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Find the shortest distance between the lines
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Find the shortest distance between the lines
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Differentiate sin(log sin x) ?
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If x = a (1 + cos θ), y = a(θ + sin θ), prove that \[\frac{d^2 y}{d x^2} = \frac{- 1}{a}at \theta = \frac{\pi}{2}\]
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Differentiate `log [x+2+sqrt(x^2+4x+1)]`
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Differentiate the following with respect to x:
\[\cot^{- 1} \left( \frac{1 - x}{1 + x} \right)\]
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Find the equation of the curve passing through the point (0, 2) given that the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5.
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The slope of the tangent to the curve at any point is the reciprocal of twice the ordinate at that point. The curve passes through the point (4, 3). Determine its equation.
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The decay rate of radium at any time t is proportional to its mass at that time. Find the time when the mass will be halved of its initial mass.
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Experiments show that radium disintegrates at a rate proportional to the amount of radium present at the moment. Its half-life is 1590 years. What percentage will disappear in one year?
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A wet porous substance in the open air loses 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.
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Show that the lines `(x+1)/3=(y+3)/5=(z+5)/7 and (x−2)/1=(y−4)/3=(z−6)/5` intersect. Also find their point of intersection
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Find the distance of the point (−1, −5, −10) from the point of intersection of the line `vecr=2hati-hatj+2hatk+lambda(3hati+4hatj+2hatk) ` and the plane `vec r (hati-hatj+hatk)=5`
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Using properties of determinants prove the following: `|[1,x,x^2],[x^2,1,x],[x,x^2,1]|=(1-x^3)^2`
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Using properties of determinants, show that ΔABC is isosceles if:`|[1,1,1],[1+cosA,1+cosB,1+cosC],[cos^2A+cosA,cos^B+cosB,cos^2C+cosC]|=0`
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Using the properties of determinants, prove the following:
`|[1,x,x+1],[2x,x(x-1),x(x+1)],[3x(1-x),x(x-1)(x-2),x(x+1)(x-1)]|=6x^2(1-x^2)`
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Using properties of determinants, prove that
`|((x+y)^2,zx,zy),(zx,(z+y)^2,xy),(zy,xy,(z+x)^2)|=2xyz(x+y+z)^3`
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If ` f(x)=|[a,-1,0],[ax,a,-1],[ax^2,ax,a]| ` , using properties of determinants find the value of f(2x) − f(x).
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f(x) = 4x2 + 4 on R .
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