Recent questions tagged kvquestionbank2012

Using vector method, find the angle between two diagonals of a cube?

asked Feb 5, 2013 by meena.p

1 answer

Using vector method prove that : $a)\;\sin (A+B)=\sin A\cos B+\cos A\sin B$

asked Feb 5, 2013 by meena.p

1 answer

In a triangle ABC, prove that $\Large\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$

asked Feb 5, 2013 by meena.p

1 answer

If $\Large\frac{1}{a},\frac{1}{b},\frac{1}{c}$are the $\large p^{th},q^{th}and\;r^{th}\;$ terms of an AP and $ \bar{u}=(q-r)\bar{i}+(r-p)\bar{j}+(p-q)\bar{k}\;and\;\bar{v}=\large\frac{1}{a}\bar{i}+\frac{1}{b}\bar{j}+\frac{1}{c}\bar{k}$ then prove that $\bar{u}\; and\;\bar{v}$ are orthogonal vectors.

asked Feb 4, 2013 by meena.p

1 answer

Prove that the area of a paralellogram with diagonals $\bar{a}\;and\;\bar{b}\;is\;\frac{1}{2}|\bar{a} \times \bar{b}|$

asked Feb 4, 2013 by meena.p

1 answer

Let $ \bar{a},\bar{b},\;and\;\bar{c}$ be three vectors such that $|\bar{a}|=3,|\bar{b}|=4,|\bar{c}|=5$ and each one of them being perpendicular to sum of the other two, find $ |\bar{a}+\bar{b}+\bar{c}|$

asked Feb 4, 2013 by meena.p

1 answer

L and M are the mid-points of sides BC & DC of a paralellogram ABCD. Prove that $ \overline{AL}+\overline{AM}=\frac{3}{2}\overline{AC}$

asked Feb 4, 2013 by meena.p

1 answer

Vectors $ 2\bar{i}-\bar{j}+2\bar{k}\;and\;\bar{i}+\bar{j}-3\bar{k}$ act along two adjacent sides of a parallelogram. Find the angle between the diagonals of the parallelogram.

asked Feb 4, 2013 by meena.p

1 answer

What is the area of the parallelogram having diagonals $ 3\bar{i}+\bar{j}-2\bar{k}\;and\;\bar{i}-3\bar{j}+4\bar{k}$?

asked Feb 4, 2013 by meena.p

1 answer

Show that in a triangle ABC the perpendicular of the point $ C $ from the line joining $ the\:points\:A\:and\:B\:is\;\Large\frac{|\bar{a} \times \bar{b}+\bar{b} \times \bar{c}+\bar{c} \times \bar{a}|}{|\bar{b}-\bar{a}|}$(Hint : use area of triangle =$\frac{1}{2}$bh)

asked Feb 4, 2013 by meena.p

0 answers

If $ \hat{a}\;and\;\hat{b}$ are unit vectors inclined at an angle $\theta$,then prove that\[(a)\;\cos \frac{\theta}{2}=\frac{1}{2}|\hat{a}+\hat{b}|\qquad(b)\;\tan \frac{\theta}{2}=\frac {|\hat{a}-\hat{b}|}{|\hat{a}+\hat{b}|}\]

asked Feb 4, 2013 by meena.p

1 answer

Prove that angle in a semi-circle is a right angle.

asked Feb 4, 2013 by meena.p

1 answer

If $ \bar{a}\;and\;\bar{b}$ are vectors,prove that $ |\bar{a} \times \bar{b}|^2+(\bar{a}.\bar{b})^2=|\bar{a}|^2.|\bar{b}|^2$

asked Feb 4, 2013 by meena.p

1 answer

Prove Cauchy - Schawarz inequality : $(\bar{a}.\bar{b})^2\leq|\bar{a}|^2.|\bar{b}|^2$

asked Feb 4, 2013 by meena.p

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Prove the triangle inequality $\bigg|\overrightarrow{a}+\overrightarrow{b}\bigg|\leq\bigg|\overrightarrow{a}\bigg|+\bigg|\overrightarrow{b}\bigg|$

asked Feb 4, 2013 by meena.p

1 answer

If $ \overrightarrow {a}+ \overrightarrow {b}+ \overrightarrow {c}=\overrightarrow 0$ show that $ \overrightarrow {a} \times \overrightarrow {b}=\overrightarrow {b} \times \overrightarrow {c}=\overrightarrow {c} \times \overrightarrow {a}$

asked Feb 4, 2013 by meena.p

1 answer

If $\bar{\alpha}=3\bar{i}-\bar{j}\;and\;\bar{\beta}=2\bar{i}+\bar{j}-\bar{k}.$ Express $ \bar{\beta}$ as a sum of two vectors $ \bar{\beta_1}\;and\;\bar{\beta_2},\;where \; \bar{\beta_1}$ is parallel to $ \bar{\alpha}\;and\; \bar{\beta_2}$ is prependicular to $ \bar{\alpha}$

asked Feb 4, 2013 by meena.p

1 answer

Find the unit vector perpendicular to both the vectors\[\bar{a}=4\bar{i}-\bar{j}-3\bar{k}\;and\;\bar{b}=2\bar{i}+2\bar{j}-\bar{k}\]

asked Feb 4, 2013 by meena.p

0 answers

Newtons law of cooling states that the rate of change of the temperature T of an object is proportional to the difference between T and the (constant) temperature t of the surrounding medium. We can write it as \[\frac{dT}{dt}=-K(T-t),K>0\; constant.\]A cup of coffee is served at $ 185^\circ F$ in a room where the temperature is $65^\circ F .$ Two minutes later the temperature of the coffee has dropped to $155^\circ F.$(log 3/4=0.144, log3 = 1.09872). Find the time required for coffee to have $105^\circ F$ temperature.

asked Feb 4, 2013 by meena.p

0 answers

A curve passing through the point (1,1) has the property that the perpendicular distance of the origin from the normal at any point P of the curve is equal to the distance of p from the x-axis. Determine the equation of the curve.

asked Feb 4, 2013 by meena.p

0 answers

The surface area of a balloon being inflated, changes at a rate proportional to time t. If initially its radius is 1 unit and after 3 seconds it is 2 units, find the radius after time 't'.

asked Feb 4, 2013 by meena.p

1 answer

A wet porous substance in the open air losses its moisture at a rate proportional to the moisture content. If a sheet hung in the wind loses half of its moisture during the first hour, when will it have lost 95% moisture, weather conditions remaining the same.

asked Feb 4, 2013 by meena.p

0 answers

A population grows at the rate of 8% per year. How long does it take for the population to double?

asked Feb 4, 2013 by meena.p

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When the interest is compounded continuously, the amount of money invested increases at a rate proportional to its size. If Rs.1000 is invested at 10% compounded continuously, in how many years will the original investment double itself?

asked Feb 4, 2013 by meena.p

0 answers

Solve: \[\frac{xdx+ydy}{xdy-ydx}=\sqrt{\frac{a^2-x^2-y^2}{x^2+y^2}}\] (Hint : take $ x=r\cos \theta,y=r \sin \theta,$ so that $x^2+y^2=r^2)$

asked Feb 4, 2013 by meena.p

0 answers

Solve:\[\frac{dy}{dx}=\frac{x^2+y^2+1}{2xy},\;where\; y(1)=1\]

asked Feb 4, 2013 by meena.p

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Solve:\[(x^3-3xy^2)dx=(y^3-3x^2y)dy\]

asked Feb 4, 2013 by meena.p

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Solve:\[\frac{dy}{dx}=\frac{y^2+y+1}{x^2+x+1}=0.\]

asked Feb 4, 2013 by meena.p

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Solve the differential equation :\[(x^2y+y^2x)dy=(x^3+y^3)dx\]

asked Feb 4, 2013 by meena.p

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Solve:\[\sqrt{1-y^2}dx=[\sin^{-1}y-x]dy\]

asked Feb 4, 2013 by meena.p

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Solve:\[(x^3-3xy^2)dx=(y^3-3x^2y)dx\]

asked Feb 4, 2013 by meena.p

0 answers

Solve:\[(x^3y^3+xy)\frac{dy}{dx}=1\]

asked Feb 4, 2013 by meena.p

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Solve:\[\frac{dy}{dx}=\cos ^3x\sin^4x+x\sqrt {2x+1}\]

asked Feb 4, 2013 by meena.p

0 answers

Solve:\[\bigg[x\sqrt {x^2+y^2}-y^2\bigg]dx+xydy=0\]

asked Feb 4, 2013 by meena.p

0 answers

Solve:\[ydx-(x+2y)^2dy=0\]

asked Feb 4, 2013 by meena.p

1 answer

Solve:\[x\frac{dy}{dx}=y(log y-log x+1)\]

asked Feb 4, 2013 by meena.p

0 answers

Solve:\[(x+y)^2\frac{dy}{dx}=a^2\]

asked Feb 4, 2013 by meena.p

0 answers

Solve:\[(x+y+1)\frac{dy}{dx}=1\]

asked Feb 4, 2013 by meena.p

0 answers

Show that $ Ax^2+By^2=1$ is a solution of the diff. equation. $ x\bigg[y\bigg(\Large\frac{d^2y}{dx^2}\bigg)+\bigg(\frac{dy}{dx}\bigg)^2\bigg]=y \frac{dy}{dx}$

asked Feb 4, 2013 by meena.p

0 answers

Form the differential equation of circles represented by \[(x-\alpha)^2+(y-\beta)^2=r^2\; by \;eliminating\;\alpha\;and\;\beta\]

asked Feb 4, 2013 by meena.p

0 answers

Write the order and degree of the diff. equation $ \bigg(\large\frac{d^2y}{dx^2}\bigg)^{2/3}=\bigg(y+\frac{dy}{dx}\bigg)^{1/2}$

asked Feb 4, 2013 by meena.p

0 answers

Form the differential equation of the family of curves represented by the equation $ (x+a)^2-2y^2=a^2$

asked Feb 4, 2013 by meena.p

0 answers

Write the order and degree of the differential equation.$ \bigg[1+(y')^2\bigg]^{3/2}=y''$

asked Feb 4, 2013 by meena.p

0 answers

Compute the area bounded by the lines $ x+2y=2,y-x=1\; and\;2x+y=7$

asked Feb 4, 2013 by meena.p

0 answers

Draw the rough sketch of $ y=\sin 2x $ and determine the area enclosed by the lines $ x=\pi/4\; and\;x=\Large\frac{3\pi}{4}$

asked Feb 4, 2013 by meena.p

0 answers

Find the area of the region enclosed between the circles $ x^2+y^2=1\;and\;\bigg(x-\frac{1}{2}\bigg)^2+y^2=1$

asked Feb 4, 2013 by meena.p

1 answer

Sketch the graph of $ y=|x-1|$ and evaluate \[\int _{-2}^4 |x-1|dx\]

asked Feb 4, 2013 by meena.p

0 answers

Find the area bounded by the curve $ y=\sin x,$ x=axis and between $x=0,x=\pi$

asked Feb 4, 2013 by meena.p

0 answers

Find the area of the smaller region bounded by the ellipse $ \Large\frac{x^2}{a^2}+\frac{y^2}{b^2}$=1 and the straight line $ \Large\frac{x}{a}+\frac{y}{b}$=1

asked Feb 4, 2013 by meena.p

0 answers

Find the area of the region bounded by the circle $ x^2+y^2=16$ and the line y = x in the first quadrant.

asked Feb 4, 2013 by meena.p

0 answers

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