Evaluate: ∫0πxsinx1 + cos2x dx
Solve the following differential equation:(x2 − y2) dx + 2xy dy = 0 given that y = 1 when x = 1
Solve the following differential equation:
dydx = x ( 2y - x )x ( 2y + x), if y = 1 when x = 1
cos2 x dydx + y = tanx
If a→ = i^ + j^ + k^ and b→ = j^ - k^, find a vector c→ such that a→ × c→ = b→and a→.c→ = 3
If a→ + b→ + c→ = 0 and a→ = 3, b→ = 5 and c→ = 7, show that the angle between a→ and b→ is 600.
Find the shortest distance between the following lines:
x - 31 = y - 5-2 = z - 71 and x + 17 = y + 1-6 = z + 11
Find the point on the line x + 23 = y + 12 = z - 32 at a distance 3 2 from the point(1, 2, 3).
Let x + 23 = y + 12 = z - 32 = λx = -2 + 3λ, y = -1 + 2λ, z = 3 + 2λTherefore, a point on this line is: { ( -2 + 3λ ), (-1 + 2λ ), ( 3 + 2λ ) }The distance of the point { ( -2 + 3λ ), (-1 + 2λ ), ( 3 + 2λ ) } from point ( 1, 2, 3 ) = 32∴ -2 + 3λ - 12 + -1 + 2λ - 22 + 3 + 2λ - 32 = 32⇒ -3+ 3λ 2 + -3 + 2λ 2 + 2λ 2 = 18⇒ 9+ 9λ 2 - 18λ + 9 + 4λ 2 - 12λ + 4λ 2 = 1817λ 2 - 30λ = 0λ = 0, λ = 3017When λ = 3017,
x = -2 + 3λ = -2 + 3 3017 = -2 + 9017 = 5617y = -1 + 2λ = 1 + 2 3017 = 1 + 6017 = 4317z = 3 + 2λ = 3 + 2 3017 = 51 + 6017 = 11117thus, when λ = 3017, the point is 5617, 4317, 11117 and when λ = 0, the point is ( -2, -1, 3 ).
A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability distribution of the number of successes.
Using integration find the area of the region bounded by the parabola y2 = 4x and the circle 4x2 + 4y2 = 9.