x^2}{a^2}\;\;\frac{y^2}{b^2}=\normalsize 1$,then prove that $a^2\cos^2\alpha+b^2\sin^2\alpha=p^2$.

asked Jan 11, 2013 by sreemathi.v

0 answers

A telephone company in a town has 500 subscribers on its list and collects fixed charges of Rs300/- per subscriber per year.The company proposes to increase the annual subscription and it is believed that for every increase of Re1/- one subscriber will discontinue the service.Find what increase will bring maximum profit?

asked Jan 11, 2013 by sreemathi.v

0 answers

Find the points of local maxima,local minima and the points of inflection of the function $f(x)=x^5-5x^4+5x^3-1$. Also find the corresponding local maximum and local minimum values.

asked Jan 11, 2013 by sreemathi.v

1 answer

If the sum of the lengths 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 $\Large \frac{1}{3}$

asked Jan 11, 2013 by sreemathi.v

1 answer

Prove that $f(x)=\sin x+\sqrt 3\cos x$ has maximum value at $x=\Large\frac{1}{6}$.

asked Jan 11, 2013 by sreemathi.v

1 answer

At what point,the slope of the curve $y=-x^3+3x^2+9x-27$ is maximum?Also find the maximum slope.

asked Jan 11, 2013 by sreemathi.v

1 answer

Show that $f(x)=\tan^{-1}(\sin x+\cos x)$ is an increasing function in$\bigg(0,\Large \frac{1}{4}\normalsize \bigg).$

asked Jan 11, 2013 by sreemathi.v

1 answer

Show that for a$\quad 1,f(x)=\sqrt 3\sin x-\cos x-2ax+b$ is decreasing in R.

asked Jan 11, 2013 by sreemathi.v

1 answer

Show that $f(x)=2x+\cot^{-1}x+log\bigg(\sqrt {(1+x^2)}-x\bigg)$ is increasing in R.

asked Jan 11, 2013 by sreemathi.v

1 answer

Show that the line $\Large \frac{x}{a}+\frac{y}{b}\normalsize=1$,touches the curve $y=b.e^{\Large \frac{-x}{a}}$ at the point where the curve intersects the axis of y.

asked Jan 11, 2013 by sreemathi.v

1 answer

At what points on the curve $x^2+y^2-2x+4y+1=0$,the tangents are parallel to the y-axis?

asked Jan 11, 2013 by sreemathi.v

1 answer

Find the equation of the normal lines to the curve $3x^2-y^2=8$ which are parallel to the line x+3y=4.

asked Jan 11, 2013 by sreemathi.v

1 answer

Prove that the curves $y^2=4x$ and $x^2+y^2-6x+1=0$ touch each other at the point (1,2).

asked Jan 11, 2013 by sreemathi.v

1 answer

Find the angle of intersection of the curves $y=4-x^2$ and $y=x^2$.

asked Jan 11, 2013 by sreemathi.v

1 answer

Find the co-ordinates of the point on the curve $\sqrt x\quad \sqrt y=4$ at which tangent is equally inclined to the axes.

asked Jan 11, 2013 by sreemathi.v

1 answer

Prove that the curves $xy=4$ and $x^2+y^2=8$ touch each other.

asked Jan 11, 2013 by sreemathi.v

1 answer

Find the condition that the curves $2x=y^2$ and $2xy=k$ intersect orthogonally.

asked Jan 11, 2013 by sreemathi.v

1 answer

x and y are the sides of two squares such that $y=x-x^2$.Find the rate of change of the area of second square with respect to the area of first square.

asked Jan 11, 2013 by sreemathi.v

1 answer

The volume of a cube increases at a constant rate.Prove that the increase in its surface area varies inversely as the length of the side.

asked Jan 11, 2013 by sreemathi.v

1 answer

A swimming pool is to be drained for cleaning .If L represents the number of litres of water in the pool seconds after the pool has been plugged off to drain and $L=200(10-t)^2$.How fast is the water running out at the end of 5 seconds?What is the average rate at which the water flows out during the first 5 seconds?

asked Jan 11, 2013 by sreemathi.v

1 answer

A man, 2m tall, walks at the rate of $1\frac{2}{3}m/s$ towards a street light which is $5\frac{1}{3}m$ above the ground.At what rate is the tip of his shadow moving?At what rate is the length of the shadow changing when he is $3\frac{1}{3}m$ from the base of the light?

asked Jan 11, 2013 by sreemathi.v

1 answer

Find the approximate volume of metal in a hollow spherical shell whose internal and external radii are 3cm and 3.0005cm,respectively.

asked Jan 11, 2013 by sreemathi.v

1 answer

FInd the approximate value of $(1.999)^5$.

asked Jan 11, 2013 by sreemathi.v

1 answer

Find an angle $\theta$,$0$ <$\;\theta<\Large \frac{1}{2}$,which increases twice as fast as its sine.

asked Jan 11, 2013 by sreemathi.v

1 answer

Two men A and B start with velocities v at the same time from the junction of two roads inclined at $45^\circ$ to each other.If they travel by different roads,find the rate at which they are being separated.

asked Jan 10, 2013 by sreemathi.v

1 answer

A kite is moving horizontally at a height of 151.5 meters If the speed of kite is 10m/s,how fast is the string being let out,when the kite is 250m away from the boy who is flying the kite?The height of boy is 1.5m

asked Jan 10, 2013 by sreemathi.v

1 answer

If the area of a circle increases at a uniform rate,then prove that perimeter varies inversely as the radius.

asked Jan 10, 2013 by sreemathi.v

1 answer

A spherical ball of salt is dissolving in water in such a manner that the rate of decrease of the surface.Prove that the radius is decreasing at a constant rate.

asked Jan 10, 2013 by sreemathi.v

1 answer

The length x of a rectangle is decreasing at the rate of 5 cm/minute and the width y is increasing at the rate of 4 cm/minute. When x = 8 cm and y = 6 cm, find the rate of change of (a) the perimeter, (b) the area of the rectangle.

asked Jan 4, 2013 by thanvigandhi_1

1 answer

Choose the correct answer in the points on the curve $9y^2 = x^3$, where the normal to the curve makes equal intercepts with the axes are

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Choose the correct answer in the normal to the curve $x^2 = 4y$ passing $(1,2)$ is

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Choose the correct answer in the normal at the point $(1,1)$ on the curve $2y + x^2 = 3$ is

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Choose the correct answer in the line $y = mx + 1$ is a tangent to the curve $y^2 = 4x$ if the value of $m$ is

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Choose the correct answer in the slope of the tangent to the curve $x = t^2 + 3t – 8, y = 2t^2 – 2t – 5$ at the point $(2,– 1)$ is

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Choose the correct answer in a cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic metre per hour. Then the depth of the wheat is increasing at the rate of

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height $h$ and semi vertical angle $\alpha$ is one-third that of the cone and the greatest volume of cylinder is $\large \frac{4}{27}$$\pi h^3 \: tan^2 \: \alpha $

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is $\large\frac{2R}{\sqrt 3}$ . Also find the maximum volume.

asked Nov 28, 2012 by thanvigandhi_1

2 answers

Let $f$ be a function defined on $[a, b]$ such that $f' (x) > 0$, for all $x \: \in (a, b)$. Then prove that $f$ is an increasing function on $(a, b).$

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius $r$ is $ \large\frac{4r}{3}$

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Find the absolute maximum and minimum values of the function $(f)$ given by $ f (x) = cos^2 x + sin\: x, x\: \in [0, \pi]$

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Find the points at which the function $f$ given by $f (x) = (x – 2)^4 (x + 1)^3$ has a (i) local maxima, (ii) local minima and the (iii) point of inflexion.

asked Nov 28, 2012 by thanvigandhi_1

1 answer

A point on the hypotenuse of a triangle is at distance $a$ and $b$ from the sides of the triangle.Show that the maximum length of the hypotenuse is $ (a^{\large\frac{2}{3}} + $$b^{\large\frac{2}{3}})^{\Large\frac{3}{2}}$

asked Nov 28, 2012 by thanvigandhi_1

1 answer

A window is in the form of a 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.

asked Nov 28, 2012 by thanvigandhi_1

1 answer

The sum of the perimeter of a circle and square is $k$, where $k$ is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.

asked Nov 28, 2012 by thanvigandhi_1

1 answer

A tank with rectangular base and rectangular sides, open at the top is to be constructed so that its depth is $2 m$ and volume is $8 m^3$. If building of tank costs Rs 70 per sq metres for the base and Rs 45 per square metre for sides. What is the cost of least expensive tank?

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Find the maximum area of an isosceles triangle inscribed in the ellipse $\large\frac{x^2}{a^2} + \frac{y^2}{b^2} $$= 1 $ with its vertex at one end of the major axis.

asked Nov 28, 2012 by thanvigandhi_1

2 answers

Find the intervals in which the function $f$ given by $f(x) = x^3 + \large\frac{1}{x^3}$$, x \neq 0$ is $(i) \;increasing\qquad (ii) \;decreasing.$

asked Nov 28, 2012 by thanvigandhi_1

1 answer

Find the intervals in which the function $f$ given by $ f(x) =\large \frac{4sin\: x - 2x - xcos \: x}{2+cos\:x}$ is (i) increasing (ii) decreasing.

asked Nov 28, 2012 by thanvigandhi_1

1 answer

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