Recent questions tagged exemplar

Let the $f:R\rightarrow R$ be defined by $f(x)=2x+\cos x,$then

asked Jan 11, 2013 by sreemathi.v

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The slope of tangent o the curve $x=t^2+3t-8,y=2t^2-2t-5$ at the point (2,-1) is :\[(A)\;\frac{22}{7}\quad(B)\;\frac{6}{7}(C)\;{-6}{7}\quad(D)\;-6\]

asked Jan 11, 2013 by sreemathi.v

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The tangent to the curve $y=e^{2x}$ at the point $(0,1)$ meets x-axis at:

asked Jan 11, 2013 by sreemathi.v

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The points at which the tangents to be curve $y=x^3-12x+18$ are parallel to x-axis are:

asked Jan 11, 2013 by sreemathi.v

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The equation of tangent to the curve $y(1+x^2)=2-x,$where it crosses x-axis is \[\begin{array}{1 1}(A)\;x+5y=2 & (B)\;x-5y=2\\(C)\;5x-y=2 & (D)\;5x+y=2\end{array}\]

asked Jan 11, 2013 by sreemathi.v

1 answer

If $y=x^4-10$ and if x changes from $2$ to $1.99$,what is the change in y

asked Jan 11, 2013 by sreemathi.v

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If the curve $ay+x^2=7$ and $x^3=y,$ cut orthogonally at $(1,1)$ then the value of a is

asked Jan 11, 2013 by sreemathi.v

1 answer

The equation of normal to the curve $3x^2-y^2=8$ which is parallel to the line x+3y=8 is\begin{array}{1 1}(A)\;3x-y=8 & (B)\;3x+y+8=0\\(C)\;x+3y-8=0 & (D)\;x+3y=0\end{array}

asked Jan 11, 2013 by sreemathi.v

1 answer

The curve $y=x^\frac{1}{5}$ has at

asked Jan 11, 2013 by sreemathi.v

1 answer

A ladder, 5 meter long,standing on a horizontal floor,leans against a vertical wall.If the top of the ladder slides downwards at the rate of 10cm/sec,then the rate at which the angle between the floor and the ladder is decreasing when lower end of ladder is 2 meters from the wall measure in "radian/seconds" is?

asked Jan 11, 2013 by sreemathi.v

1 answer

The sides of an equilateral triangle are increasing at the rate of 2cm/sec.The rate at which the area increases ,when side is 10cm is

asked Jan 11, 2013 by sreemathi.v

1 answer

The sum of the surface area of a rectangular parallelepiped with sides x,2x and $\Large\frac{x}{3}$ and a sphere is given to be constant.Prove that the sum of their volume is minimum,if x is equal to three times the radius of the sphere.Also find the minimum value of the sum of their volumes.

asked Jan 11, 2013 by sreemathi.v

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A metal box with a square base and vertical sides is to contain 1024$cm^3$.The material for the top and bottom costs Rs 5/$cm^2$ and the material for the sides costs Rs2.50/$cm^2$.Find the least cost of the box.

asked Jan 11, 2013 by sreemathi.v

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AB is a diameter of a circle and C is any point on the circle.Show that the area of $\bigtriangleup ABC$ is maximum,when it is isosceles.

asked Jan 11, 2013 by sreemathi.v

1 answer

If the sum of the surface areas of cube and a sphere is constant.What is the ratio of an edge of the cube to the diameter of the sphere,when the sum of their volumes is minimum?

asked Jan 11, 2013 by sreemathi.v

1 answer

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

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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

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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

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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

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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

Let A and B be two events.If P(A|B)=P(A),then A is______of B.

asked Jan 10, 2013 by sreemathi.v

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Let X be a random variable taking values $x_1,x_2,.....,x_n$ with probabilities $p_1,p_2,......,p_n$,respectively.Then var(X)=________.

asked Jan 10, 2013 by sreemathi.v

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If X follows binomial distribution with parameters n=5,p and P(x=2)=9,P(X=3),then p=__________.

asked Jan 10, 2013 by sreemathi.v

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If A and B are such that \[P(A'\cup B')=\frac{2}{3} \;and\;P(A\cup B)=\frac{5}{9},\]\[then\;P(A')+P(B')=\text{__________}.\]

asked Jan 10, 2013 by sreemathi.v

1 answer

If A and B are two events such that $P(A|B)=p,P(A)=p,P(B)=\frac{1}{3} \quad$ and $\quad P(A\cup B)= \frac{5}{9},$ then $\; p=\text{_____________}.$?

asked Jan 10, 2013 by sreemathi.v

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