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Differentiate the following with respect to x:
\[\cot^{- 1} \left( \frac{1 - x}{1 + x} \right)\]
Concept: undefined >> undefined
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.
Concept: undefined >> undefined
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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.
Concept: undefined >> undefined
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.
Concept: undefined >> undefined
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?
Concept: undefined >> undefined
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.
Concept: undefined >> undefined
If y = (sec-1 x )2 , x > 0, show that
`x^2 (x^2 - 1) (d^2 y)/(dx^2) + (2x^3 - x ) dy/dx -2 = 0`
Concept: undefined >> undefined
If y = sin-1 x + cos-1x find `(dy)/(dx)`.
Concept: undefined >> undefined
Solve the differential equation: (1 + x2) dy + 2xy dx = cot x dx
Concept: undefined >> undefined
Show that the height of a cylinder, which is open at the top, having a given surface area and greatest volume, is equal to the radius of its base.
Concept: undefined >> undefined
If `log (x^2 + y^2) = 2 tan^-1 (y/x)`, show that `(dy)/(dx) = (x + y)/(x - y)`
Concept: undefined >> undefined
If `"y" = (sin^-1 "x")^2, "prove that" (1 - "x"^2) (d^2"y")/(d"x"^2) - "x" (d"y")/(d"x") - 2 = 0`.
Concept: undefined >> undefined
Solve the differential equation : `"x"(d"y")/(d"x") + "y" - "x" + "xy"cot"x" = 0; "x" != 0.`
Concept: undefined >> undefined
Using elementary row operations, find the inverse of the matrix A = `((3, 3,4),(2,-3,4),(0,-1,1))` and hence solve the following system of equations : 3x - 3y + 4z = 21, 2x -3y + 4z = 20, -y + z = 5.
Concept: undefined >> undefined
Solve the following differential equation :
`"dy"/"dx" + "y" = cos"x" - sin"x"`
Concept: undefined >> undefined
Find:
`int_(-pi/4)^0 (1+tan"x")/(1-tan"x") "dx"`
Concept: undefined >> undefined
If y = `(sin^-1 x)^2,` prove that `(1-x^2) (d^2y)/dx^2 - x dy/dx -2 = 0.`
Concept: undefined >> undefined
Let R be the equivalence relation in the set Z of integers given by R = {(a, b): 2 divides a – b}. Write the equivalence class [0]
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If f = {(5, 2), (6, 3)}, g = {(2, 5), (3, 6)}, write f o g
Concept: undefined >> undefined
If f = {(5, 2), (6, 3)} and g = {(2, 5), (3, 6)}, write the range of f and g
Concept: undefined >> undefined
