Recent questions tagged kvquestionbank2012

Find the interval in which the function $f(x)=x^4-4x^3+4x^2+15$ is increasing or decreasing.

asked Jan 29, 2013 by meena.p

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Show that $ f(x)=3x^5+40x^3+240x$ is always increasing on R.

asked Jan 29, 2013 by meena.p

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Show that the function $f(x)=e^{2x}$ is strictly increasing on R.

asked Jan 29, 2013 by meena.p

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Water is passed into an inverted cone of base radius 5 cm and depth 10 cm at the rate of $\Large\frac{3}{2}$ c.c/sec. Find the rate at which the level of water is rising when depth is 4 cm.

asked Jan 29, 2013 by meena.p

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Gas is escaping from a spherical balloon at the rate of $900 cm^3$/sec. How fast is the surface area, radius of the balloon shrinking when the radius of the balloon is 30cm?

asked Jan 29, 2013 by meena.p

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The surface area of a spherical bubble is increasing at the rate of $ 2 cm^2$/sec. Find the rate at which the volume of the bubble is increasing at the instant if its radius is 6 cm.

asked Jan 29, 2013 by meena.p

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The volume of a spherical balloon is increasing at the rate of $25\; cm^3$/sec. Find the rate of change of its surface area at the instant when the radius is 5 cm.

asked Jan 29, 2013 by meena.p

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A balloon which always remains spherical is being inflated by pumping in gas at the rate of $900\; cm^3$/sec. Find the rate at which the radius of the balloon is increasing when the radius of the balloon is 15 cm

asked Jan 29, 2013 by meena.p

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An edge of a variable cube is increasing at the rate of 5 cm/sec. How fast is the volume of the cube is increasing when edge is 10cm long?

asked Jan 29, 2013 by meena.p

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A man 160 cm tall walks away from a source of light situated at the top of the pole 6m high at the rate of 1.1 m/sec. How fast is the length of the shadow increasing when he is 1m away from the pole.

asked Jan 29, 2013 by meena.p

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Sand is pouring from a pipe at the rate of 12cm3/sec. The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone in increasing when the height is 4 cm?

asked Jan 29, 2013 by meena.p

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A point on the hypotenuse of a right angled triangle is at distances a and b from the sides. Show that the length of the hypotenuse is at least $ \bigg(a^{2/3}+b^{2/3}\bigg)^{3/2}$

asked Jan 29, 2013 by meena.p

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

asked Jan 29, 2013 by meena.p

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Find the maximum slope of the curve $ f(x)=2x+3x^2-x^3-27$

asked Jan 29, 2013 by meena.p

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Given the sum of the perimeters of a square and a circle show that the sum of their areas is least when the side of the square is equal to the diameter of a circle.

asked Jan 29, 2013 by meena.p

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Show that the semi vertical angle of the cone of maximum volume and of given slant height is $ \tan ^{-1} \sqrt 2$

asked Jan 29, 2013 by meena.p

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Prove that the volume of the largest cone that can be inscribed in a sphere of radius R is $\Large\frac{8}{27}$ of the volume of the sphere.

asked Jan 29, 2013 by meena.p

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Show that the rectangle of maximum area that can be inscribed in a circle of radius 'r' cms is a square of side $\sqrt 2 r$

asked Jan 29, 2013 by meena.p

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Of all the rectangles each of which has perimeter 40 metres find one which has maximum area.Find the area also.

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Show that the height of the right circular cylinder of maximum volume that can be inscribed in a given right circular cone of height $h\; is\; h/3$

asked Jan 29, 2013 by meena.p

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Show that the volume of the greatest cylinder which can be inscribed in a cone of height h and semi vertical angle $ 30^\circ\;is\;\large\frac{4}{81}\pi h^3$

asked Jan 29, 2013 by meena.p

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A rectangular window is surmounted by an equilateral triangle. Given that the perimeter is 16 cm. Find the width of the window so that the maximum amount of light may enter.

asked Jan 29, 2013 by meena.p

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Find the area of the greatest isoscles triangle that can be inscribed in a given ellipse $ \Large\frac{x^2}{a^2}+\frac{y^2}{b^2}$=1 with its vertex coinciding with one extremity of the major axis.

asked Jan 29, 2013 by meena.p

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A window is in the form of a rectangle above which there is a semicircle. If the perimeter of the window is P cm. Show that the window will allow the maximum possible light only when the radius of the semi circle is $\large\frac{p}{\pi+4}cm$

asked Jan 29, 2013 by meena.p

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

asked Jan 29, 2013 by meena.p

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Find the absolute maximum and absolute minimum value of $ f(x)=2 \cos x+x \qquad x\in[0,\pi]$

asked Jan 29, 2013 by meena.p

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

asked Jan 29, 2013 by meena.p

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Show that the right circular cone of least curved surface area and given volume is $\sqrt 2 $ times the radius of the base.

asked Jan 29, 2013 by meena.p

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Show that the semi vertical angle of a right circular cone of given total surface area and maximum volume is $\sin^{-1} \frac{1}{3}$

asked Jan 29, 2013 by meena.p

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Show that the semi vertical angle of a right circular cone of maximum volume and given slant height is $\tan^{-1}\sqrt 2$

asked Jan 29, 2013 by meena.p

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Find the points on the curve $ y=x^3-3x^2+2x$ at which the tangent to the curve is parallel to the line $ y-2x+3=0$

asked Jan 29, 2013 by meena.p

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Prove that the curves $ y^2=4ax\;and\;xy=c^2$ cut at right angles if $ c^4=32a^4$

asked Jan 29, 2013 by meena.p

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Find the equation of all lines having slope -1 and that are tangents to the curve $ y=\Large\frac{1}{x-1}, \normalsize x\neq1$

asked Jan 29, 2013 by meena.p

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Find the point on the curve $ y=x^3-11x+5 $ at which the tangent has equation $y=x-11$

asked Jan 29, 2013 by meena.p

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Find the equation of tangent to the curve $\sqrt x+\sqrt y=\sqrt a $ at the point $(x_1,y_1)$ and show that the sum of its intercepts on the axes is constant.

asked Jan 29, 2013 by meena.p

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Find the interval in which the function f(x) is increasing or decreasing $ f(x)=x^3-12 x^2+36x+17 $

asked Jan 29, 2013 by meena.p

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A cone is circumscribed about a sphere of radius r. Show that the volume of the cone is maximum when its semi vertical angle is $\sin^{-1}\bigg(\Large\frac{1}{3}\bigg)$

asked Jan 29, 2013 by meena.p

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What angle is formed by the y-axis and the tangent to the parabola $ y=x^2+4x-17$ at the point $ p(5/2,-3/4)?$

asked Jan 29, 2013 by meena.p

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If $ P_1\;and\; P_2 $are the lengths of the perpendiculars from origin on the tangent and normal to the curve $ x^{2/3}+y^{2/3}=a^{2/3} $respectively Prove that $ 4P_1^2+P_2^2=a^2$

asked Jan 29, 2013 by meena.p

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Find the equation of normal to the curve $ y=(1+x)^y+\sin^{-1}(\sin^2x)$ at x=0

asked Jan 29, 2013 by meena.p

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Find the interval in which the following functions are increasing or decreasing\[(a)\;y=log \bigg(x+\sqrt{1+x^2}\bigg)\qquad(b)\;y=\frac{10}{4x^3-9x^2+6x}\]

asked Jan 29, 2013 by meena.p

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Determine the points of maxima and minima of the function $ f(x)= \frac{1}{8}log x-bx+cx^2\; where\; b \geq 0$

asked Jan 29, 2013 by meena.p

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The least value of $ f(x)=\tan -1(\sin x+cos x)$ strictly increasing is \[(a)\;\bigg(\frac{\pi}{4},\frac{\pi}{2}\bigg)\qquad(b)\;\bigg(0,\frac{\pi}{2}\bigg)\qquad(c)\;0\qquad(d)\;none\; of\; these.\]

asked Jan 29, 2013 by meena.p

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The least value of 'a' such that the function $ f(x)=x^2+ax+1 $ is strictly increasing on (1,2) is \[(a)\;-2\qquad(b)\;2\qquad(c)\;\frac{1}{2}\qquad(d)\;-\frac{1}{2}\]

asked Jan 29, 2013 by meena.p

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The function $ f(x)=-\Large\frac{x}{2}\normalsize+ \sin x $ is always increasing in \[(a)\;\bigg(-\frac{\pi}{2},\frac{\pi}{2}\bigg)\qquad(b)\;\bigg(0,\frac{\pi}{4}\bigg)\qquad(c)\;\bigg(\frac{\pi}{4},\frac{\pi}{2}\bigg)\qquad(d)\;\bigg(-\frac{\pi}{3},\frac{\pi}{3}\bigg)\qquad\]

asked Jan 29, 2013 by meena.p

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If the slope of the tangent is zero at $ (x_1,y_1)$then the equation of the tangent at $(x_1,y_1)$ is \[y_1=mx_1+c \qquad(b)\;y_1=mx_1\qquad(c)\;y-y_1\qquad(d)\;y=0\]

asked Jan 29, 2013 by meena.p

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If $ \Large\frac{dy}{dx} \normalsize =0$ then the tangent is \[(a)\;Parallel to x-axis\qquad(b)\;parallel to y-axis\qquad(c)\;Perpendicular to x-axis\qquad\]\[(d)\;perpendicular to y-axis\]

asked Jan 29, 2013 by meena.p

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The tangents to the curve $ y=x^3+6 $ at the points (-1,5) and (1,7) are \[(a)\;Perpendicular\qquad(b)\;parallel\qquad(c)\;coincident\qquad(d)\;none\; of\; these\]

asked Jan 29, 2013 by meena.p

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The values of a and b for which the function$f(x)= \left\{ \begin{array}{1 1} ax+1 & \quad x\leq3 \\ bx+3 & \quad x>3 \end{array} \right. $ is continuous at x=3 are \[(a)\;3a+2b=5 \qquad(b)\;3a=2+3b\qquad(c)\;3,2\qquad(d)\;none\;of\;these.\]

asked Jan 29, 2013 by meena.p

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'C' on LMV for $f(x)=x^2-3x$ in [0,1] is \[(a)\;0 \qquad(b)\;\frac{1}{2}\qquad(c)\;-\frac{1}{2}\qquad(d)\;does\;not\;exist\]

asked Jan 29, 2013 by meena.p

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